Heat and mass transfer

In this revised edition of our book we retained its concept: The main emphasis is placed on the fundamental principles of heat and mass transfer and their application to practical problems of process modelling and the apparatus design.

The diﬀerent types of heat transfer

Heat conduction

Heat conduction refers to the energy transfer between adjacent molecules in a material caused by a temperature difference In metals, this process is enhanced by free electrons that facilitate energy transfer In solids that do not allow radiation, heat conduction is the sole method of energy transfer Conversely, in gases and liquids, heat conduction occurs alongside energy transport mechanisms such as convection and radiation.

Understanding heat conduction in solids and fluids can be complex, but we can simplify our approach by focusing on key thermodynamic concepts Instead of delving deeply into theoretical frameworks, we will concentrate on practical applications involving temperature, heat flow, and heat flux These fundamental quantities are adequate for addressing most relevant conduction issues in engineering and technology.

The transport of energy in a conductive material is described by the vector ﬁeld ofheat ﬂux q˙ = ˙q(x, t) (1.2)

In continuum theory, the heat flux vector indicates both the direction and magnitude of energy flow at a specific position, denoted by the vector x, and may also vary with time t The heat flux, represented as ˙q, quantifies the heat transfer d ˙Q through a surface element dA, expressed by the equation d ˙Q = ˙q(x, t)ndA = |q|˙ cosβ dA.

The unit vector normal to the surface, denoted as Herenis, forms an angle β with the heat flow vector ˙q, as illustrated in Fig 1.1 The heat flow d ˙Q reaches its maximum when ˙q is perpendicular to the differential area dA, resulting in β = 0 This relationship highlights that the dimension of heat flow is energy per unit time, commonly referred to as thermal power.

Fig 1.1: Surface element with normal vector n and heat ﬂux vector ˙ q

SI unit J/s = W Heat ﬂux is the heat ﬂow per surface area with units J/s m 2 W/m 2

The transport of energy by heat conduction is due to a temperature gradient in the substance The temperatureϑ changes with both position and time All temperatures form a temperature ﬁeld ϑ=ϑ(x, t)

Steady temperature fields remain constant over time, while unsteady or transient temperature fields exhibit significant changes as time progresses At any given moment, all points within a body sharing the same temperature, denoted as ϑ, can be visualized as interconnected by an isothermal surface or isotherm This surface effectively divides the body into regions with temperatures higher and lower than ϑ The most significant temperature variation occurs perpendicular to the isotherm, represented by the temperature gradient gradϑ = ∂ϑ.

The gradient vector, represented by ∂z e^z, is perpendicular to the isotherm at a given point and indicates the direction of the steepest temperature increase In this context, e^x, e^y, and e^z denote the unit vectors along the three coordinate axes.

Fig 1.2: Point P on the isotherm ϑ = const with the temperature gradient grad ϑ from (1.4) and the heat

Heat flow in conductive materials is driven by temperature gradients, indicating a direct proportionality between the cause and effect This relationship can be expressed mathematically as q˙ = −λ gradϑ, where q˙ represents the heat flux and λ denotes the thermal conductivity.

J B Fourier's fundamental law of heat conduction, established in 1822, states that heat transfers from areas of higher temperature to lower temperature, in accordance with the second law of thermodynamics This principle is represented in the equation where the negative sign indicates the direction of heat flow The thermal conductivity, denoted as λ and dependent on temperature and pressure, is a key material property that influences this process.

Thermal conductivity (λ) is influenced by temperature (ϑ), pressure (p), and the composition of mixtures For isotropic materials, thermal conductivity is a scalar, indicating that heat conduction ability is position-dependent but direction-independent While most materials are assumed to be isotropic, some, like wood, exhibit directional thermal conductivity, conducting heat better across their fibers than along them In non-isotropic materials, λ becomes a second-order tensor, causing the heat flux vector (˙q) and temperature gradient (gradϑ) to form an angle In contrast, isotropic substances always have the heat flux vector perpendicular to isothermal surfaces, leading to the relationship d ˙Q = −λ(gradϑ)ndA = −λ∂ϑ for heat flow through a surface element (dA) in any direction.

Here∂ϑ/∂nis the derivative ofϑwith respect to the normal (outwards) direction to the surface element.

Table 1.1: Thermal conductivity of selected substances at 20 ◦ C and 100 kPa

Thermal conductivity, measured in watts per Kelvin meter (W/K m), is a crucial property in the field of heat transfer While the pressure dependence of thermal conductivity is relevant primarily for gases and liquids, its temperature dependence is typically minimal and can often be disregarded For detailed information, refer to the comprehensive tables of thermal conductivity available in Appendix B.

1 Jean Baptiste Fourier (1768–1830) was Professor for Analysis at the Ecole Polytechnique in Paris and from 1807 a member of the French Academy of Science His most important work

Published in 1822, "The Analytic Theory of Heat" is the first extensive mathematical framework for understanding heat conduction It introduces the "Fourier Series," a crucial tool for addressing boundary value problems in transient heat conduction.

Metals exhibit exceptionally high thermal conductivities, while non-conductive solids have significantly lower values Liquids and gases demonstrate particularly low thermal conductivities, as illustrated in Table 1.1 Additionally, the low thermal conductivity of foamed insulating materials is attributed to their structure, which comprises numerous small, gas-filled spaces surrounded by a solid with inherently low thermal conductivity.

Steady, one-dimensional conduction of heat

This section focuses on steady heat conduction, an essential concept, in flat plates, hollow cylinders, and hollow spheres It assumes that heat flows in a single direction: perpendicular to the plate surface and radially in the cylinder and sphere Consequently, the temperature field is determined by just one geometrical coordinate, a principle known as one-dimensional heat conduction.

Fig 1.3: Steady, one dimensional conduction a Temperature profile in a flat plate of thickness δ = r 2 − r 1 , b Temperature profile in a hollow cylinder (tube wall) or hollow sphere of inner radius r 1 and outer radius r 2

In this analysis, the position coordinate is represented by r, with isothermal surfaces defined as r = const, indicating that temperature ϑ is a function of r (ϑ = ϑ(r)) We establish constant temperature values, where ϑ = ϑ W1 at r = r1 and ϑ = ϑ W2 at r = r2 These temperatures are predetermined, allowing us to derive a relationship between the heat flow in the system.

To determine the heat transfer rate \( \dot{Q} \) through flat or curved walls, we need to consider the temperature difference \( \vartheta_{W1} - \vartheta_{W2} \) For this example, we assume \( \vartheta_{W1} > \vartheta_{W2} \), indicating that heat flows from the hotter to the cooler side In steady-state conditions, the heat flow \( \dot{Q} \) remains constant across the inner and outer surfaces, as well as along each isothermal surface \( r = \text{const} \), since no energy is stored within the wall.

Fourier’s law gives the following for the heat ﬂow

In a flat wall, the area remains constant regardless of the radius, meaning A = A1 = A2 When thermal conductivity is uniform, the temperature gradient (dϑ/dr) is also constant, resulting in a linear temperature profile However, this linearity does not apply to cylindrical or spherical geometries, nor when thermal conductivity varies with temperature.

In these more general cases (1.7) becomes

A(r) and after integrating over the wall thicknessδ=r 2 −r 1

− ϑ W2 ϑ W1 λ(ϑ) dϑ= ˙Q r 2 r 1 dr A(r) From the mean value theorem for integration comes

Heat flow is directly proportional to the temperature difference between two surfaces, similar to how voltage drives current in an electric circuit This temperature difference serves as the driving force for heat transfer, characterized by thermal conductance, which is represented as λ m A m /δ.

R cond := δ λ m A m (1.9) thethermal resistance In analogy to electric circuits we get

Q˙ = (ϑ W1 −ϑ W2 )/R cond (1.10) Theaverage thermal conductivitycan easily be calculated using λ m := 1 (ϑ W2 −ϑ W1 ) ϑ W2 ϑ W1 λ(ϑ) dϑ (1.11)

In many cases the temperature dependence ofλcan be neglected, givingλ m =λ.

This assumption is generally suﬃcient for the region ϑ W1 ≤ ϑ≤ ϑ W2 as λ can rarely be measured with a relative error smaller than 1 to 2%.

Theaverage areaA m in (1.8) is deﬁned by

A 1 =A 2 for a ﬂat plate 2πLr for a cylinder of lengthL 4πr 2 for a sphere.

The average area \( A_m \) is calculated by averaging the surface areas \( A_1 = A(r_1) \) and \( A_2 = A(r_2) \) For a flat plate, the arithmetic mean is used; for a cylinder, the logarithmic mean is applicable; and for a sphere, the geometric mean is utilized.

2(A 1 +A 2 ) For the thermal resistance to conduction it follows

The wall thickness for the cylinder (tube wall) and sphere is δ=r 2 −r 1 = 1

2(d 2 −d 1 ) so thatR cond can be expressed in terms of both diameters d 1 and d 2

The temperature proﬁle in each case shall also be determined We limit ourselves to the caseλ= const WithA(r) from (1.14) and integrating

−dϑQ˙ λ dr A(r) the dimensionless temperature ratio is ϑ(r)−ϑ W2 ϑ W1 −ϑ W2 ⎧ ⎪

⎪ ⎩ r 2 −r r 2 −r 1 ﬂat plate ln (r 2 /r) ln (r 2 /r 1 ) cylinder

Fig 1.4: Steady temperature proﬁle from

(1.17) in a ﬂat, cylindrical and spherical wall of the same thickness δ and with r 2 /r 1 = 3

As already mentioned the temperature change is linear in the ﬂat plate The cylinder has a logarithmic, and the sphere a hyperbolic temperature dependence on the radial coordinates.

Figure 1.4 illustrates the temperature profile in walls of uniform thickness, highlighting that the most significant deviation from the linear model occurs at the point r = r m At this point, the cross-sectional area A(r) reaches its maximum value, A(r m) = A m, as defined by equation (1.15).

Example 1.1: A ﬂat wall of thickness δ = 0.48 m, is made out of ﬁreproof stone whose thermal conductivity changes with temperature With the Celsius temperature ϑ, between

1 − bϑ (1.18) where λ 0 = 0.237 W/K m and b = 4.41 ã 10 −4 K −1 The surface temperatures are ϑ W1 =

750 ◦ C and ϑ W2 = 150 ◦ C The heat ﬂux ˙ q = ˙ Q/A and the temperature proﬁle in the wall need to be calculated.

From (1.8) the heat ﬂux is ˙ q = λ m δ (ϑ W1 − ϑ W2 ) (1.19) with the average thermal conductivity λ m = 1 ϑ W2 − ϑ W1 ϑ W2 ϑ W1 λ(ϑ) dϑ = λ 0 b (ϑ W1 − ϑ W2 ) ln 1 − bϑ W2

1 − bϑ W1 Putting as an abbreviation λ(ϑ Wi ) = λ i (i = 1, 2) we get λ m = ln(λ 1 /λ 2 ) 1 λ 2 − 1 λ 1

The average thermal conductivity λ m can be calculated using the λ values for both surfaces.

The calculation of the mean thermal conductivity, λ m, is derived from the square of the geometric mean divided by the logarithmic mean of two values, λ 1 = 0.354 W/K m and λ 2 = 0.254 W/K m, resulting in λ m = 0.298 W/K m Consequently, the heat flux is determined to be ˙ q = 373 W/m² Assuming a linear variation of λ with temperature leads to the conclusion that λ m equals 1.

The thermal conductivity value of 2 (λ 1 + λ 2 ) is calculated to be 0.304 W/K m While this figure is 1.9% higher than the exact measurement, it remains a valuable approximation, as the discrepancy falls within the acceptable uncertainty range for thermal conductivity measurements.

To calculate the temperature proﬁle in the wall we will use (1.7) as the starting point,

− λ(ϑ) dϑ = ˙ q dr , and with x = r − r 1 this gives

1 − bϑ W1 = ˙ qx With ˙ q from (1.19) and λ m from (1.20) it follows that ln 1 − bϑ

(1.21) for the equation to calculate the temperature proﬁle in the wall.

The steady temperature profile, represented as ϑ = ϑ (x/δ), is illustrated in Fig 1.5 for a flat wall exhibiting temperature-dependent thermal conductivity, as defined by equation (1.18) The deviation ∆ϑ indicates the difference between the actual temperature profile and the linear model applicable when thermal conductivity (λ) remains constant, as shown on the right-hand scale.

Figure 1.5 illustrates the temperature profile ϑ(x) and the deviation ∆ϑ(x) from the linear temperature distribution between ϑ W1 and ϑ W2 At elevated temperatures, the thermal conductivity is higher, resulting in a reduced temperature gradient compared to lower temperatures, where the thermal conductivity λ(ϑ) decreases It is essential that the heat flux ˙ q remains constant across the wall, as represented by the equation ˙ q = − λ(ϑ) dϑ/dx Consequently, lower thermal conductivity values are balanced by steeper temperature gradients.

Convective heat transfer Heat transfer coeﬃcient

In a flowing fluid, energy transfer occurs through both heat conduction and the fluid's macroscopic movement At any given position within the fluid, heat conduction takes place due to a temperature gradient, while energy in the form of enthalpy and kinetic energy also moves across that area This combined process is known as convective heat transfer, which integrates thermal conduction with the energy transfer from the flowing fluid.

Heat transfer between a solid wall and a fluid, such as in a heated tube with cold gas flowing inside, is crucial in engineering applications The fluid layer adjacent to the wall significantly influences the heat transfer rate This phenomenon is described by the boundary layer, a concept introduced by L Prandtl in 1904, which plays a vital role in fluid dynamics, particularly in the context of heat and mass transfer.

In the boundary layer, the velocity parallel to the wall transitions from zero at the wall to nearly the maximum value found in the core fluid Additionally, the temperature within the boundary layer varies from the wall temperature (ϑ W) to the fluid temperature (ϑ F) at a certain distance from the wall.

Heat transfer occurs between a wall and a fluid due to the temperature difference, with heat flowing from the wall into the fluid when the wall is hotter Conversely, if the fluid temperature exceeds that of the wall, the fluid cools as heat is absorbed by the wall The heat flux at the wall, denoted as ˙q W, is influenced by the temperature and velocity fields within the fluid, making its evaluation complex and challenging for calculations To simplify this, the relationship is expressed as ˙q W = α(ϑ W − ϑ F), where α represents the local heat transfer coefficient, defined as α = ˙q W / (ϑ W − ϑ F).

The introduction of the heat transfer coefficient, α, replaces the unknown heat flux, ˙q W, leading some researchers to view this addition as unnecessary However, the application of heat transfer coefficients remains essential in thermal analysis.

Ludwig Prandtl (1875–1953) served as a Professor of Applied Mechanics at the University of Göttingen from 1904 until his passing He was the Director of the Kaiser-Wilhelm-Institut for Fluid Mechanics starting in 1925 Prandtl's groundbreaking boundary layer theory, along with his research on turbulent flow, wing theory, and supersonic flow, represents essential advancements in the field of modern fluid mechanics.

In fluid dynamics, the velocity (w) and temperature (ϑ) profiles are influenced by the distance from the wall (y), with δ and δt indicating the thickness of the velocity and temperature boundary layers, respectively Understanding the transfer coefficients is essential, as knowing the thermal diffusivity (α) allows for straightforward answers to key questions in convective heat transfer Specifically, it enables the calculation of heat flux (˙q W) for a specified temperature difference (ϑ W − ϑ F) and determines the temperature difference required to achieve a certain heat flux between the wall and the fluid.

To understand the relationship between the heat transfer coefficient and the temperature field in a fluid, it is essential to examine the region close to the wall (as y approaches 0) In this area, the fluid typically adheres to the wall, with the exception of very dilute gases, resulting in a zero velocity Consequently, energy transfer occurs solely through heat conduction.

So instead of (1.22) the physically based relationship (Fourier’s law) is valid: ˙ q W =−λ

The thermal conductivity of the fluid at the wall temperature, denoted as λ(ϑ W), plays a crucial role in determining the heat flux ˙q W, which is derived from the temperature gradient of the fluid at the wall According to the definition provided, the heat transfer coefficient α can be expressed as α = -λ.

From this it is clear thatαis determined by the gradient of the temperature profile at the wall and the difference between the wall and fluid temperatures Therefore,

The fluid temperature ϑ varies with distance from the wall (y), and understanding this temperature field is essential for calculating the heat transfer coefficient using the ratio λ/α as a subtangent Additionally, the velocity field within the fluid significantly influences the temperature distribution.

So, in addition to the energy balances from thermodynamics, the equations of ﬂuid motion from ﬂuid mechanics furnish the fundamental relationships in the theory of convective heat transfer.

A graphical representation of the ratio λ/α illustrates the distance from the wall where the tangent to the temperature profile intersects the ϑ = ϑ F line This ratio is comparable to the thermal boundary layer thickness, which will be analyzed in sections 3.5 and 3.7.1, and is typically slightly larger than λ/α A thin thermal boundary layer suggests efficient heat transfer, while a thicker layer results in lower values of α.

The temperature of the fluid far from a wall, denoted as ϑ F, is crucial for defining the local heat transfer coefficient In external flow scenarios, ϑ F represents the free flow temperature, often written as ϑ ∞, indicating a state where the fluid is minimally affected by heat transfer from the body Conversely, in internal flow situations, such as fluid flowing through a heated tube, the temperature at various points within the channel is influenced by heat transfer from the wall In this context, ϑ F is defined as the cross-sectional average temperature, serving as a key characteristic for energy transport along the channel axis This definition connects the heat flow from the wall, characterized by the heat transfer coefficient α, to the energy carried by the flowing fluid.

Fig 1.8: Temperature proﬁle in a channel cross section Wall temperature ϑ W and average ﬂuid temperature ϑ F

To define ϑ F we will take a small section of the channel, Fig 1.9 The heat flow from the wall area dA to the fluid is d ˙ Q = α (ϑ W − ϑ F ) dA (1.26)

From the ﬁrst law of thermodynamics, neglecting the change in kinetic energy, we have d ˙ Q =

The flow of heat results in a change in the enthalpy flow of the fluid The average fluid temperature is defined in a way that allows the enthalpy flow to be expressed effectively.

Fig 1.9: Energy balance for a channel section (left); fluid velocity w and temperature ϑ profiles in channel cross section (right) as the product of the mass flow rate

The specific enthalpy h(ϑ F) corresponds to the average temperature ϑ F, also known as the adiabatic mixing temperature This temperature represents the state of the fluid when all components in a cross-section are mixed adiabatically within a container, maintaining a constant temperature of ϑ F According to the first law of thermodynamics, the enthalpy flow ˙ H of the unmixed fluid entering the adiabatic container must equal the enthalpy flow ˙ M h(ϑ F) of the fluid exiting the container, as indicated by the implicit definition in equation (1.28).

To calculate the adiabatic mixing temperature ϑ F the pressure dependence of the speciﬁc enthalpy is neglected Then setting h(ϑ) = h 0 + [c p ] ϑ ϑ 0 (ϑ − ϑ 0 ) and h(ϑ F ) = h 0 + [c p ] ϑ ϑ F

0 as the average speciﬁc heat capacity of the ﬂuid between ϑ and the reference temperature ϑ 0 at which h(ϑ 0 ) = h 0 , we get from (1.28) ϑ F = ϑ 0 + 1

For practical calculations a constant speciﬁc heat capacity c p is assumed, giving ϑ F = 1

The adiabatic mixing temperature (ϑ F) serves as a crucial link between the local heat transfer coefficient (α) and the enthalpy flow across each cross-section This relationship is articulated through the equation d ˙ Q = α (ϑ W − ϑ F ) dA = ˙ M c p dϑ F, demonstrating how heat transfer and temperature variations are interconnected Notably, the adiabatic mixing temperature differs from the integrated average of the cross-sectional temperature (ϑ m).

Fig 1.10: Average fluid temperature ϑ F , wall temperature ϑ W and local heat transfer coefficient as functions of the axial distance z, when heating a fluid in a tube of length L

Both temperatures are only equal if the velocity at each point in the cross section is the same, i.e in plug ﬂow with w = const.

Determining heat transfer coeﬃcients Dimensionless numbers

Understanding the temperature field within a fluid is essential for calculating the heat transfer coefficient This calculation can only be performed when the velocity field is established Exact values for the heat transfer coefficient can typically be derived from fundamental partial differential equations in straightforward scenarios, such as fully developed laminar flow in tubes or parallel flow over a flat plate with a laminar boundary layer However, turbulent flow situations necessitate the use of simplified models.

ﬂow, and the more complex problems such as nucleate boiling cannot be handled theoretically at all.

An essential method for determining heat transfer coefficients is through experimental measurement By assessing heat flow or flux alongside wall and fluid temperatures, one can calculate the local or mean heat transfer coefficient To accurately address the heat transfer problem, it is crucial to vary all influencing factors during these measurements, including geometric dimensions such as tube length and diameter, characteristic flow velocity, and fluid properties like density, viscosity, thermal conductivity, and specific heat capacity.

In experimental research, the number of variables typically ranges from five to ten, necessitating multiple trials to isolate the effect of a single property For instance, when examining one property with five different values while keeping other variables constant, the total number of experiments required escalates exponentially, resulting in a staggering number of trials, such as 15,625 when considering six variables This extensive requirement can lead to significant time and financial costs However, employing similarity or model theory can substantially decrease the number of experiments needed By utilizing dimensionless variables and groups, similarity theory demonstrates that temperature and velocity fields can be expressed independently of unit systems This approach allows researchers to scale these fields using characteristic values, thus transforming similar fields into one another through changes in reference quantities, ultimately streamlining the experimental process.

Velocity and temperature fields are similar only when the corresponding dimensionless groups align These dimensionless numbers incorporate geometric factors, critical temperature differences, velocities, and properties of the heat transfer fluid Notably, the number of dimensionless quantities is much smaller than the total relevant physical quantities, leading to a significant reduction in the number of experiments needed Instead of varying individual quantities, the focus is on altering the values of the dimensionless numbers to explore their functional relationships.

Using dimensionless variables at the onset of solving heat transfer problems enhances clarity in theoretical solutions Introducing these variables early simplifies both the evaluation and representation of the solution Additionally, minimizing the number of independent variables through dimensionless variables and groups streamlines the problem-solving process.

The analysis of heat transfer begins with the partial differential equations governing velocity and temperature To simplify these equations, space coordinates, velocity components, and temperature are converted into dimensionless forms by normalizing them with characteristic values of length, velocity, and temperature This transformation results in new partial differential equations that utilize dimensionless variables and groups, which are products of the characteristic quantities and fluid properties, including density, viscosity, and thermal conductivity.

This procedure illustrates the relationship between the local heat transfer coefficient (α) and the temperature field, as demonstrated by equation (1.25) The characteristic length, represented as L0 (for instance, the tube diameter), defines the dimensionless distance from the wall, denoted as y+ = y/L0 To make the temperature (ϑ) dimensionless, it is divided by a characteristic temperature difference (∆ϑ0) Since only temperature differences or derivatives are relevant, the dimensionless temperature is derived by subtracting a reference temperature (ϑ0) from ϑ, resulting in ϑ+ = ϑ - ϑ0.

The choice ofϑ 0 is adapted to the problem and ﬁxes the zero point of the dimensionless temperatureϑ + Then (1.25) gives α=−λ

The Nusselt number is defined as the product of the heat transfer coefficient (α), the characteristic length (L₀) of the specific problem, and the thermal conductivity (λ) of the fluid This dimensionless expression applies to both sides of equation (1.38).

The numerical values and subsequent figures are named in honor of distinguished researchers, specifically W Nusselt, and are represented by the first two letters of their surnames in mathematical formulas.

Wilhelm Nusselt (1882–1957) was appointed Professor of Theoretical Mechanical Engineering at the Technische Hochschule, Karlsruhe in 1920 and later taught at the Technische Hochschule, Munich from 1925 to 1952 He is renowned for his 1915 publication, "The Fundamental Laws of Heat Transfer," where he introduced dimensionless groups, marking a significant advancement in the field Nusselt's research also encompassed critical areas such as heat transfer in film condensation, cross current heat transfer, and the relationship between heat and mass transfer during evaporation.

The calculation of α is closely linked to the determination of the Nusselt number, which is influenced by the dimensionless temperature field To clarify the dimensionless numbers that define the dimensionless temperature ϑ+, we will first compile a list of relevant physical quantities This approach will precede the use of fundamental differential equations, which will be addressed in Chapter 3.

The dimensionless temperatureϑ + from (1.37) depends on the dimensionless space coordinates x + :=x/L 0 , y + :=y/L 0 , z + :=z/L 0 and a series of other dimensionless numbersK i : ϑ + =ϑ + x + , y + , z + , K 1 , K 2 , (1.40)

Dimensionless numbers play a crucial role in heat transfer, particularly in the interaction between a flowing fluid and the inner wall of a tube characterized by its diameter (d) and length (L) One significant dimensionless number is the ratio L/d, which helps identify geometrically similar tubes While geometric parameters are important, this discussion will focus on dimensionless numbers that influence velocity and temperature fields, independent of geometry.

The velocity field of a fluid is influenced by the characteristic length \( L_0 \), entry velocity \( w_0 \), density, and viscosity \( \eta \) While density affects frictionless flow, viscosity is crucial for frictional flow and boundary layer development Additionally, the thermal properties of the fluid, specifically thermal conductivity \( \lambda \) and specific heat capacity \( c_p \), play a significant role in determining the temperature field, particularly in relation to the characteristic temperature difference \( \Delta \theta_0 \) The specific heat capacity connects the fluid's enthalpy to its temperature.

With this we have seven quantities, namely

The temperature field, along with the heat transfer coefficient and the Nusselt number, relies on the parameters L, w, η, ∆ϑ, λ, and c_p Dimensionless groups K_i are formed from power products of these seven quantities, utilizing appropriately selected exponents to establish their relationships.

The dimensions of seven key physical quantities can be expressed as a power product of the four fundamental dimensions: length (L), time (T), mass (M), and temperature (Θ) These dimensions are essential for accurately describing thermodynamics and heat transfer For instance, density is defined as the ratio of mass to volume, represented by the dimension mass divided by length cubed, which is expressed as dim = M/L³.

Thermal radiation

All matter emits energy in the form of thermal radiation due to its positive thermodynamic temperature This energy release occurs as internal energy converts into electromagnetic waves, which can be absorbed, reflected, or transmitted when they encounter other matter The absorbed radiation transforms into internal energy, facilitating a unique heat transfer known as radiative exchange Notably, electromagnetic waves can traverse a vacuum, enabling heat transfer over long distances, such as the significant energy flow from the sun to the earth In gases and liquids, thermal radiation is partially transparent, resulting in volumetric emission and absorption, while in solids, these processes occur primarily at the surface, with radiation being absorbed within a thin layer Thus, it is more accurate to refer to radiating and absorbing surfaces rather than solid bodies when discussing thermal radiation.

The emission of heat radiation has an upper limit determined solely by the thermodynamic temperature (T) of the radiating body The maximum heat flux from the surface of this body is expressed by the equation ˙ q s = σT^4, where σ represents the Stefan-Boltzmann constant.

The Stefan-Boltzmann law, discovered by J Stefan in 1879 and later derived by L Boltzmann in 1884, is rooted in the electromagnetic theory of radiation and the second law of thermodynamics This law features a universal constant, the Stefan-Boltzmann constant (σ), valued at approximately σ = (5.67040±0.00004) × 10⁻⁸ W/m² K⁴.

Josef Stefan (1835–1893) was appointed Professor of Physics at the University of Vienna in 1863, where he conducted significant research and published extensively on topics such as heat conduction, fluid diffusion, ice formation, and the relationship between surface tension and evaporation His meticulous analysis of previous experiments led him to propose the T⁴ law, which describes the emission of heat from hot bodies.

11 Ludwig Boltzmann (1844–1906) gained his PhD in 1867 as a scholar of J Stefan in Vienna.

He served as a physics professor in Graz, Munich, Leipzig, and Vienna, specializing in the kinetic theory of gases and its connection to the second law of thermodynamics In 1877, he established the essential relationship between a system's entropy and the logarithm of the number of molecular distributions that constitute the macroscopic state of that system.

A black body is defined as an ideal emitter with maximum emissive power, or heat flux, denoted as ˙q s, which cannot be exceeded by any other body at the same temperature This type of body also acts as an ideal absorber, capturing all incident radiation In contrast, real radiators have their emissive power described using a correction factor, leading to the definition of emissivity ε(T) in the equation ˙q = ε(T)σT^4, where ε(T) is always less than or equal to 1 The emissivity of a material is influenced not only by its composition but also by surface conditions, such as roughness, with various values for ε compiled in Table 1.3.

Table 1.3: Emissivity ε(T ) of some materials

Wood, oak 293 0.90 Iron, shiny corroded 423 0.158

When radiation interacts with a surface, it can be reflected, absorbed, or transmitted, represented by the dimensionless quantities reflectivity (r), absorptivity (a), and transmissivity (τ) These properties are influenced not only by the material itself but also by the type of radiation and its wavelength distribution Importantly, the relationship between these quantities is expressed by the equation r + a + τ = 1.

Most solid bodies are opaque, they do not allow any radiation to be transmitted, so withτ = 0 the absorptivity from (1.60) isa= 1−r.

Chapter 5 will provide a detailed exploration of thermal radiation absorption, including the relationship between emission and absorption as described by Kirchhoff’s law This principle states that a good emitter of radiation is also an effective absorber In the case of the ideal radiator, known as a black body, both absorptivity (a) and emissivity (ε) reach their maximum value of one A black body not only absorbs all incident radiation (a = 1) but also emits more radiation than any other type of radiator, in accordance with the Stefan-Boltzmann law.

Radiative exchange

In heat transfer, the exchange of radiation between two bodies at different temperatures is crucial, as both the hotter and colder bodies emit electromagnetic waves that facilitate energy transfer The net heat flow from the hotter body to the cooler one is of primary interest, though evaluating this flow poses challenges Factors such as the presence of other bodies in the environment, reflection of radiation, and absorption by the medium between the two bodies complicate the process This complexity is particularly evident in scenarios like gas radiation, which must be accounted for in applications such as heat transfer within a furnace.

Fig 1.11: Radiative exchange between a body at temperature T and black surroundings at temperature

In this article, we will examine a basic case of radiative exchange involving a radiator with area A and temperature T, situated in an environment at temperature TS The medium between the radiator and the surroundings is assumed to be completely transparent to radiation, which closely resembles the behavior of atmospheric air Additionally, the surroundings are treated as a black body, with an absorption coefficient of aS = 1, allowing for a simplified analysis of radiation transfer.

The heat ﬂow emitted by the radiator

The equation Q˙ em = A ε σ T^4 describes how a radiator interacts with its black surroundings, where it absorbs some of the black radiation emitted by the environment at temperature T The remaining radiation is reflected back, contributing to the heat absorbed by the radiator.

The equation Q˙ ab = A a σ T S^4 describes the relationship between the absorptivity (a) of a radiator at temperature (T) and the black body radiation from the surroundings at temperature (T S) It is important to note that absorptivity is influenced not only by the material properties of the absorbing surface but also by the characteristics of the incident radiation source In this case, the incident radiation is the black body radiation, which is fully defined by the surrounding temperature (T S).

The net ﬂow of heat ˙Q, from the radiator to the surroundings enclosing it is

A common assumption in thermal radiation analysis is to treat the radiator as a grey radiator, which simplifies calculations significantly This approximation posits that the absorptivity of a grey radiator remains constant, regardless of the type of incident radiation, and is always equal to its emissivity, expressed as a=ε.

For a grey radiator in black surroundings (1.61) simpliﬁes to

The difference in the fourth power of the emitter's temperature and the receiving body's temperature is a key factor in radiative heat exchange This temperature relationship is prevalent in various problems related to radiative heat transfer involving grey radiators.

In various applications, it is essential to consider heat transfer by convection alongside radiative heat transfer For instance, when a radiator emits heat into a cooler room, radiative heat exchange occurs between the radiator and the room's walls, while heat is simultaneously transferred to the air through convection These two heat transfer mechanisms operate in parallel, necessitating the addition of convective and radiative heat flows to determine the total heat exchanged The heat flux can be expressed as ˙q = ˙q conv + ˙q rad, or ˙q = α(T−T A) + εσ(T^4 − T S^4), where α represents the heat transfer coefficient for convective heat transfer to air at temperature T A, and equation (1.62) is utilized to evaluate ˙q rad.

NormallyT A ≈T S , allowing the convective and radiative parts of heat transfer to be put together This gives ˙ q= (α+α rad ) (T−T S ) (1.64) The radiative heat transfer coeﬃcient deﬁned above, becomes from (1.63) α rad =εσT 4 −T S 4

T−T S =εσ(T 2 +T S 2 ) (T+T S ) (1.65) This quantity is dependent on the emissivity ε, and both temperatures T and

The introduction of α rad enables a comparison between the effects of radiation and convection in heat transfer Given that ε is always less than or equal to 1, it is clear that there is an upper limit to the radiative component of heat transfer.

In a large room with still air at a temperature of 18 °C, a poorly insulated horizontal pipe with an outer diameter of 0.100 m and a surface temperature of 44 °C experiences heat loss To determine the heat loss per unit length of the pipe, it is assumed that the pipe behaves as a grey radiator with an emissivity of 0.87, while the room's walls are considered black surroundings, also at 18 °C.

The tube gives oﬀ heat to the air by free convection and to the surrounding walls by radiation Then from (1.64) with ϑ S = ϑ A comes

Q/L ˙ = πd ( ˙ q conv + ˙ q rad ) = πd (α m + α rad ) (ϑ W − ϑ A ) , (1.66) where α m is the mean heat transfer coeﬃcient for free convection The heat transfer coeﬃcient for radiation, using the data given and (1.65) is α rad = εσ T W 4 − T A 4

For heat transfer by free convection from a horizontal pipe S.W Churchill and H.H.S Chu [1.3] give the dimensionless relationship

(1.67) which has the same form as Eq (1.56), that of N u m = f (Gr, P r) According to (1.57) the Grashof number is

The expansion coefficient β for air temperature ϑ A is calculated as β = 1/T A = 0.00344 K −1, treating air as an ideal gas To account for temperature-dependent material properties, the values of ν, λ, and P r must be calculated at the mean temperature ϑ m = 1/2 (ϑ W + ϑ A) At a mean temperature of 31 °C, the calculated values are ν = 16.40 × 10 −6 m²/s, λ = 0.0265 W/Km, and P r = 0.713.

Using (1.67) the Nusselt number is N u m = 18.48, out of which comes the heat transfer coeﬃcient α m = N u m λ d = 18.48 0.0265 W/Km

0.100 m = 4.90 W m 2 K Eq.(1.66) gives the heat loss as ˙ Q/L = 85.4 W/m.

The heat transfer coefficients for free convection to air and radiative exchange, α m and α rad, are approximately equal, indicating they transport similar amounts of heat In contrast, for forced convection, α m can be significantly larger—by one to two orders of magnitude—depending on flow velocity, while α rad remains constant and is typically negligible when compared to α m.

Overall heat transfer

The overall heat transfer coeﬃcient

In the scenario depicted in Fig 1.12, a flat or curved wall divides two fluids with temperatures ϑ 1 and ϑ 2, where ϑ 2 is lower than ϑ 1 At steady state, heat transfer, denoted as ˙Q, occurs from fluid 1 to fluid 2 due to the temperature gradient (ϑ 1 − ϑ 2) This heat flow passes from fluid 1 through the wall, which has an area A 1 and is maintained at a temperature ϑ W1 The heat transfer coefficient, represented as α 1, plays a crucial role in this process, as established in section 1.1.3.

Q˙ =α 1 A 1 (ϑ 1 −ϑ W1 ) (1.68) For the conduction through the wall, according to section 1.1.2

The mean thermal conductivity of the wall, denoted as λ m, is determined by its thickness, δ, and the average area, A m, calculated from the relevant equation Additionally, a similar relationship to the heat transfer equation exists for the transfer of heat from the wall to fluid 2.

Fig 1.12: Temperature proﬁle for heat transfer through a tube wall bounded by two ﬂuids with temperatures ϑ 1 and ϑ 2 < ϑ 1

The unknown wall temperatures ϑ W1 and ϑ W2 , can be eliminated from the three equations for ˙Q This means that ˙Qcan be calculated by knowing only the ﬂuid temperaturesϑ 1 andϑ 2 This results in

The overall heat transfer coefficient \( k \) for a reference area \( A \) is defined by equation (1.71) According to equation (1.72), the product \( kA \) can be determined using previously introduced quantities related to convective heat transfer and conduction.

The product kA is essential in understanding thermal insulation, as indicated by Eqs (1.71) and (1.72) Providing the value of k without specifying the area A renders it meaningless, since k varies based on the chosen area In practice, specific values for k are often utilized, such as those outlined in German building regulations (Norm DIN 4108), which establish minimum k values for house walls to ensure adequate insulation This specification of k is implicitly linked to a defined area, such as the surface area of flat walls or the outer surface of tubes.

In this article, we focus on the product kA, which typically shows minimal variation from A1 or Am For simplicity, we will refer solely to kA, and in rare instances, specific values for k will be provided alongside the corresponding area.

The overall heat transfer resistance, represented as (1/kA), consists of individual resistances from each transfer process in series This includes the convective transfer resistance between fluid 1 and the wall, the conduction resistance within the wall, and the convective transfer resistance between the wall and fluid 2 This series approach parallels electrical circuits, where total resistance is the sum of individual resistances Consequently, the heat flow, ˙Q, encounters three resistances: the boundary layer resistance in fluid 1, the conduction resistance in the wall, and the boundary layer resistance in fluid 2.

The temperature drop caused by thermal resistances mirrors the behavior of voltage drop in electrical resistors, increasing with higher resistance and stronger current This relationship is evident in the equations from (1.68) to (1.72).

From this the temperature drop in the wall and the boundary layers on both sides can be calculated To ﬁnd the wall temperatures the equations ϑ W1 =ϑ 1 − kA α 1 A 1 (ϑ 1 −ϑ 2 ) =ϑ 1 − Q˙ α 1 A 1 (1.74) and ϑ W2 =ϑ 2 + kA α 2 A 2 (ϑ 1 −ϑ 2 ) =ϑ 2 +

For theoverall heat transfer through a pipe, (1.72) can be applied, when it is taken into account that a pipe of diameter dand lengthL has a surface area of

A=πdL Then from (1.72) it follows with A m from (1.15) that

(1.76) whered 1 is the inner andd 2 the outer diameter of the pipe.

Multi-layer walls

The analogy of electrical circuits is applied to analyze overall heat transfer in multi-layer walls, commonly used in technical applications An example is the incorporation of an insulating layer made from a material with low thermal conductivity The temperature profile of such a wall, illustrated in Fig 1.13, demonstrates that the heat transfer resistance for each layer in series accumulates, resulting in the total heat transfer resistance for the entire wall.

In curved walls, the average area of a layer, denoted as A mi, is determined by calculating the inner and outer areas of the section To assess the overall resistance to heat transfer, it is assumed that each layer is in close contact with its neighbor, resulting in negligible temperature differences between them.

The temperature profile for overall heat transfer through a three-layer flat wall made of different materials must account for thermal contact resistance, akin to the contact resistance found in electric circuits.

The temperature difference between adjacent layers, represented as ϑ i − ϑ i+1, is directly proportional to the heat flow and the resistance to conduction, similar to the voltage drop across a resistor in an electrical circuit Consequently, this relationship can be expressed as ϑ i − ϑ i+1 = δ i λ mi A mi.

Using ˙Qfrom (1.71) and (1.77),ϑ i −ϑ i+1 is fairly simple to calculate The surface temperaturesϑ W1 andϑ W2 are calculated in exactly the same way as before using (1.74) and (1.75)

In tubes which consist of several layers e.g the actual tube plus its insulation, (1.77) is extended to

Thei-th layer is defined by the diameters d i and d i+1, with the first and last layers in contact with the fluid potentially accumulating dirt or scale over time This buildup can create additional conductive resistance, hindering effective heat transfer during prolonged operation.

Overall heat transfer through walls with extended surfaces

The overall resistance to heat transfer (1/kA) is calculated by summing all resistances related to convective heat transfer and conduction The largest resistance significantly influences the value of (1/kA), particularly when other resistances are minimal Enhancing wall insulation can be achieved by adding a thick layer of material with high resistance to conduction (δ/λ m A m), effectively improving thermal performance.

To optimize heat transfer in applications like heat exchangers, it is crucial to minimize high convective resistance (1/αA), primarily caused by a low heat transfer coefficient (α) Despite increasing flow velocity, α shows minimal improvement, necessitating an alternative approach The most effective solution is to enhance the heat transfer area (A) by incorporating extended surfaces such as fins, rods, or pins on the less efficient heat transfer side This method can significantly increase the original area, sometimes by factors of 10 to 100, thereby improving overall heat transfer performance.

Fig 1.14: Examples of extended surface, a straight ﬁns, b pins, c circular ﬁns

The overall heat transfer resistance does not decrease uniformly; while increasing the heat transfer area can enhance performance, it also introduces additional conduction resistance Heat absorbed by the fluid near the fin tip must be conducted from the fin base to the tip, necessitating a temperature difference that serves as a driving force Consequently, fins and other extended surfaces maintain an average temperature lower than that of the base material, resulting in reduced effectiveness This leads to a smaller temperature difference for heat transfer between the fins and the fluid compared to that between the base material and the fluid.

To calculate the eﬀectiveness of extended surfaces we consider Fig 1.15 The

Fig 1.15: Temperature in a finned wall along the line AB ϑ f is the average temperature of the fin. heat flow ˙Qinto Fluid 2 is made up of two parts:

The heat ﬂow ˙Q g removed from the surfaceA g of the ﬁnless base material (ground material) is

The heat transfer rate from the fin to the fluid is expressed by the equation Q˙ g = α g A g (ϑ W2 − ϑ 2), where α g represents the heat transfer coefficient The temperature gradient within the fin decreases from ϑ 0 at the base (x = 0) to ϑ h at the tip (x = h) The average fin temperature, denoted as ϑ f, plays a crucial role in determining the heat flow Q˙ f that is transferred to the fluid across the fin surface area A f.

Q˙ f =α f A f (ϑ f −ϑ 2 ) , whereα f is the (average) heat transfer coefficient between fin and fluid.

If the ﬁn was at the same temperature all over as at its baseϑ 0 , then the heat ﬂow would be given as

Q˙ f0 =α f A f (ϑ 0 −ϑ 2 ) The effectiveness of the fin is described by thefin efficiency η f :Q˙ f

Q˙ f0 = ϑ f −ϑ 2 ϑ 0 −ϑ 2 (1.81) and it then follows that

The fin efficiency, represented by the equation Q˙ f = α f η f A f (ϑ 0 − ϑ 2), is always less than one and is influenced by the interplay between conduction processes within the fin and convective heat transfer Key factors affecting fin efficiency include the fin's geometry, thermal conductivity (λ f), and the heat transfer coefficient (α f) A detailed discussion on these elements will be provided in section 2.2.4.

The temperature at the fin base (ϑ0) differs from the surface temperature (ϑW2) where fins are absent The heat flux entering the fin base is considerably greater than the flux from the base material into the fluid This temperature drop beneath the fin leads to the development of a periodic temperature profile in the base material, as illustrated in Fig 1.16.

As a simpliﬁcation this complicated temperature change is neglected, such that ϑ 0 =ϑ W2 , (1.83)

The periodic temperature profile along line CD indicates that the average temperature at the fin base (ϑ0) and the average temperature of the surface between the fins (ϑW2) are assumed to be isothermal This simplification, however, can lead to an overestimation of heat flow, potentially calculating values up to 25% higher than actual measurements This phenomenon was initially highlighted by researchers O Krischer and W Kast, and further validated by E.M Sparrow and D.K Hennecke, along with additional studies by E.M Sparrow.

L Lee [1.6] In many cases this error will be less than 5%, particularly if the ﬁns are thick and placed very close together We assume (1.83) to be valid and from (1.80) and (1.82), using this simpliﬁcation, we obtain

The fin is only effective over a reduced surface area, η f A f, rather than its total surface area A f, due to fin efficiency Typically, A f is less than A g, making the second term in the first bracket of equation (1.84) the most significant, even though η f is less than 1 Consequently, it is reasonable to approximate α g as α f without significant error.

The overall resistance of a ﬁnned wall, taking into account (1.68), (1.69) and (1.71), is given by

The overall heat transfer for finned walls can be determined using the same formulas as for unfinne walls, with the key difference being that the surface area of the fins is replaced by the fin area multiplied by the fin efficiency Here, δ represents the thickness, λ denotes the average thermal conductivity, and A refers to the average area of the wall without fins.

In this example, we analyze a pipe constructed from an aluminum alloy with a thermal conductivity of 205 W/K m The pipe features an inner diameter of 22 mm and an outer diameter of 25 mm Inside the pipe, water at a temperature of 60 °C flows, while outside, air at 25 °C flows perpendicularly to the pipe's axis The heat transfer coefficients are given as 6150 W/m² K for the water and 95 W/m² K for the air The objective is to calculate the heat flow per unit length of the pipe, denoted as ˙Q/L.

The overall heat transfer resistance (1/kA) is primarily influenced by the convection resistance on the pipe's exterior, which is typically the highest value This resistance can be effectively minimized by the addition of fins.

In this study, annular disc fins with an outer diameter of 60 mm, a thickness of 1 mm, and a separation of 6 mm were selected The number of fins is determined by the formula n = L/t f, where L represents the length of the tube Additionally, the analysis includes the finless outer surface of the tube.

A g = π d 2 (L − n δ f ) = π d 2 L (1 − δ f /t f ) and the surface area of the ﬁns is

The negligible contribution of the narrow surface at the top of the fin, which has a width of δ f and is only slightly warmer than the surrounding air, allows us to focus on the overall heat transfer In this context, the surface area of the finned tube significantly exceeds that of a tube without fins, enhancing its efficiency in heat transfer.

A 0 = πd 2 L, is increased by a factor of (A g + A f )/A 0 = 10.75.

The fin surfaces are not completely effective for heat transfer Taking α f = α 2 and a fin efficiency of η f = 0.55 gives according to (1.86)

The primary resistance to heat transfer occurs on the outer surface of the tube; however, the addition of fins has notably decreased this resistance The heat flow rate is measured at Q/L ˙ = 1472 W/m By increasing the surface area by a factor of 10.75, the heat flow has been enhanced by a factor of 5.75.

Heating and cooling of thin walled vessels

The principles governing steady heat flow can be utilized to address transient heat transfer issues, specifically in determining the temperature variations over time in a thin-walled vessel containing a liquid during heating and cooling processes This approach requires two key simplifications to achieve accurate results.

1 The temperature of the liquid inside the vessel is the same throughout, it only changes with time: ϑ F =ϑ F (t).

2 The heat stored in the vessel wall, or more precisely the change in its internal energy, can be neglected.

In most cases, free or forced convection caused by an agitator in a vessel results in a uniform temperature throughout the liquid However, this uniformity only holds true when the heat capacity of the liquid significantly exceeds that of the vessel walls This scenario typically occurs in the heating and cooling of liquids within thin-walled vessels, but it does not apply to vessels containing gases, which may have thick or well-insulated walls.

When both assumptions hold true, the fluid's temperature remains spatially uniform at all times, allowing for the prediction of wall temperature using steady-state equations In a flat vessel wall, temperature varies linearly, with the linear gradient shifting over time.

Fig 1.17: Temperature proﬁle for the cooling of a thin walled vessel

The cooling process involves the transfer of heat flow \( Q(t) \) from the liquid at temperature \( \dot{ϑ}_F(t) \) through the vessel wall to the surrounding environment, which maintains a constant temperature \( ϑ_S \).

The overall heat transfer coefficient (k) can be determined using equation (1.72) According to the first law of thermodynamics, the heat (Q) exiting the liquid leads to a decrease in the internal energy (U_F) of the fluid contained within the vessel.

Here,M F is the mass and c F is the speciﬁc heat capacity of the liquid, which is assumed to be constant.

The ordinary diﬀerential equation for the liquid temperature follows on from (1.87) and (1.88): dϑ F dt + kA

M F c F (ϑ F −ϑ S ) = 0 The solution with the initial conditions ϑ F =ϑ F0 at time t= 0 becomes, in dimensionless form ϑ + F := ϑ F −ϑ S ϑ F0 −ϑ S = exp

The temperature of the liquid decreases exponentially from its initial value (ϑ F0) to the surrounding temperature (ϑ S) As illustrated in Fig 1.18, the temperature plots vary based on different decay times, represented by t 0 := M F c F /kA This decay time is depicted in Fig 1.18 as the subtangent to the curve at any given moment, particularly at time t = 0.

Fig 1.18: Liquid temperature ϑ + F variation over time according to (1.89) with t 0 from (1.90) during the cooling of a vessel

Theheating of the vessel contents shall begin at timet = 0, when the whole container is at the same temperature as the surroundings: ϑ F =ϑ S for t= 0 (1.91)

A time-dependent heat flow, denoted as ˙Q H(t), is introduced to the liquid for all t ≥ 0 The heat loss through the thin walls of the vessel is calculated using equation (1.87) These factors are incorporated into the balance equation dU F/dt = −Q(t) + ˙˙Q H(t), leading to the differential equation dϑ F/dt + kA.

M F c F Its general solution, with the initial condition from (1.91), is ϑ F =ϑ S + exp(−t/t 0 ) t 0

If the heat load ˙Q H is assumed to beconstant, it follows from (1.92) that ϑ F =ϑ S +

After a long period of time has elapsed (t→ ∞), the temperature of the liquid reaches the value ϑ F∞ =ϑ S +

Q˙ H kA Then the heat ﬂow added just counterbalances the heat loss through the wall ˙Q from (1.87): a steady state is reached.

Heat exchangers

Types of heat exchanger and ﬂow conﬁgurations

The double pipe heat exchanger, depicted in Fig 1.19, features a straightforward design comprising two concentric tubes In this configuration, fluid 1 circulates through the inner pipe while fluid 2 moves through the annular space between the tubes This heat exchanger can operate under two distinct flow regimes: counter-current, where the fluids flow in opposite directions (illustrated in Fig 1.19a), or co-current, as shown in Fig 1.19b.

Figure 1.19 illustrates the cross-sectional mean values of fluid temperatures ϑ 1 and ϑ 2 throughout the heat exchanger The entry temperatures are marked with one dash, while the exit temperatures are indicated with two dashes In every cross-section, fluid 1 is consistently hotter than fluid 2 (ϑ 1 > ϑ 2) In a countercurrent flow configuration, the two fluids exit the tube at opposite ends, allowing for the possibility that the exit temperature of the hot fluid may be lower than that of the cold fluid (ϑ 1 < ϑ 2) This condition is valid as long as ϑ 1 remains greater than ϑ 2 Conversely, cocurrent flow does not permit significant cooling of fluid 1 or substantial heating of fluid 2, as both fluids exit at the same end of the heat exchanger.

In a double-pipe heat exchanger, fluid temperatures ϑ 1 and ϑ 2 demonstrate that ϑ 1 is always greater than ϑ 2, highlighting the superiority of countercurrent flow over cocurrent flow This advantage means that certain heat transfer tasks achievable in countercurrent flow cannot be accomplished in cocurrent flow Furthermore, as discussed in section 1.3.3, a countercurrent heat exchanger requires a smaller surface area to transfer the same amount of heat compared to a cocurrent exchanger, provided both flow regimes are suitable for the task Consequently, cocurrent flow is rarely utilized in practical applications.

The shell-and-tube heat exchanger, depicted in Fig 1.20, is widely recognized as the most commonly utilized design in practical applications In this system, one fluid circulates through the numerous parallel tubes that form a tube bundle, which is encased by another fluid.

Fig 1.20: Shell-and-tube heat exchanger (schematic)

Fig 1.21: Shell-and-tube heat exchanger with baﬄes

A plate heat exchanger with a shell design facilitates crossflow, where the second fluid circulates around the outside of the tubes While countercurrent flow is possible, it is limited to the ends of the heat exchanger where the shell-side fluid enters or exits The incorporation of baffles enhances this process by directing the shell-side fluid to flow perpendicular to the tube bundle, resulting in improved heat transfer coefficients compared to flow along the tubes In the spaces between the baffles, the fluid experiences crossflow rather than countercurrent or cocurrent flow.

Pure crossflow is a key feature of flat plate heat exchangers, where both fluid temperatures change perpendicularly to the flow direction Each fluid element experiences a unique temperature change, starting from a uniform entry temperature to an individual exit temperature This configuration is commonly utilized in shell-and-tube heat exchangers, particularly when one of the fluids is a gas In such systems, the gas flows crosswise around the tube rows, while the other fluid, typically a liquid, circulates inside the tubes.

Fig 1.23: Fluid temperatures ϑ 1 = ϑ 1 (x, y) and ϑ 2 = ϑ 2 (x, y) in crossﬂow

Fig 1.24: Coiled tube heat exchanger (schematic)

Fig 1.25: Regenerators for the periodic heat transfer between the gases, air and nitrogen (schematic)

ﬁns to the outer tube walls, cf 1.2.3 and 2.2.3, increases the area available for heat transfer on the gas side, thereby compensating for the lower heat transfer coeﬃcient.

A simple heat exchanger design features a coiled tube within a vessel, such as a boiler One fluid circulates through the tube, while another fluid is contained in the vessel, which can either flow through or remain stationary during heating or cooling The vessel typically includes a stirrer that enhances fluid mixing, thereby improving heat transfer efficiency to the coiled tube.

Various specialized designs for heat exchangers exist beyond the basic types, which are not covered in this article The three fundamental flow regimes—countercurrent, cocurrent, and crossflow—can be combined in multiple configurations, resulting in intricate calculation methods.

Heat exchangers typically involve two fluids flowing simultaneously, separated by a wall that facilitates heat transfer from the hotter to the colder fluid Known as recuperators, these devices differ from regenerators, which utilize packing materials like brick lattices or packed beds of stones and metal strips to allow gas passage In a regenerator, hot gas transfers heat to the packing material, storing it as internal energy, before the cold gas passes through, extracting heat and exiting at a higher temperature Continuous operation necessitates at least two regenerators, enabling one gas to be heated while the other cools, resulting in periodic changes in exit temperatures through alternating gas flows.

Regenerators play a crucial role in industrial processes, serving as air preheaters in blast furnaces and as heat exchangers in low-temperature gas liquefaction plants One notable design, the Ljungström preheater, utilizes rotating packing material to efficiently preheat air in firing equipment and gas turbine plants In this setup, warm exhaust gas from combustion is cooled to maximize energy recovery, enhancing overall efficiency in energy-intensive operations.

The regenerator theory, primarily developed by H Hausen, involves complex time-dependent calculations For those interested in exploring this topic further, it is recommended to refer to Hausen's summary and the VDI-Wärmeatlas.

General design equations Dimensionless groups

Figure 1.26 illustrates a heat exchanger scheme where the temperatures of the two fluids are represented as ϑ 1 and ϑ 2, with the assumption that ϑ 1 is greater than ϑ 2 This setup indicates that heat is transferred from fluid 1 to fluid 2, with entry temperatures marked by a single dash and exit temperatures by double dashes.

The first law of thermodynamics applies to both fluids, where heat transfer results in an increase in enthalpy for the colder fluid (fluid 2) and a decrease in enthalpy for the warmer fluid (fluid 1).

Q˙ = ˙M 1 (h 1 −h 1 ) = ˙M 2 (h 2 −h 2 ) , (1.93) where ˙M i is the mass flow rate of fluid i The specific enthalpies are calculated

Fig 1.26: Heat exchanger scheme, with the mass

The flow rates (˙M_i), entry temperatures (ϑ_i), exit temperatures (ϑ_i), entry enthalpy (h_i), and exit enthalpy (h_i) of both fluids (i = 1, 2) are determined at the respective entry and exit points These temperatures are averaged across the relevant tube cross-section and can be calculated as outlined in section 1.1.3, which discusses adiabatic mixing temperatures It is important to note that Equation (1.93) is applicable only to heat exchangers that are adiabatic in relation to their environment, a condition that will be consistently assumed.

In a heat exchanger, two fluids flow without experiencing a phase change, meaning they neither boil nor condense The minor variation in specific enthalpy due to pressure is disregarded, highlighting the significance of temperature dependence The mean specific heat capacity between two temperature points, denoted as \( c_{pi} = h_i - h_i / \theta_i - \theta_i \) for \( i = 1,2 \), can be derived from the relevant equations.

As an abbreviation theheat capacity ﬂow rateis introduced by

Temperature variations in fluids are interconnected, as dictated by the first law of thermodynamics, and are inversely proportional to the ratio of their heat capacity flow rates.

Heat flow (˙Q) is transferred from fluid 1 to fluid 2 in a heat exchanger due to the temperature difference (ϑ1 - ϑ2) This transfer must overcome the overall resistance to heat transfer, represented as 1/kA The term kA, known as the transfer capability of the heat exchanger, is a key characteristic of the system, calculated from the transfer resistances in the fluids and the conduction resistance of the wall between them While kA is often treated as a constant, variations in fluid properties and flow conditions can lead to changes in the fluid heat transfer coefficient (k) In such cases, it is necessary to calculate k and kA at different points within the heat exchanger to determine an appropriate mean value that accurately reflects the heat exchanger's characteristic transfer capability.

Before calculating heat exchanger design, it's essential to understand the key influencing factors These factors can be streamlined using dimensionless groups, leading to a more manageable analysis Ultimately, relevant design relationships will be established As illustrated in Fig 1.27, there are seven critical quantities that impact heat exchanger design, with the effectiveness of the heat exchanger primarily defined by its transfer capability, kA.

Fig 1.27: Heat exchanger with the seven quantities which aﬀect its design

In a heat exchanger, three key temperature differences significantly influence the performance: (ϑ1 - ϑ1), (ϑ2 - ϑ2), and (ϑ1 - ϑ2) These differences, along with the heat capacity flow rates (˙W1 and ˙W2) and the overall heat transfer coefficient (kA), form the essential parameters for analyzing heat exchange efficiency Consequently, the focus is narrowed down to six critical quantities that impact the system's effectiveness.

These belong to only two types of quantity either temperature (unit K) or heat capacity ﬂow rate (units W/K) According to section 1.1.4, that leaves four (= 6−

2) characteristic quantities to be deﬁned These are the dimensionless temperature changes in both ﬂuids ε 1 := ϑ 1 −ϑ 1 ϑ 1 −ϑ 2 and ε 2 := ϑ 2 −ϑ 2 ϑ 1 −ϑ 2 , (1.97) see Fig 1.29, and the ratios

The Number of Transfer Units (NTU) represents the dimensionless transfer capability of a heat exchanger We recommend defining N i as this capability, while N 2 refers to the ratio of the two heat capacity flow rates.

Fig 1.29: Plot of the dimensionless ﬂuid temperatures ϑ + i = (ϑ i − ϑ 2 ) / (ϑ 1 − ϑ 2 ) over the area and illustration of ε 1 and ε 2 according to (1.97) or its inverse

The four groups in (1.97) and (1.98), are not independent of each other, because applying the ﬁrst law of thermodynamics gives ε 1

The relationship which exists between the three remaining characteristic quantities

The operating characteristic of the heat exchanger, represented by F(ε 1 , N 1 , N 2 ) = 0 or F(ε 1 , N 1 , C 1 ) = 0, is influenced by the flow configuration This characteristic is determined by analyzing the temperature patterns of both fluids, which will be explored in greater detail in the subsequent sections.

Heat exchanger design mainly consists of two tasks:

1 Calculating the heat ﬂow transferred in a given heat exchanger.

2 Design of a heat exchanger for a prescribed performance.

In the first scenario, the values for ˙W 1, ˙W 2, and kA are provided To determine the heat flow ˙Q transferred, the temperature changes in both fluids must be calculated using equation (1.96) Given the characteristic numbers N 1 and N 2, or N 1 and C 1, this problem can be solved promptly if the operating characteristic in equation (1.102) can be explicitly expressed for ε 1, resulting in ε 1 = ε 1 (N 1, C 1).

The dimensionless temperature changeε 2 of the other ﬂuid follows from (1.101).

To design a heat exchanger, it is essential to determine the overall heat transfer coefficient, kA This can be achieved by knowing either the temperature variations in both fluids or the heat capacity flow rates along with the temperature change in one of the fluids An explicit solution for the operating characteristics N1 or N2 is required for accurate calculations.

N 1 =N 1 (ε 1 , C 1 ) This gives for the transfer capability kA=N 1 W˙ 1 =N 2 W˙ 2

Figure 1.30 illustrates the operating characteristic of a heat exchanger with a specific flow configuration, highlighting the solutions for both heat transfer and design calculations In many instances, obtaining an explicit solution for ε 1 and N 1 is challenging, even when an analytical expression exists In such cases, it is advisable to utilize a diagram akin to Figure 1.30 Additional information can be found in section 1.3.5.

The schematic representation in Fig 1.30 illustrates the operating characteristics of a heat exchanger with a constant heat capacity (C i) The point N is designated for heat transfer calculations, where the effectiveness (ε i) is dependent on the operating conditions (N i, C i) Additionally, point A serves as the reference for the design parameters of the heat exchanger.

N i = N i (ε i , C i ) The determination of the mean temperature diﬀerence Θ for point

The introduction of the heat capacity flow rate \( \dot{W}_i \) in the context of boiling and condensing fluids reveals that at constant pressure, a pure substance undergoing these phase changes does not experience a temperature change, leading to an infinite heat capacity (\( c_{pi} \to \infty \)) Consequently, this results in \( \epsilon_i = 0 \) and \( \dot{W}_i \to \infty \), which simplifies the heat exchanger calculations by reducing the relationship to two key variables, \( \epsilon \) and \( N_i \), instead of three.

N of the other ﬂuid, which is neither boiling nor condensing.

Countercurrent and cocurrent heat exchangers

The operating characteristic F(ε i , N i , C i ) = 0 for a countercurrent heat exchanger is determined by examining the temperature distribution in both fluids These findings can also be applied to the less significant case of a cocurrent heat exchanger.

In a countercurrent heat exchanger, temperature changes are influenced by the z-coordinate along the flow of fluid 1 By applying the first law of thermodynamics to a segment of length dz, the heat transfer rate, d˙Q, from fluid 1 to fluid 2 can be expressed as d˙Q = -M˙1 c p1 dϑ1 = -W˙1 dϑ1 and d˙Q = -M˙2 c p2 dϑ2 = -W˙2 dϑ2 To eliminate d˙Q, we utilize the overall heat transfer equation, d˙Q = k(ϑ1 - ϑ2) dA, which can be rewritten as d˙Q = kA(ϑ1 - ϑ2) dz.

Fig 1.31: Temperature pattern in a countercurrent heat exchanger

Table 1.4: Equations for the calculation of the normalised temperature variation ε i , the dimensionless transfer capability N i and the mean temperature diﬀerence Θ in counter and cocurrent heat exchangers

Meaning of the characteristic numbers: ε 1 = ϑ 1 − ϑ 1 ϑ 1 − ϑ 2 , ε 2 = ϑ 2 − ϑ 2 ϑ 1 − ϑ 2

L (1.111) for the temperature changes in both ﬂuids.

The temperatures of the two fluids, ϑ 1 and ϑ 2, will not be directly calculated from the differential equations Instead, the focus will be on determining the variation in the temperature difference, ϑ 1 − ϑ 2 By subtracting equation (1.111) from (1.110) and dividing by the temperature difference, we derive that the change in the temperature difference is proportional to the difference in N values, expressed as d(ϑ 1 − ϑ 2)/(ϑ 1 − ϑ 2) = (N 2 − N 1) dz.

Integrating this diﬀerential equation betweenz= 0 andz=Lleads to ln(ϑ 1 −ϑ 2 ) L (ϑ 1 −ϑ 2 ) 0 = lnϑ 1 −ϑ 2 ϑ 1 −ϑ 2 =N 2 −N 1 (1.113)

Fig 1.32: Operating characteristic ε i = ε i (N i , C i ) for countercurrent ﬂow from Tab 1.4 which gives ln1−ε 1

The operating characteristic of a countercurrent heat exchanger is represented by the implicit equation 1−ε 2 =N 2 −N 1, which remains unchanged when exchanging the indices 1 and 2 By utilizing the ratios of heat capacities C 1 and C 2 = 1/C 1, explicit equations are derived as ε i =f(N i , C i ) and N i =f(ε i , C i ) for both fluids, where i= 1,2 These explicit formulas for the operating characteristics are detailed in Table 1.4 Notably, if the heat capacity flow rates are equal, the equations maintain a consistent form.

W˙ 1 = ˙W 2 , and becauseC 1 =C 2 = 1, it follows that ε 1 =ε 2 =ε and N 1 =N 2 =N , and with a series development of the equations valid forC i = 1 towards the limit ofC i →1, the simple relationships given in Table 1.4 are obtained.

Figure 1.32 illustrates the operating characteristic ε i = f(N i , C i ) in relation to N i, with C i serving as a parameter It is evident that the normalized temperature change ε i increases steadily as N i rises, indicating an enhancement in transfer capability kA As N i approaches infinity, the limiting value is reached.

In a heat exchanger, if the coefficient of performance (IfC) is less than or equal to 1, the efficiency (ε i) is characterized by the normalized temperature change of the fluid with the smaller heat capacity flow The effectiveness of the heat exchanger can be optimized by increasing the heat transfer area (A), allowing for a minimal temperature difference between the two fluids, though this is only achievable at one end of the countercurrent exchanger.

In thermodynamics, the condition W˙ 1 = W˙ 2, where C 1 = C 2 = 1, indicates that an infinitely small temperature difference can be achieved across a heat exchanger by increasing the surface area This ideal scenario of reversible heat transfer between two fluids is only possible when W˙ 1 equals W˙ 2 in a heat exchanger designed for high transfer efficiency.

The function ε i = f(N i , C i ) is essential for calculating the outlet temperature and transfer capability of a heat exchanger To size a heat exchanger for a specific temperature change in the fluid, the alternative operating characteristic N i = N i (ε i , C i ) is utilized, as detailed in Table 1.4.

In acocurrent heat exchangerthe direction of ﬂow is opposite to that in Fig. 1.31, cf also Fig 1.20b In place of (1.108) the energy balance is d ˙Q= ˙M 2 c p2 dϑ 2 = ˙W 2 dϑ 2 , which gives the relationship d(ϑ 1 −ϑ 2 ) ϑ 1 −ϑ 2 =−(N 1 +N 2 ) dz

The equation L (1.115) replaces (1.112), indicating that the temperature difference between the two fluids decreases along the flow direction, as stated in (1.114) By integrating (1.115) from z=0 to z=L, we derive the relationship lnϑ 1 −ϑ 2 ϑ 1 −ϑ 2 =−(N 1 +N 2 ) This leads to the implicit operating characteristic expressed as ln [1−(ε 1 +ε 2 )] =−(N 1 +N 2 ) =−ε 1 +ε 2 Θ (1.116) This equation can be solved for ε i and N i, resulting in the functions listed in Table 1.4 As N i approaches infinity, the normalized temperature variation approaches a limiting value.

With cocurrent ﬂow the limiting value of ε i = 1 is never reached except when

C i = 0, as will soon be explained.

The performance and sizing calculations for a heat exchanger can be conducted using the mean temperature difference, Θ, as outlined in section 1.3.2 In countercurrent flow, the difference N2 - N1 is substituted with Θ, ε1, and ε2, leading to the expression Θ = Θ(ε1, ε2), which is detailed in Table 1.4.

Fig 1.33: Temperature in a condenser with cooling of superheated steam, condensation and subcooling of the condensate (ﬂuid 1) by cooling water

The mean temperature difference in a countercurrent heat exchanger is represented by (1.117), which calculates the logarithmic mean of the temperature differences between the two fluids at each end of the system For cocurrent flow, the normalized mean temperature difference Θ is provided in Table 1.4 By substituting the defining equations for ε 1 and ε 2 into (1.117), we can derive further insights into the heat exchange process.

So ∆ϑ m is also the logarithmic mean temperature diﬀerence at both ends of the heat exchanger in cocurrent ﬂow.

We will now compare the two ﬂow conﬁgurations ForC i = 0 the normalised temperature variation in Table 1.4 is ε i = 1−exp(−N i ) and the dimensionless transfer capability

The equation N i = −ln(1−ε i ) indicates that the flow configuration, whether countercurrent or cocurrent, does not affect the heat exchange process when one of the substances boils or condenses However, in a condenser scenario where superheated steam is first cooled to its condensation temperature and then fully condensed before cooling the condensate further, the situation becomes more complex In such cases, it is inappropriate to simplify the system to a single heat exchanger, as the interactions and temperature changes require a more detailed analysis beyond just the inlet and outlet temperatures.

The ratio of transfer capabilities in cocurrent and countercurrent flows is influenced by the heat capacity flow rate, with significant variations observed during the cooling of steam and condensate This relationship is depicted in Fig 1.34 and is further analyzed in relation to parameters ε i and C i ϑ i (where i = 1, 2), as shown in Fig 1.33.

In the condensation process, while W˙ 1 has a finite value, it becomes infinite To analyze this, the exchanger is conceptually divided into three units in series Energy balances help determine the two unknown temperatures, ϑ 2a and ϑ 2b, which correspond to the cooling and condensation sections, and the condensation and sub-cooling parts, respectively These temperatures lead to the calculation of dimensionless temperature differences, ε ia, ε ib, and ε ic, for the cooler, condenser, and sub-cooler sections Subsequently, the dimensionless transfer capabilities, N ia, N ib, and N ic, can be derived from established relationships From these values, the overall heat transfer coefficients, k j, are used to calculate the areas of the three sections, A j (where j = a, b, c), culminating in the total transfer area of the exchanger.

ForC i >0 the countercurrent conﬁguration is always superior to the cocurrent.

Cocurrent flow systems have limitations in their heat transfer capabilities, as certain temperature changes (ε i) can only be achieved under specific conditions.

1 +C i ln [1−ε i (1 +C i )] is positive This is only the case for ε i < 1

Crossﬂow heat exchangers

Before delving into pure crossflow, we first analyze the operating characteristics of a simplified crossflow scenario, where only one fluid undergoes lateral mixing In this configuration, the temperature of the first fluid, denoted as ϑ 1, is solely dependent on the position coordinate x, while the temperature of the second fluid varies with both x and y As illustrated in Fig 1.36, the laterally mixed fluid is indicated by the index 1, with its temperature changing only in the flow direction, following the relationship ϑ 1 = ϑ 1(x) We assume ideal lateral mixing, meaning that ϑ 1 remains constant in the y-direction, a condition that is closely approximated when fluid 1 passes through a single row of tubes.

Fig 1.37: Crossﬂow with one tube row as a realisation of the one side laterally mixed crossﬂow

In a one-sided laterally mixed crossflow, temperature variations are observed, where ϑ 1 represents the temperature of the laterally mixed fluid and ϑ 2 denotes the temperature of the perpendicular fluid flow This configuration, illustrated in Fig 1.36, features a single row of tubes, highlighting the dynamics of the mixed fluid in this crossflow setup.

1 in the tubes does not have to be the ﬂuid with the higher temperature, as was assumed before.

To find the temperatures of both fluids, ϑ 1 = ϑ 1 (x) and ϑ 2 = ϑ 2 (x, y), we analyze the surface element dA = dx dy as illustrated in Fig 1.36 The heat flow from fluid 1 to fluid 2 is expressed as d ˙ Q = [ϑ 1 (x) − ϑ 2 (x, y)] k dx dy The total heat transfer area is A = L 1 L 2 By utilizing the dimensionless coordinates x + := x/L 1 and y + := y/L 2, we can rewrite the heat transfer equation as d ˙ Q = [ϑ 1 (x + ) − ϑ 2 (x + , y + )] kA dx + dy.

The application of the first law of thermodynamics to fluid 2 flowing over the surface element dA results in a second relationship for the heat transfer rate, expressed as d ˙ Q The mass flow rate of fluid 2 is defined as d ˙ M 2 = ˙ M 2 dx/L 1 = ˙ M 2 dx +, leading to the equation d ˙ Q = ˙ M 2 dx + c p2 ϑ 2 + ∂ϑ 2.

Temperature variations within a strip of size L 2 dx during laterally mixed crossflow are illustrated in Fig 1.38, where ϑ m2 (x) represents the average temperature of fluid 2 in the y direction The relationships (1.122) and (1.123) are utilized to determine this temperature The resulting solution, expressed as ϑ 2 (x + , y + ) = ϑ 1 (x + ) − ϑ 1 (x + ) − ϑ 2 e −N 2 y + (1.125), still includes the unknown temperature ϑ 1 (x + ) of the laterally mixed fluid.

To determine ϑ 1 (x +), the first law of thermodynamics must be applied to fluid 1 As fluid 1 flows through a strip of width dx, it transfers a heat flow d ˙ Q ∗ to fluid 2, which differs from d ˙ Q, as illustrated in Fig 1.38 The relationship for d ˙ Q ∗ is as follows:

With x + from (1.121) it follows that

A second relationship for d ˙ Q ∗ is the equation for the overall heat transfer: d ˙ Q ∗ = ϑ 1 x +

The average temperature of fluid 2 in the y direction, denoted as ϑ 2 (x +, y +), is crucial for the overall heat transfer across the strip area L 2 dx This relationship leads to the ordinary differential equation dϑ 1/dx = -N 1 (ϑ 1 - ϑ m2), which is essential for determining the temperature ϑ 1 (x +).

Using (1.125) and (1.128) the average temperature of ﬂuid 2 can be calculated, giving ϑ m2 (x + ) = ϑ 1 (x + ) − 1

Which then yields from (1.129) the diﬀerential equation dϑ 1 dx + = − N 1

Integration between x + = 0 and x + = 1 delivers ϑ 1 − ϑ 2 ϑ 1 − ϑ 2 = exp

The temperature ratio on the left hand side agrees with (1 − ε 1 ).

The operating characteristic for one side laterally mixed crossﬂow is ε 1 = 1−exp

The temperature variation in the fluid during lateral mixing is dependent on the parameter C1 and the dimensionless transfer capability N1 By explicitly resolving the operating characteristic for N1, we can derive meaningful insights into the system's behavior.

The required value for kA can be calculated from the equation C1 ln [1 + C1 ln(1−ε1)] The temperature variation of the fluid flowing perpendicular to the tube row is determined using the formula ε2 = (ϑm2 - ϑ2) / (ϑ1 - ϑ2) = C1 ε1 In this equation, ϑm2 represents the mean outlet temperature, which can also be obtained by integrating the equation over x+.

Crossﬂow with a single row of tubes was dealt with by D M Smith [1.14] in

In 1968, H Schedwill expanded the work on the extension of tube rows, resulting in more complex equations compared to the single row case (n=1) The average outlet temperature, denoted as ε1, rises with an increasing number of tube rows For detailed equations, refer to sources [1.8] and [1.16].

An increase in the number of tube rows in series approaches a pure crossflow scenario, where the temperatures of both fluids vary with the dimensionless coordinates x+ and y+ The heat transfer through a surface element with dimensions dA = dx dy x + dy + can be determined using similar reasoning as in previous equations, leading to the relationship d ˙Q = -W˙ 1 ∂ϑ 1.

(1st law applied to ﬂuid 1), d ˙Q= ˙W 2 ∂ϑ 2

∂y + dx + dy + (1st law applied to ﬂuid 2), and overall heat transfer d ˙Q=kA(ϑ 1 −ϑ 2 ) dx + dy + Elimination of d ˙Qyields the two coupled diﬀerential equations

∂y + =N 2 (ϑ 1 −ϑ 2 ) (1.134b) for the temperaturesϑ 1 = ϑ 1 (x + , y + ) andϑ 2 = ϑ 2 (x + , y + ) These have to fulﬁll the boundary conditions ϑ 1 (0, y + ) =ϑ 1 and ϑ 2 (x + ,0) =ϑ 2 (1.135)

W Nusselt [1.17] used a power series to solve this problem With ξ:=N 1 x + = (kA/W˙ 1 )(x/L 1 ) (1.136a) and η:=N 2 y + = (kA/W˙ 2 )(y/L 2 ) (1.136b) the solution has the form ϑ 1 (ξ, η) ⎛

N 1 ξ=0 ϑ 2 (ξ, N 2 ) dξ the dimensionless temperature changes of both ﬂuids are given by ε i = 1

The ε-Θ diagram illustrates various flow configurations, including countercurrent, pure crossflow, laterally mixed crossflow, and cocurrent systems In these configurations, the symmetry of the problem allows for consistent relationships between variables for both i = 1 and i = 2 Although the equation for N i cannot be explicitly solved, H Martin [1.8] provides a concise computer program that effectively calculates the mean temperature difference (Θ = ε i / N i) and consequently ε i.

ForN i → ∞, pure crossﬂow delivers the limits

As the limit approaches infinity, the dimensionless temperature change, ε_i, behaves differently based on the flow configuration: it equals 1 for C_i ≤ 1 and 1/C_i for C_i > 1, consistent with countercurrent flow For C_i = 0, ε_i is determined by ε_i = 1 - e^(-N_i), aligning with previous findings This indicates that when one fluid is boiling or condensing, the temperature change of the other fluid remains unaffected by the flow configuration (cocurrent, countercurrent, or crossflow) Although temperature changes in crossflow systems are notably lower than in countercurrent exchangers, they outperform cocurrent flow A comparison of these configurations reveals that for a constant ratio of C_1 = 0.5, a prescribed temperature difference of ε_1 = 0.65 necessitates different dimensionless transfer capabilities: countercurrent (N_1 = 1.30), crossflow (N_1 = 1.50), and cocurrent (N_1 = 2.44) In a heat exchanger with N_1 = 3.0, the maximum dimensionless temperature change achieved is 0.874 for countercurrent flow, followed by 0.816 in crossflow, and 0.660 in cocurrent systems.

In a motor vehicle, the cooler functions as a crossflow heat exchanger, where the cooling medium flows through parallel finned tubes while air moves perpendicular to these tubes Given a volumetric flow rate of the cooling medium at 1.25 dm³/s, with a density of 1.015 kg/dm³ and a mean specific heat capacity of 3.80 kJ/kgK, the air enters the cooler at a temperature of 20.0 °C, with a flow rate of 1.100 m³/s, a density of 1.188 kg/m³, and a specific heat capacity of 1.007 kJ/kgK The cooler has a transfer capability of 0.550 kW/K and needs to transfer a heat flow of 28.5 kW to the air Using the energy balance equation, the temperatures of the motor cooling medium and the air exiting the cooler can be determined.

The relationship between heat transfer and temperature change can be expressed as Q ˙ = ˙ W 1 (ϑ 1 − ϑ 1) = ˙ W 2 (ϑ 2 − ϑ 2) The exit temperature of the air, ϑ 2, can be calculated using the formula ϑ 2 = ϑ 2 + ˙ Q/ W ˙ 2 Additionally, the temperature change in the cooling medium is given by ϑ 1 − ϑ 1 = ˙ Q/ W ˙ 1 By applying the defining equation for the normalized temperature change ε 1, the entry temperature of the cooling medium is determined to be ϑ 1 = ϑ 2 + (ϑ 1 − ϑ 1) ε 1 = ϑ 2 +.

W ˙ 1 ε 1 , (1.142) from which follows the cooling medium outlet temperature ϑ 1 = ϑ 1 − Q/ ˙ W ˙ 1 (1.143)

The operating characteristic for this case of crossﬂow with a single row of tubes (one side laterally mixed crossﬂow) according to (1.131), gives the value of ε 1 required in (1.142): ε 1 = 1 − exp

To evaluate the equations, the heat capacity ﬂow rates are calculated using the given data for the cooling medium,

W ˙ 1 = ˙ V 1 ρ 1 c p1 = 1.25 dm 3 s ã 1.015 kg dm 3 ã 3.80 kJ kgK = 4.821 kW

K From which the dimensionless numbers

C 1 N 1 = N 2 = kA/ W ˙ 2 = 0.418 are found From the operating characteristic according to (1.144) ε 1 = 0.0890 follows. This then yields the values for the temperatures of the cooling medium from (1.142) and (1.143) to be ϑ 1 = 86.4 ◦ C and ϑ 1 = 80.5 ◦ C

A relatively high temperature level of the cooling medium favourable for the working of the motor, is achieved The exit temperature of the air, from (1.141) is ϑ 2 = 41.7 ◦ C.

Operating characteristics of further ﬂow conﬁgurations Diagrams 63

In addition to countercurrent, cocurrent, and crossflow configurations, various other flow arrangements are utilized in industrial applications These alternative configurations have been extensively studied by numerous researchers, as highlighted in the compilation by W Roetzel.

B Spang [1.16] The operating characteristics F(ε i , N i , C i ) = 0 are often complex mathematical expressions, so it seems reasonable to represent the results graphically.

W Roetzel and B Spang [1.18] discussed the possibility of representing the operating characteristics in a graph, and came up with a clearly arranged diagram which can be found in the VDI-W¨armeatlas [1.16] This square shaped graph consists of two parts which are separated by the diagonal running from the bottom left to the top right hand corner, Fig 1.40 The axes of the graph are the two dimensionless temperature changesε 1 andε 2 from (1.97) The area above the diagonal contains lines of constant dimensionless transfer capabilityN 1 =kA/W˙ 1 = const as well as a host of straight lines that run through the origin, according to

The graph features only the endpoints marked along its edges to maintain clarity, without additional lines cluttering the visual Each point located in the triangle above the diagonal represents a specific operating state, allowing for the values of ε1, N1, and C1 to be easily extracted The corresponding value of ε2 can be determined by referencing the point on the abscissa.

In the triangular area below the diagonal, the operating characteristic is depicted as F(ε 2, N 2, C 2) = 0 The lines representing equal transfer capability, N 1 = N 2, converge at the diagonal where C 1 = C 2 = 1, exhibiting a kink due to equation (1.145) Notably, the line N 1 = N 2 approaching infinity does not display a kink at this juncture, indicating symmetrical flow conditions.

The ε 1, ε 2-diagram, as presented by W Roetzel and B Spang, illustrates configurations with constant N 1 and N 2 lines, such as counter and crosscurrent flow, where the operating characteristics for indices i=1 and i=2 exhibit symmetry about the diagonal In contrast, asymmetric flow scenarios, like laterally mixed crossflow, do not maintain this symmetry It is crucial to verify the indices for each fluid to avoid confusion in the results derived from the graph For comprehensive guidance on the design and construction of heat exchangers, refer to the VDI-Wärmeatlas.

The diﬀerent types of mass transfer

Diﬀusion

The calculation of mass transfer by diﬀusion requires several deﬁnitions and relationships which will be outlined in the following.

The composition of mixtures can be characterised in diﬀerent ways For a quantitative description the following quantities have to be introduced.

Themass fractionξ A is the massM A of component A over the total massM in a volume element within a phase 13 : ξ A := M A

The sum of all the mass fractions is

The mole fraction x˜ A is the number of moles N A of component A over the total number of molesN in the mixture in a given phase: ˜ x A := N A

The sum of all the mole fractions is

The molar concentration of substance A is deﬁned by c A :=N A /V (1.148)

13 The letter K under the summation sign means that the sum is taken over all the componentsK.

The molar concentration of the mixture is c:=N/V =

K c K , which gives for the mole fraction of component A ˜ x A =c A /c

For ideal gasesc A = p A /R m T and c = p/R m T, is valid, where p A = ˜x A p is the partial pressure of component A and R m = 8.31451 J/(mol K) is the molar gas constant.

These quantities in a composition are not independent from each other To ﬁnd a relationship between the mass and mole fractions, we multiply (1.147) by the molar mass ˜M A =M A /N A of component A This gives ˜ x A M˜ A =M A /N

Summation over all the components yields the average molar mass ˜M =M/N:

M K /N=M/N = ˜M (1.149) This then gives the following relationship between the mole and mass fractions ξ A = M A

In the reverse case, when the mass fractions are known, the mole fractions come from ˜ x A M˜

M˜ A ξ A , (1.151) in which the average molar mass is found from the mass fractions and the molar masses of the components as

In each volume element, the average particle velocities of different substances can vary, leading to overlapping convection and relative movement of particles, a phenomenon known as diffusion The average velocity of particles in substance A is represented by the vector \( w_A \) To describe diffusion, we introduce the relative velocity \( w_A - \omega \), with \( \omega \) being a yet-to-be-defined reference velocity The diffusional flux of substance A, measured in mol/m²·s, is defined as \( j_A = c_A (w_A - \omega) \).

The reference velocityωcan be chosen to be the velocity w at the centre of gravity of the mass This is deﬁned as the average mass velocity of a volume element: w:=

The diffusional flux can be expressed as j A = c A (w A−w), where c A represents the concentration of component A and w A is the weight of component A By multiplying this equation by the molar mass ˜M A, we derive the relationship c A M˜ A = A j AM˜ A = j ∗ A = A (w A−w), where j ∗ A denotes the mass-based diffusional flux of component A, measured in kg/m²·s This leads to further insights based on equations (1.154) and (1.155).

The reference system using the average mass velocitywfrom (1.154) is called the centre of gravity system The momentum and energy balances for this system are easily formulated.

The average molar velocityucan also be used as a further reference velocity.

The associated diﬀusional ﬂux (SI units mol/m 2 s) is u j A:=c A (w A−u) (1.158)

K c K = c it follows from (1.157) and (1.158) that

The particle reference system is defined by the average molar velocity, and various other reference systems and velocities are documented in the literature Additionally, the diffusional flux from one system can be effectively transferred to another, as demonstrated in the following example.

Example 1.6: The diﬀusional ﬂux of component A in a ’-reference system j A = c A ( w A − ω ) is given for a reference velocity ω = &

K ζ K w K , where for the “weighting factors” ζ K ,

The diﬀusional ﬂux of component A j A = c A ( w A − ω ) in -reference system with a reference velocity ω =

K ζ K = 1 has to be calculated The general relationship between the diﬀusional ﬂuxes j A and u j A has to be derived.

K (ζ K − ζ K ) w K It follows further from j A = c A ( w A −ω ) that the velocity is w A = j A /c A + ω Therefore j A − j A = c A &

K ζ K j K c K is yielded To change from the particle to the gravitational system we put in j A = j A , ζ A = ξ A and j A = u j A which give j A − u j A = − c A

Correspondingly the conversion of the diﬀusional ﬂux from the gravitational to the particle system gives u j A − j A = − c A

In a mixture of two substances, components A and B, these relationships are j A − u j A = − c A ξ A c A u j A + ξ B c B u j B

With u j A = − u j B and ξ A /c A − ξ B /c B = (V /M )( ˜ M A − M ˜ B ) it follows that j A − u j A = − c A V

Then because ˜ M = ˜ x A M ˜ A + ˜ x B M ˜ B the equation given above simpliﬁes to j A = u j A

The diﬀusional ﬂux of component A is proportional to the concentration gradient gradc A For the time being we will limit ourselves to a mixture of two components

Diffusion in a mixture of two components, A and B, occurs along a single coordinate axis, such as the y-axis According to Fick's first law, the diffusive flux can be expressed as \( u_j^A = -D_{AB} \frac{dc_A}{dy} \), where \( D_{AB} \) is the diffusion coefficient measured in square meters per second (m²/s) This equation holds true under the assumption that the molar concentration of the mixture remains constant, which is typically the case in constant pressure, isothermal mixtures of ideal gases When this assumption is relaxed, the general equation for the diffusion of two substances is given by \( u_j^A = -cD_{AB} \frac{d\tilde{x}_A}{dy} \) In a gravitational system, the corresponding relationship for diffusive flux in the y-direction is represented as \( j_A^* = -D_{AB} \frac{d\xi_A}{dy} \).

In a multicomponent mixture consisting ofN components the diﬀusional ﬂuxj ∗ A of component A is given by [1.23] j ∗ A N K=A K=1

From this equation (1.161) is obtained for the special case ofN= 2.

Example 1.7: It has to be shown that (1.161) is equivalent to (1.160).

The diﬀusional ﬂuxes u j A and j ∗ A in the y-direction are linked by j ∗ A = j A M ˜ A = u j A M ˜ B M ˜ A

M ˜ (see the solution to example 1.6) And with that j A ∗ = − c D AB ∂˜ x A

Where c = N/V = / M, and by diﬀerentiation of ˜ ˜ x A = ˜ M ξ A / M ˜ A with ˜ M = ˜ x A M ˜ A +

14 Adolph Fick (1829–1901), Professor of Physiology in Z¨ urich and W¨ urzburg, discovered the fundamental laws of diﬀusion.

Then from the equation given above j A ∗ = − D AB ∂ξ A

By exchanging the indices A and B in (1.161) the diﬀusional ﬂux for component

B in the binary mixture can be found As the sum of the two diﬀusional ﬂuxes disappears, according to (1.156), it follows that j A ∗ +j B ∗ =−D AB dξ A dy −D BA dξ B dy = 0

The diffusion coefficients of components A and B are equal, as indicated by the relationship dξ A /dy = −dξ B /dy, given that ξ A + ξ B = 1 This means that the diffusion coefficient of component A through component B is identical to that of component B through component A, allowing us to simplify notation by using D instead of D AB = D BA Typical diffusion coefficients range from 5 × 10^-6 to 10^-5 m²/s in gases.

Diffusion coefficients vary significantly across different states of matter, with values ranging from 10−10 to 10−9 m²/s in liquids and 10−14 to 10−10 m²/s in solids In gases, molecules move more freely, resulting in higher diffusion coefficients compared to liquids, which are in turn greater than those found in solids Consequently, diffusion occurs at a much slower rate in solids than in liquids, while gases exhibit the fastest diffusion rates.

The diffusional flux can be determined using Fick’s law when the concentration values are available Conversely, if the flux is already known, the concentration field can be derived through the integration of Fick’s law For illustration, consider a solid material where component A is to be extracted using a liquid solvent.

Fig 1.41: Diﬀusion through a quiescent liquid ﬁlm

The concentration of component A at the interface between the solid and liquid film is denoted as c A0, while the bulk flow concentration is represented by c Aδ Assuming a constant concentration, c = N/V, and considering that material movement occurs solely in the y-direction, the mass transfer direction can be simplified The molar flux from the solid to the liquid is expressed as ˙ n A = c A w A = u j A + c A u, where u j A = -D(d c A /dy) and u = x A w A + x B w B Given that the solvent's velocity, w B, in the y-direction is zero and with a negligible concentration of substance A in the solvent, the molar flux simplifies to ˙ n A = u j A = -D(d c A /dy).

At steady state, the influx of substance A into the liquid film equals its outflux, indicating that the rate of change of A remains constant (˙n A = const) Assuming a constant diffusion coefficient (D), integrating the relevant equation reveals that the rate of substance A is proportional to the difference in concentration between the bulk (c A0) and the film (c Aδ), expressed as ˙n A = -D(c A0 - c Aδ).

In a solid sphere of radiusr 0 , which is surrounded by a liquid film of thickness δ, the diffusional flux in the radial direction, according to (1.163), is ˙ n A = u j A =−Ddc A dr (1.164)

At steady state the molar ﬂow which diﬀuses through the spherical shell is constant,

N˙ A = ˙n A 4πr 2 =−Ddc A dr 4πr 2 , (1.165) and with that d ˙N A dr = 0 = d dr

Assuming D = const and taking into account the boundary conditions c A (r r 0 ) =c A0 andc A (r=r 0 +δ) =c Aδ the concentration proﬁle c A −c A0 c Aδ −c A0 = 1/r−1/r 0

1/(r 0 +δ)−1/r 0 (1.166) can be found by integration It corresponds to the temperature profile for steady conduction in a hollow sphere according to (1.17) The diffusional flow, eq (1.165), found by differentiation is

Diﬀusion through a semipermeable plane Equimolar diﬀusion

In the previous example, we analyzed the diffusion of component A from a solid into a solvent, assuming low convection velocity (u) and low concentration (c A), resulting in a negligible convection flow (c A u) However, this assumption is often not valid Consider a liquid A in a cylindrical vessel evaporating into a quiescent gas B The liquid level remains constant at y = y1, allowing us to treat it as stationary At the liquid's surface, the concentration of A is c A (y = y1) = c A1, which can be determined using the thermal equation of state for ideal gases: c A1 = p A1 / R m T, where p A1 is the saturation partial pressure of A at the liquid's temperature The solubility of gas B in liquid A is minimal, making this a suitable approximation for water and air A gaseous mixture of A and B flows over the top of the cylinder at y = y2, with concentration c A (y = y2) = c A2 The molar flux of component A along the y-axis is expressed as ˙ n A = c A w A = u j A + c A u, where u equals ˜x A w A + ˜x B w B Since gas B is stagnant, w B = 0, leading to u = ˜x A w A This scenario exemplifies diffusion through a semipermeable plane, with the water surface acting as the semipermeable barrier, allowing only water to pass through.

1−x˜ A u j A , (1.168) which replaces eq (1.163) for small mole fractions ˜x A of the dissolved substance.

In steady conditions d ˙n A /dy= 0, and with Fick’s law (1.160), it follows that d dy cD

The binary diffusion coefficient, denoted as HereD = D AB, is a key parameter in understanding gas mixtures For ideal gas mixtures maintained at constant pressure and temperature, the concentration (c) remains constant, represented by the equation c = N/V = p/R m T The diffusion coefficient exhibits minimal variation with changes in mixture composition, allowing it to be treated as a constant This leads to a differential equation that describes the concentration profile effectively.

= 0 , which because ˜x B = 1−x˜ A can also be written as d dy

Fig 1.42: Diﬀusion of component A in a gas mixture of components A and B or d dy dln˜x B dy = d 2 ln˜x B dy 2 = 0 (1.170)

It has to be solved under the boundary conditions ˜ x B (y=y 1 ) = ˜x B1 = 1−x˜ A1 and ˜ x B (y=y 2 ) = ˜x B2 = 1−x˜ A2 The solution is ˜ x B ˜ x B1 x˜ B2 ˜ x B1 y−y 1 y 2 −y 1

To verify the correctness of the solution, we can take logarithms of the equation (1.171), resulting in the expression ln x˜ B ˜ x B1 = y−y 1 y 2 −y 1 lnx˜ B2 ˜ x B1, which simplifies upon differentiating twice with respect to y Differentiating (1.171) yields the diffusive flux uj A =−cDd˜x A dy ˜x B dy x˜ B y 2 −y 1 lnx˜ B2 ˜ x B1 The corresponding molar flux is derived from equation (1.168), resulting in ˙ n A = 1.

1−x˜ A u j A = 1 ˜ x B u j A to be ˙ n A = cD y 2 −y 1 lnx˜ B2 ˜ x B1 = cD y 2 −y 1 ln1−x˜ A2

1−x˜ A1 (1.172) The average mole fraction of component B betweeny=y 1 andy=y 2 is ˜ x Bm = 1 y 2 −y 1 y 2 y 1 ˜ x B dy

After insertion of ˜x B from (1.171) and integration this yields ˜ x Bm = x˜ B2 −x˜ B1 ln(˜x B2 /˜x B1 ) (1.173)

The average mole fraction is the logarithmic mean of the two values ˜x B1 and ˜x B2

It is therefore possible to write (1.172) as ˙ n A = cD ˜ x Bm ˜ x B2 −x˜ B1 y 2 −y 1 = cD

Fig 1.43: Mass transfer in distillation

Fig 1.44: Equimolar diﬀusion between two containers

For ideal gases, the equations can be expressed using partial pressures \( p_B \) and total pressure \( p \) The flow rate of component A, \( \dot{n}_A \), is given by the equation \( \dot{n}_A = \frac{p_D}{R_m T} (y_2 - y_1) \ln \frac{p_{B2}}{p_{B1}} = \frac{p_D}{R_m T} (y_2 - y_1) p_{Bm} (p_{B2} - p_{B1}) \), where \( p_{Bm} \) represents the logarithmic mean partial pressure defined as \( p_{Bm} = \frac{p_{B2} - p_{B1}}{\ln \left(\frac{p_{B2}}{p_{B1}}\right)} \).

Equimolar counter diffusion occurs during the distillation of binary mixtures, where liquid descends and vapor ascends within a distillation column The colder liquid promotes the condensation of the least volatile component, while the vapor primarily consists of more volatile components According to Trouton’s rule, the molar enthalpy of vaporization remains approximately constant for all components When a specific amount of the least volatile component condenses, an equal number of moles of the more volatile component evaporates from the liquid At the phase boundary between liquid and vapor, the relationship between the concentrations is expressed as c A w A = −c B w B, with the reference velocity u being zero The molar flux to the phase boundary can be represented by ˙ n A = c A w A = u j A = −c D d˜x A dy.

Convective and diﬀusive ﬂows are in agreement with each other.

In a scenario where two containers hold different gases connected by a thin pipe, equimolar counter diffusion occurs when both gases maintain identical pressure and temperature, adhering to the thermal equation of state for ideal gases.

We consider a volume of gasV in the pipe between the two containers The thermal equation of state for this volume is p= (N A +N B )R m T /V

At steady state the total pressure is invariant with time Therefore dp dt = ( ˙N A + ˙N B )R m T /V = 0 which means that ˙N A =−N˙ B

Convective mass transfer

Mass transfer between a flowing fluid and another substance, or between two minimally miscible substances, is influenced by the materials' properties and the flow type In forced convection, external forces such as compressors or pumps drive the flow, facilitating mass transfer Conversely, mass transfer occurs in free convection when density changes due to variations in pressure or temperature.

Following on from the definition equation (1.23) for the heat transfer coefficient, the molar flow transferred to the surface (index 0) is described by

N˙ A0 :=β c A∆c A (1.177) and the molar ﬂux is given by ˙ n A0 = ˙N A0 /A=β c ∆c A (1.178)

The mass transfer coefficient (βc), measured in SI units of m/s or m³/(s m²), quantifies the volumetric flow transferred per unit area and is defined by the concentration difference (∆cA) An appropriate concentration difference, such as the difference between the wall and the surface of a liquid film (cA0 - cAδ), is crucial for accurate mass transfer assessments The coefficient is influenced by factors including flow type (laminar or turbulent), material properties, system geometry, and the concentration difference When a fluid flows over a stationary surface for substance exchange, a thin layer forms near the surface where flow velocity decreases to zero, resulting in low convective mass transfer and a predominance of diffusive mass transfer.

Fig 1.45: Mass transfer from a porous body into a gas ﬂow

According to Fick's law, the diffusive component is proportional to the concentration gradient at a stationary surface, which correlates to the concentration difference ∆c A Consequently, the mass transfer coefficient can be effectively defined as uj A0 := -cDd˜x A dy.

The diﬀusional ﬂux u j A0 is that at the surface (index 0) Therefore β= −(cDd˜x A /dy) 0

The relationship between molar flux and diffusional flux is defined by the equation c A0 w A0 = ˙n A0 = u j A0 + c A0 u 0 = u j A0 + ˜x A0 n˙ 0 In scenarios where convective flow approaches zero (˙n 0 →0), the equation simplifies to c A0 w A0 = ˙n A0 = u j A0, allowing for the calculation of molar flux as ˙ n A 0 = β∆c A, utilizing the mass transfer coefficient under the same condition.

The concentration profile and molar flux differ from the values calculated under the assumption of zero molar flux (˙n 0 → 0) For instance, in a scenario where a porous body is submerged in liquid A (such as water) while gas B (like alcohol) flows over it, liquid A evaporates into gas B, and gas B simultaneously mixes with liquid A within the porous structure Notably, the convective flux, represented by the equation c A0 w A0 + c B0 w B0 = c 0 u 0 = ˙n 0, does not vanish entirely at the surface of the porous body Consequently, the molar flux of A at the surface can be expressed by the equation ˙n A0 = u j A0 + c A0 u 0 = u j A0 + ˜x A0 n˙ 0, where c A0 u 0 is equivalent to ˜x A0 c 0 u 0 = ˜x A0 n˙ 0.

Fig 1.46: Concentration profile c A (y) in mass transfer with a vanishing convective flow ( ˙ n 0 → 0) and with finite convection flows ( ˙ n 0 > 0)

Mass transfer into the (gaseous) phase

Fig 1.47: Concentration proﬁle c A (y) for vanishing convective ﬂow ( ˙ n 0 →

0) and for ﬁnite convective ﬂows ( ˙ n 0 < 0) Mass transfer out of the (gaseous) phase

The concentration profile is significantly altered by finite convective flux compared to the profile with negligible convective flux (˙n 0 → 0), with greater convective flux leading to larger deviations When the convective flux is directed toward the wall (˙n 0 < 0), such as during vapor condensation from a mixture, the concentration profile becomes steeper due to this flux Concurrently, the material transported out of the phase by diffusion increases, while diffusive transport into the phase decreases The mass transfer coefficient for negligible convection flux (˙n 0 → 0) differs from that with finite convective flux, indicating that β(˙n 0 > 0) < β(˙n 0 → 0) and β(˙n 0 < 0) > β(˙n 0 → 0) The notation β(˙n 0 = 0) = β• can be used to represent the mass transfer coefficients, where the superscript dot signifies that the convective flow (˙n 0) is non-zero Consequently, the molar flux can be calculated as ˙n A0 = β• ∆c A + ˜x A0 n˙ 0.

At the limit ˙n 0 → 0, of vanishing convective ﬂow or by equimolar diﬀusion, we obtainβ • =β =β c , as shown by comparing (1.181) and (1.178).

The simplest mass transfer coefficient to analyze occurs in the absence of convective flows This coefficient is influenced by factors such as the flow velocity of gas B over the porous body, the physical properties of the gas, and the geometric shape of the porous material For specific geometries, these variables can be integrated into dimensionless numbers, as will be discussed later.

The Sherwood number (Sh) is defined as the ratio of the mass transfer coefficient (β) to the product of a characteristic length (L), such as the length of a plate over which gas flows, and the diffusion coefficient (D) Additionally, the Schmidt number (Sc) is represented by the ratio of kinematic viscosity (ν) to the diffusion coefficient (D), while the Reynolds number (Re) is defined using average velocity (w) and kinematic viscosity (ν) These relationships, often derived from experiments or by solving partial differential equations, form the foundation of mass transfer theory Relevant literature provides insights into these relationships for various practical geometries and flow regimes In many cases, the mass transfer coefficient (β •) can be accurately estimated by applying a correction factor (ζ), which is typically close to 1 in practical applications.

15 Thomas Kilgore Sherwood (1903–1976) completed his PhD under the supervision of Warren

K Lewis, after whom the Lewis number was named, in 1929 at the Massachussetts Institute of Technology (MIT), Boston, USA The subject of his thesis was “The Mechanism of the Drying of Solids” He was a professor at MIT from 1930 until 1969 His fundamental work on mass transfer in ﬂuid ﬂow and his book “Absorption and Extraction” which appeared in 1937 made him famous worldwide.

Ernst Schmidt (1892–1975) initially pursued civil engineering at Dresden and Munich before shifting his focus to electrical engineering He served as an assistant to O Knoblauch in the applied physics laboratory at the Technische Hochschule in Munich In 1925, he became a professor at the Technische Hochschule in Danzig, now known as Gdansk, Poland.

In 1937, he became the director of the Institute for Engine Research at the Aeronautics Research Establishment in Braunschweig and later served as a professor at the Technische Hochschule in Braunschweig In 1952, he succeeded W Nusselt as the chair for Thermodynamics at the Technische Hochschule in Munich His scientific contributions include solutions to the unsteady heat conduction equation, investigations of temperature fields in natural convection, and methods for visualizing the thermal boundary layer He was the first to utilize the number that is now named after him in a paper discussing the analogy between heat and mass transfer.

Mass transfer theories

Film theory

Film theory, originating from the work of Lewis and Whitman in 1924, describes the mass transfer of a substance A from a static solid or liquid surface to a flowing fluid B, illustrated by a flat plate The concentration of A decreases from c A0 at the plate surface to c Aδ in the fluid, occurring within a thin film of thickness δ near the wall In this model, concentration and velocity are assumed to vary only in the y-direction, remaining constant over time and in other coordinate directions This results in a steady flow characterized by a constant molar flux of A, expressed as ˙n A = c A w A If this were not true, it would lead to a change in concentration over time or flow in other directions, which contradicts the assumptions of film theory Consequently, the principle d ˙n A / dy = 0 holds true within this framework.

In the analysis of vanishing convective flux, the equation ˙n = ˙n A + ˙n B = 0 indicates that the contributions from A and B balance each other Given that ˙n A can be expressed as u j A + ˜x A n˙ = u j A = -cDd˜x A/dy, we derive the second derivative d²˜x A/dy² = 0 under the assumption of constant values for cD Consequently, the concentration profile within the film is linear, represented by the equation ˜x A - x˜ A0 / ˜x Aδ - x˜ A0 = y / δ.

Fig 1.48: Concentration profile at the surface over which a fluid is flowing

On the other hand for vanishing convection at the wall, ˙n 0 → 0, according to (1.181) ˙ n A0 =β(c A0 −c Aδ ) =−cD d˜x A dy

The film thickness (δ) is typically unknown, making it challenging to calculate the mass transfer coefficient (β) directly However, commonly used values can be sourced from relevant literature, which facilitates an approximation of the film thickness using equation (1.189) In film theory, the mass transfer coefficient (β) is proportional to the diffusion coefficient (D) when convection flux approaches zero (˙n 0 → 0).

When a finite convective flux of species A is allowed, the condition d˙n A/dy = 0 still holds The molar flux of species A can be expressed as ˙n A = u j A + ˜x A n˙, which simplifies to ˙n A = -cD d˜x A/dy + ˜x A n˙ Assuming that the diffusion coefficient cD remains constant, this leads to the formulation of a second-order differential equation.

The equation \( cDd \frac{d^2 \tilde{x}_A}{dy^2} + \dot{n} \tilde{x}_A \frac{dy}{dy} = 0 \) must be solved with the boundary conditions \( \tilde{x}_A(y = 0) = \tilde{x}_{A0} \) and \( \tilde{x}_A(y = \delta) = \tilde{x}_{A\delta} \) By rearranging the equation, we obtain \( \frac{d \ln \tilde{x}_A}{dy} = n \dot{c} D \), where \( \tilde{x}_A = \frac{d \tilde{x}_A}{dy} \) Integrating while considering the boundary conditions results in the concentration profile \( \tilde{x}_A(y) = \frac{\tilde{x}_{A} - \tilde{x}_{A0}}{\tilde{x}_{A\delta} - \tilde{x}_{A0}} \left( \exp(n \dot{c} Dy) - 1 \right) \left( \exp(n \dot{c} D \delta) - 1 \right) \).

Fig 1.49: Stefan correction factor ζ = β • /β from ﬁlm theory

The concentration profile for vanishing convection, as described in equation (1.187), is derived from equation (1.190) This can be illustrated through a Taylor series expansion of the exponential function at ˙n = 0 The material flux transferred to the surface at y = 0 (where index 0 represents the wall) in the y-direction is expressed as ˙n A = -cDd˜x A dy.

According to ﬁlm theory this is constant and equal to the value at the wall, where ˙ n A = ˙n A0 and ˙n = ˙n 0 Diﬀerentiating (1.190) and introducing the result into (1.191) yields ˙ n A =−(˜x Aδ −x˜ A0 ) n˙ exp(nδ˙ cD)−1

+ ˜x A0 n ˙ (1.192) When this is compared to (1.183), we get β • = n/c˙ exp(nδ˙ cD)−1

In the context of negligible convection, the mass transfer coefficient can be expressed as β = D/δ By applying film theory principles, a relationship between the mass transfer coefficients β• and β is established This relationship is represented by the equation β• β = ζ = n/cβ˙ exp(n˙ cβ)−1, as illustrated in Fig 1.49.

The Stefan correction factor, denoted as ζ, is essential for calculating mass transfer using film theory To determine the mass transfer coefficient β, one must consider the role of convection: if convection is negligible, mass transfer is calculated using equation (1.181), whereas significant convection requires the use of equation (1.183) for accurate mass transfer assessment.

As a special case (1.192) is also used for the situation of single side mass transfer where ˙n A = ˙nand ˙n B = 0 It then follows from (1.192) that ˜ x A0 −x˜ Aδ

−1 or when solved for ˙n A , ˙ n A = cD δ ln1−x˜ Aδ

1−x˜ A0 = cD δ lnx˜ Bδ ˜ x B0 =cβlnx˜ Bδ ˜ x B0 (1.195) it is in complete agreement with equation (1.172), which was found earlier.

When drying a wet material using air, the focus is on calculating the moisture content instead of the mole fraction For instance, consider a solid material with water that needs to be dried by air The previously established equations and relationships can be applied, designating the water index as A and the air index as B Given that the water content in moist air is minimal, we can approximate ln(1−x˜ A) as −x˜ A Consequently, equation (1.195) can be expressed as ˙ n A = cβ(˜x A0 − x˜ Aδ).

We then want to introduce the speciﬁc humidity or moisture content, which is deﬁned by

X A :=M A /M B , whereM A is the mass of the water andM B is the mass of the dry air, and therefore using the molar mass ˜M =M/N we can also write

1−x˜ A Solving for the mole fraction for water gives ˜ x A = X A M˜ B /M˜ A

1 +X A M˜ B /M˜ A The water content at ambient pressure is of the order 20ã10 − 3 kg/kg, which makes the termX A M˜ B /M˜ A ≈0.03 1 The approximation ˜ x A =X A M˜ B /M˜ A is then valid The transferred molar ﬂux of water will be ˙ n A =cβM˜ B

M˜ A (X A0 −X Aδ ) with the mass ﬂux as ˙ m A = ˙n A M˜ A =cβM˜ B (X A0 −X Aδ ) Usingc=p/R m T and the gas constant for airR B =R m /M˜ B , we get ˙ m A = p

R B T β(X A0 −X Aδ ) (1.195a) for the mass ﬂux of water.

Boundary layer theory

Boundary layer theory, similar to film theory, posits that mass transfer occurs in a thin film adjacent to a wall Unlike film theory, boundary layer theory allows for variations in concentration and velocity across multiple coordinate axes However, since the concentration profile changes more significantly in the y-direction, it is sufficient to focus on diffusion along this axis, greatly simplifying the relevant differential equations This simplification leads to the determination of the concentration profile, from which the mass transfer coefficient (β) can be calculated Typically, the mean mass transfer coefficient (βm) is used in practical applications, defined as βm = 1/L.

This can be found, for forced ﬂow from equations of the form

Sh m =f(Re, Sc) with the mean Sherwood number Sh m = β m L/D The function for the mean Sherwood number is practically identical to that for the average Nusselt number

The Nusselt number, represented as Nu = f(Re, Pr), is a crucial parameter for characterizing fluid flow over surfaces In similar contexts, the Nusselt number can be substituted with the Sherwood number, while the Prandtl number can be replaced by the Schmidt number, as referenced in [1.29].

The determination of heat transfer coefficients using dimensionless numbers has been previously discussed in section 1.1.4 This approach is also applicable to mass transfer, exemplified by the mean Nusselt number, represented as N_u_m = α_m L/λ in forced flow conditions.

In 1.196, the quantities c, n, and m are influenced by the flow type, whether laminar or turbulent, as well as the shape of the surface or channel through which the fluid flows Consequently, the mean Sherwood number can be expressed accordingly.

The average mass transfer coefficient can be derived from equation (1.197) and can also be determined using the method for calculating average heat transfer coefficients By dividing equation (1.197) by (1.196), a relationship between heat and mass transfer coefficients is established.

Fig 1.50: Stefan correction ζ = β m • /β m for ﬁlm, boundary layer and penetration theory, from [1.30] or β m = α m c p Le m−1 (1.198)

The Lewis number (Le = a/D), which is dimensionless and approximately 1 for ideal gases, is represented by m ≈ 1/3 This allows for the general approximation β m = α m / c p (1.199) to be considered valid Table 1.5 provides various values for the Lewis number.

Table 1.5: Lewis numbers of some gas mixtures at 0 ◦ C and 100 hPa in air in hydrogen in carbon dioxide

Lewis' equations, represented by Equations (1.198) and (1.199), provide mass transfer coefficients (β m) that are applicable only under conditions of negligible convective currents When convection plays a significant role, these coefficients require correction The correction factors (ζ = β m • /β m) for transverse flow over a plate, based on boundary layer theory assumptions, are illustrated in Fig 1.50 Notably, these factors are greater than those predicted by film theory for convective flow out of the phase, while they are smaller for convective flow into the phase.

Warren Kendall Lewis (1882–1978) was a prominent chemical engineer who studied at the Massachusetts Institute of Technology (MIT) and earned his PhD in chemistry from the University of Breslau in 1908 He served as a professor at MIT from 1910 to 1948, focusing his research on key areas such as filtration, distillation, and absorption Notably, he contributed to the field with his paper titled “The Evaporation of a Liquid into a Gas,” published in Mechanical Engineering.

44 (1922) 445–448, he considered simultaneous heat and mass transfer during evaporation and showed how heat and mass transfer inﬂuence each other.

Penetration and surface renewal theories

The film and boundary layer theories assume steady transport, making them unsuitable for scenarios where material accumulates, resulting in changing concentration over time In many mass transfer systems, fluids interact for such brief periods that steady state is unattainable For instance, when air bubbles rise in water, evaporation occurs only at the bubble's surface, with the surrounding water having a contact time similar to the bubble's movement distance Consequently, mass transfer happens momentarily at specific positions Higbie's penetration theory, developed in 1935, addresses this momentary mass transfer, revealing that the mass transfer coefficient is inversely proportional to the square root of the contact (residence) time, expressed as β m = 2.

The mean mass transfer coefficient, denoted as βm, is calculated from time t=0 to time t Practical experience indicates that valuable mass transfer coefficient values can be derived by calculating contact time using the formula t=d/w, where d represents the diameter of the rising or falling bubbles or droplets, and w is the mean rise or fall velocity However, determining the contact time for a liquid descending through packing material while gas flows through it poses greater challenges.

Dankwerts’ surface renewal theory, introduced in 1951, expands on penetration theory by suggesting that fluid elements in contact have varying residence times, described by a residence time spectrum Unlike Higbie's assumption of uniform contact time throughout the apparatus, Dankwerts proposes that mass exchange occurs within individual fluid cells, where the contact time follows a distribution function This allows for the replacement of fluid elements at the contact area over time, hence the term "surface renewal theory." This concept has been effectively utilized in the absorption of gases from agitated liquids, although the specific fraction of time for surface renewal remains unknown, similar to the contact times in penetration theory.

Surface renewal theory describes potential flow patterns that facilitate contact between two liquids or between a liquid and a gas, highlighting the complexities of mass transfer processes However, traditional methods may not accurately calculate the quantities involved in these mass transfer phenomena.

Application of ﬁlm theory to evaporative cooling

Evaporative cooling is effectively demonstrated through film theory, where a solid, adiabatic, insulated wall is enveloped by a layer of water This setup allows unsaturated humid air to flow over the water film, facilitating the cooling process.

The humid air takes water vapour from the ﬁlm, by which the ﬁlm of water and the air are cooled, until a time and position constant temperature is reached.

The wet bulb temperature, which remains constant throughout the film due to the adiabatic nature of the adjoining wall, is crucial for understanding the evaporation process Resistance to mass transfer occurs solely on the gas side, allowing water to continue evaporating in the unsaturated air As the water film maintains a constant temperature, the heat required for evaporation is extracted from the air, leading to changes in temperature and partial pressure of water vapor, as illustrated in Fig 1.52 Consequently, the wet substance can be cooled to its wet bulb temperature through evaporation, which is always lower than the temperature of the humid air passing over the water surface.

The wet bulb temperature is influenced by the quantity of water evaporated from a surface into the surrounding humid air This process occurs through diffusion at the phase boundary between water and air The rate of water transfer to the air can be calculated using the equation ˙ m A = ˜M A cβln1−x˜ Aδ, which describes the relationship between the water substance and its interaction with the air.

The mole fraction of water vapor in air at the water's surface is represented as ˜x AI (1.201), while ˜x Aδ denotes the mole fraction at a considerable distance from the water surface.

Adiabatic evaporative cooling involves the interplay between temperature and the partial pressure of water vapor in the air, where the amount of dry air remains constant To better understand this process, it is beneficial to introduce the moisture content, represented as \( X_A = \frac{M_A}{M_B} \), where \( \tilde{x_A} = N_A \).

1 +X A M˜ B /M˜ A Then (1.201) can also be written as ˙ m A = ˜M A cβln1 +X AI M˜ B /M˜ A

The moisture content at the water surface and the mass flux of the water transfer remain unknowns At the water's surface, saturation is achieved, and according to thermodynamic principles, the moisture content can be determined.

X AI = 0.622 p s (ϑ I ) p−p s (ϑ I ) (1.203) wherep s is the saturation pressure of the water vapour at temperature ϑ I at the surface.

The energy balance can be expressed through an additional equation To establish this, we will analyze a specific balance region on the gas side, illustrated by the dotted lines in Fig 1.53, extending from the gas at the film's surface to a designated point.

Fig 1.53: Energy balance for evaporative cooling for a steady-state ﬂow is ˙ q I + ˙m A h I = ˙q+ ˙m A h= const

In this equation the heat ﬂuxes ˙q I and ˙qare in the opposite direction to theyaxis and therefore negative From diﬀerentiation we get d dy( ˙q+ ˙m A h) = 0

This result and the introduction of ˙q = −λdϑ/dy and dh = c pA dϑfor the temperature patternϑ(y) produces the ordinary diﬀerential equation

−λd 2 ϑ dy 2 + ˙m A c pA dϑ dy = 0 , (1.204) which has to be solved for the boundary conditions ϑ(y= 0) =ϑ I and q˙ I =−λ dϑ dy y=0

, which gives ϑ−ϑ I =− q˙ I ˙ m A c pA exp m˙ A c pA λ y

The temperatureϑ δ at a large distance y=δ away from the surface is therefore ϑ δ −ϑ I =− q˙ I ˙ m A c pA exp m˙ A c pA λ δ

When the evaporation rate approaches zero, heat transfer occurs solely through conduction perpendicular to the flow of humid air In this scenario, the heat flux at the water surface can be expressed as ˙ q I =α(ϑ I −ϑ δ ) ( ˙m A →0).

The heat transfer coeﬃcientαis deﬁned, just as in section 1.1.3, by this equation.

On the other hand with the limit ˙m A →0 we obtain, from (1.206): ϑ δ −ϑ I =− q˙ I ˙ m A c pA

In the comparison with equation (1.207), we find that α equals λ divided by δ Substituting this into equation (1.206), we observe that the heat flow to the water surface facilitates evaporation This leads to the expression ˙ q I = −m˙ A ∆h v, where ∆h v represents the enthalpy of vaporization of water at temperature ϑ I Consequently, equation (1.206) can be rearranged to yield ˙ m A c pA α = ln.

To determine the unknown quantities ˙m A and ϑ I, we can utilize the mass balance from equation (1.202), providing a second equation for solving these variables By substituting equation (1.202) into equation (1.209), we can eliminate the mass flux ˙m A, resulting in a relationship for the wet bulb temperature, expressed as ϑ δ − ϑ I = ∆h v c pA.

In cases of small evaporation rates ˙m A →0 this can be simpliﬁed even more We can then write (1.202) as

M˜ A cβ+ . Therefore when the evaporation rate is small, we get ˙ m A = ˜M B cβ X AI −X Aδ

With this, equation (1.208) and ˙q I =−m˙ A ∆h v , we get the following relationship for small evaporation rates ˙m A →0 ϑ δ −ϑ I = ˜M B cβ α ∆h v X AI −X Aδ

The wet bulb temperatureϑ I , can be obtained from (1.210) or (1.211), and then the moisture content X AI , which is determined by the temperature ϑ I , can be found by using (1.203).

The wet bulb temperature is essential for calculating the moisture content of humid air, denoted as X Aδ, using specific equations The wet bulb hygrometer, also known as Assmann’s Aspiration Psychrometer, employs this method through two thermometers One thermometer has a porous covering and is submerged in water, ensuring its bulb remains wet, while air is blown over it, allowing it to reach the wet bulb temperature (ϑ I) The second thermometer, referred to as the dry thermometer, measures the surrounding air temperature (ϑ δ) By utilizing these temperature readings, the moisture content of the humid air can be determined.

Example 1.8: The dry thermometer of a wet bulb hygrometer shows a temperature ϑ δ of

To calculate the moisture content and relative humidity of air at a temperature of 30 °C with a wet thermometer reading of 15 °C and a total pressure of 1000 mbar, we can utilize Lewis' equation, which provides a value of 1.30 for the ratio of specific heats The enthalpy of vaporization of water at 15 °C is 2466.1 kJ/kg, while the specific heat capacity of air is 1.907 kJ/kgK The molar mass of water is 18.015 kg/kmol, and the molar mass of dry air is 28.953 kg/kmol.

According to (1.203) with p S (15 ◦ C) = 17.039 mbar, X AI = 0.622 ã 17.039/(1000 − 17.039) = 1.078 ã 10 −2 It then follows from (1.210) that

This gives X Aδ = 5.189 ã 10 −3 Almost exactly the same value is obtained using (1.211), where X Aδ = 5.182 ã 10 −3 The relative humidity is ϕ = p A p As = X Aδ

100042.41 = 0.195 = 19.5% (This value can also be found from a Mollier’s h 1+X , X-diagram).

Overall mass transfer

Heat transfer typically occurs through a solid barrier, while mass transfer involves the direct exchange of components between phases in contact without a solid wall For instance, when examining the transfer of component A from a gaseous mixture of A and B into a liquid C, where only A dissolves, the mole fractions of A in both phases are represented Component A faces three resistances during this transfer: within the gas phase, at the interface, and within the liquid phase However, the mass transfer at the liquid-gas interface is rapid compared to the slower movement within the phases, allowing the interface resistance to be overlooked and assuming equilibrium exists there This process is described by the two-film theory of mass transfer, which assumes equilibrium at the interface However, this assumption breaks down in the presence of chemical reactions, surfactants, or when large mass flow rates occur, as these conditions lead to rapid mass exchange that disrupts equilibrium.

At the phase boundary under equilibrium conditions, the mole fractions ˜y AI and ˜x AI are related by the equation p AI = ˜y AI p=f(˜x AI ) It is assumed that near the phase boundary, mass transfer occurs primarily through diffusion, with negligible convective flows This assumption enables the application of the mass transfer equation (1.181) for both phases at the boundary.

The relationship between mole fractions and molar concentration during mass transfer from gas to liquid is defined by (c A /c) G = ˜y A and (c A /c) L = ˜x A, where G represents gas and L represents liquid The mass flux in the gas phase can be expressed as ˙ n AI = (βc) G (˜y Am −y˜ AI ), while in the liquid phase it is ˙ n AI = (βc) L (˜x AI −˜x Am ) This leads to the equation ˜ y Am −y˜ AI = n˙ AI, highlighting the fundamental connection between mole fractions and mass transfer rates in both phases.

The mole fractions ˜y AI and ˜x AI at the interface are interdependent, as described by the thermodynamic relationship in equation (1.212) Due to the low solubility of gases in liquids, the mole fraction ˜x AI is typically very small Consequently, Henry's law allows for a linear approximation represented by ˜y AI = k H ˜x AI /p (1.217), where the Henry coefficient for binary mixtures is solely temperature-dependent, denoted as k H (ϑ) Additionally, the mole fraction ˜x Am in the liquid phase can be correlated with the equilibrium mole fraction ˜y Aeq in the gas phase using Henry's law: ˜y Aeq = k H ˜x Am /p (1.218) This establishes a relationship between the gas and liquid phases, allowing the transformation of equation (1.216) into ˜y AI - ˜y Aeq = n˙ AI k H.

Adding (1.215) and (1.219) gives ˜ y Am −y˜ Aeq = ˙n AI

This will then be written as ˙ n AI =k G (˜y Am −y˜ Aeq ) , (1.220) with the gas phase overall mass transfer coeﬃcient

18 Named after William Henry (1775–1836), a factory owner from Manchester, who ﬁrst put forward this law in 1803.

By substituting the mole fraction ˜y Am with the expression ˜y Am = k H x˜ Aeq and eliminating the mole fraction ˜y AI, we derive an equivalent relationship as shown in equation (1.220) This leads to the formulation ˙ n AI = k L (˜x Aeq − x˜ Am ) (1.222), which incorporates the overall mass transfer coefficient for the liquid phase.

The resistance to mass transfer is comprised of the individual resistances from both gas and liquid phases, allowing for the possibility of neglecting one phase's resistance when analyzing mass transfer Overall mass transfer coefficients can only be derived if the phase equilibrium is represented by a linear function, typically relevant in gas absorption processes due to the low solubility of gases in liquids, as described by Henry’s law While ideal liquid mixtures can be represented by Raoult’s law, such cases are rare in practice Consequently, the significance of calculating overall mass transfer coefficients is considerably less than that of overall heat transfer coefficients in heat transfer studies.

Mass transfer apparatus

Material balances

A material balance is essential for sizing mass transfer equipment, regardless of the design approach Just as an energy balance correlates the temperatures of fluid flows in a heat exchanger, a mass balance provides the concentrations of the fluids involved.

An absorption column serves as a prime example of a process where component A from an ascending gas mixture is absorbed by a descending liquid.

In the study of contact between two immiscible phases, it is essential to establish balance equations that remain applicable regardless of the system being analyzed, whether it be a packed column, falling film, or plate column The initial composition of the gas under consideration is a carrier gas.

In the column, the gas phase consists of component G and the absorbed component A, while the liquid phase is comprised of solvent L and the absorbed component A The total molar flow rate of the rising gas is represented by ˙N G + ˙N GA, whereas the downward flowing liquid has a total molar flow rate of ˙N L + ˙N LA Since the molar flow rates ˙N G and ˙N L remain constant throughout the column, the composition of both streams can be effectively characterized by the mole ratios of the components to their respective solvent or carrier gas.

In a countercurrent absorber, the average mole ratio of component A in the liquid phase (˜X) and in the gas phase (˜Y) is critical for understanding the material balance across the column This balance is established between the inlet (i) and the outlet (e) of the liquid, ensuring efficient separation and absorption processes.

The composition changes in gas and liquid phases are influenced by the molar flow rate ratios of the respective streams A material balance can be established for a control volume between stage e and any other stage b within the column.

In a column, the average liquid mole ratio (˜X) is consistently linked to a corresponding average gas mole ratio (˜Y) This relationship is illustrated by a straight line in the ˜Y, X-diagram, as depicted in Equation (1.225).

In rectification, a linear relationship exists between the compositions of two phases, similar to that in absorption columns The process involves boiling a multicomponent mixture, where the generated vapor ascends countercurrent to the descending condensate As the colder condensate encounters the vapor, components with higher boiling points condense and transfer their enthalpy to the more volatile components, which vaporize This results in vapor that is enriched with more volatile components while the less volatile components remain in the liquid phase.

The continuous rectification process involves a rectifying column (R), condenser (C), and boiler (V) as illustrated in Fig 1.56 The liquid mixture, referred to as the feed, enters the boiler (V) at a molar flow rate of ˙N F, where heat (˙Q V) is applied to produce vapor This vapor ascends through the column and exits at the top, where it is directed to the condenser (C) to remove heat (˙Q C) and achieve complete condensation A portion of the condensate is recycled back into the column as reflux, descending countercurrently and engaging in heat and mass transfer with the ascending vapor The remaining condensate, known as the distillate, is collected as the product stream (˙N D) To maintain a constant composition in the boiler, a separate stream (˙N B) is continuously extracted from the column's bottom, referred to as the bottom product.

The mole fractions of the feed, bottom product, and distillate play a crucial role in establishing the flow rate ratios between feed and product In a control volume extending from the top of the column to any cross-section b, the material balance can be expressed using the mole fraction of volatile components in the vapor, denoted as ˜y, and in the liquid, represented by ˜x.

As shown previously both equations yield a linear relationship between ˜yand ˜x: ˜ yN˙ L

Concentration proﬁles and heights of mass transfer columns

The global mass balance connects the average composition of one phase to that of another, yet local composition changes remain unclear To uncover these changes, a mass balance similar to that used for a differential element in a heat exchanger is necessary By integrating this balance, the composition pattern throughout the entire exchanger can be established Once the concentration variations in the column are identified, the column height required for a specific outlet concentration of a component can be calculated This process will be illustrated through examples involving an absorber and a packed rectification column.

In a plate column, calculations for separation do not need to account for the mixing of phases, as each plate achieves a state of equilibrium Consequently, each volume element corresponds to an equilibrium stage, allowing the column height to be determined by the number of stages necessary for a specific separation This process is fundamentally thermodynamic rather than reliant on mass transfer principles, which is why a distillation column can be sized without requiring detailed knowledge of mass transfer laws.

A packed column contains packing material that complicates the measurement of the phase interface area crucial for mass transfer Consequently, only the product of the mass transfer coefficients and the interface area is determined The interface area (A_I) is connected to the volume of the empty column (V_K), with the interface area per volume (a*) defined as a* = A_I / V_K Here, V_K is calculated as V_K = A_K Z, where A_K represents the cross-sectional area of the empty column and Z is its height.

Fig 1.57: Mass transfer in an absorber

In an absorber, we analyze a control volume situated between two cross sections, separated by a distance dz The transfer of material to the liquid through the interface area dA is represented by the equation ˙n AI a ∗ A K dz, which reflects the reduction of component A in the gas phase by ˙N G d ˜Y Concurrently, the liquid phase experiences an increase in component A, indicated by −N˙ G d ˜Y, which is equivalent to ˙N L d ˜X Therefore, it is established that ˙n AI a ∗ A K dz = −N˙ G d ˜Y.

The molar flux of component A, denoted as ˙n AI, is derived under the assumption that the concentrations of component A in both liquid and vapor phases are minimal, allowing for the substitution of mole fractions ˜y Am and ˜y Aeq with mole ratios Consequently, it can be expressed that k G ( ˜Y −Y˜ eq )a ∗ A K dz=−N˙ G d ˜Y, leading to the relationship dz= −N˙ G (k G a ∗ )A K d ˜Y.

The relationship between Y˜ and Y˜ eq (1.229) aligns with the heat transfer dynamics observed in countercurrent flow, as described in equation (1.112) When performing integration, it is essential to account for the variations in the mole ratio ˜Y eq at the phase boundary throughout the column's length, which subsequently affects the average mole ratio ˜Y The balance equation (1.225) provides a foundation for understanding these changes.

N˙ L ( ˜Y −Y˜ i ) + ˜X i Then using Henry’s law, ˜Y eq =k H + X˜ with k H + =k H /pit follows that

This allows ˜Y −Y˜ eq to be expressed as a linear function of the mole ratio ˜Y such that

=aY˜ −b (1.231) Equation (1.229) can then be transformed into dz= −N˙ G (k G a ∗ )A K d ˜Y aY˜ −b (1.232)

The mole ratio ˜Y(z) varies with the height of the column and can be determined through integration By applying equation (1.225) from the material balance, we can find the corresponding values for ˜X, the mole ratio in the liquid, at each height level.

Y˜, the mole ratio in the gas, can be calculated It then follows from (1.232) that z= −N˙ G (k G a ∗ )A K

(k G a ∗ )A K a lnaY˜ −b aY˜ e −b or with (1.231) we get zN˙ G (k G a ∗ )A K 1−k + H N˙ G /N˙ L ln( ˜Y −Y˜ eq ) o

The necessary column heightZcan be found from this equation, when proceeding to the liquid inlet cross sectioniat the top of the column

The meaning of the mole ratio variations ( ˜Y −Y˜ eq ) e and ( ˜Y −Y˜ eq ) i are shown in Fig 1.58.

Fig 1.58: Equilibrium and balance lines for an absorber

The material balance from equation (1.228) serves as the foundation for analyzing the concentration profile in a packed column used for rectifying a gaseous mixture In this analysis, mole fractions are utilized instead of mole ratios The transfer of components from the gas phase to the liquid phase is represented by the term −N˙ G d˜y, where ˙N G indicates the molar flow rate of the gas mixture, differing from the carrier gas in the previous equation Consequently, equation (1.228) is updated to ˙ n AI a ∗ A K dz=−N˙ G d˜y (1.235).

By substituting the abbreviations ˜y Am = ˜y and ˜y Aeq = ˜y eq into the molar flux equation (1.220), we derive the expression dz = −N˙ G (k G a ∗)A K d˜y / (˜y − ˜y eq), which aligns with equation (1.229) Upon integration, we obtain the concentration profile ˜y(z) over the height z, expressed as z = −N˙ G (k a ∗)A K ˜y / (˜y − ˜y eq).

The required height Z for the column is determined by integrating to the equilibrium point ˜y= ˜y e This integral is typically evaluated numerically due to the complexity of the mole fraction ˜y eq, which is a function of the liquid mole fraction ˜x and is influenced by the balance equation (1.227) The determination of mole fractions at equilibrium is addressed within the thermodynamics of phase equilibria.

In a packed column, a washing solution made of high boiling hydrocarbons is introduced at the top to effectively eliminate benzene from the ascending air Initially, the benzene concentration at the column's base is 3%, and the goal is to reduce this by 90% by the time the air reaches the top The washing solution itself has a benzene molar ratio of 0.3%.

Raoult's law, expressed as y A = (p As /p) x A, applies to the solubility of benzene (substance A) in a washing solution, with the saturation pressure of pure benzene at 30 °C measured at p As = 159.1 mbar The total pressure in the system is p = 1 bar The benzene-free washing solution is introduced into the column at a flow rate of ˙ N L = 2.5 mol/s, while benzene-free air flows at ˙ N G = 7.5 mol/s The column has an inner diameter of d K = 0.5 m, and the mass transfer coefficients are (βc) G a ∗ = 139.3 mol/m³ s and (βc) L a ∗ = 3.139 mol/m³ s.

How high does the column have to be to meet the desired air purity conditions?

Due to the low benzene content, Raoult’s law allows for the substitution of mole fractions with mole ratios, expressed as ˜ Y A = k H + X ˜ A, where k H + equals p As/p, which is 0.1591 This relationship is derived from equation (1.221).

1 139.3 + 0.1591 3.139 m 3 s mol k G a ∗ = 17.28 mol/(m 3 s) The benzene content of the washing solution leaving the column, from (1.224) is

Y ˜ eqe = k + H X ˜ e = 0.1591 ã 0.084 = 1.336 ã 10 −2 The required column height according to (1.234)is calculated to be

Exercises

The outer wall of a room, constructed from brickwork with a thermal conductivity of λ = 0.75 W/K m, has a thickness of 0.36 m and a surface area of 15.0 m² With surface temperatures of 18.0 °C on one side and 2.5 °C on the other, the heat loss (˙Q) through the wall can be calculated If the wall were instead made of concrete blocks with a lower thermal conductivity of λ = 0.29 W/K m and a reduced thickness of 0.25 m, the heat loss (˙Q) would decrease significantly, demonstrating the impact of material choice and wall thickness on thermal performance.

1.2: The thermal conductivity is linearly dependent on the temperature, λ = a + b ϑ. Prove equation (1.12) for the mean thermal conductivity λ m

1.3: The steady temperature proﬁle ϑ = ϑ(x) in a ﬂat wall has a second derivative d 2 ϑ/dx 2 > 0 Does the thermal conductivity λ = λ(ϑ) of the wall material increase or decrease with rising temperature?

A copper wire with a diameter of 1.4 mm and a specific electrical resistance of 0.020 × 10⁻⁶ Ω m is encased in a plastic insulation that is 1.0 mm thick, featuring a thermal conductivity of 0.15 W/K m To ensure safety, the outer surface of the insulation is kept at a constant temperature of 20 °C It is crucial to determine the maximum current that can be passed through the wire without allowing the inner temperature of the insulation to exceed safe limits.

A heat flow of 17.5 W is produced within a hollow sphere with an inner diameter of 0.15 m and an outer diameter of 0.25 m, featuring a thermal conductivity of 0.68 W/K m Given that the outer surface temperature of the sphere is 28 °C, the temperature at the inner surface can be calculated using these parameters.

A flat body with a thickness of h and thermal conductivity λ is shaped like a right-angled triangle, with each of its short sides measuring l The temperature distribution within this body is described by the equation ϑ(x, y) = ϑ 0 + ϑ 1.

In the defined region where 0 ≤ x ≤ l and 0 ≤ y ≤ x, the highest temperature (ϑ max) and the lowest temperature (ϑ min) can be identified in relation to the given temperatures ϑ 0 and ϑ 1, with the assumption that 0 < ϑ 1 < ϑ 0 To analyze the heat distribution, it is essential to calculate the temperature gradient (grad ϑ) and the heat flux vector (˙ q), identifying the point where the magnitude of the heat flux (| q| ˙) is maximized Furthermore, the heat flow through the three boundary surfaces—y = 0, x = l, and y = x—must be calculated to demonstrate that the amount of heat entering the triangular region is equal to the amount exiting, ensuring energy conservation within the system.

In a saucepan with a diameter of 18 cm, water boils at 100.3 °C while the base, heated electrically, reaches 108.8 °C with a thermal power input of 1.35 kW To determine the heat transfer coefficient of the boiling water, we calculate it based on the temperature difference between the heated base and the boiling water, which is 8.5 °C (ϑ W - ϑ s).

In the thermal boundary layer, the temperature profile can be represented by a parabolic equation ϑ(y) = a + by + cy², with its apex located at y = δt For a thermal boundary layer thickness of δt = 11 mm and a thermal conductivity of λ = 0.0275 W/K·m for air, the local heat transfer coefficient, α, can be determined based on these parameters.

1.9: In heat transfer in forced turbulent ﬂuid ﬂow through a tube, the approximation equation

To calculate the ratio of heat transfer coefficients (α W /α A) for water and air at a mean temperature of 40 °C, use the Nusselt number (N u) and Reynolds number (Re) equations, where the characteristic length is the inner diameter of the tube (d) Ensure that both fluids are analyzed at the same flow velocity (w) and diameter (d) Refer to the material properties listed in Table B1 for water and Table B2 for air in the appendix to complete the calculations.

A long hollow cylinder of length L generates heat due to radioactive decay, with a heat flow per length of ˙ Q/L = 550 W/m Constructed from a steel alloy with a thermal conductivity (λ) of 15 W/K m, the cylinder has an inner diameter of 20 mm and a wall thickness of 10 mm Heat is dissipated solely from its outer surface into space at a temperature of 0 K through radiation, with an emissivity (ε) of 0.17 The Celsius temperatures of the inner surface (ϑ i) and the outer surface (ϑ o) can be calculated, along with the radiative heat transfer coefficient (α rad).

The house wall consists of three layers: an inner plaster measuring 1.5 cm with a thermal conductivity of 0.87 W/K m, a perforated brick wall 17.5 cm thick with a thermal conductivity of 0.68 W/K m, and an outer plaster of 2.0 cm also with a thermal conductivity of 0.87 W/K m The heat transfer coefficients are 7.7 W/m² K for the interior and 25 W/m² K for the exterior To determine the heat flux through the wall, which separates an indoor temperature of 22.0 °C from an outdoor temperature of -12.0 °C, calculations must be performed to find the temperatures at the two wall surfaces, denoted as ϑ W1 and ϑ W2.

To minimize heat loss through the house wall, an insulating board with a thickness of 6.5 cm and thermal conductivity of 0.040 W/K m will be used, along with an outer facing layer that has a thickness of 11.5 cm and a thermal conductivity of 0.79 W/K m The objective is to calculate the heat flux and the surface temperature of the inner wall.

A steam flow at 600 °C passes through a steel alloy tube with an inner diameter of 0.25 m and an outer diameter of 0.27 m, exhibiting a thermal conductivity of 16 W/K m The internal heat transfer coefficient is 425 W/m² K The tube is insulated with a 0.05 m thick rock wool layer, which has a thermal conductivity that decreases with temperature, expressed as λ₂(ϑ) = 0.040 - 0.0005ϑ An additional mineral fiber hull, 0.02 m thick, surrounds the insulation, with an external heat transfer coefficient of 30 W/m² K at an ambient temperature of 25 °C.

The average thermal conductivity of the mineral fibre hull is λ m3 = 0.055 W/K m To determine the heat loss per unit length (˙ Q/L) of the tube, we need to calculate the thermal performance based on this conductivity Additionally, it is essential to verify if the temperature of the mineral fibre hull remains below the maximum allowable limit of ϑ max = 250 ◦ C.

In a scenario where a cylindrical drink can with a diameter of 64 mm and a height of 103 mm is removed from a fridge at an initial temperature of 6 °C and placed in a room at 24 °C, the temperature of the can after 2 hours is calculated to be approximately 19.2 °C Additionally, it takes about 1.5 hours for the can's temperature to reach 20 °C The heat transfer resistance is primarily on the can's exterior, with a heat transfer coefficient of 7.5 W/m²K, while the drink's density is 1.0 x 10³ kg/m³ and its specific heat capacity is 4.1 x 10³ J/kgK.

1.15: Derive the equations for the proﬁle of the ﬂuid temperatures ϑ 1 = ϑ 1 (z) and ϑ 2 = ϑ 2 (z) in a countercurrent heat exchanger, cf section 1.3.3.

In a heat pump system, outside air acts as the heat source, cooling from 10.0 °C to 5.0 °C while heating a fluid from -5.0 °C to 3.0 °C The fluid has a mass flow rate of 0.125 kg/s and a mean specific heat capacity of 3.6 kJ/kgK This heat transfer process will be analyzed across three flow configurations: countercurrent, crossflow with one row of tubes, and counter crossflow with two rows of tubes and two passageways running in opposite directions The operational characteristics of these configurations are detailed in the referenced study.

Determine the required heat transfer capability kA for the heat exchangers with these three ﬂow conﬁgurations.

Fig 1.59: Counter crossﬂow heat exchanger with two rows of tubes and two passageways

1.17: A component A is to be dissolved out of a cylindrical rod of length L and radius r 0

It diffuses through a quiescent liquid film of thickness δ which surrounds the rod Show that the diffusional flux is given by

N ˙ A = D c A0 − c Aδ ln(r 0 + δ)/r 0 2πL , when D = const and a low concentration of the dissolved component in the liquid ﬁlm is assumed.

The heat conduction equation

Derivation of the diﬀerential equation for the temperature ﬁeld

To solve a heat conduction problem, it is essential to determine the temperature field ϑ=ϑ(x, t) as a function of both space and time This allows for the calculation of the heat flux vector field ˙q using Fourier’s law, expressed as q˙(x, t) =−λgradϑ(x, t), enabling the determination of heat flux at any point within the material The temperature field is derived by solving the heat conduction equation, which is based on the first law of thermodynamics applied to a closed system—a defined region within the conductive body This region, with volume V and surface area A, leads to a power balance described by dU/dt = ˙Q(t) + P(t), where the change in internal energy U over time is influenced by the heat flow ˙Q and the power P crossing the region's surface.

Fig 2.1: Region of volume V in a thermal conductive body Surface element dA of the region with the outward normal n

In the study of heat conduction within a solid body, the minor changes in density due to temperature and pressure variations can be overlooked Consequently, the model of an incompressible body, where density remains constant, is employed Under this assumption, the relationship dU/dt = d/dt holds true.

The specific internal energy \( u \) of an incompressible body is dependent on its temperature, and can be expressed through the equation \( du = c(ϑ) dϑ \), where \( c(ϑ) \) represents the specific heat capacity By integrating over the volume of the region, we derive the relationship \( dU = dt \).

∂t dV (2.4) for the change with time of the internal energy in the region under consideration.

To calculate the heat flow \( \dot{Q} \) across a surface in a given region, we consider a differential surface element \( dA \) with an outward-facing normal vector \( n \) The heat flow entering the region through this surface element is expressed as \( d\dot{Q} = -q \cdot n \, dA \).

Heat flow is considered positive when it enters a body, as indicated by the positive heat flux vector ˙q directed into the region Conversely, the outward-facing normal vector n results in a negative scalar product, necessitating the introduction of a minus sign to ensure that the heat flow into the region remains positive By integrating all individual heat flow rates d ˙Q from the equation, the total heat flow ˙Q can be determined.

According to Gauss’ integral theorem, the integral over a region's area is transformed into a volume integral of the divergence of ˙q The power \( P \) entering the region consists of two components: the volume power \( P_V \), which induces a change in volume, and the dissipated power \( P_{diss} \) In an incompressible body, \( P_V \) is zero, while \( P_{diss} \) includes electrical power that, due to the material's resistance, dissipates as heat, known as ohmic or resistance heating Additionally, energy-rich radiation, such as γ-rays, can penetrate solid materials and be absorbed, contributing to an increase in the internal energy of the body.

The power of this dissipative and therefore irreversible energy conversion inside the region is given by

W˙ (ϑ,x, t) dV , (2.7) where ˙W is the power per volume, the so-calledpower density In a body, with a speciﬁc electrical resistancer e =r e (ϑ), which has a current ﬂowing through it,

The power density is represented by the equation W(ϑ, ˙x, t) = r e(ϑ)i² e, where i e denotes the electrical flux Its SI units are A/m² When combined with the specific electrical resistance, measured in Ωm, the resulting units for ˙W are ΩmA²/m⁴, which simplify to VA/m³ or W/m³.

The body's internal energy conversion processes, including dissipative and irreversible mechanisms, function as internal heat sources, contributing to an increase in internal energy Similar to external heat, this internal heat is generated during chemical or nuclear reactions, which convert chemical or nuclear energy into thermal energy While most reactions, except for less common endothermic ones, serve as heat sources in solids, we will focus solely on the heat generated without accounting for changes in chemical composition Consequently, the impact of these reactions is treated as an internal heat source contributing to power density, allowing us to assume that material properties, such as thermal conductivity and specific heat, vary only with temperature and not with composition.

By substituting the results of dU/dt from equation (2.4), ˙Q from equation (2.6), and P from equation (2.7) into the power balance equation (2.2) of the first law, and consolidating all the volume integrals, we obtain a comprehensive expression for the energy dynamics in the system.

This volume integral only disappears for any chosen balance region if the integrand is equal to zero This then produces c(ϑ)∂ϑ

In the last step of the derivation we make use of Fourier’s law and link the heat ﬂux ˙q according to (2.1) with the temperature gradient This gives us c(ϑ)∂ϑ

The differential equation for the temperature field in a quiescent, isotropic, and incompressible material is expressed as ∂t = div [λ(ϑ) gradϑ] + ˙W(ϑ,x,t) This equation incorporates temperature-dependent material properties, c(ϑ) and λ(ϑ), and accounts for heat sources in the thermally conductive body through the power density ˙W.

The heat conduction equation, in its general form, involves several simplifying assumptions that lead to specific differential equations tailored for particular problems A key simplification is the assumption of constant material properties, such as thermal conductivity (λ) and specific heat capacity (c) The resulting linear partial differential equations will be discussed in the following section, along with additional straightforward cases.

– geometric one-dimensional heat ﬂow, e.g only in thex-direction in cartesian coordinates or only in the radial direction with cylindrical and spherical geometries.

The heat conduction equation for bodies with constant

In deriving the heat conduction equation, we assumed an incompressible body with constant properties, neglecting the temperature dependence of thermal conductivity (λ) and specific heat capacity (c) These assumptions are necessary to achieve a mathematical solution, referred to as the "exact" solution Section 2.1.4 will address the solution options for materials with temperature-dependent properties.

With constant thermal conductivity the diﬀerential operator div [λ(ϑ) gradϑ] in (2.8) becomes the Laplace operator λdiv gradϑ=λ∇ 2 ϑ , and the heat conduction equation assumes the form

The constant which appears here is the thermal diﬀusivity a:=λ/c (2.10) of the material, with SI units m 2 /s.

In the two most important coordinate systems, cartesian coordinatesx,y, z, and cylindrical coordinatesr,ϕ,zthe heat conduction equation takes the form

W˙ c (2.12) respectively With spherical coordinates we will limit ourselves to a discussion of heat ﬂow only in the radial direction For this case we get from (2.9)

The simplest problem in transient thermal conduction is the calculation of a temperature ﬁeldϑ=ϑ(x, t), which changes with time and only in thex-direction.

A further requirement is that there are no sources of heat, i.e ˙W ≡0 This is known aslinearheat ﬂow, governed by the partial diﬀerential equation

The equation provides a clear understanding of thermal diffusivity and the heat conduction equation It indicates that the rate of temperature change over time at any point in a conductive material is proportional to its thermal diffusivity This property significantly influences the speed at which temperature variations occur As highlighted in Table 2.1, metals exhibit not only high thermal conductivities but also elevated thermal diffusivity values, suggesting that temperatures in metals fluctuate rapidly.

Table 2.1: Material properties of some solids at 20 ◦ C λ c 10 6 a

The differential equation (2.14) links temperature changes over time at a specific point to the curvature of temperature in the surrounding area This relationship allows for the identification of three distinct scenarios When the curvature is positive (∂²ϑ/∂x² > 0), it indicates a rise in temperature due to more heat influx from the right than outflow to the left, resulting in energy storage and an increase in temperature over time Conversely, a negative curvature (∂²ϑ/∂x² < 0) signifies a decrease in temperature, while a curvature of zero (∂²ϑ/∂x² = 0) indicates a constant temperature, representing a steady-state condition.

The heat conduction equation, characterized as a second-order linear partial differential equation of parabolic type, is influenced by the thermal power ˙W, which may be either linearly dependent or independent of temperature ϑ This equation has been the subject of extensive research and discussion in the 19th and 20th centuries, leading to the development of reliable solution methods, which will be explored in section 2.3.1 Numerous closed mathematical solutions are documented in the authoritative work by H.S Carslaw and J.C Jaeger.

Steady-state temperature ﬁeldsare independent of time, and are the end state of a transient cooling or heating process It is then valid that∂ϑ/∂t = 0, from

Fig 2.2: Importance of the curvature for the temperature change with time according to

The differential equation for a steady-state temperature field with heat sources is represented as ∇²ϑ + (˙W/λ) = 0, known as the Poisson differential equation In this context, thermal conductivity is the sole material property affecting the temperature distribution When there are no heat sources present (˙W ≡ 0), this equation simplifies to the potential equation or Laplace differential equation.

The Laplace operator ∇² is represented in various forms across different coordinate systems, as shown in equations (2.11), (2.12), and (2.13) For steady-state temperature fields, the differential equations (2.15) and (2.16) are classified as linear and elliptical, provided that ˙W remains constant or varies linearly with respect to ϑ This distinction necessitates alternative solution methods compared to transient conduction, where the governing differential equations are parabolic in nature.

Boundary conditions

The heat conduction equation only determines the temperature inside the body.

To accurately define the temperature field, it is essential to implement and satisfy various boundary conditions in the differential equation's solution These conditions encompass an initial-value requirement related to time, along with specific local conditions that must be adhered to at the body's surfaces Ultimately, the temperature field is governed by the differential equation in conjunction with these boundary conditions.

The initial-value conditions establish a specific temperature for each location in the body at a designated time, initiating timekeeping with the condition ϑ(x, y, z, t= 0) = ϑ 0 (x, y, z) The initial temperature profile ϑ 0 (x, y, z), such as a uniform temperature in a cooling body, evolves throughout the transient conduction process.

The localboundary conditionscan be divided into three diﬀerent groups At the surface of the body

1 the temperature can be given as a function of time and position on the surface, the so-called 1st type of boundary condition,

2 the heat ﬂux normal to the surface can be given as a function of time and position, the 2nd type of boundary condition, or

3 contact with another medium can exist.

A given surface temperature is the most simple case to consider, especially when the surface temperature is constant In the case of a prescribed heat ﬂux ˙q, the condition ˙ q =−λ∂ϑ

The condition ∂n (2.18) must be satisfied at every point on the surface, where the derivative is calculated in the outward normal direction, and λ represents the conductivity value at the surface temperature Adiabatic surfaces are commonly encountered, and with ˙q = 0, we derive important implications for thermal behavior.

To effectively address conduction problems, it is essential to adhere to the condition of symmetry within the body By focusing on a section of the body that is limited by one or more adiabatic planes of symmetry, one can simplify the analysis and enhance the efficiency of problem-solving.

When a thermally conductive body contacts another medium, various boundary conditions arise based on whether the medium is a solid or fluid and its material properties In cases where the other medium is a solid, the heat flux at the interface between the two bodies remains consistent for both According to the established principles, at the interface, it is valid that λ (1).

The temperature curve exhibits a kink at the interface, indicating a larger temperature gradient in the body with lower thermal conductivity, as shown in Fig 2.3 a Equation (2.21) is applicable only when the two bodies are firmly joined; otherwise, contact resistance leads to a minor temperature jump, illustrated in Fig 2.3 b This resistance can be quantified using a contact heat transfer coefficient, α ct, which modifies the original equation.

=α ct ϑ (1) I −ϑ (2) I , (2.22) is valid With constantα ct the temperature drop at the interface is proportional to the heat ﬂux.

Fig 2.3: Temperature at the interface between two bodies 1 and 2 in contact with each other a no contact resistance, b contact resistance according to (2.22)

When a thermally conductive body is surrounded by a fluid, a boundary layer forms within the fluid The heat transfer into the fluid can be expressed with the equation ˙ q=α(ϑ W −ϑ F ), where α represents the heat transfer coefficient This heat flux must be conducted to the surface of the body, leading to the establishment of a specific boundary condition.

The thermal conductivity of the solid at the wall is represented by λ, while Eq (2.23) establishes a linear relationship between the wall temperature (ϑ W) and the slope of the temperature profile at the surface, known as the third type of boundary condition The derivative ∂ϑ/∂n indicates the normal direction outward from the surface For solving the heat conduction problem, the fluid temperature (ϑ F), which may vary over time, and the heat transfer coefficient (α) are essential When α is significantly large, the temperature difference (ϑ W − ϑ F) becomes minimal, allowing the boundary condition to simplify to a prescribed temperature (ϑ W = ϑ F) This linear boundary condition holds only if α remains constant with respect to ϑ W or (ϑ W − ϑ F), which is crucial for accurately solving heat conduction problems In scenarios such as free convection, where α varies with (ϑ W − ϑ F), or in cases involving radiation heat transfer, the linearity of the boundary condition is compromised.

The temperature profile at the boundary condition indicates that the tangent to the temperature curve at the solid surface intersects the guide-point R at the fluid temperature ϑ F, positioned at a distance of s = λ/α = L 0 /Bi from the surface Additionally, the subtangent in the fluid boundary layer is defined as s u = λ F /α = L 0 /N u In scenarios where ˙q is influenced by T W 4, a closed-form solution for thermal conduction becomes unfeasible, necessitating the use of numerical methods, which will be elaborated on in section 2.4.

In a vessel containing fluid, the temperature (ϑ F) is often considered spatially constant due to convection or mixing, which equalizes temperatures throughout the fluid The time-dependent change in temperature (ϑ F = ϑ F(t)) is influenced by the heat transfer from the vessel wall to the fluid, affecting the fluid's internal energy and resulting in variations in temperature Consequently, the heat flux is represented as ˙ q = − λ.

(thermal conduction in the vessel wall) and also ˙ q = c F M F

To analyze the heating of the fluid, we consider the specific heat capacity (c F) and the mass (M F) of the fluid in contact with the vessel wall (area A) The boundary condition is defined by the equation c F M F A dϑ F dt + λ, which relates the heat transfer dynamics within the system.

This is to be supplemented by a heat transfer condition according to (2.23), or if a large heat transfer coeﬃcient exists by the simpliﬁed boundary condition ϑ W = ϑ F

Temperature dependent material properties

When the temperature dependence of material properties, represented as λ(ϑ) and c(ϑ), is significant, the heat conduction equation becomes essential for addressing conduction problems This results in a non-linear problem that is typically solvable only in rare instances By applying the divergence theorem, we derive the expression div [λ(ϑ) gradϑ] = λ(ϑ) div gradϑ + (dλ/dϑ) grad²ϑ from the heat conduction equation, leading to the relationship c(ϑ)∂ϑ.

∂t =λ(ϑ)∇ 2 ϑ+dλ dϑgrad 2 ϑ+ ˙W (2.25) as the heat conduction equation The non-linearity is clearly shown in the ﬁrst two terms on the right hand side.

The introduction of the transformed temperature Θ, defined as Θ = Θ 0 + 1 λ 0 ϑ ϑ 0 λ(ϑ) dϑ, simplifies Equations (2.8) and (2.25) In this equation, λ 0 represents the thermal conductivity at the reference temperature ϑ 0, which is associated with the transformed temperature Θ 0 This transformation facilitates a clearer understanding of the thermal behavior described by the equations.

∂t and grad Θ = λ λ 0 gradϑ Following on from (2.8), we obtain c(ϑ) λ 0 λ(ϑ)

The equation ∂t = ∇²Θ + W˙λ₀ aligns with the heat conduction equation for constant material properties, though thermal diffusivity (a) varies with temperature (ϑ or Θ) It is noted that a(ϑ) changes less significantly with temperature than thermal conductivity (λ), allowing a(ϑ) in the equation to be treated as approximately constant This allows for the application of solutions for constant material properties to cases with temperature-dependent properties by substituting ϑ with Θ However, a key limitation exists: only boundary conditions with specified temperature or heat flux can be applied, as the heat transfer condition is not preserved through this transformation This transformation is especially useful in steady-state heat conduction problems, where the temperature-dependent term vanishes due to ∂Θ/∂t = 0 Consequently, solving the Poisson or Laplace equations can proceed directly, provided boundary conditions for temperature or heat flux are defined, excluding the heat transfer condition.

In many instances involving temperature-dependent material properties, a closed-form solution is unattainable Therefore, a numerical solution becomes essential, which will be explored in detail in section 2.4.

Similar temperature ﬁelds

Introducing dimensionless variables simplifies the representation of physical relationships by significantly reducing the number of influencing variables, as demonstrated in section 1.1.4 For thermal conduction, these dimensionless variables are readily identifiable due to the explicit formulation of the differential equations and boundary conditions.

The starting point for the derivation of the dimensionless numbers in thermal conduction is the diﬀerential equation

The equation W˙ c encompasses all essential terms, including time dependence and local temperature field variations, along with the power density ˙W of heat sources By utilizing a dimensionless position coordinate and a dimensionless time, we define x + := x/L 0 and t + := t/t 0.

In the study of heat conduction, L₀ represents the characteristic length of the conductive body, while t₀ denotes a yet-to-be-determined characteristic time interval We define a dimensionless temperature, ϑ⁺, as (ϑ - ϑ₀)/∆ϑ₀, where ϑ₀ is the reference temperature and ∆ϑ₀ is the characteristic temperature difference relevant to the problem The heat conduction equation can then be expressed in its dimensionless form.

The characteristic time is chosen to be t 0 =L 2 0 /a The dimensionless time is then t + =at/L 2 0 (2.30a)

This dimensionless timet + is often called theFourier number

The Fourier number is not a conventional dimensionless number with a fixed value for a specific problem; rather, it serves as a dimensionless time variable that only assumes fixed values at designated times.

Now introducing the characteristic power density

W˙ 0 = ∆ϑ 0 c/t 0 =λ∆ϑ 0 /L 2 0 (2.31a) as a reference quantity, and with

W˙ + = ˙W /W˙ 0 = ˙W L 2 0 /λ∆ϑ 0 (2.31b) as the dimensionless heat source function, we obtain from (2.28)

The heat conduction equation (2.32) utilizes the dimensionless heat source function ˙W + from (2.31b), which does not necessitate the introduction of additional dimensionless numbers Instead, the dimensionless parameters that define the position, time, and temperature dependence of ˙W + serve this purpose Although the initial condition (2.17) and the homogeneous boundary condition for a prescribed temperature do not inherently create dimensionless numbers, careful selection of ϑ 0 and ∆ϑ 0 in (2.28) is essential to achieve an optimal dimensionless temperature ϑ + within the range of 0≤ϑ + ≤1 When the heat flux ˙q W is defined as a boundary condition, it results in the dimensionless relationship ˙ q + W =− from (2.18).

The dimensionless function ˙q W + incorporates parameters that characterize the spatial and temporal variability of the heat flux ˙q W, excluding the scenario where ˙q W equals zero These dimensionless parameters play a crucial role in the analysis of heat conduction problems, similar to how the parameters in the heat source function ˙W + from (2.31b) are utilized as dimensionless numbers in solutions.

Finally making the heat transfer condition (2.23) dimensionless, gives

A new dimensionless number appears here and it is known as theBiot 1 number

The temperature field in a fluid during heat transfer is influenced by the Biot number, which shares a similar structure to the Nusselt number A key distinction is that the Biot number incorporates the thermal conductivity of the solid, while the Nusselt number involves the thermal conductivity of the fluid The Nusselt number acts as a dimensionless indicator of the heat transfer coefficient, aiding in its assessment, whereas the Biot number characterizes the thermal conduction boundary conditions within a solid Specifically, the Biot number is the ratio of a characteristic length to the subtangent of the temperature curve in the solid, in contrast to the Nusselt number, which compares a characteristic length to the subtangent of the temperature profile in the fluid's boundary layer.

Jean Baptiste Biot (1774–1862) was appointed Professor of Physics at the Collège de France in Paris in 1800 He conducted research on the cooling of heated rods starting in 1804 and published the differential equation for the temperature profile in 1816, though he did not provide a detailed derivation In 1820, he, along with F Savart, discovered the Biot-Savart law, which describes the magnetic field strength around a conductor carrying an electric current.

Fig 2.5: Inﬂuence of the Biot number Bi = αL 0 /λ on the temperature proﬁle near to the surface a small Biot number, b large Biot number

The Biot number can be understood as the ratio of specific thermal conduction resistance (L0/λ) of a solid to the specific heat transfer resistance (1/α) at its surface.

A small Biot number indicates that thermal conduction resistance within a body is significantly lower than the heat transfer resistance at its boundary, resulting in a minor temperature difference within the body compared to the temperature difference between the wall and fluid Conversely, large Biot numbers lead to greater internal temperature variations, as illustrated in the cooling process examples in Fig 2.5 In cases of very large Biot numbers, the difference between the wall and fluid temperatures becomes minimal.

, and for Bi → ∞ , according to (2.34) we get ϑ + W − ϑ + F

→ 0 The heat transfer condition (2.34) can be replaced by the simpler boundary condition ϑ + W = ϑ + F

In quiescent solid bodies, temperature fields can be expressed in a dimensionless format using the equation ϑ + =ϑ + x + , y + , z + , t + ,W˙ + ,q˙ + W , Bi, N Geom This equation incorporates dimensionless variables and introduces dimensionless geometric numbers, denoted as N Geom Notable examples of these geometric numbers include the height-to-diameter (H/D) ratio of thermally conductive cylinders and the ratios L 2 /L 1.

The function in equation (2.36) is influenced by the geometry and specific conditions of the heat conduction problem, with the edge lengths L1 to L3 of a rectangular body being significant Typically, not all dimensionless variables in (2.36) are present simultaneously; for instance, in steady-state heat conduction, the time variable t disappears, and in bodies devoid of heat sources, the term ˙W becomes zero Additionally, when only prescribed temperatures are applied as boundary conditions, the terms ˙q, W, and Bi in (2.36) are absent.

In heat conduction problems, the use of dimensionless representation and the combination of influencing quantities into dimensionless numbers is less critical compared to the determination of heat transfer coefficients Therefore, in subsequent sections, we will often avoid making the heat conduction problem dimensionless, instead presenting solutions in a dimensionless form through appropriate combinations of variables and influencing factors.

Steady-state heat conduction

Geometric one-dimensional heat conduction with heat sources

In section 1.2.1, we explored one-dimensional geometric heat conduction without internal heat sources, where temperature relied solely on one spatial coordinate We derived equations for steady heat flow through various shapes, including flat walls, hollow cylinders, and hollow spheres This discussion will now expand to include thermally conductive materials with internal heat sources, such as electrical conductors dissipating energy from flowing current and cylindrical or spherical fuel elements in nuclear reactors, where heat from nuclear fission is conducted to the fuel surfaces.

The heat conduction equation reveals that the sole material property influencing the steady-state temperature field, where ∂ϑ/∂t equals zero, is thermal conductivity, denoted as λ=λ(ϑ) For the sake of simplification, we assume that λ remains constant.

The differential equation for the steady temperature field is represented as ∇²ϑ + (˙W/λ) = 0, where ˙W denotes the thermal power generated per volume, which may vary with temperature ϑ or position In this analysis, we focus on one-dimensional heat flow, with temperature changes occurring solely along the position coordinate r, applicable even in Cartesian coordinates The Laplace operator varies across Cartesian, cylindrical, and spherical coordinate systems, and integrating these three cases yields the equation d²ϑ/dr² + n r dϑ/dr +

W˙ (r, ϑ) λ = 0 (2.38) as the decisive diﬀerential equation withn= 0 for linear heat ﬂow (plate), n= 1 for the cylinder andn= 2 for the sphere.

Solving the ordinary diﬀerential equation (2.38) for constant power density

2 (1 +n)λ (2.39) for the three cases, plate (n = 0), cylinder (n = 1) and sphere (n = 2) Here c 0 and c 1 are constants, which have to be adjusted to the boundary conditions.

As an example we will look at the simple case of heat transfer to a ﬂuid with a temperature ofϑ=ϑ F taking place at the surfacer=±R We then have

=α[ϑ(R)−ϑ F ] (2.40) and due to symmetry dϑ dr r=0

The condition (2.41) requires thatc 1 = 0 Then from (2.40) the constant c 0 can be found, so that the solution to the boundary value problem is ϑ(r) =ϑ F +

W˙ 0 R 2 /λ , (2.42) the dimensionless distance r + =r/R (2.43) from the central plane or point and with the Biot number

In the bodies a parabolic temperature proﬁle exists, with the highest temperature atr + = 0, Fig 2.6.

If the heat ﬂux ˙q W at the surface (r=R) of the three bodies is given instead of the power density ˙W 0 , then from the energy balance,

Fig 2.6: Dimensionless temperature proﬁle ϑ + according to (2.45) in a plate

(n = 2) for Bi = αR/λ = 4.0 withV as the volume andAas the surface area, we obtain the simple relationship ˙ q W = ˙W 0 V

Equation (2.45) for the temperature proﬁle then has a form independent of the shape of the body: ϑ(r) =ϑ F +q˙ W R

The considerations for temperature-dependent thermal conductivity, λ = λ(ϑ), apply when the heat flux, ˙ q W, at the wall (r = R) is calculated based on the given shape of the body and ˙ W 0 The surface temperature, ϑ W, can be derived from the equation ϑ W = ϑ F + ˙ q W /α To account for variations in λ, we introduce the transformed temperature, Θ, defined as Θ = Θ W + (1/λ(ϑ W)) ∫ ϑ ϑ W λ(ϑ) dϑ This leads to a differential equation for Θ, expressed as d²Θ/dr² + (n/r) dΘ/dr +

W ˙ 0 λ (ϑ W ) = 0 , whose form agrees with (2.38) This is solved under the boundary conditions r = 0 : dΘ/ dr = 0 , r = R : Θ = Θ W (corresponding to ϑ = ϑ W ) The solution is the parabola Θ − Θ W = W ˙ 0 R 2

For the calculation of the highest temperature ϑ max in the centre of the body, with r = 0 we obtain Θ max − Θ W =

2λ(ϑ W ) and then with (2.48) for ϑ max the equation ϑ max ϑ W λ(ϑ) dϑ = λ (ϑ W ) (Θ max − Θ W ) = W ˙ 0 R 2

M Jakob [2.2] dropped the presumption that ˙ W = ˙ W 0 = const and considered heat development rising or falling linearly with the temperature The ﬁrst case occurs during the heating of a metallic electrical conductor whose electrical resistance increases with temperature.

Example 2.1: A cylindrical fuel rod of radius r = 0.011 m is made of uranium dioxide (UO 2 ) At a certain cross section in the element the power density is ˙ W 0 = 1.80 ã

10 5 kW/m 3 ; the surface temperature has a value of ϑ W = 340 ◦ C The maximum temperature ϑ max in the centre of the element is to be calculated The thermal conductivity of

UO 2 , according to J H¨ ochel [2.3], is given by λ(T ) W/K m = 3540 K

, which is valid in the region 300 K < T < 3073 K.

The maximum (thermodynamic) temperature T max is obtained from (2.50) with n = 1 and ϑ = T :

With the surface temperature T W = ϑ W + 273 K = 613 K this yields

W m From which the transcendental equation ln T max + 57 K

− 0.1412 is obtained, with the solution T max = 2491 K, and correspondingly ϑ max = 2281 ◦ C This temperature lies well below the temperature at which UO 2 melts, which is around 2800 ◦ C.

Longitudinal heat conduction in a rod

When a rod-shaped object, such as a bolt or pillar, is heated at one end, heat transfers along its length and dissipates into the environment through its outer surface This process resembles the heat conduction problem associated with heat release from fins, which will be discussed in the following section Additionally, various measurement techniques exist for determining thermal conductivity, relying on the comparison of temperature drops in rods composed of different materials.

Fig 2.7: Temperature proﬁle in a rod of constant cross sectional area A q

In this analysis, we examine a rod of length L with a constant cross-sectional area A_q and circumference U, where one end is maintained at a constant temperature ϑ_0 through heat input Heat flows along the rod's axis and is dissipated to the environment via its outer surface Due to the small cross-sectional area, the temperature is uniform across the entire cross-section, varying only along the x-coordinate For a rod segment with volume ΔV = A_q Δx, the heat transfer to the surroundings, which is at a constant temperature ϑ_S, occurs through the outer surface of the rod.

This release of heat has the same eﬀect as a heat sink in the rod material with the power density

Incorporating the heat source into the linear heat flow differential equation (2.38) yields the modified equation: \( \frac{d^2 \vartheta}{dx^2} - \alpha U \lambda A q (\vartheta - \vartheta_S) = 0 \) (2.51) This equation is essential for determining the temperature distribution along the rod.

Assuming a constant heat transfer coeﬃcientαand putting as an abbreviation m 2 =αU/λA q , (2.52) gives the general solution of (2.51) as ϑ(x) =ϑ S +c 1 exp(−mx) +c 2 exp(mx)

The constants of integrationc 1 ,c 2 ,C 1 andC 2 are determined from the boundary conditions.

At the left end of the rod (x = 0), the temperature is consistently maintained at ϑ = ϑ₀, while varying boundary conditions are applied at the opposite end (x = L) The fixed temperature condition at x = 0 indicates that the dimensionless temperature, defined as ϑ⁺ := (ϑ - ϑₛ) / (ϑ₀ - ϑₛ), should be utilized, resulting in the range of 0 ≤ ϑ⁺ ≤ 1.

We will now consider a rod which stretches into the surroundings Fig 2.7, and releases heat through its end surface (x=L) Here

The equation Q˙ L =−λA q dϑ dx =α L A q (ϑ−ϑ S ) for x=L indicates that the heat transfer coefficient α L at the end surface of the rod may differ from α at the outer surface Following calculations reveal the temperature profile within the rod, expressed as ϑ + (x) = cosh[m(L−x)] + (α L /mλ) sinh[m(L−x)] / [cosh[mL] + (α L /mλ) sinh[mL]] This relationship highlights the complexity of heat distribution along the rod and the influence of varying coefficients on the overall temperature gradient.

The heat ﬂow ˙Qreleased to the environment is equal to the heat ﬂow through the rod cross section atx= 0:

When α L = 0, a significant simplification of the equations occurs, applicable when the rod's end is insulated or when heat loss through the small cross-sectional area A q is negligible Consequently, the temperature profile is expressed as ϑ + (x) = cosh[m(L−x)] / cosh(mL).

The temperature at the free end falls to ϑ + (L) = ϑ L −ϑ S ϑ 0 −ϑ S = 1 cosh(mL) (2.58)

The heat released is yielded from (2.56) to be

The functions (coshmL) − 1 and tanhmLare represented in Fig 2.8.

To apply equations (2.57) to (2.59) for α L = 0, the original rod of length L with heat release at x = L must be conceptually replaced by an insulated rod of length L + ∆L, where the additional length ∆L accounts for the insulation at x = L + ∆L.

∆Lis determined such, that the heat ﬂow ˙Q L released via the end surface is now released via the additional circumferential areaU∆L For small values of ∆Lit is approximately valid that

Q˙ L =α L A q (ϑ L −ϑ S ) =αU∆L(ϑ L −ϑ S ) From which the corrected lengthL C of the replacement rod is obtained as

This value is to be used in place ofLin equations (2.57) to (2.59).

In this analysis, we examine the scenario where the temperature at the end of the rod, denoted as x = L, is fixed at ϑ L Through a series of calculations, we derive the temperature profile along the rod, represented by the equation ϑ + (x) = sinh[m(L − x)] sinh(mL) + ϑ + (L) sinh(mx) sinh(mL) This equation illustrates how the temperature varies along the length of the rod based on the specified boundary condition.

To calculate the heat released by the rod from x = 0 to x = L, it's essential to determine the heat flow in the x-direction across the two cross sections at x = 0 and x = L This involves analyzing the heat transfer at any given cross section.

Q(x) = ˙ − λA q dϑ dx = λA q m (ϑ 0 − ϑ S ) sinh(mL) cosh[m(L − x)] − ϑ + (L) cosh(mx)

Fig 2.8: Characteristic functions for calculation of the overtemperature at the free end of the rod according to (2.58) and of the heat

Fig 2.9: Temperature profile for a rod, whose ends are maintained at the given temperatures ϑ 0 and ϑ L a at x = L heat flows out ( ˙ Q L > 0), b at x = L heat flows in ( ˙ Q L < 0)

From which we obtain, with m according to (2.52)

Q ˙ 0 = αλA q U (ϑ 0 − ϑ S ) sinh(mL) cosh(mL) − ϑ + (L) and

The heat released between the two ends via the outer surface is then

In this example, we analyze a cylindrical steel bolt with a thermal conductivity of 52.5 W/K m, a diameter of 0.060 m, and a length of 0.200 m, which extends from a plate at a temperature of 60.0 °C The bolt transfers heat to the surrounding air, which is at a temperature of 12.5 °C, through its outer surface and free end, with a heat transfer coefficient of 8.0 W/m² K The objective is to calculate the heat flow (˙Q₀) transferred to the air and the temperature (ϑL) at the free end of the bolt.

For the heat ﬂow from (2.56) with U = πd and A q = πd 2 /4, we obtain

1 + (α/mλ) tanh (mL) Thereby according to (2.52) mL = 2

This gives ˙ Q 0 = 13.3 W The temperature of the free bolt end is yielded from (2.55) with x = L to be ϑ + = 1 cosh(mL) + (α/mL) sinh(mL) = 0.8047 which produces ϑ L = ϑ S + (ϑ 0 − ϑ S ) ϑ + L = 50.7 ◦ C

If the heat released via the free end of the bolt is neglected the value for the heat ﬂow

Q ˙ 0 = 12.7 W obtained from (2.59) is too low The temperature ϑ L is found to have the larger value ϑ L = 51.8 ◦ C Using the corrected bolt length

4 = 0.215 m according to (2.60) we find that the values for Q ˙ 0 from (2.59) and ϑ L from (2.57) agree with the exact values to three significant figures.

The temperature distribution in ﬁns and pins

Increasing the surface area for heat transfer, particularly on the side with the lower heat transfer coefficient, can enhance the efficiency of heat exchange between two fluids This can be achieved by adding fins or pins However, the effectiveness of this area enlargement is limited due to the temperature gradient present within the fins, which is necessary for heat conduction from the fin base Consequently, the average temperature difference crucial for heat transfer to the fluid is less than that at the base of the fins.

In order to describe this effect quantitatively, thefin efficiency was introduced in section 1.2.3 Its calculation is only possible if the temperature distribution in the

fin is known, which we will cover in the following Results for the fin efficiencies for different fin and pin shapes are given in the next section.

In order to calculate the temperature distribution some limiting assumptions have to be made:

1 The fin (or pin) is so thin that the temperature only changes in the direction from fin base to fin tip.

2 The ﬁn material is homogeneous with constant thermal conductivityλ f

3 The heat transfer at the ﬁn surface will be described by a constant heat transfer coeﬃcientα f

4 The temperatureϑ S of the ﬂuid surrounding the ﬁn is constant.

5 The heat ﬂow at the tip of the ﬁn can be neglected in comparison to that from its sides.

The assumptions regarding the constant α f over the surface of the fin are generally valid, except for variations in the heat transfer coefficient Research conducted by S.-Y Chen and G.L Zyskowski, L.S Han and S.G Lefkowitz, as well as H.C Ünal, has explored the impact of this varying heat transfer coefficient.

The temperature distribution in fins can be modeled by a second-order differential equation, derived from analyzing a volume element taken from any fin or pin with a thickness of ∆x This analysis involves applying the principles of energy balance to the element.

The equation Q(x) = ˙ ˙ Q(x + ∆x) + ∆ ˙ Q S (2.63) illustrates that the heat conducted into the element at point x must account for the heat transferred to x + ∆x and the heat flow ∆ ˙ Q S, which is exchanged through the surface ∆A f with the surrounding fluid at temperature ϑ S By defining the overtemperature as Θ(x) = ϑ(x) − ϑ S (2.64) and considering ∆A f as the area through which heat is dissipated from the fin element, we derive the expression for ∆ ˙ Q S.

∆ ˙ Q S = α f ∆A f Θ(x + ε∆x) , 0 ≤ ε ≤ 1 According to Fourier’s law, the heat ﬂow through the cross sectional area A q (x)

Q(x) = ˙ − λ f A q (x) dΘ dx (2.65) is transported by conduction From this we obtain (Taylor series at point x)

∆x + O(∆x 2 ) , where O(∆x 2 ) indicates that the rest of the terms is of the order of ∆x 2 Putting these relationships for the three heat ﬂows into the balance equation (2.63) gives

Division by ∆x, yields for the limit ∆x → 0 d dx

− α f λ f dA f dx Θ = 0 (2.66) as the desired diﬀerential equation for the overtemperature Θ(x).

This differential equation encompasses all types of extended surfaces, provided that the specified assumptions are satisfied The various fin or pin geometries are represented by the terms A_q = A_q(x) for the cross-sectional area and A_f = A_f(x) for the fin surface area responsible for heat dissipation.

So for a straight ﬁn of width b perpendicular to the drawing plane in Fig 2.11, with a proﬁle function y = y(x) we obtain the following for the two areas

Fig 2.10: Energy balance for a volume element of thickness ∆x and surface area ∆A f of any shape of ﬁn or pin with a cross sectional area A q (x)

Fig 2.11: Straight ﬁn of width b (perpendicular to drawing plane) with any proﬁle function y = y(x)

Fig 2.12: Annular ﬁn on a tube (outer radius r 0 ) with proﬁle function y = y(r)

The narrow sides of the ﬁn are not considered in A f In addition to this the diﬀerence between

A f and its projection on the plane formed by the ﬁn width b and the x-axis will also be neglected.

A thin ﬁn is presumed This leads to the diﬀerential equation d 2 Θ dx 2 + 1 y dy dx dΘ dx − α f λ f y Θ = 0 (2.67)

This has to be solved for a given fin profile y = y(x) under consideration of the boundary conditions at the fin base, Θ = Θ 0 = ϑ 0 − ϑ S for x = 0 , (2.68) and the condition dΘ dx = 0 for x = h (2.69)

This corresponds to the ﬁfth assumption, the neglection of the heat released at the tip of the ﬁn.

For thin annular ﬁns, as in Fig 2.12, with r as the radial coordinate we obtain

A q (r) = 4πry(r) and A f (r) = 2π(r 2 − r 2 0 ) , where y = y(r) describes the fin profile The differential equation for annular fins is found from (2.66) to be y d 2 Θ dr 2 + y r + dy dr dΘ dr − α f λ f Θ = 0 (2.70)

The solution has to fulfill the boundary conditions Θ = Θ 0 = ϑ 0 − ϑ S for r = r 0 at the fin base and dΘ/dr = 0 for r = r 0 + h at the tip of the fin.

The solutions to the differential equations (2.67) and (2.70) for various profile functions y = y(x) and y = y(r) were initially presented by D.R Harper and W.B Brown in 1922, followed by E Schmidt in 1926 In 1945, K.A Gardner conducted a comprehensive study of all profiles leading to differential equations with generalized Bessel function solutions R Focke introduced and solved the differential equation for cone-shaped pins with different profiles in 1942 For a detailed overview of temperature distributions in extended heat transfer surfaces, refer to the work of D.Q Kern and A.D Kraus.

Fig 2.13: Temperature distribution in a straight ﬁn with a rectangular proﬁle

(ﬁn thickness δ f ) as a function of x/h with mh as a parameter according to (2.72)

This article discusses a straight fin with a rectangular profile, characterized by its thickness, denoted as δ_f The profile function is represented as y(x) = δ_f / 2 From the established equation (2.67), we derive a straightforward differential equation: d²Θ/dx² - 2α_f λ_f δ_f Θ = 0.

The solution for heat conduction in the axial direction of a rod, represented by equation (2.72), is derived from the abbreviation m '2α f λ f δ f and satisfies the boundary conditions outlined in equations (2.68) and (2.69) When considering the heat release at the tip of the fin, a corrected fin height can be introduced, leading to the adjustment h C = h + ∆h = h + δ f /2 Additionally, equation (2.72) remains applicable for a pin with a constant cross-sectional area A q, provided that δ f is defined as A q /2U, where U represents the circumference of the cross-sectional area.

Figure 2.13 illustrates the temperature profile as a function of the dimensionless coordinate x/h for various values of the parameter mh, which integrates the fin's dimensions, thermal conductivity, and heat transfer coefficient For practical applications, it is recommended to select values within the range of 0.7 < mh < 2 In contrast, long fins with mh > 2 experience a rapid temperature drop, resulting in a significant portion of the fin transferring minimal heat due to insufficient overtemperature Conversely, values of mh < 0.7 suggest that extending the fin could facilitate a considerably larger heat flow.

To achieve an optimal value for mh, it is essential to maximize the heat flow (˙Q f) released from the fin while maintaining a constant volume of the fin material The heat flow conducted through the fin base at x=0 determines the efficiency of this process.

The fin width is determined by the dimensions of the cooling apparatus, influencing the heat transfer rate (˙Q f) The condition d ˙Q f /dh= 0 leads to the equation tanh(mh) = 3mh cosh²(mh) = 3mh (1−tanh²(mh)), with the solution mh = 2α f λ f δ f h = 1.4192 By applying this condition, the optimal height (h) of the fin is established, maximizing heat flow for a specified fin volume (V f) Additionally, similar analyses for annular fins with varying profile functions can be found in the work of A Ullmann and H Kalman.

E Schmidt [2.8] determined the fin shape, which for a specific thermal power, required the least material The profile of these fins is a parabola, with its vertex at the tip of the fin These types of pointed fins are difficult to produce, which is why fins used in practice have either rectangular, trapezoidal or triangular cross sections.

Fin eﬃciency

The efficiency (η_f) of a straight fin with a rectangular profile, as described by equation (2.77), and a triangular profile, according to equation (2.79), is analyzed in relation to the parameter mh from equation (2.78) This efficiency is defined as the ratio of the actual heat flow (˙Q_f) emitted by the fin to the total heat flow.

The heat flow from the fin, represented by Q˙ f0 = α f A f (ϑ 0 − ϑ S) = α f A f Θ 0, indicates the amount of heat that would be released if the temperature along the fin matched its base temperature, ϑ 0, instead of the lower mean temperature, ϑ f In this context, ϑ S refers to the fluid temperature, as noted in section 1.2.3 This heat flow from the fin corresponds to the heat flow ˙Q(x = 0) that is conducted from the fin base into the fin itself.

HereA q0 is the cross sectional area at the ﬁn base.

The heat flow ˙Q f for a straight fin with a rectangular profile was calculated in the last section Using (2.74) andA q0 =b δ f it follows from (2.76) that η f =tanh(mh) mh (2.77)

The eﬃciency of a straight rectangular ﬁn only depends on the dimensionless group mh '2α f λ f δ f h (2.78)

Fig 2.14 showsη f according to (2.77) For the optimal value determined in 2.2.3 ofmh = 1.4192 for the ﬁn with the largest heat release at a given volume the eﬃciency isη f = 0.627.

The efficiencies of various fin shapes can be determined through their temperature distribution Figure 2.14 illustrates the efficiency (η_f) as a function of the product of the fin parameter (m) and heat transfer coefficient (h) for a straight fin with a triangular profile, where δ_f represents the thickness at the fin's base The efficiency curve for this triangular fin resembles that of a straight rectangular fin Consequently, the expression for η_f that involves Bessel functions can be substituted with a similar function, as shown in equation (2.79): η_f = tanh(ϕmh) / (ϕmh).

The correction factorϕis given by ϕ= 0.99101 + 0.31484tanh (0.74485mh) mh , (2.80) from which we obtain a reproduction of the exact result, which for values of mh 0.5, the deviation from the exact values is limited to ±1%.

Disk fins, commonly shaped as squares, rectangles, or hexagons, are often attached to tubes, with multiple tubes connected using fin sheets that allow for passage These fins require consideration of two-dimensional temperature fields, as their thermal performance is influenced by multiple coordinates To estimate the efficiency (η f) of these disk fins, calculations can be based on formulas (2.81) or (2.82) for an annular fin of equivalent surface area For rectangular fins, the heat transfer coefficient is given by h = s1 s2 / π - r0 = 0.564 √(s1 s2) - r0, while for hexagonal fins, the side length is denoted as sh.

2π s − r 0 = 1.211s − r 0 (2.84) to put into the equations Somewhat more accurate values have been found by Th.E Schmidt [2.14] with h = 0.64 s 2 (s 1 − 0.2s 2 ) − r 0 (2.85)

Disk fins, represented by rectangular disks on a tube, can be arranged with both rectangular and hexagonal configurations E.M Sparrow and S.H Lin provided analytical approximations for square and hexagonal fins, yielding precise values for fin efficiency (η f) through extensive calculations For cases where the larger side of the rectangle (s 1) is equal to the smaller side (s 2), the simplified equations prove accurate when the height-to-radius ratio (h/r 0) is greater than or equal to 0.5 Additionally, H.D Baehr and F Schubert experimentally validated the efficiency of square disk fins using an electrical analogy method, confirming the accuracy of the approximation equations.

Geometric multi-dimensional heat ﬂow

Calculating plane and spatial steady temperature fields in multi-dimensional heat flow is considerably more complex than previous scenarios where temperature varied in just one coordinate direction This complexity arises particularly in finding solutions to the Laplace equation for plane temperature fields that do not involve heat sources.

The equation ∂y² = 0 can be derived using various mathematical techniques By employing the product or separation formula ϑ(x, y) = f₁(x) * f₂(y), we can simplify it into two ordinary differential equations that are straightforward to solve Meeting the boundary conditions becomes much simpler when the temperature field is analyzed within a rectangle aligned with the x and y axes For further examples of this method, refer to S Kakaác's book.

Conformal mapping is a valuable solution method for addressing regions with complex geometries, as highlighted by H.S Carslaw and J.C Jaeger, along with U Grigull and H Sandner While simple solutions are achievable with constant temperature or adiabatic boundary conditions, heat transfer conditions are generally treated as approximations An illustrative case is K Elgeti's investigation, which calculated the heat released from a pipe embedded in a wall Additionally, the method of superposition of heat sources and sinks is significant for calculating complex temperature fields between isothermally bounded bodies, akin to the singularity method used in fluid mechanics for potential flow calculations around various body contours.

An application of this method will be shown in the next section.

2.2.5.1 Superposition of heat sources and heat sinks

In section 1.1.2 we calculated the temperature distributionϑ=ϑ(r) in a hollow cylinder of lengthL The heat ﬂow in the radial direction is

The heat flow, represented by the equation Q˙ = 2πLλ(ϑ−ϑ 0 )/ln (r 0 /r), describes the temperature variation at a distance r from the axis of a cylinder, where ϑ is the temperature at that distance and ϑ 0 is the temperature at the radius r 0 This equation illustrates that the heat flow can be viewed as the intensity of a linear heat source aligned with the z-axis, creating a temperature field that solely depends on the radial distance, expressed as ϑ(r) = ϑ 0 +.

2πLλln (r 0 /r) , (2.87) with temperatures which approach inﬁnity forr→0.

Fig 2.16: Superposition of a linear heat source at point Q and a linear heat sink at point S

A linear or cylindrical heat source is positioned on the x-axis at point Q, located at a distance h from the origin, while a heat sink with strength (−Q) is situated at point S, at a distance (−h) from the origin This setup aims to analyze the plane temperature field created by the superposition of the heat source and sink within a thermally conductive material.

At any particular point P in Fig 2.16, the temperature is ϑ = ϑ 0Q + Q ˙

The equation Q ˙ = (2πLλ ln r S) / r Q (2.88) describes the relationship between the distances from a source and a sink, denoted as r Q and r S, respectively An isotherm, represented by ϑ = const, indicates a line where the distance ratio k = r S / r Q remains constant (2.89) Along the y-axis, where r S equals r Q, the isotherm corresponds to k = 1 and temperature ϑ = ϑ 0 Points to the right of the y-axis (x > 0) exhibit a distance ratio k greater than 1, resulting in temperatures above ϑ 0, while points to the left (x < 0) have a distance ratio k between 0 and 1, leading to temperatures below ϑ 0.

We will now show that all isotherms ϑ = ϑ 0 form a set of circles with centres which all lie on the x-axis According to Fig 2.16 k 2 = r S 2 r Q 2 = (x + h) 2 + y 2 (x − h) 2 + y 2 , is valid and from that follows x − k 2 + 1 k 2 − 1 h

This is the equation for a circle with radius

The equation R = 2k k² - 1 h describes a circle whose center M is positioned on the x-axis at a distance m = (k² + 1) / (k² - 1) h from the origin For temperatures above a certain threshold (ϑ > ϑ₀), the isotherms are located to the right of the y-axis due to the condition k > 1 As temperature increases (indicating a rise in k), the radii of these circles decrease, causing the center to shift closer to the source point Q, as illustrated in Fig 2.17 Conversely, isotherms with temperatures below this threshold (ϑ < ϑ₀) are represented as circles to the left of the y-axis, indicating that as temperatures drop, these circles contract around the sink S.

From (2.91) and (2.92) the following is obtained by squaring: h 2 = m 2 − R 2 (2.93)

By forming the ratio m/R, a quadratic equation for k with the solutions k = m

The positive root produces distance ratios greater than one (k > 1), making it applicable to circles located to the right of the y-axis Conversely, the other root results in distance ratios less than one (k < 1), which correspond to circles positioned to the left of the y-axis.

The isotherms described by equations (2.88), (2.89), and (2.90), illustrated in Fig 2.17, facilitate the calculation of steady-state heat conduction between two isothermal circles, such as between two tubes maintaining constant surface temperatures ϑ 1 and ϑ 2 This relationship results in the expression ϑ 1 − ϑ 2 =.

Fig 2.17: Net of isotherms from (2.88) and (2.90) for diﬀerent distance ratios k according to (2.89); Q heat source, S heat sink

This is the heat ﬂow between the isotherms ϑ = ϑ 1 and ϑ = ϑ 2 It is inversely proportional to the resistance to heat conduction (cf 1.2.2)

2πLλ (2.96) which depends on the position of the two isothermal circles (tubes) Thereby three cases must be distinguished:

1 k 1 > k 2 > 1 Two eccentric tubes, in Fig 2.18 a, with axes a distance e away from each other The larger tube 2 with surface temperature ϑ 2 surrounds tube 1 with ϑ 1 > ϑ 2

2 k 1 > 1; k 2 = 1 A tube with radius R, lying at a depth m under an isothermal plane e.g. under the earth’s surface Fig 2.18 b.

3 k 1 > 1; k 2 < 1 Two tubes with radii R 1 and R 2 , whose axes have the separation s > R 1 +R 2 They both lie in an extensive medium, Fig 2.18 c.

It is rather simple to calculate the resistance to heat conduction between a tube and an isothermal plane as shown in Fig 2.18 b With k 2 = 1 and k 1 according to (2.94) we obtain from (2.96)

If at the plane surface heat transfer to a ﬂuid with temperature ϑ 2 is to be considered, this can be approximated by calculating R cond with the enlarged distance m ∗ = m + λ/α

At a depth of λ/α, the surface is not isothermal, exhibiting a slight temperature maximum directly above the tube, which aligns with physical expectations K Elgeti [2.21] discovered an exact solution to this phenomenon, revealing that the approximation is remarkably precise Significant discrepancies from the exact solution are observed only when αR/λ is less than 0.5 and for values of m/R less than 2.

Fig 2.18: Three arrangements of tubes with isothermal surfaces a two eccentric tubes, one inside the other, b a tube and an isothermal plane, c two tubes with distance between their centres s > R 1 + R 2

The conduction resistance calculation between two eccentric tubes requires the radii R1 and R2, along with the eccentricity e, defined as the distance between the centers (e = m2 - m1) The distances m1 and m2 can be expressed using these parameters According to the relationship m2² - m1² = R2² - R1², the values of m1 and m2 can be derived from the given radii.

(m 2 − m 1 ) 2 = e 2 From these two equations we obtain m 1 = R 2 2 − R 1 2 − e 2

This gives the thermal conduction resistance as

2eR 2 or with the use of the addition theorem for inverse hyperbolic cosine functions

In the special case of concentric circles (e = 0) the result

R cond = 1 2πLλ ln (R 2 /R 1 ) already known from 1.1.2 will be obtained.

For the third case, two tubes in an extended medium as in Fig 2.18 c, we express m 1 and m 2 through the tube radii R 1 and R 2 as well as the distance s = m 1 + m 2 This gives, with (2.99) m 1 = s 2 −

The resistance to heat conduction, with ln k 2 = ln

If the tubes have equal radii R 1 = R 2 = R, we obtain from (2.101)

The combination of multiple heat sources and sinks enables the calculation of intricate temperature fields W Nusselt [2.22] innovatively substituted the tubes of a ceiling-embedded radiation heating system with linear heat sources, allowing for an accurate determination of the temperature distribution within the ceiling.

The heat ﬂow ˙Q, from one isothermal surface at temperatureϑ 1 to another isothermal surface at temperatureϑ 2 , can be calculated according to the simple relationship

The heat transfer rate (Q˙) can be expressed as the difference in temperature (ϑ1 - ϑ2) divided by the thermal conduction resistance (R_cond), assuming that R_cond is known This resistance is inversely proportional to the thermal conductivity (λ) of the material situated between two isothermal surfaces.

The equation Q˙ = λS(ϑ1 − ϑ2) introduces the shape coefficient S, which is determined solely by the geometric configuration of the two isothermal surfaces involved in the conduction of heat (˙Q) This shape coefficient has dimensions of length.

In two-dimensional temperature distributions, a dimensionless shape coefficient, known as the shape factor (SL = S/L), can be defined for bodies with a length L perpendicular to the temperature-dependent coordinate plane This concept was illustrated in the previous section with examples of plane temperature fields For a tube with radius R and length L, positioned at a depth m beneath an isothermal surface, the shape factor can be calculated using equations (2.97) and (2.104).

The shape factors for the tube arrangements shown in Figs 2.18 a and 2.18 c can be found in the same manner from equations (2.100) and (2.102) respectively.

In order to calculate the shape coefficientSin general, the heat flow ˙Qhas to be determined by integration of the local heat flux ˙ q =−λ∂ϑ

∂n on the isothermal surfacesA 1 and A 2 The heat ﬂow fromA 1 toA 2 is given by

In which the surface normalsn 1 andn 2 are directed into the conductive medium. Equation (2.105) gives the shape coeﬃcient as

This relationship enablesS to be calculated from the known temperature ﬁeld.

Transient heat conduction

Solution methods

The solution of a transient heat conduction problem can be found in three diﬀerent ways:

1 by a closed solution of the heat conduction equation, fulﬁlling all the boundary conditions,

2 by a numerical solution of the differential equation (with boundary conditions) using either a finite difference or finite element method.

3 by an experimental method implementing an analogy process.

To achieve a closed-form solution using mathematical functions, it is essential to assume that the material properties remain constant with temperature This assumption, as discussed in section 2.1.2, results in the formulation of a partial differential equation.

To derive an alineardiﬀerential equation, the analysis is restricted to scenarios involving conduction without internal heat sources (˙W ≡ 0) or where the power density ˙W is assumed to be independent of or linearly dependent on temperature (ϑ) Furthermore, the boundary conditions must be linear, necessitating a constant or time-dependent heat transfer coefficient (α) that does not vary with temperature, as outlined in the heat transfer condition.

In the 19th and 20th centuries, various linear initial and boundary condition problems were addressed using classical solution methods, including variable separation, the superimposition of heat sources and sinks, and Green’s theorem Recently, the Laplace transformation has emerged as a key method for solving transient heat conduction problems Sections 2.3.4 to 2.3.6 will apply the classical separation of variables theory, while a brief introduction to the Laplace transformation will be provided, as it is not widely known among engineers Section 2.3.3 will demonstrate the applications of the Laplace transformation for easily solvable problems, and a comprehensive discussion of mathematical solution methods and results can be found in the standard work by H.S Carslaw and J.C Jaeger.

The numerical solution of transient heat conduction problems is crucial, especially when dealing with temperature-dependent material properties or complex geometries and boundary conditions In such scenarios, numerical methods often become the sole viable approach for problem-solving The advent of computers has significantly expanded the application possibilities for these numerical solutions, which will be explored in section 2.4.

Experimental analogy procedures utilize the similarity between different transient transport processes, particularly electrical conduction, and the heat conduction equation, allowing insights from one model to be applied to the other This approach is grounded in the concept of analogous processes, as detailed by U Grigull and H Sandner However, due to significant advancements in computer technology, the practical relevance of this method has diminished, leading to a reduced focus on analogy methods in contemporary applications.

The Laplace transformation

The Laplace transformation is an effective method for solving the linear heat conduction equation with linear boundary conditions, allowing for special solutions at specific times or positions within a thermally conductive body This approach enables the analysis of temperature fields without needing to determine their complete time and spatial dependence An introductory overview of the Laplace transformation and its application to heat conduction problems can be found in H.D Baehr's work.

In H Tautz's book, the significance of the Laplace transformation in one-dimensional heat flow is emphasized, as it simplifies the solution of partial differential equations to that of linear ordinary differential equations This article focuses on calculating the temperature distribution, denoted as ϑ=ϑ(x, t) By multiplying ϑ by the factor e^(-st), where s is a complex quantity with frequency dimensions, and integrating from t = 0 to t → ∞, we derive a new function u(x, s) = L{ϑ(x, t)} = ∫₀^∞ ϑ(x, t) e^(-st) dt.

The Laplace transform of temperature, denoted as L{ϑ}, is dependent on both sand and x In theorems, we use the symbol L{ϑ}, while u represents L{ϑ} in practical problem-solving contexts.

2 In some books the Laplace parameter s is denoted by p, as in H.S Carslaw and J.C Jaeger[2.1] and in H Tautz [2.26].

Table 2.2: Some general relationships for the Laplace transform L { ϑ } = u

=sL {ϑ} −ϑ 0 =s u−ϑ 0 (x) withϑ 0 (x) = lim t→+0ϑ(x, t),( Initial temperature proﬁle )

5 IfL {ϑ(t)}=u(s) andkis a positive constant, then

The convolution theorem, expressed as L{f1(t)} = L{f2(t)}, is particularly relevant for time-dependent boundary conditions In this context, the object function is represented as L{ϑ}, where ϑ serves as the transformed function With dimensions of frequency, the Laplace transformation effectively converts ϑ from the time domain to the frequency domain.

The application of the Laplace transformation necessitates several key theorems, which are compiled in Table 2.2 without accompanying proofs Additionally, a correspondence table of functions of ϑ and their respective Laplace transforms, denoted as u, is essential This table is created through the evaluation of the defining equation (2.111).

So for example, we obtain for ϑ(x, t) = f (x) e −ct the Laplace transform u(x, s) =

We have therefore the general correspondence u(x, s) •−◦ ϑ(x, t) and in our example f (x) s + c •−◦ f (x) e −ct , where for the special case of f (x) ≡ 1 and c = 0

Table 2.3 includes crucial correspondences essential for solving the heat conduction equation For more comprehensive tables of correspondences, refer to the literature sources [2.1] and [2.26] to [2.28].

In order to explain the solution process we will limit ourselves to linear heat ﬂow in thex-direction and write the heat conduction equation as

Applying the Laplace transformation results in the ordinary differential equation \( \frac{d^2 u}{dx^2} - s a u = -\frac{1}{a} \theta_0(x) \), where \( \theta_0(x) = \theta(x, t=0) \) denotes the initial temperature distribution at time \( t=0 \) If \( \theta_0(x) = 0 \), a homogeneous linear differential equation is formed, leading to the solution \( u(x, s) = c_1 \exp\left(-\frac{s}{a} x\right) \).

To determine the integration constants \( c_1 \) and \( c_2 \) (or \( C_1 \) and \( C_2 \)), it is essential to fit the function \( u \) to the specified boundary conditions, which are also analyzed using a Laplace transform Additionally, if the initial condition \( \theta_0 = 0 \) is to be considered, the solution provided for the homogeneous differential equation must be complemented by a particular solution to the inhomogeneous equation.

To obtain the temperature distribution ϑ(x, t) that satisfies the initial and boundary conditions, the Laplace transform u(x, s) must first be determined The simplest way to perform the inverse transformation is by consulting a correspondence table, such as Table 2.3, where the desired temperature distribution can be directly extracted However, if u(x, s) is not listed in such tables, the Laplace transformation theory provides an inversion theorem for finding the solution This approach results in a complex integral that can be evaluated using Cauchy’s theorem, producing the temperature distribution as an infinite series of exponentially decaying functions over time For this discussion, we will focus on scenarios where the inverse transformation can be effectively carried out using correspondence tables.

Table 2.3: Table of some correspondences u(s, x) is the Laplace transform of ϑ(t, x) The abbreviations p = s/a and ξ = x/2 √ at should be recognised. u(s, x) ϑ(t, x)

2 1 s ν+1 , ν >−1 t ν Γ(ν+ 1), Γthis is the Gamma function, see below

6 e − px s erfcξ,complementary error function , see below

*a πte −ξ 2 −hae hx+ah 2 t erfc ξ+h√ at

10 e −px p(p+h) ae hx+ah 2 t erfc ξ+h√ at

1 h erfcξ−e hx+ah 2 t erfc ξ+h√ at

Special values of the Gamma function: Γ(1) = 1 ; Γ(n) = (n − 1)! = 1 ã 2 ã 3 ã (n − 1) , (n = 2, 3, ) Γ( 1 2 ) = √ π ; Γ(n + 1 2 ) = √ π 1 ã 3 ã 5 ã (2n − 1)

Values for the complementary error functions erfcξ and ierfc ξ can be found in Table 2.5 of section 2.3.3.1, with a corresponding plot displayed in Fig 2.22 The inversion theorem is discussed in the literature references [2.1], [2.25], and [2.26] Additionally, the inverse transformation can be performed numerically using various algorithms, as noted in reference [2.29].

The solution of transient heat conduction problems using the Laplace transformation consists of three steps:

1 Transformation of the diﬀerential equation with the initial and boundary conditions into the frequency region (ϑ→u, t→s)

2 Solution of the diﬀerential equation for the Laplace transformuconsidering the (transformed) boundary conditions.

3 Inverse transformation to the time region (u→ϑ, s→t) using a correspondence table (u•−◦ϑ) or the general inversion theorem.

Two advantageous properties of the Laplace transformation should be mentioned.

When only the temperature change over time at a specific point in a thermally conductive body is needed, the total Laplace transform u(s, x) does not require back-transformation for the entire body Instead, one can simplify the process by fixing the position variable in u as constant and back-transforming the simplified function u(s) to obtain ϑ(t) This approach not only streamlines the calculation for the desired temperature change but also allows for solutions at small time values, particularly useful at the onset of heat conduction processes, yielding straightforward results To achieve this, the Laplace transform u must be expressed as a series that converges for large s values, and a term-by-term back-transformation using a correspondence table will result in a series for ϑ that converges for small t values An example illustrating this solution procedure is provided for clarity.

In a scenario involving a flat wall of thickness δ with a constant initial temperature ϑ₀, a sudden temperature increase to ϑₛ occurs at the surface x = δ at time t = 0, while the opposite surface at x = 0 remains adiabatic As heat flows from the heated surface into the wall, the temperature within the wall gradually rises over time, with the left surface x = 0 experiencing the slowest rate of temperature increase The objective is to calculate the temperature at this point, represented as ϑ(x = 0, t).

Fig 2.19: Heating one side of a ﬂat wall, by a sharp rise in the surface temperature at x = δ from ϑ 0 to ϑ S

This transient heat conduction problem can be used as a model for the following real process A ﬁre resistant wall is rapidly heated on its outer side (x = δ) as a result of a ﬁre.

The focus of our study is the temperature increase over time at the wall's boundary located at x = 0 By assuming an adiabatic surface at this point, we observe a more rapid temperature rise than what would typically occur in real-world scenarios This conservative assumption ensures that our analysis remains on the safe side.

To address the problem, we assume that heat flows solely in the x-direction and that the thermal diffusivity \( a \) of the wall remains constant By defining the overtemperature \( \Theta(x, t) := \vartheta(x, t) - \vartheta_0 \), we can simplify the analysis without relying on dimensionless quantities until the results are presented, providing a clearer understanding of the procedure Consequently, the heat conduction equation can be expressed in a more manageable form.

∂x = 0 at x = 0 (adiabatic surface) and Θ(δ, t) = Θ S = ϑ S − ϑ 0 at x = δ

The application of the Laplace transformation results in the ordinary differential equation d²u/dx² - s a u = 0, which corresponds to the Laplace transform u = u(x, s) of the overtemperature Θ(x, t) Initial conditions have been incorporated, and the Laplace transformation of the boundary conditions produces du/dx = 0 at x = 0, along with u(δ, s) = Θ S s at x = δ.

The general solution of the diﬀerential equation (2.115) is u(x, s) = C 1 cosh px + C 2 sinh px , where the abbreviation p = s/a has been used From the boundary condition (2.116) it follows that C 2 = 0 and (2.117) gives

C 1 cosh pδ = Θ S s The required Laplace transform is therefore u(x, s) = Θ S s cosh px cosh pδ

The semi-inﬁnite solid

This article examines transient conduction in a thick plate, specifically at the free surface x=0, which is surrounded by the environment while extending infinitely in other directions An example of a semi-infinite solid is the ground at the earth's surface adjacent to the atmosphere Additionally, finite thickness bodies can be approximated as infinitely wide during the initial stages of heating or cooling at the surface x=0, enabling the application of straightforward results to these scenarios.

2.3.3.1 Heating and cooling with diﬀerent boundary conditions

The desired temperature distribution ϑ = ϑ(x, t) has to fulﬁll the diﬀerential equation

∂x 2 , t≥0, x≥0, (2.120) and the simple initial condition ϑ(x, t= 0) =ϑ 0 = const

In addition ϑfor x → ∞ has to be bounded At the surface x = 0 diﬀerent boundary conditions are possible These are shown in Fig 2.21:

– a jump in the surface temperature to the value ϑ S , which should remain constant fort >0, Fig 2.21a;

– a jump in the temperature of the surroundings to the constant valueϑ S ϑ 0 fort >0, so that heat transfer with a heat transfer coeﬃcient ofαtakes place, Fig 2.21c.

Fig 2.21: Heating of a semi-inﬁnite body with diﬀerent boundary conditions at the surface

(x = 0) a jump in the surface temperature to ϑ S , b constant heat ﬂux ˙ q 0 , c heat transfer from a ﬂuid at ϑ = ϑ S

For the calculation of the temperature distribution in all three cases we introduce the overtemperature Θ :=ϑ−ϑ 0 as a new dependent variable Then instead of (2.120) we have

We apply the Laplace transformation to these equations and obtain the ordinary diﬀerential equation already discussed in section 2.3.2 d 2 u dx 2 −s au= 0 for the Laplace transformuof Θ with the solution u(x, s) p

As Θ and thereforeumust be bounded forx→ ∞ the second exponential term exp(+px) in (2.114) disappears.

Table 2.4 outlines the boundary conditions at x = 0 for the three scenarios depicted in Fig 2.21, along with their Laplace transforms and the corresponding expressions for the constant C The inverse transformation of the Laplace transform u, as per equation (2.123), is straightforward when utilizing the values from Table 2.4 and matching them with the relevant correspondences in Table 2.3 We will now proceed to examine the various temperature distributions in detail.

Witha jump in the surface temperaturefromϑ 0 toϑ S (Case a) the temperature distribution looks like (No 6 from Table 2.3) Θ Θ S = ϑ−ϑ 0 ϑ S −ϑ 0 = erfc x

Table 2.4: Boundary conditions at the free surface x = 0 of a semi-inﬁnite solid as in Fig 2.21 with their Laplace transforms and the constants C yielded from Eq (2.123)

Boundary condition Transformed Constant C Case at x = 0 boundary condition in Eq (2.123) a Θ = ϑ S − ϑ 0 = Θ S u = Θ S s C = Θ S s b − λ ∂Θ

It is dependent on the dimensionless variable combination ξ = x

The error function complement, denoted as erfcξ, is defined by the integral √π ∫ e^(-w²) dw, while the error function is referred to as erfξ Values for erfcξ can be found in Table 2.5, with more comprehensive tables available in references [2.30] to [2.32] For detailed series developments and equations related to erfξ and erfcξ, consult the work of J Spanier and K.B Oldham [2.28] For larger arguments where ξ exceeds 2.6, the asymptotic approximation for erfcξ is given by erfcξ ≈ exp(-ξ²).

The functions erfξ and erfcξ are illustrated in Figure 2.22 The limit erfc(ξ = 0) = 1 signifies the temperature ϑ = ϑ S, which is achieved at x = 0 due to the defined boundary condition and within the body as time approaches infinity (t → ∞) Conversely, the limit erfc(∞) = 0 corresponds to the temperature ϑ = ϑ 0, representing the initial condition at t = 0.

The heat ﬂux at depthxat timetis obtained from (2.124) to be ˙ q(x, t) =−λ∂ϑ

It is usual here to introduce the material dependentthermal penetration coeﬃcient b: λc=λ/√ a , (2.128)

Fig 2.22: Error functions erf ξ and erfc ξ from (2.126) along with integrated error function ierfc ξ according to (2.134) which gives ˙ q(x, t) = b(ϑ S −ϑ 0 )

The entering heat ﬂux is proportional to the material propertyband decays with time ast − 1/2 During the time intervalt= 0 tot=t ∗ heat

The equations presented are applicable for cooling processes as well, where the condition ϑ S < ϑ 0 holds true In this scenario, the signs in equations (2.129) to (2.131) are reversed By rearranging equation (2.124), the temperature distribution can be expressed as ϑ−ϑ S / ϑ 0 −ϑ S = erfξ= 2.

If the surface of the semi-inﬁnite solid is heated with constant heat ﬂux q˙ 0 (Case b), the temperature distribution, from No 7 in Table 2.3 is found to be Θ(x, t) =ϑ(x, t)−ϑ 0 =q˙ 0 λ 2√ atierfcξ = 2q˙ 0 b

Table 2.5: Values of the error function complement erfc ξ from (2.126) and the integrated error function ierfcξ in the form √ π ierfc ξ = e −ξ 2 − √ π ξ erfc ξ ξ erfc ξ √ π ierfc ξ ξ erfcξ √ π ierfc ξ ξ erfcξ √ π ierfc ξ

0.50 0.47950 0.35386 1.25 0.07710 0.03879 2.5 0.00041 0.00013 0.55 0.43668 0.31327 1.30 0.06599 0.03246 2.6 0.00024 0.00007 0.60 0.39614 0.27639 1.35 0.05624 0.02705 2.7 0.00013 0.00004 0.65 0.35797 0.24299 1.40 0.04771 0.02246 2.8 0.00008 0.00002 0.70 0.32220 0.21287 1.45 0.04031 0.01856 2.9 0.00004 0.00001 is also called the integrated error function It is shown in Fig 2.22 and in Table 2.5 With ierfc(0) =π − 1/2 the surface temperature is obtained as ϑ(0, t) =ϑ 0 +q˙ 0 b

Initially, the temperature rises rapidly but slows down over time Materials with high thermal penetration coefficients can absorb significant heat flows, resulting in a slower increase in surface temperature compared to those with low coefficients.

In the scenario where heat transfer occurs from a fluid at temperature ϑ=ϑ S to the free surface at x=0, as outlined in Case c from Table 2.4, the corresponding temperature can be derived from correspondence table No 11 in Table 2.3 The solution is expressed in a dimensionless format using the variables ϑ + := ϑ−ϑ 0 / (ϑ S −ϑ 0), x + := x λ/α, and t + := at.

(2.138)This temperature distribution is illustrated in Fig 2.23 The heat ﬂuxes at the surface and within the body are easily calculated by diﬀerentiating (2.138) The

The temperature field ϑ + (x +, t +) in a semi-infinite solid experiencing heat transfer from a fluid to the free surface at x + = 0 is analyzed The temperature distribution for cooling, where ϑ S < ϑ 0, is expressed by the equation ϑ−ϑ S / ϑ 0 −ϑ S = 1−ϑ + (x +, t +) This relationship illustrates how ϑ + varies with respect to both x + and t +, providing insights into the cooling process as described by equation (2.138).

2.3.3.2 Two semi-inﬁnite bodies in contact with each other

In this article, we examine two semi-infinite bodies with distinct constant initial temperatures, denoted as ϑ01 and ϑ02, along with differing material properties λ1, a1 and λ2, a2 At time t = 0, these bodies come into thermal contact at the plane x = 0, leading to the establishment of an average temperature ϑm shortly thereafter Heat transfers from the body with the higher initial temperature to the one with the lower temperature, illustrating a transient conduction process This model effectively represents brief contact scenarios between two finite bodies at varying temperatures, such as when a hand or foot touches different objects or during the interaction of a heated metal body with a cooled item in reforming processes.

The heat conduction equation holds for both bodies and is

At the interface the conditions, according to section 2.1.3

Fig 2.24: Temperature pattern in two semi-inﬁnite bodies with initial temperatures ϑ 01 and ϑ 02 in contact with each other along the plane x = 0 and λ 1

(2.139) must be fulﬁlled In addition to this the initial conditions already mentioned are valid For the solution, we assume the contact temperature ϑ m at x = 0 to be independent of t.

This assumption is validated and leads to a straightforward solution to the problem, which is divided into two parts Each part's solution, discussed in section 2.3.3.1, reveals a temperature distribution in each body that evolves from the initial constant temperatures ϑ 01 and ϑ 02 when the surface temperature abruptly changes to the constant value ϑ m.

For body 1, with the following variable combination ξ 1 = x/ √

4a 1 t (2.140) we obtain, according to (2.132), the temperature ϑ 1 = ϑ m + (ϑ 01 − ϑ m ) erf ξ 1 , x ≥ 0 , (2.141) with a derivative at x = +0 of ∂ϑ 1

4a 2 t (2.143) it holds for body 2 according to (2.124) ϑ 2 = ϑ 02 + (ϑ m − ϑ 02 ) erfc( − ξ 2 ) , x ≤ 0 , (2.144) and at the boundary

Putting (2.142) and (2.145) in the boundary condition (2.139), a relationship independent of t follows: λ 1

The assumption of a time-independent contact temperature (ϑ m) is valid, as its position is influenced by the thermal penetration coefficients (b1 and b2) of the two bodies involved Specifically, ϑ m can be calculated using the equation ϑ m = ϑ 02 + (b1 / (b1 + b2)) * (ϑ 01 - ϑ 02) This contact temperature typically aligns closely with the initial temperature of the body that has the higher thermal penetration coefficient Consequently, this principle elucidates why different solids at the same temperature can feel distinctly warmer or cooler when touched.

Periodic temperature variations are prevalent in both natural and technological contexts, such as daily fluctuations in building walls and seasonal changes in the earth's crust In combustion engines, the cylinders experience significant and rapid temperature changes that affect the inner walls, potentially impacting material strength This article will explore a simplified model that approximates these temperature variations and highlights their key characteristics.

This article examines a semi-infinite solid characterized by constant material properties, λ and a, in contact with a fluid at the free surface located at x = 0 The fluid temperature, ϑ F, varies over time according to the equation ϑ F(t) = ϑ m + ∆ϑ cos(ωt), where ϑ m represents the mean temperature and ∆ϑ indicates the temperature amplitude.

This temperature oscillation is harmonic with a periodic time of t 0 and an amplitude of ∆ϑaround the mean value ϑ m At the surfacex= 0 the heat transfer condition is

We seek the temperature field ϑ=ϑ(x, t) in a body, which reaches a quasi-steady end state after a long duration, as the effects of the initial temperature distribution diminish This is described by the equation =α[ϑ F −ϑ(0, t)] (2.149), where α is a constant value.

A solution to the heat conduction problem can be achieved using the Laplace transformation, as referenced in various sources This method calculates the effect of a constant initial temperature, which diminishes over time However, due to the complexity of this approach, an alternative is considered It is anticipated that the temperature within the body experiences harmonic oscillations that become increasingly damped with depth, accompanied by a phase shift The temperature distribution can be expressed as ϑ(x, t) = ϑ m + ∆ϑ ηe − mx cos (ωt−mx−ε).

Fig 2.25: Fluid temperature over time ϑ F from (2.148) and surface temperature ϑ(0, t) from (2.153) with k = (b/α) π/t 0 = 1.0 satisﬁes the heat conduction equation (2.120), when m 2 = ω 2a= π at 0 (2.151)

The constantsηand εare yielded from the boundary condition (2.149) to

Their meaning is recognised when the surface temperature is calculated From (2.150) withx= 0 it follows that ϑ(0, t) =ϑ m + ∆ϑ ηcos (ωt−ε) (2.153)

Cooling or heating of simple bodies in one-dimensional heat ﬂow 159

The time dependent temperature ﬁeldϑ=ϑ(r, t) in a body is described by the diﬀerential equation

In the study of geometrical one-dimensional heat flow, a plate, cylinder, and sphere are analyzed with the parameter \( n \) set to 0, 1, and 2, respectively For cylinders and spheres, the radial coordinate is denoted as \( r \) It is important to note that the cylinder should be significantly longer than its diameter to ensure that axial heat flow can be ignored.

The temperature must remain independent of angular coordinates, a requirement also applicable to spheres In the case of a plate, as previously discussed in section 1.1.2, the x-coordinate is represented by r, which is the coordinate perpendicular to the two expansive boundary planes and measured from the center of the plate.

The three bodies—plate, very long cylinder, and sphere—start at a constant initial temperature (ϑ₀) at time t=0 For t > 0, their surfaces contact a fluid with a constant temperature (ϑₛ = ϑ₀) Heat transfer occurs between the body and the fluid, resulting in cooling if ϑₛ < ϑ₀ and heating if ϑₛ > ϑ₀, until the body reaches the fluid's temperature (ϑₛ), achieving a steady end-state The heat transfer coefficient (α) is constant on both sides of the plate and uniform across the surfaces of the cylinder and sphere, remaining independent of time When considering only half of the plate, the problem simplifies to unidirectional heating or cooling with the other surface insulated (adiabatic).

Under the assumptions mentioned, we obtain the boundary conditions r= 0 : ∂ϑ

Fig 2.27: Diagram explaining the initial and boundary conditions for the cooling of a plate of thickness 2R, a long cylinder and a sphere, each of radius R

At the initial condition t = 0, the temperature distribution is defined as ϑ(r) = ϑ₀ for 0 ≤ r ≤ R, as illustrated in Fig 2.27 To simplify the analysis, we introduce dimensionless variables: r⁺ = r/R and t⁺ = at/R², where R represents half the thickness of the plate or the radius of the cylinder or sphere Additionally, we define a new temperature variable as ϑ⁺ = (ϑ - ϑₛ) / (ϑ₀ - ϑₛ).

With these quantities the diﬀerential equation (2.157) is transformed into

∂r + =Bi ϑ + for r + = 1 (2.162) with the Biot number

Bi=αR/λ , cf section 2.1.5 The initial condition is ϑ + = 1 for t + = 0 (2.163)

The dimensionless temperature profile for three distinct geometries—plate, cylinder, and sphere—can be expressed as ϑ + = f n (r +, t +, Bi), where the function f n varies for each shape (n=0 for the plate, n=1 for the cylinder, and n=2 for the sphere) due to the differences in their corresponding differential equations.

Fig 2.28: Temperature ratio ϑ + = (ϑ − ϑ S )/(ϑ 0 − ϑ S ) a in cooling and b in heating

The dimensionless temperature can only assume values betweenϑ + = 0 (fort + →

∞) and ϑ + = 1 (for t + = 0) The temperature variations during cooling and heating are described in the same manner byϑ + , cf Fig 2.28 In cooling (ϑ S < ϑ 0 ) according to (2.159) ϑ(x, t) =ϑ S + (ϑ 0 −ϑ S )ϑ + r + , t + holds, and for heating (ϑ S > ϑ 0 ) it follows that ϑ(x, t) =ϑ 0 + (ϑ S −ϑ 0 ) 1−ϑ + r + , t +

In the limit where the body surface temperature approaches infinity, we find that the surface is maintained at a constant temperature, denoted as ϑ S, which corresponds to ϑ + = 0 This scenario leads to the fastest possible cooling or heating rate for a given temperature difference, ϑ 0 − ϑ S, as α approaches infinity.

In order to ﬁnd a solution to the diﬀerential equation (2.160) whilst accounting for the initial and boundary conditions, the method of separating the variables or product solution is used 3 ϑ + (r + , t + ) =F(r + )ãG(t + )

The functionsF andGeach depend on only one variable, and satisfy the following diﬀerential equation from (2.160)

F(r + )dG dt + d 2 F dr +2 + n r + dF dr +

The Laplace transformation yields consistent results, and transitioning from the frequency domain back to the time domain involves the application of the inversion theorem However, in this instance, a straightforward classical product solution is utilized to simplify the process.

The equation (2.164) demonstrates the separation of variables, with the left side depending on the dimensionless time t + and the right side relying on the position coordinate r + For the equality stated in (2.164) to hold true, both sides must equal a constant.

−à 2 This constant à is known as the separation parameter With this the following ordinary diﬀerential equations are produced from (2.164) dG dt + +à 2 G= 0 (2.165) and d 2 F dr +2 + n r + dF dr + +à 2 F = 0 (2.166)

Products of their solutions with the same value of the separation parameteràgive solutions to the heat conduction equation (2.160).

The solution of the diﬀerential equation (2.165) for the time functionG(t + ) is the decaying exponential function

The position function F varies among plates, cylinders, and spheres, yet the solution functions F must adhere to identical boundary conditions across all three geometries This is evidenced by equations (2.161) and (2.162), leading to the conclusion that dF/dr + = 0 when r + = 0 (2.168).

The boundary value problem defined by the differential equation and its corresponding boundary conditions leads to Sturm-Liouville eigenvalue problems, which are governed by a set of established theorems The solution function, F, satisfies the boundary conditions only at specific discrete values known as eigenvalues These eigenvalues are integral to the boundary value problem, with the associated solution functions referred to as eigenfunctions Key principles from the theory of Sturm-Liouville eigenvalue problems are outlined in the literature, such as in K Jänich's work.

2 The eigenvalues form a monotonically increasing inﬁnite series à 1 < à 2 < à 3 with lim i→∞à i = +∞

3 The associated eigenfunctionsF 1 , F 2 , are orthogonal, i.e it holds that b a

A i for i=j with A i as a positive constant Hereaand b are the two points at which the boundary conditions are stipulated In our problema= 0 andb= 1.

4 The eigenfunctionF i associated with the eigenvalueà i has exactly (i−1) zero points between the boundaries, or in other words in the interval (a, b). These properties will now be used in the following solutions of the boundary value problems for a plate, a cylinder and a sphere.

The function F(r + ) satisfies the differential equation (2.166) for the plate with n= 0, which is known as the differential equation governing harmonic oscillations.

It has the general solution

It follows from the boundary condition (2.168) that c 2 = 0 The heat transfer condition (2.169) leads to a transcendental equation for the separation parameter à, namely tanà=Bi/à (2.170)

The eigenvalues of the problem, which are the roots of this equation, are influenced by the Biot number As illustrated in Fig 2.29, there exists an infinite series of eigenvalues denoted as λ₁ < λ₂ < λ₃, consistent with Sturm-Liouville theory Additionally, only specific eigenfunctions are relevant to this analysis.

The equation F i = cos à i r, along with the associated eigenvalues à i, fulfills the boundary conditions (2.168) and (2.169) Consequently, the infinite series represented below, consisting of an infinite number of eigenfunctions and the time-dependent function G(t +) as per equation (2.167), is expressed as ϑ + (r +, t +) = ∑ (from i=1 to ∞).

Fig 2.29: Graphical determination of the eigenvalues according to (2.170) is the general solution to the heat conduction problem in a plate It still has to

ﬁt the initial condition (2.163) It must therefore hold that

A given function, in this case the number 1, is to be represented by the inﬁnite sum of eigenfunctions in the interval [0, 1].

The coefficient \( C_i \) is calculated by multiplying with the eigenfunction \( \cos(a_j r) \) and integrating from \( r_+ = 0 \) to \( r_+ = 1 \) Due to the orthogonality of the eigenfunctions, all terms where \( j \neq i \) vanish, leaving only the term where \( j = i \).

1 0 cos 2 à i r + dr + delivers the coeﬃcientC i with the result

With this the desired temperature distribution for cooling or heating a plate is found.

The eigenvalues (à i) and coefficients (C i) are influenced by the Biot number, with specific values for à 1 and C 1 detailed in Table 2.6 The first six eigenvalues can be found in reference [2.1] As the Biot number approaches infinity (Bi → ∞), the boundary condition ϑ + (1, t + ) = 0, which corresponds to ϑ(R) = ϑ S, is achieved with eigenvalues à 1 = π/2, à 2 = 3π/2, à 3 = 5π/2, and so forth.

Cooling and heating in multi-dimensional heat ﬂow

In geometric multi-dimensional heat ﬂow the temperature ﬁelds that have to be calculated depend on two or three spatial coordinates and must satisfy the heat conduction equation

The Laplace operator ∇²ϑ is defined for both Cartesian and cylindrical coordinate systems as described in section 2.1.2 This article revisits the transient heat conduction problem addressed in section 2.3.4, focusing on a body with a uniform initial temperature ϑ₀ that is in contact with a fluid at a constant temperature ϑₛ = ϑ₀ In this scenario, heat transfer occurs between the fluid and the body, with the constant heat transfer coefficient α playing a crucial role in the process.

In the following section we will consider two solution procedures:

– an analytical solution for special body shapes, which is a product of the temperature distributions already calculated in 2.3.4.

– an approximation method for any body shape, which is only adequate for small Biot numbers.

The latter of the two methods oﬀers a practical, simple applicable solution to transient heat conduction problems and should always be applied for suﬃciently small Biot numbers.

A cylinder of a ﬁnite length is formed by the perpendicular penetration of an inﬁnitely long cylinder with a plate In the same manner, a very long prism

Fig 2.33: Dimensions of a parallelepiped and a ﬁ- nite cylinder, with the positions of the coordinate systems for the equations (2.191) and (2.192)

Fig 2.34: Parallelepiped with adiabatic surface at y = 0 (hatched).

Heat transfer on the opposing surface with a rectangular cross-section can occur through the interaction of two plates or the intersection of three orthogonal plates The resulting spatial temperature distributions during the heating and cooling processes of such bodies are derived from the geometric one-dimensional temperature distributions of the individual bodies that intersect perpendicularly For a rectangular parallelepiped with dimensions 2X, 2Y, and 2Z, the temperature relationship can be expressed as ϑ + = ϑ−ϑ S ϑ 0 −ϑ S = ϑ + Pl x.

The solution for the plate, as described in equation (2.171), involves substituting r + with x/X, y/Y, or z/Z, and t + with at/X², at/Y², and at/Z², respectively The significance of the Biot number αR/λ is outlined in equation (2.191) It's important to note that the heat transfer coefficient must be consistent on opposite surfaces, while it can vary on pairs of surfaces oriented along the x, y, or z directions Additionally, the last factor in equation (2.191) is not applicable for a prism that is significantly elongated in the z-direction The temperature distribution for a cylinder with a height of 2Z is derived from the expression ϑ + = ϑ + Cy r.

The temperature of an infinitely long cylinder is described by equation (2.178), while the heat transfer coefficient at the shell may differ from the heat transfer coefficient at both ends of the cylinder.

When one or two parallel flat surfaces of a parallelepiped, prism, or cylinder are adiabatically insulated while heat is transferred at the other surface, the previously mentioned equations remain applicable This process involves reducing the dimension perpendicular to the adiabatic surface by half, positioning the zero point of the coordinate system at the adiabatic surface An illustrative example for a parallelepiped is shown in Fig 2.34.

The validity of relationships (2.191) and (2.192) is established in H.S Carslaw and J.C Jaeger's book, specifically on pages 33–35 Additionally, alternative product solutions for multi-dimensional temperature fields can be derived from the solutions applicable to semi-infinite solids For more comprehensive information, refer to references [2.1] and [2.18].

The heat released in multi-dimensional heat conduction can also be determined from of the product solutions According to section 2.3.4

The equation \(1 - \vartheta + m (t +)\) (2.193) represents the total heat released up to time \(t\), as referenced in (2.173) The dimensionless average temperature \(\vartheta + m\) is derived from the average temperatures obtained in section 2.3.4.3 for both the plate and the infinitely long cylinder For the parallelepiped shape, we find that \(\vartheta + m = \vartheta + m_{\text{Pl}}\).

(2.194) with ϑ + mPl from (2.174) In the same way we have for the cylinder of ﬁnite length ϑ + m = ϑ + mCy at

, (2.195) where ϑ + mCy is given by (2.180).

For large values of the dimensionless time \( t + \), the temperature distribution and average temperature can be simplified by focusing on the first term of the infinite series representing the temperature profiles \( \vartheta + Pl \) in the plate and \( \vartheta + Cy \) in the long cylinder Utilizing the equations from section 2.3.4.5 and Table 2.6, one can explicitly calculate the heating or cooling times needed to achieve a specified temperature \( \vartheta k \) at the center of the thermally conductive solids by limiting the series to their initial terms For more detailed information, refer to [2.37].

In this example, we analyze a Chromium-Nickel Steel cylinder with a diameter of 60 mm and a length of 100 mm, initially heated to 320 °C When submerged in an oil bath at 30 °C, with a heat transfer coefficient of 450 W/m²K, we aim to determine the time required for the cylinder's center temperature to drop to 70 °C Additionally, we will calculate the maximum surface temperature of the cylinder during this cooling process.

The temperature distribution in a finite-length cylinder can be expressed using equation (2.192) To determine the cooling time \( t_k \), we will focus on the first term of each infinite series, with a subsequent verification of this simplification's accuracy Consequently, we set \( \vartheta^+ = C C_y J_0\left(\frac{C_y r}{R}\right) \exp \).

− à 2 Cy at/R 2 ã C Pl cos (à Pl z/Z) exp

(2.196) with R = d/2 = 30 mm and Z = L/2 = 50 mm The eigenvalues à Cy , à Pl and the coeﬃcients C Cy , C Pl are dependent on the Biot number It holds for the cylinder that

Bi Cy = αR/λ = 0.900 , out of which we obtain from Table 2.6 (page 165) à Cy = 1.20484 and C Cy = 1.1902 For the plate

Bi Pl = αZ/λ = 1.500 is obtained and from Table 2.6 à Pl = 0.98824 and C Pl = 1.1537.

The temperature at the centre (r = 0, z = 0) of the cylinder follows from (2.196) as ϑ + k = C Cy C Pl exp

, (2.197) where ϑ + k can be calculated from the given temperatures: ϑ + k = ϑ k − ϑ S ϑ 0 − ϑ S = 70 ◦ C − 30 ◦ C

320 ◦ C − 30 ◦ C = 0.1379 The cooling time t k we are looking for is found from (2.197) to be t k = 1 a ln C Cy + lnC Pl − ln ϑ + k (à Cy /R) 2 + (à Pl /Z) 2 With the thermal diﬀusivity a = λ/c = 3.77 ã 10 −6 m 2 /s this yields t k = 304 s = 5.07 min

In order to check whether it is permissible to truncate the series after the ﬁrst term, we calculate the Fourier numbers using this value of t k : t + k

Pl = a t k /Z 2 = 0.459 According to Fig 2.31 these dimensionless times are so large that the error caused by neglecting the higher terms in the series is insigniﬁcant.

The highest surface temperatures are observed at the cylindrical surface (r = R) when z = 0 and at the center of the circular end surfaces (z = Z, r = 0) For the cylindrical surface, the temperature is calculated as ϑ (r = R, z = 0, t = t k) = 56.7 °C, derived from the equation ϑ + (r = R, z = 0, t = t k) = J 0 (à Cy) ϑ + k = 0.6687 ã 0.1379 Meanwhile, at the center of the end surfaces, the temperature is ϑ (r = 0, z = Z, t = t k) = 52.0 °C, calculated from ϑ + (r = 0, z = Z, t = t k) = cos (à Pl)ϑ + k = 0.5502 ã 0.1379 Thus, the maximum surface temperature occurs on the cylindrical surface at z = 0.

2.3.5.2 Approximation for small Biot numbers

For bodies with small Biot numbers (Bi→0), a straightforward calculation for heating or cooling is feasible This scenario occurs when the heat conduction resistance within the body is significantly lower than the heat transfer resistance at its surface At any given moment, the temperature variations inside the thermally conductive body remain minimal, while there is a substantial temperature difference between the surface and the surrounding environment.

In our analysis, we assume that the body's temperature relies solely on time rather than spatial coordinates, leading to Bi = 0 as λ approaches infinity and the heat transfer coefficient α being zero By applying the first law of thermodynamics, we can assess the temperature variation over time, where the change in internal energy is directly proportional to the heat flow across the surface, expressed as dU/dt = ˙Q.

If a body of volumeV has constant material propertiesandc, then it holds that dU dt =V du dt =V cdϑ dt WithAas the surface area of the body we obtain

Q˙ =αA(ϑ−ϑ S ) for the heat ﬂow The diﬀerential equation now follows from (2.198) as dϑ dt = αA cV (ϑ−ϑ S ) , with the solution ϑ + = ϑ−ϑ S ϑ 0 −ϑ S = exp

The temperature change over time can be expressed with the equation (2.199), satisfying the initial condition ϑ(t = 0) = ϑ₀ This approach simplifies the analysis compared to a series expansion, as it consolidates all influencing factors—such as the heat transfer coefficient, material properties, and body geometry—into a single parameter known as the decay time, t₀ = cα.

In this study, we demonstrate that equation (2.199) provides the exact solution for a Biot number (Bi) of zero and an alpha value of zero We further explore the applicability of this simplified temperature change calculation for various non-zero Biot numbers, ensuring that it maintains a high level of accuracy.

Solidiﬁcation of geometrically simple bodies

Pure substances and eutectic mixtures solidify and melt at specific temperatures, known as the eutectic temperature (ϑ E), which vary by substance and are minimally affected by pressure A prime example is water, which freezes at 0 °C under atmospheric pressure, releasing a fusion enthalpy of 333 kJ/kg During melting, heat must be supplied to the solid to provide the necessary enthalpy of fusion Solidification processes are crucial in various fields, including cryogenics, food processing, and metallurgy A key focus is the rate at which the solid-liquid boundary moves, allowing for calculations of the time required to solidify layers of specific thicknesses These processes are modeled within the realm of transient heat conduction, as the heat from the released enthalpy of fusion must be conducted through the solid.

A general mathematical solution for this type of thermal conduction problems does not exist Special explicit solutions have been found by F Neumann 4 in

In this article, we will first explore the solution proposed by J Stefan in 1865 and later expanded upon in 1891 The initial section focuses on deriving quasi-steady solutions, which operate under the assumption that heat storage within the solidified body is negligible Subsequently, we will examine enhancements to these quasi-steady solutions, taking into account the approximate heat storage within the body.

2.3.6.1 The solidiﬁcation of ﬂat layers (Stefan problem)

A solidified body is maintained at a constant temperature ϑ0, which is below the solidification temperature ϑE, with one-dimensional heat conduction assumed in the x-direction At the moving phase boundary x = s, the solid is in contact with liquid that has been cooled to the solidification temperature As the phase boundary advances and a layer of thickness ds solidifies, the enthalpy of fusion is released and must be transferred as heat to the cooled surface of the solid at x = 0.

Fig 2.35: Temperature proﬁle (for t = const) for the solidifying of a plane solid s is the distance between the phase boundary and the cooled surface x = 0

The temperatureϑ=ϑ(x, t) in the solidiﬁed body satisﬁes the heat conduction equation

∂x 2 (2.202) with the boundary conditions ϑ=ϑ 0 for x= 0 , t >0 , (2.203)

In his lectures at the University of Königsberg, F Neumann introduced a significant solution, which was first published in the German book "Die partiellen Differentialgleichungen der Physik," edited by B Riemann and H Weber, in 1912 This work, found in Volume 2 on pages 117–121, discusses the equation ϑ=ϑ E for x=s and t > 0, along with the initial condition s=0 for t=0.

At the phase interface, the energy balance λ∂ϑ

The equation ∂xdt = h E ds must be satisfied, where h E represents the specific enthalpy of fusion This leads to the determination of the phase boundary's advancement over time, indicating the solidification speed expressed as ds/dt = λ h E.

A solution of the heat conduction equation (2.202) is the error function ϑ=ϑ 0 +Cerf x

, cf 2.3.3.1; it satisﬁes the boundary condition (2.203) The condition (2.204) requires that ϑ E =ϑ 0 +Cerf s

The argument of the error function is a constant γ that is independent of time t The thickness of the solidified layer increases proportionally with the square root of time, expressed as s = γ²√(at) This relationship satisfies the initial condition Consequently, the temperature profile in the solidified layer (for x ≤ s) is given by ϑ⁺ = (ϑ− - ϑ₀) / (ϑₑ - ϑ₀) = erf(x/√).

The still unknown constant γ is yielded from the condition (2.206) for the solidiﬁcation speed It follows from (2.207) that ds dt =γ

*a t and from (2.206) and (2.208) ds dt = λ h E ϑ E −ϑ 0 erfγ e − γ 2

This gives us the transcendental equation independent oft

P h (2.209) for the determination ofγ This constant is only dependent on thephase transition number

The Stefan number (St) is defined as the ratio of two specific energies: the specific enthalpy of fusion (hE) and the difference in internal energy of a solid at temperature (ϑE) compared to absolute zero (ϑ0) The reciprocal of the Stefan number is also referred to as P.

Using a series expansion of the error function, see, for example [2.28], [2.30], enables us to expand the left hand side of (2.209) into a series which rapidly converges for small values ofγ:

From this equation we get the series γ 2 = 1

1890P h − 4 + , (2.212) which for large values of the phase transition number allows us to directly calculate γ Finally the time t required to solidify a plane layer of thickness s is yielded from (2.207) and (2.212) as t= s 2

The solidiﬁcation time increases with the square of the layer thickness and is larger the smaller the phase transition numberP h.

Neglecting the heat stored in the solidiﬁed body corresponds, due toc= 0, to the limiting case ofP h→ ∞ This is the so-called quasi-steady approximation, which gives from (2.213), a solidiﬁcation time t ∗ = h E s 2

2λ(ϑ E −ϑ 0 ) (2.214) which is always too small The relative error t−t ∗ t = 1−2P hγ 2 = 1

945P h − 3 (2.215) is shown in Fig 2.36 ForP h >6.2 the error is less than 5% It increases strongly for smaller phase transition numbers.

The problem discussed here was ﬁrst solved by J Stefan [2.39] in 1891 The solution of the more general problem ﬁrst given by F Neumann has been discussed

The relative error of the solidification time \( t^* \) was calculated using the quasi-steady approximation, assuming an initial liquid temperature \( \vartheta_{0F} > \vartheta_E \), which necessitates solving the heat conduction equation for the liquid temperature \( \vartheta_F(x, t) \) without convection The solution incorporates error functions and requires consideration of the liquid's material properties \( \lambda_F \), \( c_F \), and \( F \) For further analytical solutions related to variations of the Stefan-Neumann Problem, refer to H.S Carslaw and J.C Jaeger Additionally, generalized mathematical formulations and discussions on alternative solution methods are provided by A.B Tayler and J.R Ockendon.

For sufficiently large values of the phase transition number \( P_h \) (approximately \( P_h > 7 \)), the storage capability of the solidified layer can be neglected, allowing for a temperature profile akin to steady-state heat conduction This quasi-steady approximation enables the use of different boundary conditions than those of the exact solutions by F Neumann and J Stefan at the cooled end of the solid layer Furthermore, this method facilitates the examination of solidifying processes in cylindrical and spherical geometries, where exact analytical solutions are not available The equations for solidification times derived by R Plank and K Nesselmann are applicable due to the large phase transition number for the freezing of ice and water-containing substances, as \( h_E \) is particularly high for water.

The solidifying process occurs on a flat, cooled wall characterized by thickness δ W and thermal conductivity λ W, as illustrated in Fig 2.37 This wall is cooled by a fluid at temperature ϑ 0, with the heat transfer coefficient α playing a crucial role As a result, a solidified layer forms on the opposite side of the wall.

The temperature profile during the solidification of a flat layer, based on the quasi-steady approximation, is illustrated in Fig 2.37 In this scenario, the layer has a thickness of 's', and the liquid is presumed to be at the solidifying temperature, denoted as ϑ E.

During the time interval \(dt\), the phase interface moves a distance \(ds\), releasing fusion enthalpy \(dQ = h_E A_{ds}\), which must be transferred as heat through the already solidified layer In the quasi-steady approximation, there is overall heat transfer between the interface temperature \(\theta_E\) and the cooling medium temperature \(\theta_0\) According to the established relationship, \(dQ = \dot{Q} dt = (\theta_E - \theta_0) A_s \lambda + \delta W \lambda_W + \frac{1}{\alpha} dt\), where \(\lambda\) represents the thermal conductivity of the solidified body From these equations, we derive that \(dt = \frac{h_E \lambda (\theta_E - \theta_0)}{s + \lambda k ds}\), with \( \frac{1}{k} = \frac{\delta W}{\lambda_W} + \frac{1}{\alpha} \).

The solidiﬁcation-time for a ﬂat layer of thickness s is found by integrating (2.216) to be t= h E s 2 2λ(ϑ E −ϑ 0 )

In the Stefan problem, the boundary condition at x = 0 is defined by the temperature value ϑ = ϑ₀ From the equation (2.218), we derive the initial term of the exact solution (2.213), which is relevant as P approaches infinity, leading to the determination of the time t* When considering finite heat transfer resistance, represented by (1/k), the solidification time exceeds t*, indicating that it does not increase proportionally to s².

The solidification times for layers on cylindrical tubes and spherical surfaces can be calculated similarly This article will derive the results specifically for a layer that forms on the outer surface.

In the quasi-steady approximation, solidification occurs on the outer surface of a tube being cooled from the inside, as illustrated in Fig 2.38 When a layer of thickness ds solidifies, it releases heat represented by the equation dQ = h E 2π(R + s)L ds, where L denotes the tube's length The overall heat transfer can be expressed as dQ = ˙ Q dt = ϑ E − ϑ 0.

R + 1 kA dt is valid with

2πL(R − ∆R)α , where ∆R is the thickness of the tube wall It follows that dt = h E λ (ϑ E − ϑ 0 )

R + (R + s)β ds (2.219) with the abbreviation β = λ λ W ln R

Integration of (2.219) taking account of the initial condition s = 0 for t = 0 yields with s + = s/R the solidiﬁcation-time t = h E s 2

The relationships in Table 2.7 were found in the same manner It should be noted here that

Heat sources

Heat sources appear within a heat conducting body as a result of dissipative processes and chemical or nuclear reactions According to 2.1.2 the diﬀerential equation

The equation W˙(x, t, ϑ) (2.222) describes the temperature field under the assumption that material properties remain constant with respect to temperature and concentration The thermal power per volume, W˙(x, t, ϑ), arises from internal processes within the body The heat conduction equation (2.222) can be solved effectively when W˙ is either independent of or linearly related to temperature ϑ, with the Laplace transformation being the preferred method for this linear scenario Alternative solution methods and a variety of cases addressing different geometries and boundary conditions can be found in reference [2.1], while H Tautz's book [2.26] also presents several instances utilizing the Laplace transformation for problem-solving.

This section focuses on homogeneous heat sources, where internal heat generation is uniformly distributed throughout the entire body Following this, we will explore local heat sources, characterized by concentrated heat development at specific points or along lines within the heat-conducting material.

Assuming geometric one-dimensional heat ﬂow in the x-direction and a power density ˙W which is independent of temperature, the heat conduction equation is then

This linear non-homogeneous diﬀerential equation can easily be solved using the Laplace transformation The steps for the solution of such a problem are shown in the following example.

In a semi-inﬁnite body (x ≥ 0), a spatially constant, but time dependent power density is supposed to exist:

At time t₀, the power density W₀ is defined, and the heat release is modeled as it occurs during processes such as concrete setting This results in an initial significant release of heat that diminishes rapidly over time The semi-infinite body starts with a constant temperature of ϑ = 0 at t = 0, with the condition that the surface at x = 0 must always maintain this temperature: ϑ(0, t) = 0 for x = 0.

By applying the Laplace transformation to (2.223) and (2.224) we obtain the non-homogeneous ordinary diﬀerential equation d 2 u dx 2 − p 2 u = − W ˙ 0 λ

The solution to the homogeneous differential equation, as x approaches infinity, is given by u_hom = C exp(−px), where C is a constant Additionally, a particular solution for the non-homogeneous equation is represented as u_inh This analysis is based on the initial condition of s with p² = s/a, specifically considering the value of 2.225.

√ πt 0 s 3/2 The transformed function composed of these two parts u(x, s) = Ce −px +

W ˙ 0 √ πt 0 cs 3/2 still has to be ﬁtted to the boundary condition (2.226) This gives u(x, s) = W ˙ 0 c

The reverse transformation, with the assistance of a correspondence table, Table 2.3, provides us with ϑ(x, t) = W ˙ 0 √ t 0 c 2 √ t

(2.227) with the integrated error function from section 2.3.3.1 Introducing into (2.227) the following dimensionless variables t + := t/t 0 , x + := x/ √ at 0 , ϑ + := ϑc/( ˙ W 0 t 0 ) ,

Fig 2.40: Temperature ﬁeld ϑ + (x + , t + ) according to (2.228) in a semi-inﬁnite body with time dependent homogeneous heat sources according to (2.224) yields ϑ + x + , t +

This temperature distribution is illustrated in Fig 2.40 For very small values of x + /(2 √ t + ) it holds that √ π ierfc x + /

As time approaches infinity, the steady-state temperature is represented by the linear equation ϑ + ∞ = √ πx +, which serves as the tangent to all temperature curves at the free surface x + = 0 At this point, the heat flux remains constant over time, denoted as ˙ q 0 = − λ.

= − W ˙ 0 √ πat 0 has to be removed in order to maintain the temperature at the value ϑ + = 0.

The solutions for power densities ˙W(x, t) and boundary conditions, as outlined in equation (2.223), can be derived similarly However, for finite thickness bodies, the inverse transformation of u(x, s) typically requires the use of the inversion theorem referenced in section 2.3.2 For further details and specific examples, please refer to sources [2.1] and [2.26].

2.3.7.2 Point and linear heat sources

Local heat sources refer to the generation of heat concentrated in a confined area, often modeled as point, line, or sheet singularities A common example is an electrically heated thin wire, which can be considered a linear heat source These singularities are not only crucial for various technical applications but also hold significant theoretical importance in calculating temperature fields.

This article discusses the calculation of the temperature field generated by a point heat source within an infinitely extending body that contains a spherical hollow space of radius R The boundary condition at the surface of the sphere is defined by a constant heat flux, denoted as ˙q 0 (t), which represents the thermal power produced by the heat source located inside the hollow space.

As the limit approaches R→0 at a specific time t while maintaining a constant heat flow rate ˙Q, it results in a point heat source with strength ˙Q(t) This point heat source is positioned at the center of the sphere, designated as the origin r = 0 in the radial (spherical) coordinate system.

Fig 2.41: Inﬁnite body with respect to all dimensions with a spherical hollow space at whose surface the heat ﬂux ˙ q 0 is prescribed

The temperature field outside the spherical hollow satisfies the differential equation

Q(t) ˙ 4πR 2 for r = R and ϑ = 0 for r → ∞ , if we assume the initial temperature to be ϑ = 0 Applying the Laplace transformation leads to d 2 u dr 2 + 2 r du dr − p 2 u = 0 with p 2 = s/a The boundary conditions are du dr = − 1 4πR 2 λ L -

for r = R (2.229) and u → 0 for r → ∞ A solution which satisﬁes the last condition is u = B r e −pr The constant B is obtained from (2.229), so that u = L - Q(t) ˙

1 + pR is the desired transformed function Letting R → 0 we obtain for the point heat source at r = 0 u = L -

Q(t) ˙ 4πλr e −pr From No 4 in Table 2.3 on page 145 e −pr = L

, , such that u is obtained as the product of two Laplace transforms u = 1 4πλ L -

Which, according to the convolution theorem, No 6 in Table 2.2 (page 143), is u = 1 4πλ L

So the temperature distribution around a point heat source atr= 0, which, at timet= 0 is “switched on” with a thermal power of ˙Q(t), is yielded to be ϑ(r, t) = 1

This general solution for any time function ˙Q(t) contains special simple cases. The temperature ﬁeld for a source of constant thermal power ˙Q(t) = ˙Q 0 is found to be ϑ(r, t) Q˙ 0 4πλrerfc r

Fort→ ∞we obtain, with erfc(0) = 1, the steady-state temperature ﬁeld around a point heat source as ϑ(r) Q˙ 0 4πλr

A very fast reaction or an electrical short can cause a sudden release of energy

Q 0 at timet= 0 in a small space atr= 0 For this “heat explosion” the limit in (2.230) is set asτ →0 and the heat released is given by

(2.232) as the temperature distribution It has the boundary value of ϑ→ ∞for t= 0 andr= 0 At a ﬁxed pointr= const, wherer= 0 the temperature change with

The temperature field in an infinite body following a "heat explosion" at the origin (r = 0) is illustrated in Fig 2.42 The reference temperature is defined as ϑ₀ = Q₀ / (c r₀³), where r₀ is a chosen distance from the origin Initially, the temperature is ϑ = 0, rises to a maximum at time tₘₐₓ = r² / (6a), and subsequently returns to ϑ = 0 The temperature profile over time for various ratios of r/r₀ is depicted in Fig 2.42, according to equation (2.232).

We will begin our analysis by examining the "heat explosion" to calculate the temperature field surrounding a linear heat source located at r = 0 At time t = 0, a linear heat source of length L releases an initial heat Q0, oriented perpendicular to the r, ϕ-plane of the polar coordinate system In the absence of additional heat sources, the heat Q0 must be accounted for as an increase in the internal energy of the environment surrounding the heat source over time.

∞ 0 rϑ(r, t) dr holds, independent of timet The desired temperature distribution ϑ(r, t), must satisfy this equation.

In analogy to the point “heat explosion” we introduce the function ϑ(r, t) =f(t) exp

−r 2 4at corresponding to (2.232) and obtain

The deﬁnite integral which appears here has the value of 1/2, such that the time functionf(t) is yielded to be f(t) = Q 0 /L 4πcat = Q 0 /L

4πλt From this we obtain the desired temperature distribution ϑ(r, t) = Q 0 /L

It is easy to prove that this satisﬁes the heat conduction equation (2.157) with n= 1 (cylindrical coordinates).

The results can be generalized for a linear source at r = 0 with time-dependent thermal power ˙Q(t) During the interval from t = τ to t = τ + dτ, it releases heat ˙Q(τ) dτ, creating a corresponding temperature field By superimposing these incremental "heat explosions" over time, a temperature field analogous to the previous equation is derived, represented as ϑ(r, t) = 1.

− r 2 4a(t−τ) dτ t−τ (2.234) For the special case of constant thermal power ˙Q 0 this becomes ϑ(r, t) =− Q˙ 0

The function which is present here is known as the exponential integral

Ei(−ξ) ∞ ξ e − u u du with the series expansion

(−1) n ξ n nãn! and the asymptotic expansion (ξ1)

, cf [2.28] and [2.30] Some values of Ei(−ξ), which is always negative, are given in Table 2.8.

Table 2.8: Values of the exponential integral Ei( − ξ) ξ − Ei( − ξ) ξ − Ei( − ξ) ξ − Ei( − ξ) ξ − Ei( − ξ) ξ − Ei( − ξ)

Numerical solutions to heat conduction problems

The simple, explicit diﬀerence method for transient heat conduction

The finite difference method replaces the derivatives in the heat conduction equation with difference quotients, transforming the differential equation into a finite difference equation This approach allows for the approximation of the solution at discrete grid points in both space and time While reducing the mesh size enhances accuracy by increasing the number of grid points, it also raises computational demands Consequently, using the finite difference method requires a balance between achieving high accuracy and managing computation time effectively.

This article explores the case of transient, one-dimensional heat conduction in a geometric context, focusing on regions where material properties remain constant Specifically, we analyze the heat conduction equation within the defined range of 0 ≤ x ≤ xn.

∂x 2 (2.236) has to be solved for timest ≥t 0 , whilst considering the boundary conditions at x 0 andx n The initial temperature distribution ϑ 0 (x) fort=t 0 is given.

A grid is created along the strip defined by 0 ≤ x ≤ x_n and t ≥ t_0, utilizing a mesh size of Δx in the x-direction and Δt in the t-direction Each grid point (i, k) is identified by the coordinates x_i = x_0 + iΔx, where i = 0, 1, 2, , and t_k = t_0 + kΔt, where k = 0, 1, 2, The approximated value of ϑ at the grid point (i, k) is represented as ϑ_k^i = ϑ(x_i, t_k).

Fig 2.43: Grid for the discretisation of the heat conduction equation (2.236) and to illustrate the ﬁnite diﬀerence equation

We avoid using a different symbol, such as Θ, to denote an approximate temperature value instead of the exact temperature ϑ, which is common in mathematical literature The time level t k is represented by the superscript k without brackets to prevent any confusion with the k-th power of ϑ.

The derivatives which appear in (2.236) are replaced by diﬀerence quotients, whereby adiscretisation errorhas to be taken into account derivative = diﬀerence quotient + discretisation error

Reducing the mesh size (∆x or ∆t) leads to a decrease in discretisation error In this context, the second derivative in the x-direction at position x_i and time t_k is approximated using the central second difference quotient.

The writing ofO(∆x 2 ) indicates that the discretisation error is proportional to

By decreasing the mesh size, the error diminishes proportionally to the square of the mesh size, resulting in improved accuracy Additionally, the first derivative concerning time is approximated using the less precise forward difference quotient.

The discretisation error decreases in proportion to ∆t, leading to a numerically straightforward explicit finite difference formula By substituting equations (2.238) and (2.239) into the differential equation (2.236) and rearranging, we derive the finite difference equation: ϑ k+1 i = M ϑ k i − 1 + (1−2M)ϑ k i + M ϑ k i+1 (2.240).

The modulus or Fourier number of the difference method is represented by M = a∆t/∆x² (2.241), indicating that the discretization error of the difference equation is of order O(∆x²), as derived from ∆t = M∆x²/a Equation (2.240) is an explicit difference formula that allows for the calculation of the temperatures ϑ k+1 i at the next time level t k+1 = t k + ∆t, using three known temperatures from the current time level t = t k Starting with k = 0, where all temperatures ϑ 0 i are known from the initial temperature profile, this equation facilitates the computation of all ϑ 1 i at time t 1, which then allows for the subsequent values ϑ 2 i at time t 2, and so forth.

The equations (2.238) and (2.239) utilize Taylor series expansion to replace derivatives with difference quotients at the point (xi, tk) Additionally, the finite difference formula (2.240) is derived from an energy balance combined with Fourier’s law, incorporating heat conduction resistances ∆x/λ between grid points and energy storage in a surrounding "block" at position xi This approach is further advanced by the Volume Integral Method, which allows for the derivation of various finite difference equations.

Many finite difference formulas suffer from numerical instability, where small initial and rounding errors amplify throughout the calculation, leading to inaccurate results In contrast, a stable difference formula reduces errors during the computation, minimizing their impact on the final outcome Most difference equations exhibit conditional stability, meaning they remain stable only for specific step or mesh sizes For instance, explicit equation (2.240) is stable only when the modulus M meets certain conditions.

When ∆x is specified, the time step ∆t must be carefully selected; ignoring this requirement can lead to significant inaccuracies and ultimately render the process ineffective.

The stability condition can be derived through various methods, as detailed in the extensive discussion found in reference [2.57] A key requirement for the stability of explicit difference formulas is that all coefficients in the equation must be non-negative, as indicated in reference [2.60] This principle applies to the equation (2.240).

The inequality 1 − 2M ≥ 0 leads to the derivation of equation (2.242) To analyze stability behavior, the ε-scheme is employed, where at time t = t₀, all initial values ϑ₀ᵢ are set to zero, except for one value which is assigned ε = 0 Error propagation is then examined by applying the difference formula for subsequent time steps with k = 1, 2, and so on The ε-scheme for the difference equation (2.240) is presented in a specific format.

The finite difference formula with M = 1/2 exhibits near stability, as the error is distributed between adjacent points and diminishes gradually Furthermore, the grid is divided into two disconnected sections, similar to the difference formula represented by ϑ k+1 i = 1/2 ϑ k i−1 + ϑ k i+1.

In equation (2.243), only two temperatures at time tk are used to determine the new temperature ϑk+1 i, indicating that ϑk i does not influence ϑk+1 i The connection between the two grid sections is weak, relying solely on the given boundary temperatures Consequently, it is not advisable to use (2.240) with M = 1/2, despite it allowing for the largest time step ∆t This difference equation (2.243) serves as the foundation for the graphical methods previously employed by L Binder and E Schmidt.

Discretisation of the boundary conditions

Considering the boundary conditions in finite difference methods we will distin- guish three different cases:

– preset heat ﬂux at the boundary and

– the heat transfer condition, cf section 2.1.3.

To ensure accurate temperature representation at the boundary x = x R, grid divisions must be selected so that the boundary aligns with a constant grid line x i The left boundary is defined as x R = x 0, while the right boundary is expressed as x R = x n+1 = x 0 + (n + 1) ∆x The specified temperature ϑ(x R, t k) serves as the temperature value ϑ k 0 or ϑ k n+1 in the difference equation (2.240).

With givenheat ﬂuxq(t) the condition˙

To satisfy the boundary condition ˙q(t) (2.247), the heat flux ˙q must be directed outward, indicating a positive flow A grid is established with the boundary positioned between two grid lines, specifically at the left boundary x R = x 0 + ∆x/2, as illustrated in Fig 2.44 This configuration introduces grid points (0, k) outside the thermally conductive body, where the temperatures ϑ k 0 are calculated solely to meet the boundary condition (2.247).

Fig 2.44: Consideration of the boundary conditions (2.247) at x R = x 0 + ∆x/2 by means of the introduction of temperatures ϑ k 0 outside the body

The local derivative which appears in (2.247) is replaced by the rather exact centraldiﬀerence quotient It holds at the left hand boundary that

Atx=x 1 the diﬀerence equation (2.240) with i= 1 holds: ϑ k+1 1 =M ϑ k 0 + (1−2M)ϑ k 1 +M ϑ k 2 (2.250) Elimination ofϑ k 0 from both these equations yields ϑ k+1 1 = (1−M)ϑ k 1 +M ϑ k 2 −M∆x λ q(t˙ k ) (2.251)

This equation replaces (2.250), when the boundary condition (2.247) is to be satisﬁed at the left hand boundary.

To satisfy equation (2.247) at the right-hand boundary, the grid must be selected such that \( x_R = x_n + \Delta x/2 \) is applicable By eliminating \( \vartheta^{k}_{n+1} \) from the difference equation for \( x_n \) and the boundary condition, we derive the equation \( \vartheta^{k+1}_{n} = M \vartheta^{k}_{n-1} + (1-M) \vartheta^{k}_{n} + M \Delta x \lambda q(\dot{t}^{k}) \) (2.252).

In the context of heat transfer, when the heat flux is directed out of the body, ˙q(t k ) remains positive, as indicated in equation (2.251) The difference equations (2.251) and (2.252) apply under the condition that ˙q is equal to zero for adiabatic boundaries, which also serve as planes of symmetry within the body By strategically placing the grid so that the adiabatic plane of symmetry is positioned between two adjacent grid lines, the temperature calculations can be effectively confined to one half of the body.

If heat is transferred at the boundary to a ﬂuid at temperatureϑ S , where the heat transfer coeﬃcientαis given, then theheat transfer conditionis valid

The derivative is taken in the outward normal direction, with α and ϑ S potentially varying over time For effective discretization of equation (2.253), it is ideal for the boundary to align with a grid line, allowing for the boundary temperature to be directly applied in the difference formula To substitute the derivative ∂ϑ/∂n with the central difference quotient, grid points outside the body, such as temperatures ϑ k 0 or ϑ k n+1, are necessary These can be removed from the difference equations by leveraging the boundary condition.

At the left hand boundary (x R =x 1 ) the outward normal to the surface points in the negativex-direction Therefore from (2.253) it follows that

Fig 2.45: Consideration of the heat transfer condition (2.253), a at the left hand boundary x R = x 1 , b at the right hand boundary x R = x n

Replacing with the central diﬀerence quotient,

Bi ∗ =α∆x/λ (2.255) as the Biot number of the finite difference method Using (2.254), the temperature ϑ k 0 is eliminated from the difference equation (2.250), which assumes the form ϑ k+1 1 = [1−2M(1 +Bi ∗ )]ϑ k 1 + 2M ϑ k 2 + 2M Bi ∗ ϑ k S (2.256)

At the right hand boundary (x R =x n ), we obtain from (2.253) the boundary condition

∂x= α λ(ϑ−ϑ S ) Its discretisation leads to an equation analogous to (2.254) ϑ k n+1 =ϑ k n − 1 −2Bi ∗ ϑ k n −ϑ k S (2.257)

This is then used to eliminateϑ k n+1 from the diﬀerence equation valid forx n , which then has the form ϑ k+1 n = 2M ϑ k n−1 + [1−2M(1 +Bi ∗ )]ϑ k n + 2M Bi ∗ ϑ k S (2.258)

The analysis of the heat transfer condition (2.253) negatively impacts the stability of the explicit difference formula To ensure stability, the coefficients in the explicit difference equations (2.256) and (2.258) must remain positive, leading us to establish a necessary stability condition.

This tightens the condition (2.242) and leads to even smaller time steps ∆t.

In heat transfer analysis, an alternative grid layout, as depicted in Fig 2.44, is often utilized instead of the one shown in Fig 2.45 This involves replacing the derivative ∂ϑ/∂x at x R = x 0 + ∆x/2 with the central difference quotient from equation (2.248) The boundary temperature, denoted as ϑ (x R , t k ), is then approximated using ϑ (x R , t k ) = ϑ k 1/2 = 1.

This then gives us the relationship from (2.253) ϑ k 0 = 2 − Bi ∗

2 + Bi ∗ ϑ k S , (2.261) which, with (2.250), leads to the diﬀerence formula ϑ k+1 1 =

The discretisation error of the equation is O(∆x), contrasting with the O(∆x²) error associated with approximation (2.256) This indicates that using approximation (2.262) may result in larger errors compared to (2.256) However, it is important to note that the stability behavior is improved with this approach.

2 + 3Bi ∗ (2.263) is valid, which for large values of Bi ∗ delivers the limiting value M = 1/3, against which (2.259) leads to M → 0.

To minimize the larger discretization error, it is essential to replace the boundary temperature ϑ k 1/2 with a more precise calculation than the basic arithmetic mean By employing a parabolic curve that incorporates the three temperatures ϑ k 0, ϑ k 1, and ϑ k 2, the boundary temperature can be accurately determined as ϑ k 1/2 = 1/8.

From which, instead of (2.261) we get ϑ k 0 = 8 − 6Bi ∗

8 + 3Bi ∗ ϑ k S and from (2.250) the diﬀerence equation ϑ k+1 1 =

The equation 8 + 3Bi ∗ ϑ k S (2.264) is derived, offering a more precise solution than equation (2.262) despite its increased numerical complexity due to the boundary condition (2.250) This equation's discretization error is O(∆x²), leading to improved accuracy in the results Additionally, the corresponding stability condition is established.

This only leads to a tightening of the stability condition (2.242) for Bi ∗ ≥ 4/3 and for very large

Bi ∗ yields the limiting value M ≤ 1/4.

For small values of Bi*, the straightforward relationship outlined in equation (2.256) yields highly accurate results when using the grid shown in Fig 2.45 a However, for larger Bi* values, it is essential to apply equation (2.264) with the grid depicted in Fig 2.44, while ensuring compliance with the stability condition specified in equation (2.265).

A steel plate with thermal conductivity λ = 15.0 W/K m and diffusivity a = 3.75 × 10 −6 m²/s has a thickness of 270 mm and an initial temperature ϑ 0 At time t 0, the plate is placed in contact with a fluid at a lower constant temperature ϑ S < ϑ 0 The heat transfer coefficient at both surfaces is α = 75 W/m² K The cooling temperatures of the plate will be determined numerically, with simple initial and boundary conditions chosen to validate the accuracy of the finite difference method against the explicit solution discussed in section 2.3.3.

Due to the symmetry of the plate, only one half, measuring 135 mm in thickness, needs to be analyzed The left surface of the plate is considered adiabatic, while heat is transferred to the fluid at the right surface The grid utilized for this analysis is depicted in Fig 2.46, featuring a specific mesh size.

∆x = 30 mm The left boundary of the plate lies in the middle of the two grid lines x 0 and

Fig 2.46: Grid division for the calculation of the cooling of a plate of thickness 2δ x 1 = x 0 + ∆x; the right boundary coincides with x 5 The Biot number of the diﬀerence method is

0.030 m 15.0 W/K m = 0.15 For the modulus we choose

M = a∆t/∆x 2 = 1/3 , which satisﬁes the stability condition (2.259) The time step will be

We set ϑ S = 0 and ϑ 0 = 1.0000 The temperatures ϑ k i calculated agree with the dimensionless temperatures ϑ + (x i , t k ) in the explicit solution from (2.171).

With these arrangements the ﬁve following diﬀerence equations are valid: ϑ k+1 1 = 2

3 ϑ k 2 according to (2.251) with ˙ q(t k ) = 0, ϑ k+1 i = 1 3 ϑ k i−1 + ϑ k i + ϑ k i+1 for i = 2, 3, 4 according to (2.240) as well as ϑ k+1 5 = 2

The results for the ﬁrst 12 time steps are shown in Table 2.9 In Table 2.10 the surface temperatures ϑ k 5 and the temperature proﬁle at time t 12 = 960 s are compared with the exact solution.

Table 2.9: Temperatures in the cooling of a steel plate, calculated with the explicit diﬀerence method. k t k /s ϑ k 1 ϑ k 2 ϑ k 3 ϑ k 4 ϑ k 5

Table 2.10: Comparison of the surface temperature ϑ k 5 and the temperatures ϑ 12 i calculated using the diﬀerence method with the values of ϑ + from the exact solution according to (2.171). k ϑ k 5 ϑ + (x 5 , t k ) k ϑ k 5 ϑ + (x 5 , t k ) k ϑ k 5 ϑ + (x 5 , t k )

Despite using a coarse grid, the results align well with the explicit solution from (2.171) However, the initial 12 time steps capture only a minor portion of the cooling process, primarily due to the time step restriction ∆t imposed by the stability condition To address this limitation, transitioning to an implicit difference method is necessary.

The implicit diﬀerence method from J Crank and P Nicolson

The explicit difference method has a significant limitation due to the stability conditions, which restrict the time step size and often necessitate numerous time steps to obtain a temperature profile To overcome this limitation, an implicit difference method can be employed, which involves solving a tridiagonal system of linear equations at each time step This system features a straightforward structure, with the coefficient matrix populated only along the main diagonal and its immediate neighbors Efficient algorithms for solving tridiagonal systems are detailed in D Marsal's work and other standard numerical mathematics references.

J Crank and P Nicolson introduced a highly accurate implicit difference method that ensures stability in numerical analysis This method utilizes temperature values at two time levels, \( t_k \) and \( t_{k+1} \), while discretizing the differential equation for the intermediate time \( t_k + \Delta t/2 \) This approach allows for the precise approximation of the time derivative \( \frac{\partial \vartheta}{\partial t} \) at \( k+1/2 \) using a central difference quotient.

Utilizing an implicit difference method offers significant benefits, as it enables the use of larger time steps while necessitating a more precise approximation of the time derivative.

The second derivative (∂ 2 ϑ/∂x 2 ) k+1/2 i at time t k + ∆t/2 is replaced by the arithmetic mean of the second central diﬀerence quotients at times t k and t k+1 This produces

(2.267) With (2.266) and (2.267) the implicit diﬀerence equation is obtained

At time \( t_k \), the known temperatures are utilized to compute the three unknown temperatures at time \( t_{k+1} \) The difference equation results in a system of linear equations, represented by a tridiagonal matrix where the main diagonal consists of \( (2 + 2M) \) and the sub- and superdiagonals contain \( (-M) \) Notably, the first equation excludes the term \( -M \vartheta_{k+1}^0 \), and the last equation omits \( -M \vartheta_{k+1}^{n+1} \), as these terms are removed by applying the boundary conditions.

When the boundary temperatures are specified, the grid is designed so that x=x0 and x=xn+1 align with the boundaries Consequently, the values ϑk0, ϑk+10, and ϑkn+1, ϑk+1n+1 are predetermined, leading to the first equation of the tridiagonal system.

(2 + 2M)ϑ k+1 1 −M ϑ k+1 2 =M ϑ k 0 +ϑ k+1 0 + (2−2M)ϑ k 1 +M ϑ k 2 , (2.269) whilst the last equation will be

When the heat flux \( q(t) \) at the left boundary is specified, the grid is selected to align with the configuration shown in Fig 2.44, positioning the boundary equidistantly between \( x_0 \) and \( x_1 \) Consequently, \( \vartheta_{k0} \) and \( \vartheta_{k+1 0} \) will be eliminated using equation (2.249), resulting in a revised first equation.

The adiabatic wall represents a scenario where the heat flux is zero (˙q≡0) In this case, when the heat flux at the right edge is specified, the variables ϑ k n+1 and ϑ k+1 n+1 can be removed similarly as previously described To analyze the heat transfer condition (2.253), the grid is arranged such that the boundary aligns with x1 or xn, depending on whether (2.253) is applied at the left or right boundary Consequently, ϑ k 0 and ϑ k+1 0 are eliminated from the first equation of the tridiagonal system (2.268) by utilizing equation (2.254).

+M ϑ k 2 +M Bi ∗ ϑ k S +ϑ k+1 S (2.272) for the ﬁrst equation We eliminateϑ k n+1 and ϑ k+1 n+1 using (2.257) giving us as the last equation

The Crank-Nicolson difference method maintains stability for all values of M However, the time step size must adhere to accuracy requirements Excessively large values of M can result in finite oscillations within the numerical solution, which diminish slowly as k increases.

Example 2.7: The cooling problem discussed in Example 2.6 will now be solved using the Crank-Nicolson implicit diﬀerence method The grid divisions will be kept as that in Fig 2.46.

The tridiagonal system for each time step, which has to be solved consists of ﬁve equations and has the form

⎦ , the solution vector ϑ k+1 = ϑ k+1 1 , ϑ k+1 2 , ϑ k+1 3 , ϑ k+1 4 , ϑ k+1 5 T and the right side b =

In the ﬁrst step of the process (k = 0) all temperatures are set to ϑ 0 i = 1.000, corresponding to the initial condition ϑ(x i , t 0 ) = 1.

We chose M = 1, a time step three times larger than that in the explicit method from Example 2.6, namely ∆t = 240 s = 4.0 min The temperatures for the ﬁrst 10 time steps are given in Table 2.11.

Table 2.11: Temperatures in the cooling of a steel plate, calculated using the Crank-Nicolson method with M = a∆t/∆x 2 = 1. t k /s ϑ k 1 ϑ k 2 ϑ k 3 ϑ k 4 ϑ k 5

A comparison of temperatures at time t = 960 s reveals that the Crank-Nicolson method outperforms the explicit method, even with a time step that is three times larger.

To assess the impact of time step size on accuracy, we examined multiple values of M, specifically M = 1, 2, 5, and 10 The temperature distributions at t = 9600 seconds (or 160 minutes) are presented in Table 2.12.

Table 2.12: Comparison of the temperatures for t = 9600 s, calculated with diﬀerent modulus values M

This time is reached after 40 steps with M = 1, 20 steps for M = 2, 8 steps for M =

In the analysis of temperature variations, the results for M = 1 and M = 2 closely align with both each other and the analytical solution, indicating strong consistency However, for M = 5, deviations become more pronounced, and the findings for M = 10 are rendered ineffective due to the emergence of unphysical temperature oscillations To mitigate such oscillations, a specific condition regarding step size limitations is referenced, as noted in [2.57], p 122, which pertains to transient heat conduction problems with differing boundary conditions from the current example Applying this condition to the present scenario establishes a critical limit for step size.

It conﬁrms the result of our numerical test calculations.

Noncartesian coordinates Temperature dependent material

This article explores the difference method while accounting for temperature-dependent material properties in both cylindrical and spherical coordinates, focusing on one-dimensional heat flow in the radial direction The key differential equation governing the temperature field is presented as c∂ϑ.

The establishment of diﬀerence equations is based on the discretisation of the self-adjoint diﬀerential operator

We will now discuss this and derive the diﬀerence equations for diﬀerent coordinates systems with and without consideration of the temperature dependence of the properties of the material.

2.4.4.1 The discretisation of the self-adjoint diﬀerential operator

Utilizing a non-uniform grid for the discretization of the operator D is often beneficial, particularly in the r-direction This approach allows for smaller separations between grid lines, enabling a more accurate representation of the temperature profile's curvature at lower r-values Consequently, we will adopt a centered grid with a mesh size ∆r i that depends on the index i for the discretization of D, as illustrated in Fig 2.47 By applying central difference quotients, we can derive the necessary results.

⎦ Replacing the ﬁrst derivatives with central diﬀerence quotients gives

, m= 0,1, 2 , (2.278) is valid The thermal conductivities which appear here at the temperaturesϑ k i+ 1 orϑ k i− 1 2

To determine thermal conductivities at specific temperatures, we must utilize an appropriate mean value formulation based on the known temperatures at grid points In this discussion, we will focus on the simplified scenario where properties remain constant.

Fig 2.47: Centred grid for the discretisation of the self-adjoint differential operator D according to (2.275)

2.4.4.2 Constant material properties Cylindrical coordinates

Assuming constant thermal conductivity, we derive the difference equation for a cylinder (m=1), while the case for a sphere (m=2) is left for the reader to solve Simple difference equations for a plate (m=0) can be found in sections 2.4.1 and 2.4.3, assuming a constant mesh size of ∆r i = ∆x For a cylinder with λ = constant, we obtain the expression from equations (2.277) and (2.279).

∆r 2 i g i + ϑ k i+1 −ϑ k i −g i − ϑ k i −ϑ k i−1 , (2.280) where we have introduced the abbreviations g i + := 2 + ∆r i /r i

A further simpliﬁcation is possible if thegrid spacing∆ris assumed to beconstant. With g i + = 1 +∆r

If anexplicitdiﬀerence equation is desired, then the time based derivative in

= 1 cr i D k i has to be replaced according to (2.239) This produces the explicit diﬀerence equation ϑ k+1 i =M

1 +∆r 2r i ϑ k i+1 , (2.284) whose modulusM has to satisfy the stability criterion mentioned in 2.4.1.2

The boundary conditions are treated in the same manner as in section 2.4.2.

To transfer the Crank-Nicolson [2.65]implicit diﬀerence method, which is always stable, over to cylindrical coordinates requires the discretisation of the equation

With the time derivative from (2.266) and D k i orD i k+1 according to (2.283) we obtain the following diﬀerence equation

The equation 1 + ∆r 2r i ϑ k i+1 (2.287) incorporates the known temperatures at time t k, resulting in a tridiagonal system of linear equations that must be solved at each time step, as detailed in section 2.4.3 The temperatures ϑ k 0 and ϑ k+1 0 from the first equation, along with ϑ k n+1 and ϑ k+1 n+1 from the last equation, can be eliminated using the boundary conditions discussed in section 2.4.3.

A non-equidistant grid, cf Fig 2.47, requires the use of the discretised diﬀerential operator

D k i from (2.280) with the functions g + i and g i − according to (2.281) The explicit diﬀerential equation analogous to (2.284) has the form ϑ k+1 i = M i g − i ϑ k i−1 +

The diﬀerence method is only stable if none of the coeﬃcients in (2.288) is negative This leads to the stability condition M i ≤ (g i + + g − i ) −1 , and therefore

The stable, implicit method from Crank and Nicolson can be used without this restriction.

A generalisation of (2.286) delivers the tridiagonal linear equation system

2 + M i g + i + g − i ϑ k+1 i − M i g + i ϑ k+1 i = C i k (2.291) with the right hand side

2 − M i g + i + g i − ϑ k i + M i g + i ϑ k i+1 , (2.292) which has to be solved for each time step The modulus M i is given by (2.289).

When temperature-dependent properties are considered, a closed solution to the heat conduction equation is generally unattainable, necessitating the use of numerical methods This article illustrates how to incorporate these temperature-dependent properties through the example of a plate (m=0) Readers are encouraged to extend this approach to cylinders and spheres (m=1 or 2) using the general discretization equation By employing an equidistant grid with spacing ∆x, we can derive results from the relevant equations for the differential operator.

The thermal conductivity, denoted as λ k i ± 1/2, is evaluated at the temperature ϑ k i ± 1/2, necessitating the selection of an appropriate mean value—be it arithmetic, geometric, or harmonic—based on the thermal conductivities at the known temperatures ϑ k i and ϑ k i+1 or ϑ k i and ϑ k i−1 The choice of mean value has minimal impact when λ exhibits weak dependence on ϑ or when the step size ∆x is very small D Marsal recommends utilizing the harmonic mean for this purpose.

2 = 2λ k i λ k i±1 λ k i +λ k i±1 , (2.294) where λ k i = λ(ϑ k i ) = λ[ϑ(x i , t k )] Using the harmonic mean we obtain, from (2.294),

To simplify the process and avoid complex iterations, it is advisable to utilize an explicit difference method By substituting the time derivative with the first difference quotient, we derive the difference equation ϑ k+1 i = ϑ k i + 2a k i ∆t from the relevant equations.

Herea k i =a(ϑ k i ) is the thermal diﬀusivity at the temperatureϑ k i

The stability of the diﬀerence method is guaranteed by choosing a small enough time step ∆t such that the coeﬃcient of ϑ k i in (2.296) is positive Therefore it always has to be

Asaandλchange with each time step, a control for this inequality should be built into the computer program and should the situation arise ∆t should be reduced stepwise.

When analyzing heat transfer conditions, it is essential to consider the temperature dependence of the thermal conductivity, λ Additionally, the Biot number, Bi*, will vary with temperature, which is crucial for accurately eliminating the temperatures ϑ k 0 and ϑ k n+1 in the relevant equations This temperature dependency must be taken into account in the calculations, particularly in equations (2.254), (2.257), and (2.297) for i = 1 or i = n.

Transient two- and three-dimensional temperature ﬁelds

Determining planar or spatial temperature fields significantly increases computation time and storage requirements compared to one-dimensional problems This discussion will focus on rectangular regions using Cartesian coordinates, while cylindrical and spherical problems can be addressed through the discretization of their respective differential equations The finite difference method is often inadequate for complex geometries, particularly when second or third-type boundary conditions are imposed on irregular shapes In such scenarios, the finite element method is a more effective approach.

We will assume that the material properties are constant The heat conduction equation for planar, transient temperature ﬁelds with heat sources has the form

The temperatureϑ=ϑ(x, y, t) is to be determined in a rectangle parallel to the x- andy-axes We discretise the coordinates by x i =x 0 +i∆x , y j =y 0 +j∆y and t k =t 0 +k∆t

The temperature at an intersection point of the planar grid in Fig 2.48 will be indicated by ϑ k i,j =ϑ x i , y j , t k The corresponding equation for the thermal power density is

Fig 2.48: Planar grid for the discretisation of the heat conduction equation (2.298) in the rectangular region x 0 ≤ x ≤ x n , y 0 ≤ y ≤ y l in the case where ˙W is only dependent on the temperature

The two second derivatives in thex- ory-directions are approximated by the central diﬀerence quotient, so that

∆y 2 (2.300) are valid In order to obtain an explicit diﬀerence equation, we use the forward diﬀerence quotient for the time derivative

Using these difference quotients, we obtain, from (2.298), the explicit finite difference formula ϑ k+1 i,j = 1−2M x −2M y ϑ k i,j +M x ϑ k i − 1,j +ϑ k i+1,j

For a square grid (∆x= ∆y) we obtainM x =M y =M =a∆t/∆x 2 , and (2.302) is simpliﬁed to ϑ k+1 i,j = (1−4M)ϑ k i,j +M ϑ k i−1,j +ϑ k i+1,j +ϑ k i,j−1 +ϑ k i,j+1 +∆t c

Equations (2.302) and (2.303) allow for the explicit calculation of temperatures ϑ 1 i,j at time t 1 = t 0 + ∆t based on the initial temperature distribution ϑ 0 i,j This method facilitates the computation of subsequent temperature levels, making it relatively straightforward to program Additionally, it accommodates temperature-dependent thermal power density ˙W(ϑ), as ˙W i,j k is derived from the previously determined temperature ϑ k i,j However, the explicit difference method has a significant drawback in terms of stability, as highlighted in section 2.4.1.2, where the stability condition restricts the allowable time step by prohibiting any negative coefficients on the right-hand side of (2.302).

This leads to an even smaller time step than in the condition (2.242) for geometrical one-dimensional heat conduction.

To overcome the step size limitation of 2.304 imposed by the stability condition of the explicit difference method, an implicit method can be employed This approach involves discretizing equation (2.298) at time step k+1 and utilizing the backward difference quotient to substitute the time derivative.

∆t ϑ k+1 i,j −ϑ k i,j +O(∆t) and the difference quotients (2.299) and (2.300) formulated for t k+1 we obtain from (2.298), with the simplification that ˙W ≡0, the implicit difference equation a

The equation ∆t ϑ k+1 i,j −ϑ k i,j is formulated for all grid points (i, j) to establish a system of linear equations for the unknown temperatures at time t k+1 This system must be solved at each time step, with each equation containing five unknowns, relying on the known temperature ϑ k i,j from the previous time t k A robust solution method for this problem is provided by P.W Peaceman and H.H Rachford, known for its effectiveness in solving such equations.

“alternating-direction implicit procedure” (ADIP) Here, instead of the equation system (2.305) two tridiagonal systems are solved, through which the computation time is reduced, see also [2.53].

The boundary conditions for two-dimensional temperature fields are easily met when boundaries align with the coordinate axes Utilizing additional grid points outside the region allows for the effective application of the three types of boundary conditions However, complications arise when boundaries are not parallel to the axes The simplest condition to implement is the prescribed temperature, which can be achieved by approximating curved walls with straight lines parallel to the x and y axes, using small grid spacings Nonetheless, the discretization of the normal derivative ∂ϑ/∂n along the edges results in complex expressions that can be challenging to manage.

The discretization of the heat conduction equation can be applied to three-dimensional temperature fields, which readers are encouraged to explore The stability condition for the explicit difference formula requires smaller time steps compared to planar problems The implicit difference method's system of equations cannot utilize the ADIP method due to its instability in three dimensions Instead, a stable alternative method proposed by J Douglas and H.H Rachford is employed, resulting in tridiagonal systems However, this method incurs a greater discretization error than the ADIP method.

Steady-state temperature ﬁelds

The difference method for steady-state heat conduction problems involves discretizing a two- or three-dimensional region using a suitable grid to determine the temperature field Temperatures at grid points are calculated through difference equations that relate each temperature to its neighboring points, resulting in a system of linear equations that approximates the heat conduction equation with boundary conditions Decreasing the grid spacing enhances the accuracy of temperature approximations by increasing the number of grid points and equations While this discussion focuses on planar temperature fields, similar methods apply to three-dimensional cases For more advanced techniques and algorithms related to temperature fields, refer to the work by D Marsal.

2.4.6.1 A simple finite difference method for plane, steady-state temperature fields

Plane, steady-state temperature ﬁelds ϑ= ϑ(x, y) with heat sources of thermal power density ˙W are described by the diﬀerential equation

Asquaregrid with mesh size ∆x= ∆y is chosen for the discretisation, such that x i =x 0 +i∆x , i= 0,1,2, and y j =y 0 +j∆x , j= 0,1,2, are valid The temperature at grid point (x i , y j ) is indicated by ϑ i,j = ϑ(x i , y j ), and equally the thermal power density by ˙W i,j = ˙W(x i , y j , ϑ i,j ).

In this section, we will derive the difference equation related to (2.306) using an energy balance approach Each grid point is viewed as the center of a small block, cut from a thermally conductive material, with a square base of side length ∆x and height b The temperature ϑ i,j at the grid point (i, j) represents the average temperature of the block Heat transfer occurs from the block to its four neighboring blocks, characterized by the mean temperatures ϑ i+1,j, ϑ i,j+1, ϑ i−1,j, and ϑ i,j−1.

Fig 2.49: Block with square cross section around the point (i, j) for the derivation of the energy balance

(2.307) balance for this block contains the four heat ﬂows from Fig 2.49 and the thermal power arising from the internal heat sources ˙W i,j ∆V, in which ∆V = ∆x 2 bis the volume of the block :

Q˙ i+1 + ˙Q j+1 + ˙Q i − 1 + ˙Q j − 1 + ˙W i,j ∆V = 0 (2.307) The heat ﬂow ˙Q i+1 , from block (i+ 1, j) to block (i, j) is given by

In thermodynamics, heat flows into the block (i, j) are considered positive quantities From the energy balance equation, we derive the difference equation: ϑ i+1,j + ϑ i,j+1 + ϑ i−1,j + ϑ i,j−1 - 4ϑ i,j = -W˙ i,j ∆x² / λ.

The equation presented approximates the heat conduction in finite blocks, where each block is assigned a single discrete temperature, considering only interactions between immediate neighbors By applying the discretization equations (2.299) and (2.300) for second derivatives, we derive equation (2.308) through the standard discretization of the differential equation (2.306) The discretization error is O(∆x²), which diminishes to zero as the mesh size, or block width, decreases.

The difference equation must be formulated for each grid point within the region, while the equations for edge points will be addressed in the subsequent section This process results in a linear equation system that becomes extensive with finer grid divisions Numerical mathematics provides various methods to tackle large systems of equations, as discussed by D Marsal and G.D Smith Typically, iterative methods, particularly the Gauss-Seidel method and the Successive Over-Relaxation (SOR) method, are employed, with detailed explanations provided by D.M Young.

In a scenario where a room with a square base is encased by a wall of thickness δ, with each internal side measuring 2.5 δ, the wall maintains constant surface temperatures of ϑ i and ϑ o, where ϑ o is lower than ϑ i To determine the heat flow escaping from the room, one must calculate the effect of the temperature difference between the interior and exterior, represented as ϑ i − ϑ o.

Fig 2.50: Wall of a room with square base a Dimensions, b Coarse grid (∆x = δ/2) with four unknown (normalised) temperatures ϑ + 1 to ϑ + 4

To maintain symmetry, it is adequate to focus on one-eighth of the base, as illustrated in Fig 2.50a The target heat flow rate is constrained between two calculated limits, specifically concerning the inner wall surface.

Q ˙ i = 8 ã 1.25 ã δ ã b λ δ (ϑ i − ϑ o ) = 10 bλ(ϑ i − ϑ o ) and with the outer surface of the wall

Q ˙ o = 8 ã 2.25 ã δ ã b λ δ (ϑ i − ϑ o ) = 18 bλ(ϑ i − ϑ o ) where b is the dimension of the wall perpendicular to the drawing plane in Fig 2.50a The correct value between 10 and 18 for the shape factor

Q ˙ λb(ϑ i − ϑ o ) , see section 2.2.5.2, is obtained by calculating the temperature ﬁeld in the wall.

We use the dimensionless temperature ϑ + = (ϑ − ϑ o )/(ϑ i − ϑ o ), which lies in the region

0 ≤ ϑ + ≤ 1 A very coarse grid with ∆x = δ/2 delivers four grid points with the unknown temperatures ϑ + 1 to ϑ + 4 , cf Fig 2.50b They are calculated using the diﬀerence equations according to (2.308) with ˙ W i,j ≡ 0 ϑ + 2 + 0 + ϑ + 1 + 1 − 4 ϑ + 1 = 0 , ϑ + 3 + 0 + ϑ + 1 + 1 − 4 ϑ + 2 = 0 , ϑ + 4 + 0 + ϑ + 2 + 1 − 4 ϑ + 3 = 0 ,

The ﬁrst and fourth equations concern the symmetry of the temperature ﬁeld along the dotted line of symmetry in Fig 2.50b.

The solution of the four equations yields the temperatures ϑ + 1 = 0.4930, ϑ + 2 = 0.4789, ϑ + 3 = 0.4225, ϑ + 4 = 0.2113

With these results the heat ﬂow conducted to the outer surface of the wall is determined to be

The shape factor is determined to be S b = 12.85, indicating that the corners enhance the heat flow by 28.5% compared to the heat flow calculated using the internal wall area However, these calculations are approximations that may lack precision due to the grid's coarseness Utilizing a refined grid would yield more accurate temperature results, but it would also increase the complexity of the system by adding more difference equations For instance, reducing the mesh size to half (∆x = δ/4) results in a system comprising 24 linear equations.

2.4.6.2 Consideration of the boundary conditions

To accurately solve the system of linear equations derived from the difference equation, it is essential to incorporate boundary conditions through difference equations at the boundaries For simplicity, we assume that boundaries are aligned with the x- and y-directions, allowing curved boundaries to be approximated by straight lines parallel to these axes Achieving a sufficient level of accuracy requires a very small mesh size, ∆x When dealing with boundaries defined in polar coordinates (r, ϕ), it is advisable to express the differential equation and its boundary conditions in polar form before deriving the corresponding finite difference equations.

In the simplest scenario of boundary conditions, the boundaries must align with grid lines, as demonstrated in Example 2.8 The specified temperatures are applied in the difference equation (2.308) for the grid points on the boundaries For boundaries parallel to the x- and y-axes, this alignment is achieved by appropriately selecting the mesh size ∆x If needed, the existing square grid can be substituted with a rectangular grid featuring different spacings ∆x and ∆y Adjustments to the difference equation (2.308) can be made easily, following the guidance provided in section 2.4.6.1.

The boundary condition for a specific temperature can be effectively approximated even for curved boundaries As illustrated in Fig 2.51, a grid point (i, j) near the boundary is intersected by the straight lines x = x_i and y = y_j.

In Fig 2.51, the grid point (i, j) is positioned near the curved boundary R, where the temperature ϑ R is specified The temperatures ϑ R,n and ϑ R,n+1 are defined at the intersections The difference equation for grid point (i, j) is presented as an alternative to (2.308).

W˙ i,j (2.309) Its derivation can be found in G.D Smith [2.57].

Heat transfer conditions are effectively met when the boundary consists of straight lines aligned with the x- and y-axes For accurate modeling, the boundary must align with a grid line, allowing the difference equation to be derived from an energy balance This principle is illustrated for a grid point along the boundary at y = yj, as shown in Fig 2.52, where heat flows via conduction from adjacent blocks into the specified shaded block.

Fig 2.52: Derivation of the finite difference equation for the boundary point (x i , y j ) with heat transfer to a fluid at temperature ϑ S

Fig 2.53: Explanation of the difference equation for the boundary point a in the reflex corner, b at the outer corner with heat transfer to a fluid at temperature ϑ S

The heat ﬂow transferred from the ﬂuid at temperatureϑ S is

Q˙ S =α i ϑ S −ϑ i,j ∆x b where α i is the local heat transfer coeﬃcient at the point x i Putting the four heat ﬂows into the balance equation

2 ϑ i − 1,j +ϑ i+1,j +Bi ∗ i ϑ S −(2 +Bi ∗ i )ϑ i,j =−W˙ i,j ∆x 2 /2λ (2.310) with Bi ∗ i = α i ∆x/λ as the local Biot number This number allows us to easily account for changes in the heat transfer coeﬃcient along the boundary.

The difference equations governing heat transfer at the internal reflex corner, as illustrated in Fig 2.53a, and at the outer corner shown in Fig 2.53b, are derived using a similar approach For the internal corner, the equation can be expressed as ϑ i,j+1 + ϑ i−1,j + 1.

(2.311) and for the corner in Fig 2.53b

If no internal heat sources are present, ˙W i,j = 0 has to be put into the relationships (2.309) to (2.312).

By substituting Bi ∗ i = 0 and α i = 0 into equations (2.310) to (2.312), the relationships become applicable for anadiabatic surfaces, specifically under the condition ˙q = 0 for prescribed heat flux As discussed in section 2.4.2, this boundary condition is more accurately represented when the grid is arranged so that the boundary with the specified heat flux is positioned at a distance of ∆x/2 from the grid line, as illustrated in Fig 2.54 The energy balance for the highlighted block in Fig 2.54, which has a temperature of ϑ i,j, is also considered.

Mass diﬀusion

Remarks on quiescent systems

Mass diffusion processes are rarely observed in quiescent systems, making them less significant in practical applications compared to heat conduction.

Mass diffusion refers to the transport of molecules driven by their natural movement from one area to another within a system Similarly, heat conduction is the transfer of energy that occurs due to the random motion of particles, which is influenced by uneven temperature distribution.

In this respect a close relationship between mass diﬀusion and heat conduction exists.

Unlike heat conduction, mass diffusion allows for varying average velocities among particles within a volume element, leading to noticeable relative movement between them This results in a collective macroscopic movement of all particles, often manifesting as convection Consequently, unlike in heat conduction, quiescent systems cannot always be assumed in mass diffusion; such assumptions are valid only under specific conditions, which will be explored further.

In this article, we focus on systems where the reference velocity for determining mass diffusional flux is absent Specifically, we utilize gravitational velocity as the reference, as indicated in equation (1.154).

K w K , (2.314) and the average molar velocityuaccording to (1.157), which we will write in the following form cu=

Vanishing gravitational velocityw= 0 at every position has

In the observed system, the mass flow into a volume element equals the mass flow out, ensuring that the overall mass remains unchanged Consequently, the density within a specific volume element remains constant, although it may vary for elements located at different positions.

∂/∂x = 0 is possible but ∂/∂t = 0 Vanishing gravitational velocity w = 0 merely has the result∂/∂t= 0, but not∂/∂x= 0.

This article discusses the model of an incompressible body, characterized by a constant density throughout its movement Specifically, the density of a volume element remains unchanged, leading to the condition d/dt = 0 This scenario is applicable here, as both the total derivative and the partial derivative with respect to time equal zero, indicating that the material does not experience any changes in density during its motion.

In a porous body where hydrogen diffuses, a volume element with a density of 7.8 x 10³ kg/m³ contains a mass of 7.8 x 10⁻³ g for a 1 mm³ sample At an ambient temperature of 298 K and a partial pressure of hydrogen at 1 hPa, the ideal gas law is applied to determine the amount of hydrogen absorbed by this volume element.

The addition of hydrogen has a negligible effect on the mass of the element, allowing us to disregard any mass change Consequently, the center of mass remains stationary, with a good approximation indicating that the rate of change in position, d x /dt, is effectively zero.

In very dilute solutions, such as gas absorption in liquids, similar results are observed for mass diffusion The liquid's density, far from the critical region, significantly exceeds that of the gas, which has an exceptionally low mass Consequently, the mass of a volume element remains virtually unaffected by gas absorption, allowing for the approximation d x /dt = w = 0 to hold true.

Vanishing gravitational velocity and therefore a medium, quiescent relative to the gravitational velocity, can be presumed for the diﬀusion of gases into solids or diﬀusion in very dilute solutions.

We will now look at a system, in which the average molar velocityuaccording to (2.315) disappears This gives

The total molar flow into a volume element equals the total flow out, indicating that diffusion is equimolar Consequently, the number of moles (N) within the volume element remains constant, leading to a stable molar density.

V →0N/V a ﬁxed position is constant It can change with positionxbut∂c/∂t= 0.

In this example, we analyze a vessel containing 0.5 mol of helium (He) on the left side and 0.5 mol of argon (Ar) on the right side, with a pressure of 0.1 MPa and a temperature of 298 K Upon removing the separating wall, the gases begin to diffuse into one another while maintaining constant pressure and temperature Throughout this process, the ideal gas equation is applicable to each volume element within the vessel.

∂t = 0 , which is only possible if the amount of material ﬂowing into each volume element is equal to the amount of the other material ﬂowing out of the element:

(c w ) He = − (c w ) Ar The average molar velocity, according to (2.315) is u = 0 As the pressure, temperature and

In an equimolar diffusion of two gases, the initial center of mass is located on the side of the heavier gas, argon, as shown in point a of Fig 2.56 After mixing, a homogeneous distribution is achieved, resulting in the center of mass shifting to the midpoint of the vessel, represented as point b in the figure At this stage, the gravitational velocity is non-zero, while the average molar velocity remains at u = 0, indicating a unique interplay between the two velocities during the diffusion process.

In the context of mass diffusion of ideal gases at constant pressure and temperature, it can be assumed that the average molar velocity diminishes, and the system remains quiescent relative to this average molar velocity.

Derivation of the diﬀerential equation for the concentration ﬁeld 225

Solving a diffusion problem requires determining the mass fraction composition of a material over time and space, specifically for component A represented as ξ A = M A / M This relationship can be expressed as ξ A = ξ A (x, t) By applying Fick’s law, the diffusive flux can be calculated For a binary mixture of components A and B, the flux is given by j ∗ A = -D grad ξ A, while for a multi-component mixture, the equation expands to j ∗ A = Σ (A K=1 to N).

M˜ 2 D AK grad ˜x K (2.319) with the mole fraction ˜x A = ˜M ξ A /M˜ A according to (1.151) and the average molar mass according to ˜M from (1.154).

Fig 2.57: Input of substance A by diﬀusion and production by a chemical reaction in volume V a for the mass balance, b for the molar balance

The mass fraction ξ_A is derived from a partial differential equation based on mass balance principles To understand this, we consider a defined region of volume V and surface area A, from which we conceptually isolate a body undergoing diffusion Within this context, a surface element dA, with an outward-directed normal vector n, receives a mass flow d˙M_A = -A w_A dA The negative sign indicates that mass flow is considered positive when entering the region Here, the velocity vector w_A of substance A moves inward, while the normal vector n points outward, resulting in a negative scalar product The diffusive flux is defined as j*_A = A (w_A - w).

We will deal with a quiescent medium relative to the gravitational velocity,w= 0, so that the diﬀusional ﬂux is j ∗ A= A w A (2.321)

It follows from (2.320) and (2.321) that d ˙M A =−j ∗ A ndA Integration gives the total inlet mass ﬂow of the component

This will then be converted from the integral over the whole surface area of the region to the volume integral of the divergencej ∗ A according to Gauss’ integral theorem.

In a specific region, substance A is either generated or absorbed through a chemical reaction The production rate within a volume element is denoted as ˙Γ A, measured in kg/m³·s This rate is positive when substance A is produced and negative when it is consumed, reflecting the mass flow dynamics of the reaction.

The generation of substance A within the region is represented by the equation (V) Γ˙ A (2.323) This increase in mass over time is influenced by both the diffusion across the surface area and the mass flow resulting from chemical reactions, leading to a net accumulation of mass within the region as expressed by d dt.

Taking into account (2.322) and (2.323) the mass balance for substance A is

As the balance region can be inﬁnitely small, it holds that

A mass balance of this type is valid for each component in the mixture Therefore, there are as many mass balances as substances Addition of all the components leads to

K =is the density of a mixture ofN substances The sum of all the diﬀusional ﬂuxes is given by (1.156), i.e & N

K=1 j ∗ K= 0.As the mass of a certain substance generated per unit time in the chemical reaction can only be as much as is consumed of another substance, it follows that

The summation over all substances, according to (2.326), leads to

The density at a specific location remains constant over time in a quiescent medium with w = 0 However, this constancy applies only at fixed positions, as local density variations can occur due to differing compositions.

As the densityis independent of time, (2.325) may, because A = ξ A , also be written as

∂t =−divj ∗ A+ ˙Γ A (2.327) Introducing Fick’s law (2.318) gives

∂t = div( Dgradξ A ) + ˙Γ A (2.328) as the desired diﬀerential equation for the mass fraction ξ A (x, t) in a quiescent, relative to the gravitational velocity, isotropic binary mixture of substances A and

B A corresponding equation also exists for B However this does not need to be solved, because when the mass fractionξ A is known, the other mass fraction in a binary mixture isξ B = 1−ξ A

Putting Fick’s law (2.319) into (2.327) provides us with the corresponding equation to (2.328) for a mixture ofN components

In a quiescent body where gravitational velocity is zero (w = 0) and time derivative (∂/∂t) is also zero, the quantities D and ˙Γ A are influenced by the density (x), temperature (ϑ), and composition (ξ A) Furthermore, these quantities, as expressed in equation (2.329), not only rely on the mass fraction ξ A of component A but also on the mass fractions of other substances present Additionally, the gradient grad ˜x A is affected by the gradients of the mass fractions of other substances, derived from the relation ˜ x A = ˜M ξ A /M˜ A with 1/M˜.

K ξ K /M˜ K Therefore the diﬀusion equations (2.328) and (2.329) are, in general, non- linear The general equation (2.329), contains (2.328) as a special case withN= 2, which can be easily checked.

In a quiescent system with constant temperature and pressure, we can derive the diffusion equation by examining the molar flow of component A into the surface element dA The molar flow is represented as d ˙N A = −c A w A ndA.

The negative sign arises because the normal vector n is oriented outward from the area dA, necessitating that the molar flow into this area be considered positive The diffusional flux is defined by the equation u j A = c A (w A - u) and is equivalent to the molar flux u j A = c A w A, based on the assumption of negligible average molar velocity u Consequently, the total molar flow into the area is established.

The amount of substance A generated by chemical reaction in the region is indicated by the molar production ﬂux ˙γ A (SI units mol/m 3 s) It is

The molar flow into the region, driven by diffusion and chemical reactions, leads to a gradual increase in the amount of substance stored over time.

The material balance for substance A is then

As the balance region can be chosen to be inﬁnitely small, it holds that

A balance of this type exists for all the substances involved in the diﬀusion, so they can be added together giving

K=1 c K = c is the molar density of a mixture consisting ofN substances.

K=1 u j K= 0, cf section 1.4.1, page 72, (2.336) is transformed into the following equation valid for the sum over all components

In a chemical reaction, the total number of moles typically changes, meaning the right side of equation (2.337) does not vanish unless the moles produced equal those that are consumed By applying Fick’s law (1.160) and substituting D_AB with D, we can derive results from equation (2.335).

∂t = div(c Dgrad ˜x A ) + ˙γ A (2.338) as the diﬀerential equation for the molar concentration c A (x, t) in a quiescent, relative to the average molar velocity u, binary mixture of components A and

B A corresponding equation exists for substance B It is also possible to solve (2.338) and then determinec B , becausec B =c−c A

For a multicomponent mixture of ideal gases at constant temperature and pressure, Fick’s law in the particle reference system [2.75], can be written as u j A N K=A K=1

Putting this into (2.335) we obtain for a multicomponent mixture at constant pressure and temperature, the diﬀusion equation

The diffusion coefficient, represented as D = D(p, ϑ, x˜ A), is influenced by the molar production density, ˙γ A = ˙γ A (p, ϑ, x˜ A), where ˜x A is defined as the ratio of concentration c A to the total concentration c Additionally, these parameters are affected by the mole fraction ˜x A and the mole fractions of other substances present in the system.

Simpliﬁcations

This article focuses exclusively on binary mixtures, as the diffusion coefficients for mixtures with more than two components are frequently unknown, preventing quantitative calculations of diffusion in these cases Consequently, the diffusion equations for binary mixtures can often be significantly simplified.

In the diffusion of gas A into a solid or liquid B, the density of the volume element remains nearly unchanged due to the minimal mass of the absorbed gas compared to the volume element If substance B is initially homogeneous, its density remains constant throughout the diffusion process Therefore, it is reasonable to approximate the density as constant over time and space Additionally, studies indicate that the diffusion coefficient in dilute liquid solutions at a constant temperature can be considered approximately constant Similarly, when gas diffuses into a homogeneous, porous solid at a constant temperature, the diffusion coefficient remains nearly constant, as concentration variations are limited Thus, under these conditions, the equations governing diffusion can be simplified significantly.

The equation for vanishing average molar velocity (u = 0) can be simplified for binary mixtures of ideal gases At low pressures, typically up to 10 bar, the diffusion coefficient remains composition-independent It increases with temperature and is inversely related to pressure, resulting in constant diffusion coefficients in isobaric, isothermal mixtures Consequently, the equation is transformed for constant diffusion coefficient scenarios.

The equations (2.341) and (2.342) are equivalent to each other, because putting c A = A /M˜ A into (2.342) results in

On the other hand, because= const we can also write

By deﬁnition ˙γ A M˜ A = ˙Γ A , which shows that the two equations are equivalent For the case where no chemical reaction occurs, ˙γ A = 0, (2.342) was ﬁrst presented by A Fick in 1855 [2.76], and

∂t =D∇ 2 c A (2.343) is therefore known as Fick’s second law.

Boundary conditions

The equations (2.342) and (2.341) are of the same type as the heat conduction equation (2.11) The following correspondences hold c A =ϑ/ , D=a/ and γ˙ A = ˙/W /c

Many solutions to the heat conduction equation can also be applied to mass diffusion problems, provided that the differential equations and the corresponding initial and boundary conditions are consistent Crank's book offers numerous solutions to the relevant differential equation Similar to heat conduction, the initial condition specifies the concentration at each position within the medium at a designated time, marking the start of the timekeeping process with the equation c A (x, y, z, t= 0) = c Aα (x, y, z) Additionally, local boundary conditions can be categorized into three groups, paralleling those found in heat conduction scenarios.

1 The concentration can be set as a function of time and the positionx 0 , y 0 , z 0 on the surface of the body (boundary condition of the 1st kind) c A (x 0 , y 0 , z 0 , t) =c A0 (t) (2.345)

During the initial drying phase of porous materials, a liquid film covers the surface, and water evaporation occurs in dry air At the interface between the liquid and gas, the saturation pressure p AS (ϑ) related to the temperature ϑ is evident, leading to the equation c A0 = p AS (ϑ)/R m T.

The solubility of gas A in a liquid or solid B is significant, particularly when gas A exhibits weak solubility According to Henry’s law, the mole fraction of the dissolved gas, denoted as ˜ x A0, can be calculated using the formula ˜ x A0 = p AS(ϑ)/k H, where k H represents the Henry coefficient, which varies with temperature and pressure in binary mixtures However, the pressure effect can often be disregarded when the gas's total pressure is low enough to be treated as ideal For specific numerical values of k H(ϑ), refer to the tables provided by Landolt-Börnstein.

2 The diﬀusional ﬂux normal to the surface can be prescribed as a function of time and position (boundary condition of the 2nd kind) uj A =−c D∂˜x A

∂n , (2.346) whereby for surfaces impermeable to the material we get

3 Contact between two quiescent bodies (1) and (2) at the interface can exist (boundary condition of the 3rd kind)

If a quiescent body (1) is bounded by a moving fluid (2) a diffusion boundary layer develops in the fluid Instead of (2.348), the boundary condition

The concentration difference in the fluid, denoted as ∆c (2) A = c (2) A0 − c (2) Aδ, is measured between the interface 0 and the boundary layer edge δ The mole fraction ˜x A can be converted to concentration c A using the relationship c A = ˜x A c For ideal gases, concentration c is defined as c = N/V and can be expressed as c = p/R m T, while for real gases, it must be derived from the thermal equation of state.

When heat is transferred by conduction between two bodies in contact, their temperatures become equal, indicating that heat fluxes are balanced In contrast, at the interface of phase diffusion, a concentration jump typically occurs due to the equilibrium conditions governing mass transfer, which require the chemical potential of each component to be equal across phases This relationship can be expressed as ˜x (1) A0 = f(˜x (2) A0), demonstrating the connection between temperature, pressure, and phase equilibria, and serves as the definition for the equilibrium constant.

K := ˜x (1) A0 /˜x (2) A0 (2.351) For the solubility of gases in liquids

The relationship between the equilibrium constant and the Henry coefficient is expressed as K = k H (ϑ)/p, where k H (ϑ) denotes the Henry coefficient For a deeper understanding of these concepts, refer to textbooks on the thermodynamics of mixtures, such as [1.1] The solubility of gas A in a solid phase can be determined using the equation c A0 = L S p A0 (2.352), with solubility L S measured in SI units of mol/m³ bar Additional numerical values can be found in resources like the Landolt-Börnstein tables [2.79].

In a spherical vessel with an internal diameter of 1450 mm and a wall thickness of 4 mm, made from chromium-nickel-steel, hydrogen gas is stored at 85 °C under high pressure, with an initial partial pressure of 1 MPa Due to diffusion through the vessel wall, hydrogen escapes into the surrounding air, causing the internal pressure to decrease over time To determine the time required for the pressure to drop by 10 −3 MPa and the corresponding amount of hydrogen lost, we assume steady-state diffusion as the pressure change is gradual The solubility coefficient for hydrogen at the wall surface is L S = 9.01 mol/m³ bar, and the diffusion coefficient for hydrogen at 85 °C is D = 1.05 × 10 −13 m²/s The gaseous mixture adheres to the thermal equation of state for ideal gases, while the hydrogen concentration in the surrounding air is negligible.

The diffusion of hydrogen through the vessel wall is directly proportional to the decrease in the stored amount of hydrogen This relationship is expressed by the equation j_AA = -dN/dt = -dN_A/dt Additionally, the diffusion flux j_AA can be represented by the equation j_AA = D * δ_A * m * (c_Ai - c_Aa), where δ is the wall thickness and A_m is the geometric mean, calculated as the square root of the product of the concentrations.

A i A o ∼ = A i of the internal and external surface areas With c Ao = 0 and c Ai = L S p A we obtain j A A = D δ A m L S p A

On the other hand, for the amount of hydrogen inside the vessel we have N A = p A V /R m T This then gives

Putting in the numerical values we obtain p A (t 1 ) = 1 MPa ã exp ( − 2.915 ã 10 −11 t 1 /s) from which we get t 1 = 1

The loss of hydrogen follows from the equation of state for ideal gases

1 MPa This is 0.1 % of the initial amount of hydrogen present, which was

Steady-state mass diﬀusion with catalytic surface reaction

In previous sections, we explored steady-state one-dimensional diffusion without considering chemical reactions Now, we will examine the impact of chemical reactions, particularly in catalytic reactors, where heterogeneous reactions occur at the interface between the reacting medium and the catalyst These reactions can be modeled as boundary conditions for mass transfer problems, as they occur on the surface In contrast, homogeneous reactions occur within the medium itself, where new chemical compounds are formed from existing materials based on temperature, composition, and pressure, making each volume element a source of material production, akin to heat sources in conduction processes.

In a catalytic reactor, a chemical reaction occurs between gas A and reaction partner R, producing a new product P The reactor is designed to feed in R and gas A while removing excess gas A and product P The reactor contains spheres coated with a catalytic material, facilitating the reaction at the catalyst surface, which accelerates the process Although the complex mechanisms at the catalyst surface are often not fully understood, simplified models can be utilized At steady-state, the amount of gas generated equals the amount transported away by diffusion, making the reaction rate equal to the diffusive flux The reaction rate of gas A is primarily influenced by its concentration at the surface, expressed as ˙n A0 = -k1 c A0 for a first-order reaction.

For ann-th order reaction we have ˙ n A0 =−k n c n A0 (2.354)

Fig 2.58: Catalytic reactor Fig 2.59: Uni-directional diﬀusion with reaction at the catalyst surface

The rate constant \( k_n \) is expressed in SI units as (mol/m²·s)/(mol/m³)ⁿ, while the rate constant \( k_1 \) for a first-order reaction has SI units of m/s The superscript indices signify that the reaction occurs at the surface, and the negative sign indicates that the rate of change of substance A (\( \dot{n}_{A0} \)) is negative due to its consumption in the reaction Conversely, if substance A were being generated, the value would be positive.

Presuming that the reaction is ﬁrst order, at the catalyst surface it holds that ˙ n A0 =− c D∂x˜ A

In the region above the catalyst surface, substance A is primarily transported by diffusion in the x-direction Within a thin layer adjacent to the wall, the diffusion boundary layer, mass transport via convection is minimal Consequently, the steady one-dimensional diffusion equation, applicable in the absence of chemical reactions, can be expressed as d/dx(c D_d˜x A) = 0.

The solution must satisfy both the boundary condition (2.355) and the condition ˜ x A (x=L) = ˜x AL (2.357), where L represents the thickness of the diffusion boundary layer Assuming a constant diffusion coefficient (c D), we derive the solution as ˜ x A − x˜ AL = k 1 c A0 c D (x−L) (2.358).

The mole fraction at the wallx= 0 is found to be ˜ x A0 =−k 1 c A0 c D L+ ˜x AL =−k 1 x˜ A0 L

Furthermore, if we takec= const as an assumption, due toc A = ˜x A c, we get c A0 = c AL

1 +k 1 L/D (2.360) and therefore the reaction rate with (2.355) ˙ n A0 =− k 1 c AL

In convective mass transfer at the catalyst surface, the mass transfer coefficient β replaces the term D/L, where β is defined as β = D/L This substitution can be verified, as the equation ˙ n A0 =−k 1 c A0 can be rewritten as ˙ n A0 =−β(c A0 −c AL) By eliminating c A0, we derive the relationship ˙ n A0 =− k 1 c AL.

The negative sign in (2.361) and (2.362) indicates that the mass ﬂow of A is towards the catalyst surface.

Da 1 =k 1 L/D (2.363) is called the Damköhler 5 number for a heterogeneous first order reaction (there are further Damköhler numbers) It is the ratio of the reaction rate k 1 to the diffusion rateD/L.

In (2.361) or (2.362) two limiting case are of interest: a) Da 1 =k 1 L/Dork 1 /βare very small, becausek 1 D/Lork 1 β.Then ˙ n A0 =−k 1 c AL (2.364)

The material conversion will be determined by the reaction rate It is

“reaction controlled” According to (2.360) the concentration of substance

A perpendicular to the catalyst surface is constant, i.e c A0 =c AL b) Da 1 =k 1 L/D ork 1 /β is very large because k 1 D/Lork 1 β Then we get ˙ n A0 =−D

The material conversion is governed by diffusion, making the process "diffusion controlled." As indicated by the equation (2.360), the initial concentration of substance A (c A0) is zero, leading to the complete conversion of substance A during a rapid reaction at the catalyst surface.

Gerhard Damköhler (1908–1944) was a pioneering figure in the development of similarity theory for chemical processes, focusing on the interplay of heat and mass transfer His influential work, “The Influence of Diffusion, Flow and Heat Transport on the Yield of a Chemical Reaction,” published in Der Chemie-Ingenieur in 1937, laid the groundwork for advanced studies in chemical reaction technology.

Example 2.11: In the catalytic converter of a car, nitrogen oxide is reduced at the catalyst surface, according to the following reaction

The NO-reduction of substance A follows a first-order reaction, described by the equation ˙ n A0 = − k 1 c A0 For a 75 kW engine, the exhaust flow rate is ˙ M = 350 kg/h, with a constant molar mass of the exhaust gases at ˜ M = 32 kg/kmol Operating at a temperature of 480 °C and pressure of 0.12 MPa, the exhaust gases contain NO at a mole fraction of ˜ x AL = 10 −3, from which 80% needs to be removed Given a mass transfer coefficient β of 0.1 m/s and a rate constant k 1 of 0.05 m/s, the required catalyst surface area can be calculated to ensure efficient NO reduction.

The molar ﬂow rate of the exhaust is

3600 s/h ã 32 kg/kmol = 3.038 ã 10 −3 kmol/s The material balance around the dotted balance region, Fig 2.60, yields

Fig 2.60: For the material balance of an exhaust catalytic converter

( ˙ N x ˜ AL ) z = ( ˙ N x ˜ AL ) z+dz + ˙ n AL dA or, because ˙ N = const

Here ˙ n AL = β c (˜ x AL − x ˜ A0 ), if we presume a low mass ﬂow rate normal to the wall and we neglect the Stefan correction factor for the mass transfer coeﬃcients With c = N/V = p/R m T we get

1 − x ˜ A0 ˜ x AL dA Putting in (2.359) yields, whilst accounting for β = D/L

1 + k 1 /β dA Integration between the inlet cross section i and the exit cross section e of the catalytic converter of area A yields ln (˜ x AL ) e (˜ x AL ) i

Steady-state mass diﬀusion with homogeneous chemical reaction 238

In the context of substance A diffusing into a porous solid or quiescent fluid B, we can observe a notable example in the biological treatment of wastewater In this process, oxygen diffuses from the air or oxygen bubbles into the surrounding wastewater, where it reacts with organic pollutants, such as hydrocarbons, converting them into carbon dioxide and water through the action of microorganisms Additionally, substance B may function as a catalyst, often comprising spherical or cylindrical pellets with intricate capillaries This porous internal structure significantly increases the internal surface area, enhancing chemical conversion rates at both the outer and inner surfaces of the catalyst.

Fig 2.61: Unidirectional mass diﬀusion with homogeneous reaction

According to (2.338), for steady-state, geometric one-dimensional diﬀusion with chemical reaction we have d dx c Dd˜x A dx

In this analysis, we consider that substance A and the byproducts of the chemical reaction exist in minimal quantities within substance B, leading to a constant concentration represented by the equation c = N/V Furthermore, we establish that D remains constant throughout the process.

In the case of a porous solid, the molecular diffusion coefficient D must be substituted with the effective diffusion coefficient D_eff, which is lower due to the obstruction of molecular movement by the pores To quantify this effect, a diffusion resistance factor, denoted as à, is commonly defined as the ratio of D to D_eff.

A porous solid of volume V consists of the volume V S of the solid material and

The volume of the solid-free spaces, denoted as V_G, is calculated using the formula V_G = V - V_S The void fraction, represented by ε_p, is defined as the ratio of V_G to the total volume V Additionally, the resistance factor is influenced by the voidage of the porous material and a diversion or winding factor, denoted as à_p, which relates through the equation à = à_p / ε_p.

The following table contains several values for the void fraction and the diversion factor.

Table 2.13: Void fraction ε p and diversion factor à p of some dry materials

Material Density kg/m 3 Void fraction ε p Diversion factor à p

In a first order reaction, substance A is consumed at a rate described by the equation ˙γ A =−k 1 c A, where k 1 represents the rate constant in seconds (s −1) The negative sign indicates that substance A is being depleted, while a positive sign would imply its generation The reaction rate is influenced by the surface areas of porous solids, as larger inner and outer surfaces facilitate quicker reactions Consequently, the rate constant k 1 can be expressed as k 1 = a P k 1, where a P denotes the specific area (m 2 /m 3), reflecting the reactive area per unit volume of the porous material.

With ˜x A =c A /c, (2.366) and (2.367) are transformed, under the assumptions already given into d 2 c A dx 2 −k 1

This has to be solved under the boundary conditions shown in Fig 2.61 c A (x= 0) =c A0 (2.370) and dc A dx x=L= 0 (2.371)

In the analysis of heat conduction in porous solids, the diffusion coefficient \(D\) is replaced by the effective diffusion coefficient \(D_e\) The governing equation aligns with the earlier equation for heat conduction in rods, demonstrating similar boundary conditions of constant temperature at one end and no heat flow at the other Specifically, these conditions correspond to equations (2.370) and (2.371) Consequently, the solution derived from these equations mirrors the relationship previously established for rods, resulting in the expression \(c A - c A_0 = \cosh[m(L-x)] / \cosh(m L)\), where \(m\) is defined as \(k / \sqrt{D}\).

The amount of substance transferred is

The equation N˙ A0 = An˙ A0 = -A D dc A dx at x=0 = A D c A0 mtanh(mL) defines a relationship involving the dimensionless Hatta number, Ha, expressed as m L k 1 L 2 /D This Hatta number is significant as it quantifies the interplay between reaction and diffusion processes Specifically, the square of Ha represents the ratio of the reaction relaxation time, t R = 1/k 1, to the diffusion relaxation time, t D = L 2 /D, highlighting the dynamics of the system.

A large value for the Hatta number means that the chemical reaction is rapid in comparison with diﬀusion Substance A cannot penetrate very far into substance

B by diﬀusion, but will be converted by chemical reaction in a layer close to the surface.

Shironji Hatta (1895–1973) was a prominent professor at Tohoku Imperial University, now known as Tohoku University, located in Tokyo, Japan He conducted significant research on the absorption of gases in liquids, focusing particularly on the processes that involve simultaneous chemical reactions.

We present the pore effectiveness factor (η P) for porous bodies, defined as the ratio of the actual substance transfer rate (˙N A0) to the theoretical transfer rate (˙N A) under uniform concentration (c A0) conditions throughout the porous medium This highlights the significance of the effective diffusion coefficient (D eff), which replaces the molecular diffusion coefficient in this context.

D, was very largeD eﬀ → ∞, thenHa→0 Then according to (2.367) the reaction rate of substance A ˙ γ A =−k 1 c A0 disappears and the total amount delivered would be

N˙ A =k 1 c A0 A L (2.376) The pore eﬀectiveness factor would then be η P N˙ A0

N˙ A = D eff k 1 Lmtanh (mL) or η P =tanh (mL) mL (2.377) withm= k 1 /D eff = k 1 a p /D eff

The pore effectiveness factor applies to pores with a constant cross-sectional area and is analogous to the fin efficiency of a straight, rectangular fin For pores with varying cross-sectional shapes, the pore effectiveness factor can still be approximated using a specific equation, provided that the length L is considered as a characteristic length, as demonstrated by Aris.

L=V /A (2.378) with the volumeV of the porous body and its outer surface areaA A spherical pellet of radiusRwould, for example, haveL= (4/3)πR 3 /4πR 2 =R/3.

For small values of mL less than 0.3, the parameter ηp exceeds 0.97, indicating it is close to 1 This suggests that the composition of reaction partner A remains relatively constant throughout the pore length, with diffusion resistance being minimal compared to other resistances Additionally, a small mL value, represented as k1/D_eff, signifies either a small pore size, a slow reaction rate, or rapid diffusion.

To minimize CO emissions from furnace exhaust, the gas is directed through a catalytic reactor containing porous CuO pellets Within this reactor, CO (substance A) undergoes oxidation with O2, converting it to CO2 both inside and on the surface of the pellets.

In the reaction 2 O₂ = CO₂, it can be approximated as a first-order reaction where the rate of change of moles of species A is given by ˙n_A = -k₁ a P c A To determine the pore effectiveness factor, one must analyze the reaction dynamics Additionally, to reduce the mole fraction of CO to 1/10 of its initial value, with an initial mole fraction ˜x_A = 0.04, a calculation is needed to ascertain the required mass of CuO, considering that the molar flow rate ˙N of the exhaust gas remains constant.

The molar flow rate of the exhaust gas is 3 mol/s, with a mole fraction of carbon monoxide (CO) at 0.04 The exhaust operates at a temperature of 480 °C and a pressure of 0.12 MPa Additionally, the spherical pellets have a diameter of 5 mm and a specific area of 5 × 10^6 m²/m³, along with an effective diffusion coefficient.

The effective diffusivity (D_eff) is calculated as 5 x 10^-5 m²/s, while the reaction rate constant (k₁) is 10^-3 m/s, and the density of CuO is 8.9 x 10³ kg/m³ The mean length (mL) is derived from the equation mL = (k₁ a_P / D_eff)^(1/2) L, where L = R/3 Substituting the values gives mL = (10^-3 m/s x 5 x 10^6 m²/m³ / 5 x 10^-5 m²/s)^(1/2) (2.5 x 10^-3 m/3) = 8.33, leading to the calculation of the permeability (η_p) as η_p = tanh(mL)/(mL) = 0.12 Additionally, carbon monoxide (CO) flowing outside the pores undergoes decomposition through a chemical reaction In a differential segment dz, the amount of CO changes by d( ˙ N x ˜ A₀), which is transformed by the chemical reaction in the pores, represented as d ˙ N A₀ = ˙ γ A dV_P = -k₁ a_P c_A₀ η_p dV_P = -k₁ a_P c˜ x A₀ η_p dV_P.

V P is the volume of the pellets With that we get d( ˙ N x ˜ A0 ) = − k 1 a p c x ˜ A0 η p dV P From which, with ˙ N = const and c = p/R m T it follows that d˜ x A0 ˜ x A0

After integration between mole fraction ˜ x Ai at the inlet and ˜ x Ao at the outlet of the reactor we obtain the volume V P of the pellets

Transient mass diﬀusion

In section 2.5.3, it was demonstrated that the differential equation governing transient mass diffusion parallels the heat conduction equation, indicating that numerous mass diffusion issues can be related to analogous heat conduction scenarios This article aims to explore transient diffusion in semi-infinite solids and basic geometries such as plates, cylinders, and spheres in detail.

2.5.7.1 Transient mass diﬀusion in a semi-inﬁnite solid

This article examines the transient diffusion of a substance within a semi-infinite medium At the initial time (t=0), substance A is present in the medium at a concentration of c Aα The target concentration profile, denoted as c A = c A (x, t), adheres to a specific differential equation, assuming that the diffusion coefficient (cD) remains constant.

∂x 2 , t≥0, x≥0 (2.379) and should fulﬁll the initial condition c A (t= 0, x) =c Aα = const

In analogy to the heat conduction problem, the following conditions are possible at the surface of the body:

– a stepwise change in the surface concentration to the value c A0 , which should remain constant fort >0,

– input of a constant molar ﬂux ˙n A0 ,

– a stepwise change in the concentration in the surroundings to c AU =c A0 , so that mass transfer takes place with a mass transfer coeﬃcient β.

The solutions for heat conduction problems, as discussed in section 2.3.3, can be applied to mass diffusion issues due to the similarities in their differential equations, initial conditions, and boundary conditions The table below illustrates the corresponding quantities between heat conduction and mass diffusion.

Table 2.14: Corresponding relationships between quantities in heat conduction and diﬀusion

Heat conduction Diﬀusion ϑ c A a D t + = at/L 2 t + D = Dt/L 2

Using these correspondences allows us to write up the solutions to the diﬀusion problem from the solutions already discussed for heat conduction.

In transient diffusion in a semi-infinite solid with stepwise change in the surface concentration, we find from (2.126) c A −c Aα c A0 −c Aα = erfc x

Dt (2.380) and with (2.131) the transferred molar ﬂux ˙ n A (t, x) √D

The molar ﬂux transferred at the surfacex= 0 is ˙ n A (t, x= 0) √D

Solutions corresponding to a constant molar flux, denoted as ˙n A0, are derived from equation (2.135) at the surface Similarly, the mass transfer solution from the surface to another fluid is obtained from equation (2.140).

2.5.7.2 Transient mass diﬀusion in bodies of simple geometry with one-dimensional mass ﬂow

The time dependent concentration ﬁeld c A (x, t) in a body is determined by the diﬀusion equation which corresponds to the heat conduction equation (2.157). WithcD= const this equation is as follows

In the context of geometric one-dimensional flow in the radial direction, the parameter \( n \) takes on values of 0 for a plate, 1 for a cylinder, and 2 for a sphere The radial coordinate \( r \) is applicable for both cylinders and spheres, with the assumption that the cylinder is significantly longer than its diameter Additionally, the concentrations \( c_A \) within the cylinder and sphere are independent of the angular coordinate.

In the context of a plate, the x-coordinate is represented by r, extending from the center outward At the initial condition t=0, the concentration cA(r) is set to cAα for 0≤r≤R The boundary conditions are defined as follows: at r=0, the derivative of cA with respect to r is zero, and at r=R, the equation -D∂cA/∂r equals β(cA - cAS) By applying fundamental principles of heat conduction, we derive the results, leading to the equation c+A = cA - cAS, where cAα - cAS is also considered.

For theﬂat plate, from (2.171) and (2.172), as well as withBi D andt + D according to Tab 2.14 andr + =r/R, it follows that: c + A (r + , t + D ) ∞ i=1

1 cosà i cos(à i r + ) exp(−à 2 i t + D ) (2.384) with the eigenvaluesà i from tanà=Bi D /à (2.385)

The average concentrationc + Am follows from (2.174) c + Am (t + D ) = 2Bi 2 D

Analogous results are yielded with the equations from section 2.3.4.4, page 168, for the cylinder and the sphere.

Example 2.13: What is the equation for the average concentration c + Am (t + D ) of a sphere, if the concentration jumps from its initial value c A (r, t = 0) = c Aα to a value of c A (r =

The increase in surface concentration at r = R can occur only if the mass transfer resistance (1/βA) between the surface area and its surroundings is negligible This condition implies that β approaches infinity, leading to Bi D = βR/D AB approaching infinity, and resulting in c A0 equaling c AS Additionally, the equation c + Am can be expressed as (c Am − c A0) 0.

(c Aα − c A0 ) For a sphere we obtain, from (2.183) and (2.185) c + A (r + , t + D ) =

2 Bi D à 2 i + (Bi D − 1) 2 à 2 i + Bi D (Bi D − 1) sin à i à i sin(à i r + ) à i r + exp( − à 2 i t + D ) (2.387) with the eigenvalues à i from à cotà = 1 − Bi D (2.388)

The average concentration is, in accordance with (2.184), c + Am (t + D ) = 6Bi 2 D

For Bi D → ∞ we obtain, from (2.388), the eigenvalues à 1 = π, à 2 = 2π, à 3 = 3π and as the average concentration from (2.389) c + Am (t + D ) = 6

∞ i=1 exp( − à 2 i t + D ) à 2 i With the eigenvalues à i = iπ, i = 1, 2, 3 we can also write c + Am (t + D ) = 6 π 2

Exercises

To derive the differential equation for the temperature field ϑ = ϑ(r, t) in a hollow cylinder undergoing transient one-dimensional heat conduction in the radial direction, we begin with the energy balance for a hollow cylinder with internal radius r and thickness ∆r As we take the limit of ∆r approaching zero, we consider the material properties λ (thermal conductivity) and c (specific heat capacity) as functions of the temperature ϑ It is important to note that there are no internal heat sources present in this system This analysis leads to the formulation of a differential equation that describes the temperature distribution over time within the cylindrical structure.

In a cooled plate of thickness δ, one surface (x = 0) is insulated while the opposite surface transfers heat to a fluid at temperature ϑ F For a fixed time t ∗, the temperature profile ϑ = ϑ(x, t ∗) within the plate must reflect specific thermal conditions Near the insulated surface at x = 0, the temperature gradient must be zero, indicating no heat flow, while at the surface x = δ, the temperature should equal the fluid temperature ϑ F Given a Biot number of Bi = αδ/λ = 1.5, the temperature curve must smoothly transition between these boundary conditions, demonstrating a balance between conductive and convective heat transfer.

To analyze the thermal characteristics of the fluid on the plate surface, it is essential to sketch the temperature profile within the boundary layer This profile must consider the condition where the Nusselt number (Nu) is defined as Nu = αδ/λF, with a value of 10 Here, λF represents the thermal conductivity of the fluid, which plays a critical role in determining the heat transfer efficiency in the boundary layer.

2.3: The temperature profile in a steel plate of thickness δ = 60 mm and constant thermal diffusivity a = 12.6 ã 10 −6 m 2 /s, at a fixed time t 0 is given by ϑ + := ϑ − ϑ 1 ϑ 2 − ϑ 1 = x + − B(t 0 ) cos π(x + − 1

In the given scenario, the temperature distribution of the plate is represented as ϑ = ϑ(x, t₀) with boundary conditions where B(t₀) = 0.850, and the surface temperatures are constant at ϑ₁ = 100°C for x⁺ = 0 and ϑ₂ = 250°C for x⁺ = 1 To visualize the temperature distribution, one must draw the graph, which indicates whether the plate is being heated or cooled Additionally, it is important to identify the position x⁺ₜ where the temperature changes most rapidly over time and to determine the magnitude of this change.

To analyze the temperature profile at time \( t_0 \), we need to determine if \( x + T \) corresponds with the position \( x + \text{min} \) where the temperature reaches its minimum Additionally, we must calculate the time function \( B(t) \) while considering the initial condition \( B(t_0) = 0.850 \) Finally, we will identify the steady temperature pattern that emerges as time approaches infinity.

2.4: Heat is released due to a reaction in a very long cylinder of radius R The thermal power density increases with distance r from the cylinder axis:

The heat flux released from the cylinder, denoted as ˙ q(R), can be determined from the equation W ˙ (r) = ˙ W R (r/R) m, where m is greater than or equal to zero To ensure that the heat flux ˙ q(R) matches that of a cylinder with a uniform thermal power density ˙ W 0, it is essential to calculate the required value of ˙ W R Additionally, for a cylinder with a constant thermal conductivity λ, the overtemperature Θ(r) can be calculated using the formula Θ(r) = A, where Θ(r) represents the temperature difference between the point r and the reference point R.

In this analysis, we examine the boundary conditions where at r = 0, the derivative of temperature with respect to radius (dΘ/dr) equals zero, indicating a symmetry at the center, and at r = R, the temperature Θ equals zero, representing the outer boundary of the cylinder We then compare the maximum overtemperature, Θ max, of the cylinder with varying thermal power density to the maximum overtemperature, Θ 0 max, of a cylinder with a constant thermal power density, ˙ W 0 The ratio of these maximum overtemperatures, Θ max /Θ 0 max, is calculated for modes m = 0, 1, 2, and 3, under the condition that both cylinders release the same amount of heat.

A steel bolt with a diameter of 20 mm and thermal conductivity of 52.0 W/K m extends from an insulation layer, with one end maintained at a constant temperature of 75.0 °C The insulated length of the bolt measures 100 mm, while the exposed portion is 200 mm long, allowing heat to dissipate into the surrounding environment at a temperature of 15.0 °C The heat transfer coefficient at the bolt's outer surface and free head is 8.85 W/m² K, facilitating the heat exchange process.

To analyze the thermal behavior of a steel bolt protruding from an insulation layer, first, determine the temperatures at both ends of the bolt, denoted as ϑ 0 and ϑ L Next, calculate the heat flow, ˙ Q L, that the bolt releases to its surroundings from the exposed head end Finally, compare the obtained temperature profile results, based on equation (2.55), with those from a replacement bolt of length L C featuring an adiabatic head end, as described in equation (2.60).

To determine the efficiency (η f) of a straight fin with a rectangular cross-section, three temperatures must be measured: the base temperature (ϑ 0), the top temperature (ϑ h), and the surrounding fluid temperature (ϑ S) In this case, with ϑ 0 set at 75.0 °C, ϑ h at 40.5 °C, and ϑ S at 15.0 °C, the calculation of η f can be performed to assess the fin's performance in heat transfer applications.

To achieve a heat flow to the air that is six times larger from a brass tube with an outer diameter of 25 mm and a heat transfer coefficient of 90 W/m²K, a specific number of annular brass fins, each with a thickness of 1.5 mm and an initial height of 20 mm, must be affixed per meter of the tube Additionally, when the height of these fins is increased to 30 mm, the heat flow is expected to increase significantly compared to the original configuration with 20 mm fins.

A tube with a diameter of 0.30 m and a surface temperature of 60 °C is buried 0.80 m underground, where the thermal conductivity of the surrounding soil is 1.20 W/K m The ambient air temperature is 10 °C, and the heat transfer coefficient between the air and the earth's surface is 8.5 W/m² K To determine the heat loss per unit length of the tube, we need to calculate the heat transfer rate, denoted as ˙ Q/L.

An asphalt road coating with thermal conductivity of 0.65 W/K m, density of 2120 kg/m³, and specific heat capacity of 920 J/kg K has reached a uniform temperature of 55 °C, extending several centimeters below the surface due to prolonged sunlight exposure Following a sudden rainstorm that cools the surface temperature to 22 °C within 10 minutes, it is essential to calculate the heat released per unit area by the asphalt during the rainstorm, as well as the temperature at a depth of 3.0 cm at the end of the rain shower.

After 2.0 hours of exposure to a constant heat flux of 650 W/m² on a very thick concrete wall with thermal conductivity λ = 0.80 W/K m, density 1950 kg/m³, and specific heat capacity c = 880 J/kg K, the surface temperature reaches approximately 64.3 °C At a depth of 10 cm within the wall, the temperature is around 36.1 °C at the same time.

2.11: Solve exercise 2.10 with the additional condition that the concrete wall releases heat to air at a temperature ϑ S = ϑ 0 = 20 ◦ C, whereby the heat transfer coeﬃcient α = 15.0 W/m 2 K is decisive.

The investigation into the penetration of daily and seasonal temperature fluctuations in the ground reveals that the amplitude of daily temperature variations (∆ϑ = 10 °C, t₀ = 24 h) is negligible at a depth of 1 meter In contrast, for seasonal temperature fluctuations (∆ϑ = 25 °C, t₀ = 365 d), the temperature profile at a depth of 2 meters indicates that the highest temperature occurs on August 1st, while the lowest temperature is reached during the winter months, demonstrating the significant impact of seasonal changes on subsurface temperatures.

Preliminary remarks: Longitudinal, frictionless ﬂow over a ﬂat plate

less ﬂow over a ﬂat plate

To understand how heat and mass transfer coefficients vary with flow, we first examine longitudinal flow over a flat plate, assuming a constant average velocity and negligible fluid viscosity In real scenarios, flow velocity increases from zero at the wall to its maximum at the center, but for theoretical analysis, we consider fluids that do not adhere to the wall Although such fluids are impractical, their use allows us to derive approximate local temperature and concentration fields, enabling the calculation of local heat and mass transfer coefficients This approach simplifies the understanding of the relevant partial differential equations that will be discussed later.

A fluid at temperature ϑ α flows over a flat plate with a constant average velocity, where the plate's surface temperature is maintained at a constant value of ϑ 0 When the plate is hotter than the fluid (ϑ 0 > ϑ α), a specific temperature profile develops, while the opposite occurs when the plate is cooler (ϑ 0 < ϑ α) The temperature variations are confined to a thin layer near the wall, known as the thermal boundary layer, which increases in thickness along the flow direction This layer represents the region where temperature changes occur and is defined by a point where the temperature deviates slightly, such as 1%, from the core fluid temperature.

In the thermal boundary layer, heat is transferred to a fluid element through conduction, where it is stored as internal energy and subsequently carried away with the fluid Outside this layer, a constant temperature, denoted as ϑ α, exists in the surrounding flow An observer moving with the fluid element at a distance y from the wall will find that after a time t, their position is x = w m t For this observer, the temperature ϑ(x, y) can be simplified to ϑ(w m t, y), depending solely on time and the distance from the wall, assuming a constant velocity w m Therefore, the observer's temperature ϑ(t, y) can be effectively modeled using Fourier’s equation for transient heat conduction.

Assuming the thermal conductivity λ is constant with respect to temperature, the initial temperature ϑ α is established at position x = 0 or time t = 0 The wall temperature, denoted as ϑ 0, is specified, leading to the boundary conditions: ϑ(t = 0, y) = ϑ α and ϑ(t, y = 0) = ϑ 0.

The boundary conditions outlined in section 3.1 characterize the temperature distribution in a semi-infinite body, starting with an initial temperature of ϑ α When the surface temperature abruptly changes to a constant value of ϑ 0 = ϑ α, the implications for the temperature field are significant This phenomenon has been previously addressed in related studies.

Fig 3.1: a Temperature profile and b Thermal boundary layer in flow along a flat plate discussed in section 2.3.3.1 The solution is, cf (2.124), ϑ−ϑ α = (ϑ 0 −ϑ α ) erfc y

The local heat transfer coeﬃcientα is obtained from the energy balance at the wall ˙ q =α(ϑ 0 −ϑ α ) =−λ

The gradient present at the wall is yielded by diﬀerentiation of (3.4) to be

√πat Which then gives the heat transfer coeﬃcient as α= λ

The mean heat transfer coeﬃcient α m is the integral mean value over the plate lengthL α m = 1 L

It is twice as big as the local heat transfer coeﬃcient at the positionL As (3.6) indicates, the local heat transfer coeﬃcient falls along the length byα ∼x −1/2

At the beginning of the plate, the heat transfer coefficient is infinitely large, resulting in an extremely high heat flux However, as the length increases, the heat transfer coefficient decreases to a negligible level.

An approximation of the thicknessδ T of the thermal boundary layer is obtained by linearising the temperature increase

Then from (3.5) α≈ λ δ T The heat transfer coeﬃcient is inversely proportional to the thickness of the thermal boundary layer The resistance to heat transfer

The thermal boundary layer thickness, denoted as δ T, is proportional to the thermal diffusivity (α) and the characteristic length (λ) Initially, at the start of the plate, the thickness is minimal (δ T approaches 0 as α approaches infinity), but it increases significantly for an infinitely long plate (δ T approaches infinity as α approaches 0) According to the derived equation, the thermal boundary layer can be approximated by δ T ≈ λ α = √π.

'ax w m (3.8) increases with the square root of the lengthx.

The investigation of mass transfer involves analyzing a flat plate coated with substance A, such as naphthalene, which diffuses into an air flow along the plate To maintain the flow velocity of the fluid, it is assumed that the amount of material transferred by diffusion is negligible compared to the flowing material The convection normal to the wall is not considered significant The concentration of substance A on the plate surface, c A0, remains constant, while the concentration of the incoming fluid is c Aα, which is less than c A0 For an ideal gas, the concentration c A is expressed as c A = N A / V = p A / R m T, where c A0 is derived from the saturation pressure p A (ϑ 0 ) at the plate surface, and c Aα is based on the partial pressure of the substance.

As fluid flows, a concentration profile develops, similar to the temperature variation observed previously This concentration change occurs within a thin layer, known as the concentration boundary layer, which has a thickness of δc(x) and extends in the direction of flow For an observer within a volume element, the concentration is a function of both time and position, as described by the transient diffusion equation (2.343), assuming the diffusion coefficient remains constant regardless of concentration.

The analogous problem for heat conduction was given by (3.1) to (3.3) Corre- spondingly we obtain the solution related to (3.4) c A −c Aα = (c A0 −c Aα ) erfc y

Asc A = ˜x A c, the concentrationc A may be replaced here by the mole fraction ˜x A The local heat transfer coeﬃcient is obtained from a material balance at the wall ˙ n A0 =β(c A0 −c Aα ) =−cD

The gradient at the wall is found by diﬀerentiating (3.12) to be

√πDt Which means the mass transfer coeﬃcient is β= D

The mean mass transfer coeﬃcientβ m over the plate length L β m = 1 L

The concentration boundary layer thickness, denoted as δc, can be approximated by linearizing the concentration profile, as indicated by the equation L x=0 βdx= 2β(x=L), which shows that the value at point L is twice the local value, similar to the mean heat transfer coefficient.

∂y ≈cx˜ Aα −x˜ A0 δ c = c Aα −c A0 δ c , and leads to the expression corresponding to (3.8) δ c ≈D β =√ π

'Dx w m , (3.15) according to which the thickness of the concentration boundary layer increases with the square root of the lengthx.

The ratio of the thicknesses of the thermal and concentration boundary layers can be approximated by the Lewis number, defined as Le = a/D, as indicated in equations (3.8) and (3.15) This relationship highlights the interplay between thermal and concentration diffusion processes.

The heat and mass transfer coeﬃcients are also linked by the Lewis number due to (3.6) and (3.14) α c p β *a

As the Lewis number is of order one for ideal gases, the relationship (1.199): β=α/ c p , discussed earlier, holds.

The balance equations

Reynolds’ transport theorem

The balance equations can be simplified through the application of Reynolds’ transport theorem To derive these equations, we analyze an infinitesimally small fluid mass, denoted as dM, and track its movement within a flow field This fluid mass maintains consistent components, occupying a specific volume and surface area However, as the fluid moves, both its volume and surface area may change over time due to the alterations in its shape.

The mass M of the bounded ﬂuid volume is found from the sum of all the mass elements dM

(M) dM , which, with the density

∆V = dM dV , as a steady function of time and the position coordinate in a continuum, can also be written as

Each state quantity, including internal energy, enthalpy, and entropy, can be derived by integrating the corresponding specific state quantity For the specific quantity of state z, it is expressed as z = lim.

∆M = dZ dM, which results in

The specific quantity of statez can be represented as a scalar, vector, or tensor of any rank Similar to density, it varies with time and position, whereas the extensive quantity of stateZ is solely dependent on time.

Fig 3.2: Deformation of a closed system of volume V (t) in a ﬂow

The time derivative ofZ will now be formed dZ dt = d dt

The abbreviation for the quantity of state per volume is represented as z = Z/V, which varies with both time and position By definition, this relationship can be expressed as dZ/dt = lim.

⎥ ⎦ , in which the position dependence of Z V has not been written to prevent the expression becoming too confusing The derivative can also be written as dZ dt = lim

The volume increase with time ∆tis indicated here by ∆V(t) =V(t+ ∆t)−V(t). According to Fig 3.2 the volume element is given by dV = w i dA i ∆t 1 Which transforms the second integral into a surface integral,

Z V (t)w i dA i , and the formation of limits is superﬂuous, as the time interval ∆t is no longer present The integrand of the ﬁrst integral goes to∂Z V /∂taccording to the limits.

In sections 3.3, 3.4, 3.6, and 3.9, we will utilize tensor notation to present the balance equations clearly For a comprehensive understanding of tensor notations, please refer to Appendix A1.

Therefore, theReynolds transport theorem is obtained from (3.16) for volumina, in which we put once againZ V =z dZ dt

The equation describes the change in the extensive quantity of state Z within a volume V over time t It consists of two components: the increase in state Z within the volume and the portion of state Z that flows out with the material The outward direction of the surface element dA i indicates that a positive value signifies a flow out of the volume According to the equation, the total change in state Z for a substance amount M is determined by both the internal increase and the outflow Utilizing Gauss' theorem allows for the conversion of the surface integral into a volume integral, facilitating the calculation of dZ/dt.

The mass balance

We will now apply (3.18) to the mass as an extensive quantity of state, i.e.Z=M andz=M/M = 1 Equation (3.18) then becomes dM dt

In a closed system where the mass \( dM \) of an infinitesimally small subsystem remains constant, the overall mass of the entire system also remains constant, even as the volume \( V(t) \) changes over time This implies that the rate of change of mass, \( dM/dt \), is zero Consequently, the principle of constant mass applies even as the volume approaches zero, leading to the conclusion that the sum of the two integrands must equal zero.

∂x i = 0 the relationship equivalent to equation (3.20) is found to be d dt +∂w i

∂x i The equations (3.20) and (3.21) say that the mass is conserved in an inﬁnitesimally small volume element, and they are calledcontinuity equations.

In an incompressible ﬂuid the density remains constant, = const The continuity equation is simpliﬁed to

Example 3.1: Show, with help from (3.18) and (3.20), that the following equation is valid: dZ dt = d dt

Diﬀerentiation of the integrands on the right hand side also allows (3.18) to be written dZ dt = d dt

The application of the continuity equation (3.20) eliminates the first integral on the right side, while the integrand within the brackets of the second integral corresponds to the total differential, expressed as dz dt = ∂z.

∂x i Whereby the relationship dZ dt = d dt

V (t) dz dt dV is proved.

The continuity equation for component A in a mixture with N components is established by focusing solely on substance A At time t, substance A enters the volume V(t) and has a surface area A(t), allowing us to analyze its discharge.

The volume represented in Fig 3.2 is expanded by the equation dV = w Ai dA i ∆t, where w Ai denotes the flow velocity of substance A This transition from equations (3.16) to (3.17) and (3.18) indicates that the gravitational velocity w i is substituted with the flow velocity w Ai of substance A Additionally, for the extensive state quantity Z in (3.17), we define the mass M A, leading to the relationships z = M A / M = ξ A and z = M A / V = A, resulting in the expression dM A / dt.

The increase in substance A consists of two components: the increase occurring within the system and the amount of substance A that exits the system The production density of component A, represented as Γ˙ A, is measured in SI units of kg/m³·s for a specific volume element, quantified as dM A/dt.

The production density ˙Γ A varies with time and location, influenced by the progression of the chemical reaction within the system Understanding this variation is a key focus of reaction kinetics By applying Gauss' theorem, the surface integral in equation (3.22) can be transformed into a volume integral.

As this relationship also holds forV(t)→0, the integrands have to agree There- fore we obtain Γ˙ A = ∂ A

The component continuity equation is a mass balance that can be established for each individual component within a system Consequently, the number of equations corresponds to the number of components present By summing these equations, a total mass continuity equation is derived, based on the principle that the total mass flow rate equals zero Instead of using N separate continuity equations for a system with N components, one can utilize N−1 component continuity equations along with the total mass continuity equation for a more streamlined analysis.

The mass flux A w Ai in (3.23) can be found from the diffusional fluxj Ai ∗

A w Ai =j Ai ∗ + A w i This transforms (3.23) into Γ˙ A = ∂ A

Here the partial density A can be expressed in terms of the mass fractionξ A and the density A =ξ A , so that the left hand side of the equation is rearranged into

∂x i The term in the square brackets disappears due to the continuity equation (3.20) for the total mass The component continuity equation is then

For a binary mixture, with the introduction of the diﬀusional ﬂowj Ai ∗ according to (1.161) withD AB =D BA =D, this is transformed into

In the case of quiescent systems,w i = 0, this yields the relationship already known for transient diﬀusion, (2.341) If constant density is presumed, it follows from (3.26) that

+ ˙Γ A / (3.27) or withξ A = ˜M A c A /, in whichc A is the molar concentrationc A =N A /V,

+ ˙r A (3.28) with the reaction rate ˙r A = ˙Γ A /M˜ A (SI-units kmol/m 3 s), the amount of substance

Under the assumption of film theory, which involves steady-state mass transfer occurring solely along the coordinate axis next to the wall and a production density that approaches zero, the component continuity equation (3.25) can be reformulated to align with equation (1.186) of film theory.

Indicating the coordinate adjacent to the wall with y, under the given preconditions, (3.25) is transformed into w ∂ξ A

∂y , wherein j ∗ A is the diﬀusional ﬂow and w is the velocity in the y-direction On the other hand, as a result of the continuity equation (3.20)

∂y = 0 With that w = const, and (3.25) can also be written as

According to the definition given in equation (1.155), the relationship j ∗ A = A (w A − w) leads to the conclusion that j A ∗ + A w = A w A This indicates that the derivative ∂(A w A)/∂y equals zero, or equivalently, d ˙ M A /dy = 0 Consequently, since ˙ M A is expressed as ˜ M A N ˙ A, it follows that d ˙ N A /dy = 0, as derived from equation (1.186) This outcome can also be directly obtained from equation (3.23) by substituting ˙Γ A = 0 and d A /dt = 0 while assuming one-dimensional mass flow.

The momentum balance

In fluid dynamics, the transfer of momentum by mass elements in a flowing fluid is defined as the product of mass and velocity Specifically, a mass element, denoted as dM, moving with a velocity of wj carries momentum expressed as wj dM, which is equivalent to wj dV This relationship highlights the fundamental principle of momentum transport in fluid flow.

I j transported in a ﬂuid of volumeV(t) is therefore

Newton's second law of mechanics states that the rate of change of momentum of a body over time is equal to the resultant of all forces acting on that body, expressed mathematically as dI j/dt = F j.

Applying the transport theorem gives, if we putz=w j into (3.17),

A(t) w j w i dA i =F j (3.32) or taking into account the continuity equation (3.20) (see also example 3.1)

V (t) dw j dt dV =F j (3.33) with dw j dt = ∂w j

The forces acting on a body can be categorized into two types: body forces, which are proportional to the mass of the body, and surface forces, which are proportional to the surface area.

The body experiences various forces that influence all its particles, primarily due to extensive force fields A prominent example is the Earth's gravitational field, which exerts an acceleration, denoted as g, on every molecule Consequently, the gravitational force acting on a fluid element with mass ∆M can be quantified.

It is proportional to the mass of the ﬂuid element The body force is deﬁned by k j := lim

Fig 3.3: Surface force ∆F j on a surface element ∆A

Fig 3.4: Dependence of the surface forces on the orientation of the surface element ∆A and the surface force by f j := lim

In the case of gravitational force with k j = g j and f j =g j , in general it holds that f j = k j Other body forces are centrifugal forces or forces created by electromagnetic ﬁelds.

When analyzing a multicomponent mixture, it is essential to account for the varying effects of body forces on each individual component Specifically, the body force acting on component A, denoted as k Aj, is defined as the limit of the forces influencing that component.

The equation ∆M A = dF Aj dM A describes the relationship between changes in mass and force Here, dF Aj is expressed as k Aj dM A, which can be rewritten as k Aj A dV Additionally, dF j equals dF Kj, leading to the formulation dV & k Kj K, where the summation encompasses all substances K Furthermore, according to equation (3.35), dF j can also be represented as k j dM = k j dV, resulting in the conclusion that k j equals k Kj K.

If the gravitational force is the only mass force acting on the body then g j = k j = k Kj and = &

Surface forces are short-range forces that operate in the immediate vicinity of a fluid's surface These forces act directly on the surface elements of a body, denoted as ∆A, with a corresponding force ∆F j applied to it.

The stress vector, represented by the equation ∆A = dF j dA, is influenced by position, time, and orientation, particularly the normal vector to the surface element To illustrate this dependence, we can analyze flow along a flat plate A normal force acts perpendicularly on a surface element ∆A, while if the element is rotated to a position parallel to the wall, it experiences only shear stress Consequently, these two forces typically differ in magnitude.

The total force acting at time t on a ﬂuid of volume V(t) and surface area A(t) is found by integrating the body and surface forces to be

With this the momentum equation (3.33) can be written as

It means that the change in momentum over time of a ﬂuid of volumeV at time tis eﬀected by the body and the surface forces.

To calculate the stress vector \( \mathbf{t} \), we consider an infinitesimally small tetrahedral fluid element, where one surface is oriented in any direction and the others align with the coordinate axes The oblique surface has an outward normal unit vector \( \mathbf{n}_i \) and an area \( dA \), upon which the stress vector \( \mathbf{t} \) acts Specifically, the stress vector \( \mathbf{t}_1 \) acts on surface \( dA_1 \) perpendicular to the x-axis, while stress vectors \( \mathbf{t}_2 \) and \( \mathbf{t}_3 \) act on areas \( dA_2 \) and \( dA_3 \), respectively The forces on the four surface elements must maintain equilibrium, independent of the fluid element's instantaneous movement, as derived from the momentum equation applied to an infinitesimally small volume As the volume approaches zero, the volume integral diminishes more rapidly than the surface integral, ensuring local equilibrium of surface forces Thus, for the tetrahedron, we have \( \mathbf{t} \cdot dA = \mathbf{t}_1 \cdot dA_1 + \mathbf{t}_2 \cdot dA_2 + \mathbf{t}_3 \cdot dA_3 \).

The surface elements can be still be eliminated with the relationships from Fig. 3.6 of dA= 1

Fig 3.5: Equilibrium of surface forces

Fig 3.6: Relationship between the areas Fig 3.7: Stress components on the surface dA 1 and therefore dA 1 = dAOD

AD = dAcosα Then, because cosα=n 1 /n=n 1 , we can also write dA 1 = dA n 1 Correspond- ingly, it holds that dA 2 = dA n 2 , dA 3 = dA n 3

The following relationship exists between the stress vectors t=t 1n 1 +t 2n 2 +t 3n 3 (3.41)

Stress vectors t1, t2, and t3 can be expressed through three stress components, defined as forces per unit area Each stress component requires two indices: the first indicates the surface on which the stress acts, corresponding to the normal coordinate axis, while the second specifies the direction of the stress For instance, the stress components acting on an area dA1 perpendicular to the x-axis are represented as t1 = τ11 e1 + τ12 e2 + τ13 e3, t2 = τ21 e1 + τ22 e2 + τ23 e3, and t3 = τ31 e1 + τ32 e2 + τ33 e3 These components collectively form a tensor consisting of nine components, denoted as τij.

Normal stresses, denoted as τ 11, τ 22, and τ 33 (or τ ij where i=j), act perpendicular to a surface, while tangential or shear stresses (τ ij where i≠j) act parallel to it According to the principle of equilibrium, shear stresses are equal across corresponding planes, leading to a symmetrical stress tensor where τ ij = τ ji By substituting stress vectors into the stress equation, the resultant stress vector t can be expressed as a combination of normal and shear stresses acting on the surface, represented by t = (τ 11 e 1 + τ 12 e 2 + τ 13 e 3)n 1 + (τ 21 e 1 + τ 22 e 2 + τ 23 e 3)n 2 + (τ 31 e 1 + τ 32 e 2 + τ 33 e 3)n 3.

The stress vector \( t \) acting on the oblique surface of the tetrahedron can be decomposed into three components: \( t_1 \), \( t_2 \), and \( t_3 \), aligned with the coordinate axes Thus, the stress vector can be expressed as \( t = t_1 e_1 + t_2 e_2 + t_3 e_3 \).

In comparing the current relationship to the previous one, the components of the stress vector can be expressed as follows: t₁ = τ₁₁n₁ + τ₂₁n₂ + τ₃₁n₃, t₂ = τ₁₂n₁ + τ₂₂n₂ + τ₃₂n₃, and t₃ = τ₁₃n₁ + τ₂₃n₂ + τ₃₃n₃, which can be generalized as tⱼ = τⱼᵢnⁱ (where i, j = 1, 2, 3) Theoretically, the stress tensor τⱼᵢ can be represented as the sum of two tensors.

3δ ji τ kk (3.44) with theunit tensorδ ji , also called theKronecker-δ, δ ji ⎧ ⎨

From this deﬁnition it immediately follows thatδ ji =δ 11 +δ 22 +δ 33 = 3.

We call ˆτ ji thedeviatorof the tensorτ ji It is thereby characterised such that the diagonal elements, the so-called trace of the tensor, disappear, which then gives

In general a deviator is a tensor with a zero trace.

The diagonal elements represent normal stresses that neutralize each other, while the deviator plays a crucial role in the shearing of a fluid element Additionally, the term 1/3δ ij τ kk reflects hydrostatic stresses, which are normal stresses that are equal in all directions The average of the three normal stresses, τ kk, is referred to as the average pressure.

In fluid mechanics, the negative value of the shear stress tensor component τ kk arises because liquids are unable to withstand tensile stresses Consequently, while τ kk is typically negative, the average pressure remains positive This relationship can be expressed as τ ji = ˆτ ji − δ ji p ¯, highlighting the distinction between shear stress and pressure in fluid dynamics.

The energy balance

According to the ﬁrst law of thermodynamics, the internal energyU of a closed system changes due to the addition of heatQ 12 and workW 12 into the system

U 2 −U 1 =Q 12 +W 12 or in diﬀerential form dU= dQ+ dW (3.60) or dU dt = dQ dt + dW dt = ˙Q+P (3.61)

In this analysis, we will focus on a fluid element of a specified mass within a closed system, allowing us to disregard mass transport across the system boundary and eliminate the factor of diffusion.

With the help of the transport theorem (3.17), by putting inz=u, we obtain the following temporal change in the internal energy of a ﬂowing ﬂuid dU dt = d dt

The change in internal energy in a flowing fluid over time is determined by the internal energy contained within the fluid volume V(t) at time t, as well as the internal energy that exits through the surface A(t) of the fluid volume.

Heat is transferred, by deﬁnition, between a system and its surroundings, and therefore the heat fed into the system via its surface is

A(t) ˙ qdA , (3.63) in which ˙q is the heat ﬂux It is a scalar (3.63), with the normal vectorn, which indicates the orientation of the surface elements in space, can also be written as

The equation \( \dot{q}_{n_i} = -\dot{q}_i \) (3.64) represents the components of the heat flux vector \( \dot{q} \), where \( \dot{q}_1 \), \( \dot{q}_2 \), and \( \dot{q}_3 \) correspond to the heat flux over areas \( dA_1 \), \( dA_2 \), and \( dA_3 \), respectively The projections of the surface area \( dA \) onto the coordinate axes are detailed in section 3.2.3.1, highlighting that \( dA_i \) are the components of the surface vector \( n dA \) The negative sign in the heat flux equation arises from the convention that heat flux entering a system is considered positive, while the normal vector \( n_i \) of a closed area points outward.

The work performed to alter a system's internal energy is exchanged with its surroundings, flowing across the system's surface Body forces cause the displacement of each element as a whole, influencing both kinetic and potential energy changes, while not affecting internal energy unless mass exchange occurs with neighboring elements In such scenarios, individual particles move at varying speeds, and the body forces acting on them contribute to the overall power This contribution is particularly significant in multicomponent mixtures and must be considered in such analyses.

Surface forces can influence internal energy by introducing components that alter it, while others can move the fluid element without affecting its internal energy The portion of power contributing to changes in internal energy, supplied through the system's surface, can also be expressed mathematically.

Power density, denoted as ω˙ (in W/m²), is defined by a specific equation to avoid confusion with velocity, which is represented by the letter w This distinction is crucial for clarity, especially when discussing related concepts such as heat flux The equation can also be rearranged for further analysis.

The quantities ˙ωn i =−ω˙ i represent the components of the power density vector ˙ω, where ˙ω 1 denotes the power density over area dA 1, ˙ω 2 over area dA 2, and ˙ω 3 over area dA 3 The negative sign indicates that the power supplied to the system is considered positive.

The ﬁrst law (3.61), in conjunction with the expressions for internal energy (3.62), the heat ﬂow (3.65) and the power (3.68), can also be written as

Converting the surface into the volume integral using Gauss’ law and then transferring to a small volume elementV(t)→0 yields

∂x i The left hand side can also be written as

By taking into account the continuity equation (3.20), the equation above is sim- pliﬁed to

∂x i =du dt Which then gives the energy equation du dt =−∂q˙ i

In order to calculate the power density ˙ω i , we will now consider the total power

The power generated by surface forces is distinct from the drag component, which displaces the fluid element Consequently, the remaining power is responsible for altering the internal energy of the system.

Using the stress tensor t i = τ ji n j the total power from the surface forces is obtained as

On the other hand, the surface forces acting on the volume element

∂x j dV cause a movement of the system The force acting on a volume element dV dF i = ∂τ ji

∂x j dV displaces it during a short time dtalong a path dx i , such that the drag is dF i dx i dt =∂τ ji

∂x j dV w i The total drag is therefore

∂x j w i dV , and the part of the surface forces contributing to the change in internal energy will be

∂x j dV follows As (3.68) can be rearranged using Gauss’ law into

The energy balance (3.69) is given by du dt =−∂q˙ i

As we have taken the ﬂuid element to be a closed system, this equation is not valid for multicomponent mixtures and diﬀusion.

Diffusion occurs in multicomponent mixtures that are not in equilibrium, leading to the crossing of individual material flows across system boundaries As a result, it is incorrect to assume a closed system Consequently, the first law of thermodynamics is expressed as dU = dQ + dW.

For substance A, the partial specific enthalpy is defined as h A = (∂H/∂M A) T, p, M K=A, while e M ˙ A represents the mass flow rate of component A entering the system The total is calculated by summing all material flows, with the heat flow ˙ Q and power P determined using the same methodology as previously established.

Substance A ﬂowing at a velocity w Ai across the surface element is an energy carrier It increases the internal energy of a ﬂuid element moving with the centre of mass velocity w i by

In energy calculations, it's essential to use a minus sign, as energy entering a system is considered positive, while the surface vector is assigned a negative value This distinction is crucial when accounting for the energy from all feed streams interacting with the total surface.

When this is applied to a ﬂuid element, the energy equation (3.72) for a multicomponent mixture can be written as du dt = − ∂ q ˙ i

The power density ˙ ω i is defined as the power derived from surface forces, subtracting the drag forces that shift the fluid element without altering its internal energy Unlike the equation for pure substances, this equation includes an additional term due to the varying velocities w A i of different particle types Consequently, the body forces k A i acting on substance A within a volume element dV contribute to the overall power density.

A (w Ai − w i )k Ai dV = j Ai ∗ k Ai dV For all the substances, the contribution on a system of volume V (t) would be

The total power P for the change in the internal energy is made up of the power of the surface forces

∂x i dV and the body forces

V (t) j Ki ∗ k Ki dV minus the drag force

Then for the power density, it follows, due to

It is diﬀerent to the power density (3.70) for pure substances due to the presence of the term

K j ∗ Ki k Ki for the power exerted by the body forces on the individual substances The energy equation (3.73) is transformed to the following for multicomponent mixtures du dt = − ∂ q ˙ i

K h K j Ki ∗ , (3.76) which is transformed into the energy balance (3.71) for pure substances when a vanishing diﬀu- sion ﬂow j Ki ∗ = 0 exists.

The energy equation highlights that the internal energy of a fluid element is influenced by heat influx and work performed In multicomponent mixtures, an additional term accounts for energy introduced through material input The work involved includes both reversible and dissipated components, which we aim to analyze separately using an entropy balance, focusing solely on pure substances This analysis is grounded in Gibbs’ fundamental equation.

T ds dt = du dt + p dv dt = du dt − p 2 d dt (3.77)

Inserting the energy equation (3.71) and the continuity equation (3.21) yields

∂x i and according to (3.53) τ ji = ˆ τ ji − δ ji p , if we presume that the bulk viscosity disappears With that we can write the entropy balance (3.77) as ds dt = − ∂( ˙ q i /T )

∂x i Because δ ji ∂w i /∂x j = ∂w j /∂x j = ∂w i /∂x i this equation simpliﬁes to ds dt = − ∂( ˙ q i /T )

The first term in equation (3.78) signifies the entropy influx associated with heat, referred to as entropy flux The subsequent two terms illustrate the generation of entropy, with the second term arising from temperature differences in thermal conduction, while the third term is linked to mechanical energy We define φ as ˆ τ ji ∂w i.

∂x j (3.79) the viscous dissipation This is the mechanically dissipated energy (SI units W/m 3 ) per unit volume Taking (3.53) into consideration τ ji = ˆ τ ji − δ ji p the ﬁrst law (3.71) can also be written du dt = − ∂ q ˙ i

Vanishing bulk viscosity is also presupposed here Introducing the viscous dissipation into the energy equation (3.75) for multicomponent mixtures, gives du dt = − ∂ q ˙ i

Likewise (3.81) presupposes vanishing bulk viscosity The derivation of the entropy balance for mixtures can be found in Appendix A 5.

Example 3.3: Calculate the viscous dissipation for a Newtonian ﬂuid How large is the viscous dissipation for the special case of one dimensional ﬂow w 1 = w 1 (x 2 )?

For a Newtonian ﬂuid, according to (3.55) we have ˆ τ ji = η

In the case of w 1 = w 1 (x 2 ) the expression is reduced to φ = η ∂w 1

3.2.4.2 Constitutive equations for the solution of the energy equation

To solve the energy equations (3.71) and (3.75), it is essential to incorporate certain constitutive equations, specifically focusing on equation (3.71) for pure substances This requires the introduction of the caloric equation of state, represented as u = u(ϑ, v) By differentiating this equation, we derive the expression for du/∂u.

In thermodynamics, as shown, for example, in [3.2],

−p Furthermore dv=−d/ 2 This leads to du dt = c v dϑ dt −

With the continuity equation (3.21) we obtain for this du dt = c v dϑ dt +

The expression in the square brackets disappears for ideal gases For incompressible ﬂuids,= const, d/dt= 0 and∂w i /∂x i = 0 The expression is simpliﬁed in both cases to du dt = c v dϑ dt

In incompressible fluids it is not necessary to differentiate between c p and c v , c p =c v =c While in an isotropic body, the heat flux is given by Fourier’s law ˙ q i =−λ ∂ϑ

Summary

The most important balance equations derived in this chapter shall be summarised here In this summary we will use the abbreviation d dt = ∂

∂x i The following balance equations are valid forpure substances: d dt =−∂w i

The mass balance or continuity equation is represented by Equation (3.90), while Equation (3.91) denotes the momentum balance, also known as Cauchy’s equation of motion Additionally, Equation (3.92) illustrates the energy balance Since a momentum balance applies to each of the three coordinate directions (j=1,2,3), there are a total of five balance equations Furthermore, the enthalpy form presented in Equation (3.83) is equivalent to the energy balance outlined in Equation (3.92).

In multicomponent mixtures N −1 mass balances for the components have to be added to the continuity equation, see (3.25), whilst the energy balance is replaced by (3.81) or (3.87).

The system of equations still has to be supplemented by the so-called constitutive equations or material laws which describe the behaviour of the materials being investigated.

If the frequently used case of anincompressible Newtonian ﬂuid, = const, d/dt= 0, of constant viscosity is presumed, the continuity equation, momentum and energy balances are transformed into

The Navier-Stokes equation for incompressible fluids is represented by Equation (3.94) For multicomponent mixtures containing N components, it is essential to include the continuity equations for the total mass as stated in Equation (3.93), along with continuity equations for N−1 components, expressed as dA/dt = −∂jAi*.

The momentum balance (3.94) remains unchanged, whilst for the energy equation we obtain (3.89): cdϑ dt =λ ∂ 2 ϑ

Couette flow is a fluid dynamics phenomenon where a fluid is situated between two parallel, infinitely large plates In this setup, the top plate moves at a constant velocity \( w_L \), while the bottom plate remains stationary.

A good example of this type of flow is oil in a friction bearing Here we will consider an incompressible steady-state flow, whose velocity profile is given by w 1 (x 2 ) Furthermore

In a system where the upper plate maintains a temperature of ϑ L and the lower plate is at a lower temperature of ϑ 0 (where ϑ 0 < ϑ L), the temperature profile ϑ(x 2) must be calculated while considering the significant effects of viscous dissipation This analysis reveals how heat distribution varies across the medium, highlighting the importance of viscous forces in influencing the thermal gradient Understanding this temperature profile is crucial for applications where heat transfer and fluid dynamics intersect, particularly in engineering and material science contexts.

Velocities are only present along the x 1 -axis This means that only the momentum equation (3.94), for j = 1 has to be considered In this equation the left hand side disappears, as

∂/∂t = 0, ∂w/∂x 1 = 0, w 2 = w 3 = 0 Futhermore the mass force is k 1 = 0, and also

∂p/∂x 1 = 0, as the ﬂow is caused by the movement of the upper plate and not because of a pressure diﬀerence The momentum equation is reduced to

Likewise, in the energy equation (3.95), all the terms on the left hand side disappear because ∂/∂t = 0, ∂ϑ/∂x 1 = 0, w 2 = 0 This then becomes λ ∂ 2 ϑ

Integration of the energy equation yields a parabolic temperature proﬁle in x 2 ϑ(x 2 ) = − η

2 x 2 2 + a 1 x 2 + a 0 The constants a 0 and a 1 follow out of the boundary conditions ϑ(x 2 = 0) = ϑ 0 and ϑ(x 2 = L) = ϑ L With this the temperature proﬁle is found to be ϑ = ϑ 0 + η w L 2

The temperature profile depicted in Fig 3.11 illustrates the pattern over x² When the upper plate remains stationary (wₗ = 0), the resulting temperature distribution mirrors that observed in pure heat conduction between two flat plates.

For suﬃciently large values of w L the temperature proﬁle has a maximum, the position of which is calculated from dϑ/dx 2 = 0 to be

It has to be that (x 2 ) max ≤ L This criterium is met, if w L ≥

At velocities below w L ∗, energy dissipation is minimal, resulting in the fluid's maximum temperature matching that of the upper plate However, when velocities exceed w L ∗, significant energy dissipation occurs, causing the fluid in a specific region between the plates to reach temperatures higher than that of the upper plate.

Inﬂuence of the Reynolds number on the ﬂow

A general solution to the Navier-Stokes equations remains elusive due to the inherent complexities associated with their non-linear nature, particularly stemming from the inertia terms expressed as dw j/dt = ∂w j.

The Navier-Stokes equation, particularly in the context of incompressible flow, presents challenges in finding solutions, which are typically known only for specific cases These solutions arise when certain terms in the equation can be neglected, allowing for simplification To evaluate the significance of these terms, the equation is rearranged to incorporate dimensionless quantities, ensuring that the term magnitudes remain consistent regardless of the measurement system used This approach highlights the fundamental principle that the analysis and resolution of physical problems should not depend on the measurement system Utilizing standardized measures that are problem-oriented is akin to employing dimensionless groups, facilitating a clearer understanding of the equation's behavior in fluid dynamics.

In fluid dynamics, all velocities are normalized by a reference velocity \( w_\alpha \), such as the upstream velocity of a body in crossflow, while lengths are normalized by a reference length \( L \), exemplified by the length over which fluid flows across a body This normalization allows the formation of dimensionless quantities: \( w^+_i = \frac{w_i}{w_\alpha} \), \( p^+ = \frac{p}{w^2_\alpha} \), \( x^+_i = \frac{x_i}{L} \), and \( t^+ = \frac{t w_\alpha}{L} \) These dimensionless quantities can then be introduced into the Navier-Stokes equation, with viscosity \( \eta \) being replaced by kinematic viscosity \( \nu \), leading to a more simplified analysis of fluid behavior.

∂x + i 2 (3.99) with the already well known Reynolds number

In addition we have the dimensionless form of the continuity equation for incompressible ﬂow

The equation ∂x + i = 0, derived from (3.93) through the application of dimensionless groups and boundary conditions, fully characterizes the flow The resulting solution provides the velocity field w j + (t + , x + i) and the pressure field p + (t + , x + i).

The solution is influenced by the Reynolds number, which represents the ratio of inertia to friction forces The inertia force is proportional to w²α/L, while the friction force is related to ηwα/L² By comparing these two forces, we find that their ratio is wαL/η, which simplifies to the Reynolds number (Re).

Inertia forces increase with the square of velocity, significantly outpacing the linear increase of friction forces At high Reynolds numbers, the persistent velocity disturbances cannot be mitigated by the relatively small friction forces, leading to a shift in flow patterns at the critical Reynolds number Below this threshold, fluid particles follow distinct streamlines with disturbances quickly dissipating, resulting in laminar flow Conversely, above the critical Reynolds number, disturbances are intensified, resulting in turbulent flow, characterized by three-dimensional, unsteady motion and irregular vortex patterns In turbulent flow, velocity at a fixed point fluctuates around a mean value, with momentary values of velocity, pressure, temperature, and concentration appearing random.

The Navier-Stokes equations provide solutions for laminar flows at high Reynolds numbers, but these solutions are only valid if they remain stable against minor disturbances When the Reynolds number exceeds a critical threshold, small disturbances do not dissipate; instead, they amplify, leading to a transition from laminar to turbulent flow.

Osborne Reynolds (1842–1912) first observed flow phenomena by adding dye to the flow in a glass tube In laminar flow, a narrow thread of color forms along the tube's axis, remaining largely unchanged due to minimal molecular diffusion However, as the velocity increases and surpasses the critical Reynolds number, this thread mixes rapidly into the flow, demonstrating the transition from laminar to turbulent flow in tube dynamics.

In fluid dynamics, the critical Reynolds number (Re crit) is defined as Re crit = w m d/ν ≤ 2300, where w m is the average velocity over the cross-section and d is the tube diameter Even in highly disturbed inlet flows, the flow remains laminar as long as it stays below this critical threshold In ideal conditions, critical Reynolds numbers can reach up to 40,000; however, in most practical applications, achieving completely disturbance-free flow is unrealistic Consequently, flow is considered laminar only when Re is below 2300 As Reynolds numbers approach 2600, the flow exhibits an intermittent nature, oscillating between laminar and turbulent states, while flow becomes fully turbulent at Reynolds numbers exceeding 2600.

In transverse flow along a flat plate, the transition from laminar to turbulent flow occurs at Reynolds numbers between 3 × 10^5 and 5 × 10^5, where w α represents the initial flow velocity and L is the plate length Turbulent flows exhibit significantly enhanced heat and mass transfer compared to laminar flows.

In general, at the same time there is also an increase in the pressure drop.

Simpliﬁcations to the Navier-Stokes equations

Creeping ﬂows

Creeping flow occurs in the limiting case when Re → 0, characterized by very low velocities, low density (as seen in dilute gases), and high viscosity (typical of highly viscous fluids) This phenomenon can also be observed in scenarios involving small body dimensions, such as flow around dust particles or fog droplets In these conditions, viscous forces dominate over inertial forces, making the term (1/Re)(∂²w + j/∂x + 2i) negligible However, the pressure gradient ∂p + /∂x + j remains significant and cannot be overlooked.

The relationship ∆p w α 2 = ∆p/L w 2 α /L illustrates that inertia forces are minimal when compared to pressure terms in the denominator The pressure term can be disregarded only if the pressure drop is considerably less than the inertia forces To determine the applicability of this condition, one must first solve the momentum and continuity equations By omitting the inertia forces in the analysis, the non-linear terms are eliminated, resulting in a linear differential equation upon converting back to dimensioned quantities.

In hydrodynamic lubrication theory, the flow is considered incompressible, leading to the continuity equation ∂w j /∂x j = 0 Consequently, the pressure is governed by the potential equation These principles are fundamental to understanding oil flow in bearings.

Frictionless ﬂows

Assuming a frictionless flow with η = 0 leads to 1/Re = 0, which eliminates the friction term from equation (3.99) After transforming to dimensional quantities, we derive the corresponding results.

This isEuler’s equation It contains, in the special case of one-dimensional steady- state ﬂow, the relationship w 1 dw 1 dx 1 =−dp dx 1 which when integrated w 2 1

Bernoulli’s equation, expressed as 2 + p = const, neglects the influence of gravity In contrast, the Navier-Stokes equation is of second order, while Euler’s equation consists solely of first order terms Consequently, Euler’s equation requires one less boundary condition to be satisfied upon integration, given its lower order compared to the Navier-Stokes equation.

In real fluid flows, the velocity at the wall is zero due to the no-slip condition; however, in this scenario, a finite velocity at the wall is observed because the assumption of no friction is made.

Boundary layer ﬂows

In the Navier-Stokes equation, the friction term becomes significant for large Reynolds numbers (Re → ∞) when accurately modeling flow near the wall and adhering to the no-slip condition This critical region, where friction forces cannot be overlooked in comparison to inertia forces, is typically confined to a very thin layer adjacent to the wall.

Fig 3.12: Velocities in a boundary layer the velocity increase and therefore also the shear stress are very large, as is clearly illustrated in Fig 3.12 The shear stress τ 21 =η∂w 1

The velocity boundary layer, located near the wall, experiences significant inertia and friction forces, while these forces become negligible at greater distances from the wall This region is crucial for understanding momentum, heat, and mass transfer in fluid flows, which is the focus of boundary layer theory The boundary layer thickness, denoted as δ(x₁), is estimated by examining the balance of inertia and friction forces along the x₁-axis The characteristic inertia term is represented by w²α/x₁, where the characteristic length corresponds to the distance from the leading edge In the x₁-direction, specific friction terms are relevant for the flow with velocity w₁(x₁, x₂) and can be expressed as η∂²w₁.

Whilst the ﬁrst term on the right hand side is of magnitudeηw α /x 2 1 , the order of magnitude of the second term isηw α /δ 2

The boundary layer is significantly thinner relative to the plate length, making the friction term within the boundary layer crucial at a sufficient distance from the plate's front, denoted as δ x 1.

Equating the inertia and friction forces leads to the relationship \( w \alpha 2 x_1 \approx \eta w \alpha \delta^2 \), resulting in \( \delta \approx \frac{x_1}{Re_{x_1}} \) with \( Re_{x_1} = \frac{w \alpha x_1}{\nu} \) The boundary layer thickness increases with \( \sqrt{x_1} \) and is thinner at higher Reynolds numbers In the confined boundary layer, friction forces are significant, whereas outside this layer, they are negligible, allowing Euler’s equation to apply in the external region.

The boundary layer equations

The velocity boundary layer

At high Reynolds numbers, the flow can be divided into two distinct regions: the outer frictionless flow governed by Euler’s equation and the boundary layer where friction forces become significant compared to inertia forces While Euler’s equation allows us to determine the velocity profile in the outer region based on a specific pressure field, understanding flow resistance requires knowledge of the velocity profile within the boundary layer Here, the velocity transitions from zero at the wall to the asymptotic velocity of the outer region, and similarly, temperature and concentration values shift from wall conditions to those of the outer flow.

In this analysis, we will assume that the velocities, temperatures, and concentrations in the outer region are known while focusing on a steady-state, two-dimensional flow The influence of body forces is considered negligible When flow occurs along a curved wall, it can still be treated as two-dimensional, provided that the wall's radius of curvature is significantly larger than the boundary layer's thickness In this scenario, the curvature has minimal impact on the thin boundary layer, allowing it to develop similarly to flow on a flat wall, while the wall's curvature primarily affects the outer flow and its pressure distribution.

In this article, we will discuss boundary layer coordinates, where the coordinate x₁ = x aligns with the surface of the body, and x₂ = y is oriented perpendicular to it We will assume an initial velocity profile denoted as wα(y), with its integral mean value represented as wm.

In fluid dynamics, density changes in flowing liquids are typically negligible, except in extreme cases like the rapid opening and closing of valves in piping systems Consequently, for most flowing liquids, the assumption d/dt→0 is valid, allowing us to utilize the continuity and momentum equations for incompressible flow Similarly, in gases, density changes remain minor as long as the Mach number, defined as Ma = w m /w S, is less than 1, particularly in scenarios involving reversible, adiabatic flow.

Fig 3.13: Boundary layer in transverse ﬂow along a body equation of statep=p(, s) that dp ∂p

:=w 2 S wherew S is the velocity of sound So, for a reversible, adiabatic ﬂow, it holds that dp=w S 2 d

As no technical work is executed, we have dw t = 0 =vdp+ d w 2 2

, when we neglect changes in the potential energy With that it follows that vdp=vw 2 S d=w S 2 d/ and d

=−Ma 2 dw w dw/w≤1 is valid for the relative velocity change The relative density variation d/ is small for Ma 0, here |d ¯w x /dy|(d ¯w x /dy) is written as (dw x /dy) 2 With the abbreviations ¯ w + x = w¯ x w τ and y + = w τ y ν we obtain

The following solutions are yielded by integration a) In the laminar sublayer,y →0 and therefore y + →0, the second term is negligible, giving ¯ w x + =y + (3.158) or taking into account the deﬁnitions for ¯w + x , y + and w τ τ 0 /: τ 0 =ηw¯ x y

In the laminar sublayer, the velocity profile is represented by a straight line Conversely, in the fully turbulent region, far from the wall where y approaches infinity, the influence of the second term surpasses that of the first.

2 or d ¯w + x = 1 κ dy + y + Integration yields a logarithmic velocity proﬁle ¯ w + x = w¯ x w τ = 1 κlny + +c (3.159)

The constants κ and chave to be found by experiment Values for them have been found to beκ≈0.4 andc≈5.

The laminar sublayer is constantly evolving into the fully turbulent region, with a transition area known as the buffer layer separating the two This allows for the wall law of velocity to be divided into three distinct regions, with boundaries determined through experimentation The laminar sublayer occupies a specific portion of this transition.

0< y + 0), the conditions lead to a positive second derivative of the velocity profile at the wall This scenario aligns with Bernoulli's equation, which shows that as the core flow decelerates, it is only in this decelerating core flow that boundary layer detachment can occur.

In turbulent flow, momentum is continuously transferred to the wall-adjacent layer due to interactions between layers with varying velocities Unlike laminar flow, the kinetic energy of fluid elements near the wall diminishes at a slower rate, allowing turbulent boundary layers to remain attached longer This turbulence enhances heat and mass transfer near the wall, as the fluid flows over a larger surface area without detachment Additionally, the prolonged attachment of fluid flow reduces pressure resistance, resulting in a more efficient flow path.

The flow pattern around a cylinder in crossflow is significantly influenced by the Reynolds number At low Reynolds numbers (Re < 5), the flow remains attached to the cylinder As the Reynolds number increases to between 5 and 40, the flow begins to detach, leading to the formation of vortices When the Reynolds number reaches between 40 and 150, these vortices are periodically shed, resulting in a vortex pattern while the flow remains laminar However, as the Reynolds number continues to rise, the flow eventually transitions to a turbulent state.

In fluid dynamics, the Reynolds number (Re) plays a crucial role in determining flow characteristics For Reynolds numbers between 150 and 300, the flow is fully turbulent, while in the range of 300 to 3 × 10^5, the flow remains turbulent downstream of the cylinder Beyond Re = 3 × 10^5, the boundary layer also exhibits turbulence The wake generated is concentrated in a narrow region, characterized by fully turbulent flow without large eddies Notably, a narrow, fully turbulent vortex pattern begins to reform at approximately Re ≈ 3.5 × 10^6.

The flow significantly impacts heat and mass transfer, as illustrated by the local Nusselt number plotted against the angular coordinate from the forward stagnation point Initially, the Nusselt number decreases as the boundary layer develops, reaching a minimum around 80° At this angle, flow detachment occurs, leading to increased heat transfer along the perimeter due to effective fluid mixing from vortices For high Reynolds numbers exceeding 10^5, two minima in the local Nusselt number are observed The sharp increase between 80° and 100° signifies the transition from a laminar to a turbulent boundary layer, which initially inhibits heat transfer until about 140°, where improved fluid mixing in the wake enhances heat transfer once again.

In practice, the mean heat transfer coeﬃcient is of greatest interest It can be described by empirical correlations of the form

The mean Nusselt and Reynolds numbers are determined based on the tube diameter, with all material properties calculated at the free stream temperature (ϑ ∞), except for the Prandtl number (P r 0), which is evaluated at the wall temperature (ϑ 0) The coefficients c, m, n, and p are sourced from a study by ˇZukauskas and are detailed in Table 3.1.

Table 3.1: Constants and exponents in equation (3.210)

2 ã 10 5 to 10 7 0.023 0.8 0.4Heating the ﬂuid: p=0.25Cooling the ﬂuid: p=0.20

A single, empirical equation for all Reynolds numbers 1≤Re≤10 5 and for Prandtl numbers 0.7< P r 0).

A simple relationship exists between the wall temperatureT 0 and the heat transfer parameter, because it follows from (3.368) that

With the eigentemperature according to (3.355) we can write for this

Inserting (3.369) into (3.364) yields an alternative form of (3.364), which now contains the heat transfer parameter:

Fig 3.55: Temperature proﬁle in a com- pressible ﬂow of ideal gas Curve parameter ϑ + = (T e − T 0 )/(T e − T δ ) In heating T 0 > T e and ϑ + < 0, in cooling T 0 < T e and ϑ + > 0

The temperature profile for ideal gases at P r = 1 can be characterized by the velocity profile and the parameter ϑ + By applying Blasius’ solution as an approximation for the velocity profile, we can derive the temperature profile shown in Fig 3.55 for Ma δ = 2, plotted against the distance from the wall η + However, it is important to note that this representation provides only an approximate temperature profile due to the simplifications involved.

Calculation of heat transfer

The heat flux from a surface at a specific temperature, denoted as ϑ₀ = ϑₑ, is determined by solving the momentum and energy equations while considering dissipation This process involves the use of boundary layer coordinates, as outlined in section 3.1.1, and requires converting partial differential equations into ordinary differential equations, which can be solved numerically Eckert and Drake provided a solution for the incompressible flow of fluid along a plate with a constant wall temperature, indicating that the local heat flux from the plate can be approximated by the equation ˙q(x) = 0.332Re¹/²ₓPr¹/³λₓ(T₀ - Tₑ) for the range 0.6 ≤ Pr ≤ 10 This leads to the definition of the heat transfer coefficient α as ˙q = α(T₀ - Tₑ).

The local Nusselt number formed with this heat transfer coeﬃcient

N u x = α x λ = 0.332Re 1/2 x P r 1/3 (3.373) agrees with that found for incompressible ﬂow, (3.196), for 0.6< P r λ 2 With τ i = exp[ − κ(λ)s] from this we get, for s = 2.0 mm the spectral absorption coeﬃcients κ =

∞ for λ < λ 1 0.328 cm −1 for λ 1 ≤ λ ≤ λ 2 17.1 cm −1 for λ > λ 2 This then gives the pure transmissivity of the thicker glass sheet s = 4.0 mm as τ i =

0.001 for λ > λ 2 and ﬁnally the desired spectral transmissivity as τ λ =

The total transmissivity is then τ = 0.805 [F(0, λ 2 T S ) − F (0, λ 1 T S )] + 0.001 [1 − F(0, λ 2 T S )] = 0.652

A glass sheet with double the thickness absorbs more solar radiation, permitting 65% of the incident light to pass through, compared to 70% with a thinner sheet The spectral reflectivity of the thicker sheet can be expressed as r λ = (5.100).

It is increased between λ 1 and λ 2 by the multiple reﬂection at the edges, so that in total6.7% instead of 4.26% of the solar energy is reﬂected.

Solar radiation

Extraterrestrial solar radiation

The sun is an almost spherical radiation source with a diameter of 1.392ã10 6 km.

The Earth orbits the Sun in an elliptical path, with the Sun located at one of the foci The intensity of solar radiation received by the Earth is inversely proportional to the square of the distance between the two The average distance from the Earth to the Sun is approximately 149.6 million kilometers, known as one astronomical unit (AU) The closest approach occurs on January 3rd, at a distance of 0.983 AU, while the farthest point in the orbit varies throughout the year.

7 The exact value is 1AU = 149.597 870 ã 10 6 km For the strict deﬁnition of the AU, see [5.32].

Extraterrestrial solar radiation on a surface is influenced by the polar angle β S, with Earth at a distance of 1.017 AU on July 4th The eccentricity factor, used for radiation calculations, is represented by (r 0 /r)² and can be approximated using the formula f ex = (r 0 /r)² = 1 + 0.033 cos (2πd n /365), where d n denotes the day of the year, starting from January 1st (d n = 1) to December 31st (d n = 365) For a more precise calculation, refer to the relationship provided by J.W Spencer, as noted in M Iqbal's work.

The vast distance between the Sun and Earth results in solar radiation arriving as a nearly parallel bundle of rays This unaltered radiation, unaffected by atmospheric scattering and absorption, is known as extraterrestrial radiation When this radiation strikes a surface just outside Earth's atmosphere perpendicularly at a distance of 1 astronomical unit (AU) from the Sun's center, its irradiance is referred to as the solar constant (E₀) Recent measurements by C Fröhlich and R.W Brusa have provided an updated value for this constant.

The World Meteorological Organisation (WMO) recognized the solar constant E₀ as (1367±1.6) W/m² in 1981, establishing it as the most accurate measurement Using this value, the surface temperature of the sun, Tₛ, was determined to be 5777 K, assuming it behaves as a black body.

The irradiance of extraterrestrial radiation, that falls perpendicularly onto a surface that is at the same distanceras the earth is from the sun, is given by

E n sol =E 0 (r 0 /r) 2 =E 0 f ex (5.104) withf ex from (5.103) If the direction of the sun’s rays forms the polar angle β S with the surface normal, Fig 5.39, then the irradiance will be

The sun's polar angle (β S) is influenced by the position and orientation of the irradiated surface, a relationship described by trigonometric equations found in solar radiation literature The spectral irradiance (E sol λ,n) of extraterrestrial solar radiation, which impacts a surface at a distance of 1 AU from the sun, has been established through various experiments conducted with stratospheric aircraft, with evaluations performed by C Fröhlich and C Werli at the World Radiation Centre in Davos.

Fig 5.40: Spectral irradiance E λ,n sol of extraterrestrial solar radiation falling perpendicularly on an area at a distance r 0 = 1 AU from the sun

Switzerland produced the spectrum illustrated in Fig 5.40, with numerical values sourced from M Iqbal [5.34] The peak of E λ,n sol occurs in the visible light range at approximately λ ≈ 0.45 µm, while 99% of the irradiance is concentrated within the wavelength band of λ ≤ 3.8 µm Additionally, Fig 5.40 presents the spectral irradiance data.

E λ,s of the radiation emitted by a “black” sun atT S = 5777 K The areas under the two curves (up toλ→ ∞) are equal — they each yield the solar constantE 0

—, but the spectrum of the extraterrestrial solar radiation deviates signiﬁcantly at some points, in particular atλ 5 \), the atmosphere becomes nearly opaque to short wavelength light This phenomenon accounts for the reddish-yellow hue of the sun's disc during sunrise and sunset.

Dust and small suspended water droplets form aerosols They scatter and absorb solar radiant energy, whereby the scattered proportion predominates The scattering and absorption

John William Strutt, Third Baron of Rayleigh (1842–1919), established a physical laboratory at Terling Place in Essex, England, where he published 430 scientific works covering various aspects of classical physics, particularly acoustics His notable contribution to the field includes "The Theory of Sound" (1877/78) Alongside W Ramsey, he discovered the element Argon between 1892 and 1895, earning the Nobel Prize in Physics in 1904, the same year Ramsey received the Nobel Prize in Chemistry.

The spectral transmissivity τ λ,R of the atmosphere is influenced by Rayleigh scattering, as illustrated in Fig 5.45, and varies with different relative optical masses m r,L Modeling the effects of aerosols on this transmissivity presents challenges and uncertainties A commonly utilized turbidity formula, proposed by A Ångström, is expressed as τ λ,A = exp.

Values for the exponent α ∗ range from 0.8 to 1.8, while β ∗ varies from 0 (pure atmosphere) to 0.3 (very murky atmosphere), reflecting different atmospheric conditions The relative optical aerosol mass m r,A is typically unknown due to significant variations in the size, distribution, and composition of aerosol particles, leading to the common use of m r,L from equation (5.114) as a substitute for m r,A.

Absorption occurs within specific wavelength intervals known as absorption bands, unlike scattering The primary components of the atmosphere, nitrogen (N2) and oxygen (O2), dissociate into atomic nitrogen and oxygen at altitudes exceeding 100 km These gases exhibit strong absorption, particularly at short wavelengths, with nitrogen and oxygen absorbing all radiation below 0.085 micrometers.

Direct solar radiation on the ground

The spectral transmissivity τ λ indicates the ratio of the spectral irradiance E λ,n received on the ground, which is oriented perpendicular to the radiation, to the spectral irradiance E λ,n sol from extraterrestrial solar radiation.

3τ λ,W τ λ,G E λ,n sol , (5.122) and for an area whose normal forms an angleβ S with the suns rays, it follows that

The extraterrestrial spectrum E λ,n sol and its associated pattern E λ,n are illustrated in Fig 5.48, where the upper edge of the curve indicates irradiance reduced solely by Rayleigh scattering The black-marked deteriorations highlight absorption effects from gases such as O3, O2, H2O, and CO2 Additional diagrams depicting variations in influencing factors like water vapor, ozone content, aerosol turbidity, and different optical masses can be found in M Iqbal [5.34].

In practical applications, it is often sufficient to measure the irradiance (E) at a specific ground area The most accurate way to obtain E is by integrating the spectral irradiance (Eλ) across all wavelengths, typically ranging from λ = 0.3 µm to λ = 4.0 µm To simplify this complex numerical process, alternative methods may be employed.

The spectral irradiance of extraterrestrial solar radiation and direct solar radiation at ground level in a pure, cloudless atmosphere is depicted, illustrating the effects of Rayleigh scattering and gas absorption The graph highlights the attenuation caused by Rayleigh scattering, while the dark areas represent the absorption by specific gases, including ozone and water vapor Integration and approximation formulas have been utilized for this analysis, with various contributions compiled and compared by M Iqbal.

The relationships established by R.E Bird and R.L Hulstrom enable a precise calculation of solar irradiance (E) for direct solar radiation The irradiance on a surface, where the normal makes an angle β S with the sun's rays, can be accurately determined using these formulas.

The irradiance E n of a surface normal to the sun’s rays is, according to [5.42] and [5.34],

E n = 0.975 E 0 τ R τ A τ O 3 τ W τ G , (5.125) where the transmissivities are calculated according to the following equations The relative optical mass m r,L from (5.114) is uniformly used, simpliﬁed to here m r

Aerosol scattering: τ A = 0.1245α ∗ − 0.0162 + (1.003 − 0.125 α ∗ ) exp [ − β ∗ m r (1.089 α ∗ + 0.5123)] , where α ∗ and β ∗ are the parameters from (5.120) τ A can also be given as a function of the (horizontal) visibility s h in km: τ A = exp m 0.9 r ln

The utilisation of s h replaces the estimation of the parameters α ∗ and β ∗

(1 + 139.5 h O 3 m r ) 0.3035 , where h O 3 is entered in cm, cf section 5.4.2.3.

6.385 w m r + (1 + 79.03 w m R ) 0.6828 with w in cm, cf section 5.4.2.3.

Absorption by CO 2 and other gases: τ G = exp

On September 1st at 12:00 PM Central European Summertime, the irradiance of direct solar radiation on a horizontal surface in Berlin can be calculated under cloudless conditions With an ozone thickness of 0.30 cm, a water vapor content of 2.6 cm, and a horizontal visibility of 40 km, these parameters are essential for accurate irradiance assessment.

According to Example 5.6 the sun’s polar angle is β S = 46.05 ◦ This gives, from (5.114), a relative optical mass m r,L = m r = 1.439 The transmissivities that appear in (5.125) have the following values: τ R = 0.8852, τ O 3 = 0.9811, τ W = 0.8715, τ G = 0.9861, τ A = 0.8100

It follows from (5.125) that E n = 805.8 W/m 2 ; with cos β S = 0.694, according to (5.124)

The calculated irradiance value of E = 559 W/m² represents only 59.9% of the extraterrestrial solar radiation, E sol, as shown in Example 5.6 This measurement is significantly affected by the scattering and absorption caused by aerosols However, in a less turbid atmosphere with a scale height of s h = 100 km, the aerosol optical depth τ A increases to 0.8766, resulting in an increased irradiance of E = 605 W/m².

12 The equation for τ O 3 has been simpliﬁed, compared with [5.42], without any loss of accuracy.

Diﬀuse solar radiation and global radiation

In addition to direct solar radiation, some of the radiation scattered in the atmosphere reaches the ground, known as diffuse solar radiation Both diffuse and direct solar radiation are partially reflected by the Earth's surface, with some of this reflected radiation being sent back to the ground by the atmosphere This ongoing reflection process results in an additional flow of radiation towards the ground, collectively referred to as sky-radiation It is important to differentiate sky-radiation from atmospheric counter-radiation, which is long-wave radiation emitted primarily by water vapor and CO2 in the atmosphere Sky-radiation, on the other hand, consists of short wavelengths created by the scattering of direct solar radiation, including contributions from Rayleigh scattering, aerosol scattering, and multiple reflections.

Global radiation refers to the combination of direct solar radiation and diffuse sky radiation The global irradiance (E G) measured on a horizontal surface on the ground consists of these two components.

The irradiance from direct solar radiation is represented by the first term on the right side, as derived from equations (5.124) and (5.125) Additionally, E d Ra signifies the irradiance of diffuse radiation resulting from Rayleigh scattering by air molecules, while E d Ae denotes the irradiance of diffuse radiation.

The spectral irradiance of diffuse sky radiation is analyzed across three fractions, calculated under specific conditions: airmass (m r,L) of 1.5, ozone thickness (h O 3) of 0.3 cm, and water vapor (w) of 2.0 cm Additionally, parameters α ∗ and β ∗ are set at 1.3 and 0.10, respectively The results include the spectral irradiance due to Rayleigh scattering (E λ,d Ra) and the scattering effects from aerosols (E λ,d Ae).

The spectral irradiance, E λ,d MR, resulting from multiple reflections is influenced by aerosol scattering, while E d MR specifically addresses the irradiance from these reflections In regions that are not horizontally aligned, the relationships become more complex due to the necessity of considering radiative exchanges with the surrounding environment For a comprehensive understanding of these interactions, M Iqbal's work [5.34] is recommended.

The model developed by R.E Bird and R.L Hulstrom is utilized to calculate the diffuse fraction of radiation This model considers the non-absorbed direct solar radiation as the source of scattered radiation, contributing to the overall irradiance.

E na n = 0.786 E 0 cosβ S τ O 3 τ W τ G τ A abs Here, τ A abs is the transmissivity resulting from the absorption by aerosols alone: τ A abs = 1 − (1 − ω 0 ) (1 − τ A )

The quantity ω 0 represents the ratio of energy scattered by aerosols to the total energy scattered and absorbed by them, typically estimated at around 0.9.

Under the assumption that half of the energy scattered by molecules in the atmosphere reaches the ground, according to [5.42], we obtain

1 − m r + m 1.02 r for the irradiance as a result of Rayleigh scattering For the irradiance from aerosol scattering we get

1 − m r + m 1.02 r The factor F A indicates what proportion of the energy scattered by the aerosols is scattered

The forwards, or the portion that contacts the ground, must be estimated, with R.E Bird and R.L Hulstrom suggesting a value of F A = 0.84 G.D Robinson has calculated F A for aerosols in the British Isles, and these findings are consistently replicated.

Through considerations, analogous to the explanation of multiple reﬂection in section 5.3.4, we obtain

The reflectivity of the Earth's surface for shortwave solar radiation, known as Albedo (r E), can be calculated using data on solar radiation absorptivity Additionally, the atmosphere's reflectivity (r At) is minimal and can be determined using the formula r At = 0.0685 + (1 − F A ).

With E d MR from (5.127), we obtain the following for the irradiance of global radiation on a horizontal area from (5.126)

1 − r E r At This equation is also valid for models in which E n , E d Ra , E d Ae and r At are determined using diﬀerent relationships than those in [5.42].

Example 5.8: Calculate the irradiances of the diﬀuse solar radiation and the global radiation for the case dealt with in Example 5.7 Additional assumptions are: ω 0 = 0.90,

With m r = 1.439 and τ A = 0.810 from Example 5.7, we obtain τ A abs = 0.980 With that and the other transmissivities calculated in Example 5.7, the irradiance of the non-absorbed direct solar radiation is found to be

E n na = 0.786 ã 1367 W m 2 0.694 ã 0.9811 ã 0.8715 ã 0.9861 ã 0.980 = 616 W m 2 For the Rayleigh fraction, the irradiance then follows as E d Ra = 35.0 W/m 2 With

F A = 0.45 + 0.65 cos 46.05 ◦ = 0.90 we obtain the irradiance for the aerosol fraction as E Ae d = 95.3 W/m 2

The reﬂectivity of the atmosphere is r At = 0.086, and with that the fraction due to multiple reﬂection is calculated to be

1 − 0.25 ã 0.086 = 15.1 W m 2 The irradiance of the diﬀuse sky-radiation is therefore

E Ra d + E d Ae + E MR d = 145 W/m 2 , which is 26 % of the irradiance of the direct solar radiation The global irradiance is

If we assume a less turbid atmosphere, like in Example 5.7, with τ A = 0.8766, then τ A abs = 0.987 and r At = 0.080 This yields E d Ra = 35.2 W/m 2 , E d Ae = 61.8 W/m 2 and

The measured irradiance of the diffuse radiation is 111 W/m², while the direct radiation has risen to 605 W/m², as demonstrated in Example 5.7 Consequently, the total global radiation is recorded at 716 W/m², reflecting a slight increase of 1.7% compared to measurements taken in a more turbid atmosphere.

Absorptivities for solar radiation

The spectral emissivity ε λ (λ, T) of most substances varies significantly at small wavelengths below 2 µm compared to larger wavelengths Electrical insulators typically exhibit lower spectral emissivities, while metals show slightly higher values at longer wavelengths For a diﬀuse radiating surface, this variation also applies to spectral absorptivity a λ (λ, T), as they are equal (a λ (λ, T) = ε λ (λ, T)) Consequently, it is anticipated that substances will demonstrate differing behaviors in absorbing solar radiation versus absorbing long-wave radiation from terrestrial sources.

The grey Lambert radiator model, which assumes that absorptivity equals emissivity (a(T) = ε(T)), is not applicable for solar radiation absorption, as absorptivities can significantly differ from tabulated emissivities For the absorption of predominantly short-wave solar radiation, specific measurements are necessary to determine the absorptivities (aS) Table 5.8 presents some findings from these measurements Additional data on natural surfaces, such as cornfields, various soil types, forests, snow, and ice, can be found in K.Y Kondratyew's work, where reflectivity (rS = 1 - aS), commonly referred to as albedo in meteorology, is discussed.

Table 5.8: Absorptivity a S for solar radiation and (total) emissivity ε = ε(300 K) of diﬀerent materials

Aluminium, polished 0.20 0.08 2.5 Asphalt, Road covering 0.93

Iron, galvanised 0.38 Tar paper, black 0.82 0.91 0.90 rough 0.75 0.82 0.91 Earth, ploughed 0.75

Copper, polished 0.18 0.03 6.0 Zinc white 0.22 0.92 0.24 oxidised 0.70 0.45 1.56 Black oil paint 0.90 0.92 0.98 Magnesium, polished 0.19 0.12 1.6 Marble, white 0.46 0.90 0.51

In solar technology applications, the ratio of surface area (S) to emissivity (ε) is crucial A high S/ε ratio is essential for solar collectors, as it ensures that the radiation lost to the surroundings, which is proportional to ε, remains minimal compared to the amount of solar radiation absorbed.

To maintain a low temperature on surfaces exposed to solar radiation, it is essential to minimize the ratio of solar absorptance (a S) to emissivity (ε) This can be effectively achieved by painting the surface white, which results in a solar absorptance of 0.22 and an emissivity of 0.92, yielding a favorable S/ε ratio of 0.24.

Radiative exchange

View factors

The calculation of radiative exchange between two surfaces relies on the view factor, also known as the configuration or angle factor, which quantifies how much one surface can "see" another Specifically, it measures the proportion of radiation from surface 1 that reaches surface 2 To calculate the view factor, one must first determine the radiation flow, d²Φ₁₂, emitted from surface element dA₁ that impacts surface element dA₂ The intensity of radiation from dA₁ is represented by L₁, leading to the equation d²Φ₁₂ = L₁ cosβ₁ dA₁ dω₂, where dω₂ is the solid angle subtended by dA₂ as seen from dA₁ This relationship can be expressed as d²Φ₁₂ = L₁ cosβ₁ cosβ₂ / r² dA₁ dA₂.

The photometric fundamental law states that the intensity of radiation received at a surface area (dA) diminishes with the square of the distance (r) from the radiation source Additionally, the orientation of the surface elements relative to the direct line connecting the source and receiver plays a crucial role, which is mathematically represented by a cosine function involving two polar angles (β1 and β2).

In this section, we will calculate the radiation emitted from finite surface 1 that impacts surface 2, as illustrated in Fig 5.50 We assume a constant intensity, L1, across the entirety of surface 1 By integrating equation (5.128) over both surfaces, we find that the total radiation transfer, Φ12, is equal to L1.

Fig 5.50: Geometric quantities for the calculation of the view factor

This is the radiation ﬂow emitted by 1 that falls on 2 With Φ 1 =πL 1 A 1 as the radiation ﬂow emitted by surface 1 into the hemisphere, theview factoris obtained as

The view factor quantifies the fraction of radiation emitted from surface 1 that reaches surface 2, and it is solely influenced by geometric considerations This relationship is derived under the assumption that surface 1 radiates diffusely, maintains a constant temperature, and exhibits uniform radiation properties across its entire area.

If the indices 1 and 2 are exchanged in (5.130), then

The equation \( A_1 \cos \beta_1 \int A_2 \cos \beta_2 \frac{r^2 dA_1 dA_2}{r^2} \) represents the proportion of radiation flow emitted by surface 2, which strikes surface 1 while maintaining a constant intensity \( L_2 \) Together, equations (5.130) and (5.131) establish the crucial reciprocity rule for view factors in radiative heat transfer.

This means that only one of the two view factors has to be determined by the generally very complicated integration of (5.130) or (5.131).

In an enclosed area where the view factors remain constant (L i = const), a relationship can be established between these view factors According to the radiation balance for a specific area, the sum of the view factors (Φ i1, Φ i2, , Φ in) equals the total view factor (Φ i) This relationship can be expressed mathematically by dividing the equation by Φ i, leading to the summation rule for n view factors.

Fig 5.51: Enclosure and radiation ﬂows Φ ij , emitted from the area A i

The enclosure created by concentric spherical areas 1 and 2 is significant in understanding view factors The term F ii represents the fraction of radiation emitted that strikes surface i, highlighting its relevance in radiative heat transfer Notably, F ii equals 0 only for concave surfaces, as they can "see themselves," while flat and convex surfaces have F ii equal to 0, indicating no self-radiation interaction.

An example of applying the relationships (5.132) and (5.133) can be observed in radiation within an enclosure created by two spherical surfaces, labeled as 1 and 2 In this scenario, four view factors are present: F11, F12, F21, and F22 To determine these view factors, the summation rule is utilized for the inner sphere.

In the case of a convex surface (surface 1), the view factor F 11 is zero, leading to F 12 being equal to one, indicating that all radiation emitted by surface 1 impacts the outer surface of sphere 2 The view factor F 21 can be determined using the reciprocity rule, taking into account the areas of both spheres, A 1 and A 2.

The fourth view factor is found by applying the summation rule F 21 + F 22 = 1 to the outer sphere

It is not equal to zero because part of the radiation emitted by the outer spherical surface also strikes it again.

Determining view factors can be complex, especially compared to simpler geometries In an enclosure defined by 'n' surfaces, there are a total of n² view factors However, the summation rule allows for the calculation of 'n' view factors from each surface, while the reciprocity rule enables the determination of n(n−1)/2 additional view factors Consequently, only n(n−1)/2 view factors need to be computed using the multiple integral method This number is further reduced by considering flat or convex surfaces where Fii = 0.

The multiple integral in equation (5.130) has been computed for various geometrical configurations, many of which result in complex equations that are challenging to evaluate The techniques necessary for deriving these equations are documented in the relevant literature.

R Siegel [5.45], and other authors Some examples of calculated view factors are presented in Table 5.9 A larger collection of view factors can be found in R Siegel and others [5.45] with numerous information on sources, in the VDI-W¨armeatlas [5.46] and in J.R Howell [5.47].

Calculate the view factors for the inside of a cylinder, according to Fig 5.53, with r = 0.10 m and h = 0.25 m.

Fig 5.53: Hollow cylinder with end areas 1 and 2 and body surface 3

In a three-surface enclosure, there are nine view factors, but only three require calculation as per equation (5.130) Since the end areas 1 and 2 are flat, the view factors F 11 and F 22 are both zero, necessitating the calculation of just one view factor, F 12 This view factor can be determined by evaluating the double integral from (5.130) and is referenced in Table 5.9 for two equally sized, parallel concentric circular discs, yielding a value of z = 2 + (h/r)² = 8.25.

From symmetry (or by applying the reciprocity rule), we ﬁnd that F 21 = F 12 = 0.123 and

Table 5.9: View factors F 12 for selected geometric arrangements

Two inﬁnitely long, parallel strips, with centre lines that lie vertically above one another x = b 1 /h ; y = b 2 /h

Two inﬁnitely long strips, perpendicular to each other with a common edge

Two identical, parallel rectangles lying opposite each other x = a/h ; y = b/h

Two rectangles perpendicular to each other with a common edge x = b 1 /a ; y = b 2 /a

Table 5.9: (continued) Two parallel circular disks with common central vertical x = r 1 /h ; y = r 2 /h z = 1 +

An inﬁnitely long strip and an inﬁnitely long cylinder parallel to it

Two inﬁnitely long, parallel cylinders with equal diameters x = h/2r

A sphere and a circular disk, whose central vertical goes through the centre of the sphere

Two areas on the inner side of a hollow sphere

Radiative exchange between black bodies

The calculation of radiative exchange is simplified when considering black bodies, as they absorb all incident radiation without reflection Additionally, the intensity of a black body, denoted as Ls, solely depends on its temperature, resulting in a constant intensity across an isothermal surface This characteristic is essential for accurately calculating view factors.

Fig 5.54: Radiation ﬂows Φ 12 and Φ 21 in direct radiative exchange between black bodies 1 and 2

In this analysis, we will determine the direct radiative interchange between two black bodies with arbitrary shapes, designated as surfaces A1 and A2, and uniform temperatures T1 and T2 The study will focus on the radiation emitted by one body while disregarding any radiation that does not impact the other body.

1 and incident on 2 is given by Φ 12 =A 1 F 12 σT 1 4

This energy ﬂow is absorbed by black body 2 The radiation ﬂow received and absorbed by black body 1, from body 2, is Φ 21 =A 2 F 21 σT 2 4

The net radiation ﬂow transferred by direct radiative interchange from 1 to 2 is therefore Φ ∗ 12 = Φ 12 −Φ 21 =A 1 F 12 σT 1 4 −A 2 F 21 σT 2 4

If both bodies are at the same temperatureT 1 = T 2 , then no (net) energy ﬂow will be transferred between them: Φ ∗ 12 = 0 This then yields

A 1 F 12 =A 2 F 21 , i.e the reciprocity rule, (5.132), for view factors, previously derived in another manner This gives Φ ∗ 12 =A 1 F 12 σ T 1 4 −T 2 4 =A 2 F 21 σ T 1 4 −T 2 4 (5.134)

The net radiation ﬂow transferred by direct radiative exchange between two black bodies is proportional to the diﬀerence of the fourth powers of their thermodynamic temperatures.

This article examines a hollow enclosure with walls composed of multiple parts, each exhibiting anisothermal surfaces According to H.C Hottel and A.F Sarofim, these anisothermal wall sections are referred to as zones Non-isothermal walls with varying temperatures can be represented by a series of small zones, each maintaining a distinct temperature For the purposes of this discussion, all zones are considered to be black surfaces Additionally, an opening in the enclosure is treated as a zone with a temperature that corresponds to the black radiation entering the enclosure from the outside.

To achieve a stable state, each zone must have external heat flow added or removed to balance the difference between emitted radiation and the total incident and absorbed radiation The energy balance for a specific zone, denoted as zone i, with surface area A_i and temperature T_i, is essential for maintaining this equilibrium.

By applying the reciprocity rule (5.132) the following is obtained

F ij = 1, it also holds that

The heat flow to zone i from external sources, or released to the outside when ˙Q i < 0, is the total of the net radiation flows Φ ∗ ij, as defined in equation (5.134), between zone i and the surrounding zones that form the enclosure.

A zone where ˙Q i > 0 is identified as a net radiation source, emitting more radiation than it absorbs Conversely, a zone with ˙Q i < 0 functions as a net radiation receiver, absorbing more radiation than it emits An adiabatic zone, characterized by ˙Q i = 0, is referred to as a reradiating wall, maintaining a temperature that allows it to emit radiation equal to the amount it absorbs, achieving radiative equilibrium.

The heat ﬂows for all the zones can be found from (5.136) for given temperatures If, on the contrary, some of the heat ﬂows are known, thenbalance

Fig 5.55: a Hollow enclosure bounded by black radiating edges b Illustration of the energy balance for the zone i

Fig 5.56: Enclosure formed from radiation source 1, radiation receiver 2 and reradiating walls with

Q ˙ R = 0 equations (5.136) represent a linear system of equations, from which all the unknown temperatures and heat ﬂows can be determined.

An enclosure with only three zones is often a good approximation for the case of a radiation source of area A 1 and temperature T 1 in radiative exchange with a radiation receiver of area

In a system where the temperature T2 is lower than T1, adiabatic walls contribute to radiative heat exchange These walls can be approximated as having a unified temperature, TR The reradiating walls that surround the space are treated as a single zone with temperature TR, and the net radiative heat transfer, denoted as ˙QR, is considered to be zero.

The following balance equations are valid for this hollow enclosure with three black radiating zones:

According to the reciprocity rule (5.132), it can be concluded that ˙ Q 2 = − Q ˙ 1, which is consistent with the overall balance of the enclosure The temperature of the reradiating zone can be determined from equation (5.137c).

A 1 F 1R + A 2 F 2R (5.138) and by elimination of T R 4 from (5.137a)

(5.139) with the modiﬁed view factor

The heat flow \( \dot{Q}_1 \) transferred from point 1 to point 2 is greater than the net radiation flow \( \Phi^{*}_{12} \) due to the influence of reradiating walls, as indicated by the condition \( F_{12} > F_{12} \) When the radiation source and receiver feature flat or convex surfaces (where \( F_{11} = 0 \) and \( F_{22} = 0 \)), the view factors \( F_{1R} \) and \( F_{2R} \) can be related back to \( F_{12} \), modifying the original equation.

(5.141) is obtained Only one view factor, namely F 12 is required to calculate ˙ Q 1

Example 5.10: The hollow cylinder from Example 5.9 has black radiating walls The two ends are kept at temperatures T 1 = 550 K and T 2 = 300 K The body area 3 is adiabatic,

Q ˙ 3 = ˙ Q R = 0 Calculate the heat ﬂow ˙ Q 1 and the temperature T 3 = T R of the reradiating body area, if this is taken to be an approximately isothermal area (zone).

In order to determine the heat flow ˙ Q 1 from (5.139), the modified view factor F 12 is required This can be calculated according to (5.141), because the two ends are flat With

F 12 = 0.123 from Example 5.9 and A 1 /A 2 = 1, F 12 = 0.5615 is obtained This yields the following from (5.139)

The temperature of the reradiating shell area can be found from (5.138) With F 1R =

This is the temperature at which the assumed isothermal body area emits as much energy as it absorbs In reality the temperature of the body area varies continuously between

In a radiative exchange process, different irradiance levels from sources T1 and T2 impact each annular strip of infinitesimal width, resulting in varying emissive powers and corresponding temperatures for each strip The mathematical treatment of this complex scenario, which involves continuously varying temperatures across an infinite number of infinitesimal zones, is highly intricate and detailed information can be found in reference [5.45], pages 107–132.

Radiative exchange between grey Lambert radiators

In radiative exchange scenarios where bodies cannot be assumed to be black bodies, it is essential to consider the effects of reflected radiation, particularly in hollow enclosures where multiple reflections and partial absorption occur A general solution for these problems is rarely feasible without simplifying assumptions However, when the boundary walls are segmented into isothermal zones that act as grey Lambert radiators, a more straightforward solution emerges Each zone is defined by its hemispherical total emissivity ε_i (T_i), with absorptivity a_i equal to ε_i and reflectivity r_i calculated as 1−ε_i The intensity remains constant across each zone, and under the assumption of diffuse reflection, the reflected radiation also maintains a constant intensity Consequently, the combined radiation emitted and reflected from a zone adheres to the cosine law, allowing the use of view factors to effectively describe the radiative exchange between zones.

This article explores the radiative exchange between isothermal walls within an enclosure, as depicted in Fig 5.57 While some zones have known temperatures, others have specified heat flows exchanged with the external environment The objective is to determine the heat flows of the zones with established temperatures and the temperatures of zones with defined heat flows Notably, the number of unknown variables, whether temperatures or heat flows, matches the number of zones present.

To address the radiative exchange issue, it is essential to establish energy balance equations for all zones, utilizing the net-radiation method proposed by G Poljak This approach results in a system of linear equations that, upon resolution, provides the unknown temperatures and heat flows.

The analysis of a hollow enclosure defined by isothermal surfaces, each functioning as a grey Lambert radiator, reveals key insights into thermal dynamics Focusing on scenarios with two or three zones, the electrical circuit analogy introduced by A.K Oppenheim simplifies the understanding of the relationship between the temperatures and heat flows across these zones.

5.5.3.1 The balance equations according to the net-radiation method

G Poljak suggests the introduction of a new quantity in the energy balance equation for a zone, which accounts for the radiation emitted and reflected by an isothermal surface This new quantity is derived by adding the emissive power, ensuring a comprehensive understanding of energy dynamics in the system.

M i of the surfaceiand the reﬂected portion of its irradianceE i :

H i :=M i +r i E i =M i + (1−ε i )E i (5.142) The quantityH i is called theradiosityof the surfacei, cf E.R.G Eckert [5.51].

Fig 5.58: Illustration of the energy balance for the zone i with area A i

To establish the energy balance for zonei, as depicted in Fig 5.58, we must account for the heat flow ˙Q i supplied from external sources This heat flow must compensate for the difference between the emitted and reflected radiation flow and the incoming radiation flow.

Q˙ i =A i (H i −E i ) ; (5.143) the heat ﬂow agrees with the net radiation ﬂow We now calculate the irradiance

E i from (5.142) and put it into (5.143) with the result

The emissivity ε i appearing here is normally dependent on the temperature; it has to be calculated at the temperatureT i of the zonei: ε i =ε i (T i ).

A second relationship between ˙Q i andH i is obtained, when the radiation ﬂow

A i E i incident on zone i is linked with the radiation flows emitted by the other zones The radiation flowA j H j is sent out by zonej, but only the radiation flow

A j F ji H j , multiplied by the view factor F ji , strikes zone i Therefore, the total radiation striking zoneiis

F ij H j , where the reciprocity rule (5.132) has been applied for the view factors Putting this expression into (5.143), and taking (5.133) into account, it follows that

The balance equations (5.144) and (5.145) can be formulated for each zone (i = 1, 2, , n), resulting in a total of 2n equations for the n unknown radiosities H i, along with the required values for ˙Q i and T i Prior to exploring the equation system detailed in section 5.5.3.4, the following section presents solutions for simpler scenarios involving enclosures defined by only two or three zones.

5.5.3.2 Radiative exchange between a radiation source, a radiation receiver and a reradiating wall

An enclosure consisting of three isothermal zones provides an effective model for analyzing complex radiative exchange scenarios Zone 1, at temperature T1 and emissivity ε1, acts as the radiation source, receiving an external heat flow ˙Q1 Zone 2, with a lower temperature T2 (T2 < T1) and emissivity ε2, functions as the radiation receiver The third zone, a reradiating wall at a constant temperature TR, has no heat flow (˙QR = 0) The objective is to calculate the radiative heat flow ˙Q1, which is equal to the negative of the heat flow ˙Q2 exchanged within the enclosure.

Fig 5.59: Enclosure formed by a radiation source 1, radiation receiver 2 and (adiabatic) reradiating walls R

The solution of this problem starts with the writing of the two fundamental balance equations (5.144) and (5.145) in the form

Fig 5.60: Equivalent electrical circuit diagram for (5.146) with “reﬂec- tion resistance” (1 − ε i )/A i ε i

Fig 5.61: Equivalent electrical circuit diagram for (5.147): Current branching with the “geometric resistances” (A i F ij ) −1 and

In this analogy, we compare relationships to an electrical circuit, where the "current" ˙Q i is driven by the "potential difference" between σT i 4 and H i This current flows through a "conductor" characterized by a "resistance" of (1−ε i )/A i ε i, as depicted in the equivalent electrical circuit diagram in Fig 5.60 Thus, Eq (5.146) can be understood as representing the current within this circuit framework.

”potential“H i splitting at a node into wires with the “geometric resistances” (1/A i F ij ) to the “potentials”H j , see Fig 5.61 The wire possible for

F ii = 0 is missing, as due toH j =H i no “current” ﬂows.

The radiative exchange in the enclosure from Fig 5.59 can be replaced by the circuit diagram from Fig 5.62.

As the reradiating wall has no current, ( ˙Q R = 0), the current ˙Q 1 of potential σT 1 4 flows to the nodeH 1 , where it branches off, it flows directly and via H R to

The equivalent electrical circuit diagram for radiative exchange in a hollow enclosure illustrates the potential transition from σT1^4 to σT2^4 through three resistances arranged in series The central resistance, denoted as (A1F12)^-1, is composed of three individual resistances: (A1F12)^-1, (A1F1R)^-1, and (A2F2R)^-1 This configuration highlights the impact of current branching on the overall resistance within the system.

With parallel resistances the conductances are added together; it then follows that

This relationship for F 12 agrees with (5.140), which was derived in a diﬀerent manner If both surfaces 1 and 2 are ﬂat or convex (F 11 = 0 andF 22 = 0), then

A 1 F 12 can, according to (5.141), be calculated using onlyF 12 ,A 1 and A 2 Equation (5.148) is often written in the form

Q˙ 1 =ε 12 A 1 σ T 1 4 −T 2 4 , (5.150) through which theradiative exchange factor ε 12 is deﬁned From (5.148) we get

For emissivities dependent on the temperature,ε 1 =ε 1 (T 1 ) andε 2 =ε 2 (T 2 ) should be used.

In the absence of current flow between nodes with potentials σT R 4 and H R, the resistance (1−ε R )/ε R A R becomes irrelevant, validating the equation σT R 4 = H R Consequently, the temperature T R of a reradiating wall is independent of its emissivity ε R The necessary radiosity H R for determining this temperature is derived from the energy balance.

A 1 F 1R +A 2 F 2R (5.152)The radiositiesH 1 andH 2 are obtained from (5.146) fori= 1 andi= 2, with ˙Q 1 from (5.150) and ˙Q 2 =−Q˙ 1

In the electrically heated oven designed for surface treatment of thin, square metal plates, the setup features a square base measuring 1.50 m on each side The heating elements, positioned 0.25 m above the metal plates, emit radiation with an emissivity of 0.85 and are supplied with 12.5 kW of power in two rows, while the non-insulated side walls have an emissivity of 0.70 Under steady-state conditions, the heating elements achieve a surface temperature of 750 K This setup allows for the determination of both the side wall temperature and the temperature of the covered metal plates.

Fig 5.63: a Electrically heated oven for the surface treatment of metal plates, b

Hollow enclosure for the calculation of the radiative exchange of the top half of the oven

The construction's symmetry allows us to focus solely on the oven's upper half, depicted in Fig 5.63b This upper half is defined by the heated square surface (1) with an emissivity of ε 1 = 0.85 at the top, rectangular side areas (2) with ε 2 = 0.70 that dissipate heat externally, and a metal plate (R) at the bottom, which acts as an adiabatic reradiating wall due to symmetry We will denote the nearly uniform temperatures of these surfaces as T 1, T 2, and T R, enabling us to calculate the radiative exchange within this hollow enclosure using equations (5.148) or (5.151).

The ﬁrst step is the determination of the view factors required for (5.149), F 12 , F 1R and

F 2R The easiest to calculate is the view factor F 1R between two parallel squares lying one above the other (side length a) from Table 5.9 With x = y = a/b = 6.0 we obtain, from

, the value F 1R = 0.7326 As F 11 = 0, then F 12 = 1 − F 1R = 0.2674 In order to determine

F 2R , we consider that due to symmetry F R2 = F 12 is valid Using the reciprocity rule (5.132), it then follows, with A R = A 1 = a 2 and A 2 = 4ab, that

This yields from (5.149) the modiﬁed view factor F 12 = 0.4633.

We now calculate the radiative exchange factor ε 12 from (5.151) and obtain

= 1.3234 m −2 and ε 12 = 0.3358 The temperature T 2 of the four side walls is calculated from (5.150) with ˙ Q 1 = 12, 5 kW:

This gives T 2 = 396.3 K The temperature T R of the covered metal plates is obtained from their radiosity H R as

The radiosities H 1 and H 2 are required for the calculation of H R from (5.152) They are found from (5.146) to be

It then follows from (5.152) that H R = 13.75 kW/m 2 and ﬁnally from (5.153), T R = 702 K.

5.5.3.3 Radiative exchange in a hollow enclosure with two zones

The heat flow, ˙Q 1, from radiation emitter 1 to receiver 2 is applicable to enclosures limited to these two areas In the absence of a reradiating zone, the configuration results in F 1R = 0.

F 2R = 0 from (5.149),F 12 =F 12 is obtained The heat ﬂow transferred from 1 to

Q˙ 1 =ε 12 A 1 σ T 1 4 −T 2 4 (5.154) The radiative exchange factorε 12 is yielded from (5.151) to be

These equations hold for several important, practical cases:

1 The area 1 is completely enclosed by the area 2, so thatF 12 = 1 For the radiative exchange factor we have now

The findings are particularly relevant for concentric spheres and elongated concentric cylinders, where the assumption of isothermal surfaces is more applicable However, when body 1 is positioned eccentrically within an enclosure surrounded by body 2, the surfaces typically do not remain isothermal This is due to the significantly higher radiation flow in areas where the surfaces are in close proximity compared to regions with greater distance between them.

Fig 5.64: Enclosure surrounded by body 2 with an eccentrically placed body 1

2 Surface 2 completely surrounds surface 1, (F 12 = 1), and is black: ε 2 = 1.

It now follows from (5.156) thatε 12 =ε 1 , the simple equation

Q˙ 1 =ε 1 A 1 σ T 1 4 −T 2 4 (5.157) for the transferred heat flow As the surrounding shell does not reflect any radiation, the sizeA 2 of its surface has no effect on the radiative exchange.

Protective radiation shields

Protective radiation shields, typically composed of thin foils or sheets made from highly reflective materials, are essential for minimizing radiative heat exchange between walls at varying temperatures These shields are strategically placed between the walls, and the spaces between them are usually evacuated to eliminate heat transfer through convection This multi-layer insulation method is primarily utilized in cryogenic applications, specifically for insulating containers that store extremely cold liquefied gases.

The heat ﬂux transferred between two very large, parallel, ﬂat walls, according to (5.154) and (5.158), is given by ˙ qQ˙

We will now consider the case ofN radiation shields present between the walls

The emissivity (ε S) must be consistent across both sides of the shield and applicable to all shields Due to their thinness, each shield can be assigned a uniform temperature The equations derived incorporate T Si, representing the temperature of the i-th shield.

=σ(T SN 4 −T 2 4 ) The temperatures of the shields drop out of the right hand side when all the

Fig 5.66: a Flat radiation shields between two ﬂat parallel walls 1 and 2 b Concentric radiation shields between concentric spheres or very long cylinders 1 and 2 equations are added together, giving ˙ q

The heat flux is notably decreased when protective shields are implemented, as evidenced by the comparison with the case of no shields (N=0) Table 5.10 illustrates that for an emissivity of ε S = 0.05, the ratio ˙q(N)/q(N˙ = 0) diminishes as N increases, highlighting the effectiveness of shielding Furthermore, the impact of shielding is more pronounced with higher emissivities (ε 1 = ε 2) of the outer walls.

Table 5.10: Ratio of the heat flux ˙ q(N ) for N protective shields with ε S = 0.05 to the heat flux ˙ q(N = 0) without protective shields between two flat walls with ε 1 = ε 2 = ε ε N = 1 2 5 10 20 50

We will now look at the radiative exchange between concentric cylinders or spheres The heat ﬂow transferred calculated from (5.154) and (5.156) is

Thin protective shields with uniform emissivity ε S are positioned concentrically between cylinders or spheres Utilizing a similar approach as with flat radiation protection shields, the heat flow is effectively reduced.

Here,A 1 is the surface area of the inner wall and the indexiof the shields rises from inside to outside.

Surfaces with low emissivities display mirrorlike or specular reflection instead of diffuse reflection, prompting an investigation into the impact of this assumption on heat transfer The principles governing diffuse and grey radiation emissions remain unchanged, leading to the conclusion that grey Lambert radiators exhibiting mirrorlike reflection are assumed in this analysis.

In the scenario depicted in Fig 5.67, radiation emitted from one large, flat plate consistently impacts the other plate, leading to continuous reflection until fully absorbed This process applies equally to radiation from both plates The heat transfer mechanism mirrors that of diffuse reflection, allowing equation (5.169) for heat flux to remain valid regardless of whether the walls reflect diffusely or with mirror-like properties Additionally, the relationship expressed in equation (5.170) for heat flux with N radiation shields is applicable without modification for mirror-like reflection, even when only the shields exhibit mirror-like properties while the plates reflect diffusely.

Fig 5.67: Radiation pathways for mirrorlike reﬂection between two large, ﬂat, parallel walls

Figure 5.68 demonstrates the radiation pathways between two concentric cylinders or spheres that behave like mirrors Radiation emitted from the inner area (pathway a) consistently hits the outer area and is reflected back to the inner area, maintaining the condition F12 = 1 Similar to diffuse reflection, the emitted radiation from the inner area is repeatedly reflected between the two surfaces until it is fully absorbed.

The radiation emitted by outer area 2 behaves differently, as it either strikes the inner surface or bypasses it, returning to outer surface 2 This reflection, illustrated as pathway b in Fig 5.68, is specular and does not engage in radiative exchange between the surfaces Conversely, the radiation that follows pathway c, represented by the view factor F21 = A1 F12 / A2 = A1 / A2, impacts area 1 and is reflected between the surfaces until fully absorbed Thus, the outer surface contributes to radiative exchange solely through mirror-like reflection, effectively reducing its surface area A2 by a specific factor.

F 21 : it has the eﬀective surface areaA 2 F 21 =A 1

Therefore, if the outer surface 2 reﬂects mirrorlike, in the following equation for the heat ﬂow, from (5.148) with ¯F 12 =F 12 = 1,

In the analysis of ray pathways within mirror-like reflections between concentric spheres or elongated cylinders, it is essential to substitute the area \( A_2 \) in the third "resistance" of the denominator with the effective area \( A_1 \).

The size of the outer area 2 is immaterial for radiative exchange Eq (5.173) also holds if the inner surface 1 reﬂects diﬀusely.

N concentric, thin radiation shields with uniform emissivity ε S are positioned between the inner area 1, which may be either diffuse or mirrorlike reflecting, and the diffuse reflecting outer area 2 These shields exhibit mirrorlike reflection properties The balance equations governing this setup are described by equation (5.173).

The temperatures of the shields drop out when the equations are added together giving

If the outer area 2 exhibits mirror-like reflection, the term A2 in the denominator should be substituted with ASN, representing the effective surface area of 2 for mirror-like reflection.

In a thermal analysis of a concentric tube system, the inner tube, with an external diameter of 30 mm, has a low emissivity of 0.075 and operates at a temperature of 80 K Surrounding this tube is a second tube with an internal diameter of 60 mm, an emissivity of 0.12, and a temperature of 295 K The space between the two tubes is evacuated, necessitating an investigation into the heat flow per unit length that is transferred by radiation It is essential to consider both the limiting cases of diffuse and mirror-like reflection for the outer tube in this analysis.

With diffuse reflection, the desired heat flow is obtained from (5.171) to be

The minus sign signifies that heat is being transferred from the outside to the inside The inner tube is cooled using liquid nitrogen, resulting in a calculated radiative exchange value of ˙ Q/L = − 1.948 W/m under mirrorlike reflection conditions Since only a portion of the outer area participates in the radiative exchange, the insulation effect is enhanced.

To enhance the insulation of the nitrogen pipe, a thin radiation protective shield with a diameter of 45 mm is placed between the tubes This shield acts as a reflective barrier, exhibiting an emissivity of 0.025 The introduction of this shield significantly reduces the incident heat flow.

Eqn (5.174) with N = 1 is applied here, so that in the denominator the sum drops out:

The study reveals that the heat flow significantly decreases when using a diffuse reflecting outer cylinder, with a ratio of ˙ Q(N = 1)/ Q(N = 0) at 0.206 When the outer cylinder is replaced with a mirrorlike reflector, the heat flow changes to ˙ Q(N = 1)/L = − 0.480 W/m, resulting in a new ratio of 0.246 Although the protective shield's impact on heat flow is slightly less pronounced with specular reflection compared to diffuse reflection, the lowest absolute value of ˙ Q/L occurs when both the shield and the outer cylinder exhibit mirrorlike reflection.

Gas radiation

Absorption coeﬃcient and optical thickness

When radiation travels through an optically turbid gas, its energy diminishes due to absorption by gas molecules In this context, we focus on infrared wavelengths, where absorption occurs without significant scattering, as Rayleigh scattering is negligible for wavelengths greater than 1 µm In a gas mixture, such as one containing CO2, only specific components can absorb radiant energy, while others, like N2 and O2, remain non-absorbing.

Fig 5.69: Reduction in the spectral intensity L λ with increasing beam length s as a result of absorption by a gas

Absorption causes a reduction in the spectral intensity L λ with increasing beam lengthsof the gas, Fig 5.69 This fall in the spectral intensity as radiation passes through the distance dsis described by

L λ =k G (λ, T, p, p G ) ds , (5.175) through which thespectral absorption coeﬃcientk G of the absorbing gas is deﬁned.

The absorption characteristics of a gas are influenced by its wavelength (λ), temperature (T), pressure (p), and the partial pressure (pG), which indicates the concentration of the absorbing gas in the mixture For simplicity, we often omit the explicit notation of these dependencies.

We simply writek G instead ofk G (λ, T, p, p G ).

In anon-homogeneous gas mixturethe properties vary along the path followed by the radiation The absorption coeﬃcient therefore has an indirect dependence ons Integration of (5.175) betweens= 0 andsyields ln L λ (s)

The integral κ G, defined as \( \int_0^s k_G(\lambda, T, p, p_G) \, ds \), represents the optical thickness of a gas layer with geometric thickness s Unlike s, κ G is a dimensionless quantity that indicates the absorption strength of the gas layer A gas is considered optically thin when κ G approaches 0 When κ G equals 7, the intensity of the penetrating radiation, L λ(s), diminishes to less than 1 per thousand of its initial value, indicating that the gas effectively absorbs nearly all of the radiation.

A homogeneous gas mixture maintains constant intensive properties such as temperature (T), pressure (p), and partial pressure (p G) throughout its volume Consequently, the spectral absorption coefficient (k G) remains independent of the path length (s) As a result, the optical thickness can be expressed as κ G = k G (λ, T, p, p G)s = k G s, leading to a decrease in spectral intensity.

This equation corresponds to the law from P Bouguer (1729), according to which the spectral intensity falls exponentially along the path radiation is passing through.

The spectral absorption coefficient \( \kappa_G \) of a gas is directly proportional to its molar concentration \( c_G \) and partial pressure \( p_G \) This relationship indicates that as radiation is absorbed by individual gas molecules, the optical thickness of a homogeneous gas can be expressed as \( \kappa_G = k_G s = k_G^*(\lambda, T, p)(p_G s) \) This principle, established by A Beer in 1854, highlights the proportionality between the absorption coefficient and the product of the gas's partial pressure and the distance the light travels through the gas.

It applies to some gases, e.g CO 2 , very well, but from other gases, in particular H 2 O, it is not satisﬁed.

The spectral absorption coefficient \( k_G \) plays a crucial role in determining the radiation energy emitted by a gas volume element The radiation flow \( d^2 \Phi_{\lambda,V} \) from a homogeneous gas volume \( dV \) within a specific wavelength interval \( d\lambda \) is expressed as \( d^2 \Phi_{\lambda,V} = 4 \pi k_G L_{\lambda s}(\lambda, T) dV d\lambda = 4 k_G M_{\lambda s}(\lambda, T) dV d\lambda \).

The spectral intensity L λs and the hemispherical spectral emissive power M λs of a black body are given here by (5.50) A derivation of (5.181) can be found in R Siegel and J.R Howell[5.37], p 531.

Absorptivity and emissivity

The schematic in Fig 5.70 depicts an enclosure filled with a homogeneous gas mixture that includes an absorbent component To define and calculate the directional spectral absorptivity \( a_{\lambda,G} \), we focus on the surface element \( dA \) within the gas volume The radiation emitted from \( dA \), characterized by the spectral intensity \( L_{\lambda} \), experiences attenuation due to absorption The varying lengths of the path through the gas, influenced by direction, result in different levels of reduction in \( L_{\lambda} \).

Fig 5.70: For the calculation of the spectral absorptivity in a gas space

The spectral absorptivitya λ,G belonging to a certain direction is the ratio of the energy absorbed in the distances, to the energy emitted: a λ,G := L λ (s= 0)−L λ (s)

The ratio L λ (s)/L λ (s = 0) is the directional spectral transmissivity τ λ,G For a homogeneous gas or gas mixture, we obtain from (5.179) a λ,G (λ, s, T, p, p G ) =a λ,G (k G s) = 1−exp (−k G s) (5.183)

Directional spectral absorptivity is a state variable of the absorbing gas that varies based on the direction of radiation This directional dependence is influenced by the lengths of the beams as they traverse through the gas.

According to Kirchhoﬀ’s law, cf section 5.3.2.1., thedirectional spectral emis- sivityε λ,G of a gas is equal to its directional spectral absorptivity: ε λ,G (λ, T, p, p G , s) =ε λ,G (k G s) =a λ,G (k G s) = 1−exp (−k G s) (5.184)

Fig 5.71: Directional spectral emissivity ε λ,CO

At a temperature of 294 K and a pressure of 10.13 bar, the directional emissivity (ε λ,G) of carbon dioxide (CO2) varies with beam lengths, as illustrated in Figure 5.71 The emission bands of CO2 are distinctly observable in this context The radiation flow (d 3 Φ λ,G) received by a surface element (dA) from a solid angle (dω) can be calculated using the equation d 3 Φ λ,G = ε λ,G (k G s)L λs (λ, T) dλdωcosβdA Here, L λs represents the spectral intensity of a black body, emphasizing the relationship between emissivity, beam length, and spectral intensity.

To determine the radiation flow \( d^2 \Phi_{\lambda,G} \) impacting the area element \( dA \) from the entire gas space, it is necessary to integrate \( d^3 \Phi_{\lambda,G} \) over all solid angles associated with \( dA \) and their corresponding beam lengths This integration results in the equation \( d^2 \Phi_{\lambda,G} = \epsilon_{\lambda,G}(k_G s) \cos \beta d\omega L_{\lambda s}(\lambda, T) d\lambda dA \) Similar to the hemispherical spectral emissivity of solids, the spectral emissivity of the gas volume for radiation on the surface element \( dA \) is defined as \( \epsilon_{V \lambda,G}(k_G L_0) = \frac{1}{\pi} \epsilon_{\lambda,G}(k_G s) \cos \beta d\omega \) This spectral emissivity is influenced by the geometry of the gas space and the positioning of the surface element.

Fig 5.72: Gas space with surface element dA and associated solid angle element dω

In a gas hemisphere with radius R, the radiation emitted from a surface element dA at the center is influenced by the optical thickness κ, defined as κ G = k G L 0, where L 0 represents a characteristic length for the gas space The spectral irradiance of this surface element, due to gas radiation, is expressed as M λs = πL λs.

Integration of E λ,G over all wavelengths, taking k G =k G (λ, T, p, p G ) into account, yields the (total) irradiance of dA:

The total emissivity of a gas volume for its radiation on a surface element dA is represented by the formula ε V G = ∫₀∞ ε V λ,G (k G L 0 )M λs (λ, T) dλ This emissivity is influenced by the shape of the gas space, which is indicated by the characteristic length L 0 in the equation.

The emissivity of gas radiation is influenced by the shape of the gas space, making it distinct from the material properties of solid surfaces For instance, when considering radiation from a hemisphere of gas directed at a surface element at its center, the directional spectral emissivity \( \varepsilon_{\lambda,G} \) remains constant across all directions, as the beam length equals the radius \( R \) This leads to the relationship \( \varepsilon_{V \lambda,G} = \varepsilon_{\lambda,G}(k_G R) \) Consequently, \( \varepsilon_{V \lambda,G} \) is defined as the spectral emissivity \( \varepsilon_{\lambda,G} \) of the gas, which can be expressed as \( \varepsilon_{\lambda,G}(k_G R) = 1 - \exp(-k_G R) \) This formulation treats \( \varepsilon_{\lambda,G} \) as a gas property, despite it representing the spectral emissivity of a gas hemisphere radiating towards its center surface element.

The total emissivity of the radiating hemisphere is yielded from (5.190), with ε V λ,G =ε λ,G (k G R) from (5.191), as ε G (T, p, p G , R) = 1 σT 4

Emissivity is considered a fundamental material property, with graphs provided in the following section illustrating the emissivity (ε G) for carbon dioxide (CO2) and water vapor (H2O) According to Beer’s law, the emissivity ε G is influenced by the product of pressure (p G) and the gas constant (R), leading to the relationship ε G = ε G (T, p, p G R).

In section 5.6.4, we demonstrate that the complex determination of the emissivities ε V λ,G and ε V G for any gas space shape can be linked to the standard case of a gas hemisphere The mean beam length s m is defined such that a gas hemisphere with radius R = s m produces the same spectral irradiance on a central surface element as radiation from any gas volume shape on a corresponding surface element From equations (5.188) and (5.191), we establish that ε V λ,G (k G L 0) = ε λ,G (k G s m) = 1−exp(−k G s m) Consequently, equations (5.189) and (5.192) yield ε V G (T, p, p G , L 0) = ε G (T, p, p G , s m).

This method eliminates the need for integration across all solid angles and wavelengths By utilizing the graphs in the subsequent section, one can easily determine ε G, which facilitates the calculation of irradiance E G using equations (5.189) and (5.194), once the mean beam lengths m have been established for the specific problem.

Results for the emissivity

H.C Hottel and R.B Egbert [5.57], [5.58] have critically compared the results available for CO 2 and H 2 O from radiation measurements, and oﬀered best values of the total emissivity in graphs, in whichε CO 2 and ε H 2 O are plotted against the gas temperature T with the product (p CO 2 s m ) or (p H 2 O s m ) as curve parameters. These diagrams have formed the basis of the calculations for gas radiation from

For the past 50 years, the application of CO2 and H2O has involved uncertainties of at least 5% Updated correction factors based on recent data are available in reference [5.37], page 636 The emissivities of CO2 and H2O are essential for accurately calculating the radiation from a gas hemisphere of a specified radius.

The results obtained for the surface element in the center of the sphere can be applied to different gas space shapes by utilizing the mean beam length, s m This concept will be further elaborated in the following section.

The hemispherical total emissivity (ε CO2) of carbon dioxide at a pressure of 100 kPa is depicted in Fig 5.74, showing a slight increase in emissivity with rising pressure Notably, D Vortmeyer [5.59] introduces a complex pressure correction factor that can be disregarded for pressures below approximately 200 kPa.

As H 2 O does not follow Beer’s law,ε H 2 O has to be determined from ε H

The emissivity of water vapor (ε ∗ H 2 O) at a pressure of 100 kPa is depicted in Fig 5.75, with values extrapolated as the water vapor approaches zero To account for partial pressure corrections, refer to Fig 5.76 for the C H 2 O values For pressures exceeding 100 kPa, it is essential to multiply ε ∗ H 2 O by the appropriate pressure correction, as outlined in equation [5.59].

The hemispherical total emissivity (ε CO2) of carbon dioxide at a pressure of 1 bar is influenced by temperature (T) and is characterized by the product of the partial pressure (p CO2) and the mean beam length (s m) Notably, 1 bar is equivalent to 100 kPa or 0.1 MPa.

In mixtures of combustion gases, the simultaneous presence of CO2 and H2O results in an emissivity that is slightly lower than the sum of their individual emissivities, ε CO2 + ε H2O, when calculated at their respective partial pressures This reduction in emissivity is attributed to the overlapping absorption and emission bands of CO2 and H2O H.C Hottel and R.B Egbert established a correction factor to account for this phenomenon.

∆ε that has to be introduced into ε G = ε CO 2 + ε H 2 O − ∆ε (5.196) and plotted it in graphs These can also be found in [5.59].

Equations describing the dependency of emissivities ε CO2 and ε H2O on temperature (T), pressure (p), and gas mass density (p G s m) are essential for model design and process simulations Various researchers have developed these equations, as referenced in sources [5.60] to [5.63] and [5.37], pages 639–641 Despite their complexity, these equations yield accurate results only within specific variable ranges, making them unsuitable for unrestricted application.

2 O for water vapour at p = 1 bar, extrapolated to p H 2 O → 0, as a function of temperature T , with the product of the partial pressure p H 2 O and the mean beam length s m as parameter 1 bar = 100 kPa = 0.1 MPa

The partial pressure correction factor C H 2 O for water vapor is utilized in the equation (5.195) Emissivities for other gases such as SO 2, NH 3, and CH 4 are illustrated graphically in reference [5.59], while similar diagrams for CO, HCl, and NO 2 can be found in reference [5.48].

Emissivities and mean beam lengths of gas spaces

Gas radiation emission is influenced by the size and shape of the gas space, quantified by the irradiance produced at its surface Key equations incorporate spectral emissivity (ε V λ,G) and total emissivity (ε V G), which can be substituted with the emissivity of a gas hemisphere matching the mean beam lengths (m) of the gas space's configuration.

To determine the mean beam length, we begin by calculating the spectral emissivity (ε V λ,G) of a gas volume Subsequently, we will outline the determination of s m and present calculated values of s m for various geometries in Table 5.11 Additionally, we will derive a straightforward approximation formula to calculate s m for gas spaces not included in this collection.

The spectral emissivity ε V λ,G encompasses the radiation emitted from the entire gas space, impacting the surface element dA = dA 2 as depicted in Fig 5.77 The solid angle element dω 1 in the equation ε V λ,G = 1 π ε λ,G (k G s) cos β 2 dω 1 is associated with the beam length s and is constrained by the surface element dA 1, leading to the relationship dω 1 = cos β 1 dA 1 /s 2 Consequently, this results in the equation ε V λ,G = 1 π.

The integration across all solid angles is substituted with the integration over the surface elements dA 1 that are visible from dA 2 Typically, in equation (5.198), this integration encompasses the entire surface area A 1 of the gas space We will demonstrate the integration process using the example of a gas sphere with a diameter D = 2R, which radiates onto an element dA 2 of its surface.

According to Fig 5.78, β 1 = β 2 = β has to be put into (5.198) We choose an annular surface element dA 1 = 2πs sin β s dβ cos β = 2πs 2 sin β cos β dβ

Fig 5.77: Gas space with surface element dA 2 , which receives radiation from the solid angle element dω 1 which is bounded by the surface element dA 1

Fig 5.78: Gas sphere of radius R = D/2 and the surface elements dA 1 and dA 2

According to Fig 5.78, we have cos β = s/2R = s/D, from which sin β dβ = − ds/D follows. With that dA 1 = − 2πs ds With ε λ,G (k G s) from (5.184), we obtain out of (5.198) ε V λ,G = 2

Carrying out the integration gives ε V λ,G (k G D) = 1 − 2

The spectral emissivity of a gas sphere is influenced by its optical thickness, represented as κ G = k G D The characteristic length L 0 of the gas space, as defined previously, corresponds to the diameter D of the sphere.

The symmetry of the sphere indicates that the spectral emissivity ε V λ,G remains constant regardless of the location of the irradiated surface element dA 2 Consequently, the spectral irradiance E λ,G is uniform across the entire sphere's surface Therefore, when considering ε V λ,G (k G D), the mean irradiance E λ,G can be determined for any sized segment of the sphere.

The mean spectral irradiance \( E_{\lambda,G} \) of a finite large surface can be determined by integrating over all surface elements \( dA_2 \) that constitute the surface area \( A_2 \) Additionally, the emissivity \( \varepsilon_{V \lambda,G} \) is derived from a further integration of the relevant equation over the same surface elements, followed by dividing the result by the total area \( A_2 \).

The mean beam length (s m) of a gas space, regardless of its shape, is defined by the spectral irradiance (E λ,G) of a surface element (dA 2) being equal to that of a surface element at the center of a gas hemisphere with radius (R = s m) This relationship is detailed in section 5.6.2, which provides the spectral irradiance for the surface element in question.

By setting this expression equal to E λ,G from (5.188), the relationship (5.193) from section 5.6.2 is obtained, from which s m (k G L 0 )

(5.201) follows According to this, the mean beam length of a particular gas space is not constant, but depends on its optical thickness k G L 0 For a gas sphere s m /D = f(k G D) can be calculated exactly from (5.199).

The mean beam length \( s_m \) is dependent on the optical thickness of the gas space, complicating its application To address this, a constant mean beam length is utilized, optimized to satisfy the equation in the best approximation across all relevant optical thicknesses By maintaining a constant mean beam length \( s_m \), the integration performed over all wavelengths can be simplified, allowing the use of a specific integration that approximates the relationship, thereby enabling the calculation of irradiance \( E_G \) using \( \epsilon_G \) instead of \( \epsilon_{V_G} \).

To establish a constant mean beam length \( s \, m \), we initially examine the scenario of an optically thin gas where \( \kappa_G = k_G L_0 \rightarrow 0 \) In this context, the spectral radiation flow emitted from a volume element of the gas in all directions is represented by the equation \( d^2 \Phi_{\lambda,V} = 4k_G M_{\lambda}(s, T) dV d\lambda \).

The radiation flow emitted from the entire volume of optically thin gas remains strong and unaffected as it passes through, resulting in the expression dΦ λ = 4k G V M λs (λ, T ) dλ Consequently, this leads to the generation of mean spectral irradiance over the entire surface area A of the gas volume.

An optically thin gas hemisphere with a radius of R = s m influences the spectral irradiance as described by the equation E λ (k G s m) = lim k G s m →0 [1 − exp (1 − k G s m)] M λs (λ, T) This relationship simplifies to s ∗ m = 4V /A when the irradiance is set equal to E λ,G, indicating that s ∗ m represents the limit of s m for an optically thin gas as k G L 0 approaches zero.

Values of s ∗ m can be found easily for diﬀerent gas spaces For example a sphere has s ∗ m = 4 πD 3 /6 πD 2 = 2

3 D and for the gas layer between inﬁnitely large, ﬂat plates with separation d we get s ∗ m = 4 d

To account for finite optical thickness, the corrected optical thickness \( s_m \) is defined as \( s_m = C s^*_m = C \frac{4V}{A} \) The constant correction factor \( C \) is selected to ensure that the spectral emissivity \( \epsilon_{\lambda,G}(k_G s_m) \) of the gas sphere closely aligns with \( \epsilon_{V \lambda,G}(k_G L_0) \) within a few percent across a range of optical thicknesses, satisfying equation (5.193) on average This relationship is illustrated in Figure 5.79, which depicts the ratio \( \epsilon_{V \lambda,G}(k_G D)/\epsilon_{\lambda,G}(k_G Cs^*_m) \) against the optical thickness \( k_G D \) When \( C = 1 \), only negative deviations from the ideal value are observed, while correction factors \( C < 1 \) result in reduced deviations, achieving an optimal fit.

C = 0.96, that is for s m = 0.64 D in place of s ∗ m = (2/3)D.

The selection of the correction factor C is flexible within specific boundaries, influenced by the optical thickness range that achieves optimal alignment between ε V λ,G (k G L 0 ) and ε λ,G (k G s m ) Consequently, variations in the values of s m across different literature cases may occur.

The spectral emissivity ratio ε V λ,G (k G D) for a gas sphere with diameter D has been precisely calculated, alongside the mean beam lengths s m for various gas spaces, as detailed in Table 5.11 The analysis indicates that the correction factors C, as defined in equation (5.203), show minimal deviation from 0.9 Consequently, for gas space geometries not listed in Table 5.11, it can be approximated that s m is roughly equal to 0.9s ∗ m, resulting in a value of 3.6 V /A, as expressed in equation (5.204).

With this mean beam length, the mean irradiance E G of the total surface A of the radiating gas space of volumeV is found to be

The emissivitiesε G for CO 2 and H 2 O can be taken from Fig 5.74 to 5.76 in section 5.6.3.

Example 5.14: A hemisphere of radius R = 0.50 m contains CO 2 at p = 1 bar and

T = 1200 K Determine the mean irradiance E CO 2 of its surface and compare this value with the irradiance E CO 2 of a surface element at the centre of the sphere.

We obtain E CO 2 from (5.205), whereby ε CO 2 is taken from Fig 5.74 As no value is given for s m for a hemisphere in Table 5.11, s m is calculated approximately from (5.204) With

V = (2π/3)R 3 and A = 2πR 2 + πR 2 = 3πR 2 , this gives s ∗ m = (8/9)R and s m = 0.80R = 0.40 m With that we have p CO 2 s m = ps m = 0.40 bar ã m ε CO 2 = 0.16 is read oﬀ Fig 5.74. This yields, according to (5.205), the approximate value

Radiative exchange in a gas ﬁlled enclosure

Calculating radiative exchange in a gas-filled enclosure is more complex than in scenarios without an absorbing, self-radiating gas This article explores two straightforward cases involving isothermal gas that exchanges radiation with uniformly heated boundary walls Additionally, we will highlight advanced methods for addressing more challenging radiative exchange issues.

An isothermal gas at temperature T_G is contained within walls that are isothermal at a lower temperature T_W The walls are treated as ideal black bodies, eliminating the need to consider reflection Heat flow, denoted as ˙Q_GW, occurs from the gas to the colder walls through radiative exchange To sustain the gas's temperature T_G, an equivalent amount of energy must be supplied, which can be achieved through a combustion process occurring within the gas space.

The radiation ﬂow emitted by the gas generates a mean irradiance

E G =ε G (T G , p, p G , s m )σT G 4 at the walls, cf (5.205) The black walls with an areaA W absorb the radiation ﬂowA W E G completely The radiation ﬂow emitted by them

The radiation flow A W M s (T W ) = A W σT W 4 is partially absorbed while passing through the gas The unabsorbed portion then impacts the walls, where it is absorbed again Since the walls maintain a uniform temperature T W, this part of the radiation does not contribute to the heat flow ˙Q GW.

In the context of gas absorptivity, \( a_G \) represents the absorptivity of a gas for radiation emitted by a black body at temperature \( T_W \) It is important to note that since the gas does not behave as a grey radiator, \( a_G \) differs from the emissivity \( \varepsilon_G \), except in the specific case where \( T_W = T_G \) Research conducted by H.C Hottel and R.B Egbert has established values for \( a_G \) for CO2 and H2O through absorption measurements, linking these values to emissivity as outlined in equation 5.6.3.

Emissivities are derived from Figures 5.74 to 5.76, considering a wall temperature (T W) and the product (p G s m) adjusted by the factor (T W /T G) For pressures exceeding 1 bar, it is essential to apply the pressure corrections outlined in section 5.6.3 to the emissivity of carbon monoxide (ε CO).

When a gas-filled enclosure is surrounded by grey walls with emissivity ε W (T W), it is essential to account for the energy reflected by these walls Typically, rough, oxidized, and dirty surfaces have emissivities that are close to one, significantly impacting thermal energy interactions.

In furnace applications, the proportion of reflected radiation is often negligible The heat flow (˙Q GW) transferred from the gas to the walls is reduced by a factor between ε W and 1 compared to the heat flow calculated for black walls When ε W exceeds 0.8, the approximation suggested by H.C Hottel and A.F Sarofim becomes applicable.

When analyzing the radiation reflected by walls, the exchange of radiation can be computed similarly to the method outlined in section 5.5.3 As stated in equation (5.143), the heat emitted by the walls to the external environment can be determined through this calculation.

The net flow of radiation, denoted as Q˙ GW, is represented by the equation Q˙ GW = A W (E W − H W), where E W signifies the mean irradiance and H W represents the mean radiosity of the area A W The mean irradiance E W is composed of two components: the fraction generated by gas radiation, E G = ε G σT G^4, and the portion emitted by the walls that is not absorbed by the gas, calculated as (1−a G)H W.

The radiosity H W includes, according to (5.142), the emissive power M W ε W σT W 4 of the wall area and the reﬂected fraction of the irradianceE W :

Solving the two equations, (5.211) and (5.212), forE W andH W , and by putting the result into (5.210), gives

The equation presented is not exact due to the restrictive assumptions made, particularly regarding the geometry of gas spaces, such as spheres or long cylinders, which exhibit nearly constant values of E W and H W on their boundary walls The term (1 − a G )H W in equation (5.211) fails to consider that the radiation passing through the gas comprises various spectral distributions, including grey radiation emitted and reflected by the walls after multiple interactions with the absorbing gas Thus, applying a uniform absorptivity a G is inaccurate However, when the reflected fractions are minimal (indicating a high ε W), using the absorptivity derived from equations (5.207) and (5.208) serves as a reasonable approximation, surpassing the assumption of a grey gas where a G equals ε G.

K Elgeti has presented a precise calculation of radiative exchange, extensively modeling the absorption of reflected radiation in the gas space Additionally, another approach for analyzing spectral absorption bands is detailed in section 17-7 of reference [5.37].

Example 5.15: A cylindrical combustion chamber with diameter D = 0.40 m and length

In a combustion chamber with a length of 0.95 m, the gas temperature reaches 2000 K and the pressure is 1.1 bar The partial pressures of carbon dioxide (CO2) and water vapor (H2O) are measured at 0.10 bar and 0.20 bar, respectively The chamber walls are maintained at a temperature of 900 K, with an emissivity of 0.75 To determine the heat flow transferred from the high-temperature gas to the chamber casing, calculations must consider these parameters, focusing on the thermal interactions between the gas and the chamber walls.

The equations presented in this section facilitate the calculation of the average heat flux across the entire surface of the gas space, as indicated by equation (5.213), represented as ˙ q GW =.

This provides us with an approximate value for the heat ﬂow transferred from the gas to the chamber casing

The mean beam length has to be determined for the calculation of ε G and a G Its limit for an optically thin gas is s ∗ m = 4 V

According to Table 5.11, the ratio C = s m /s ∗ m for a cylinder with L/D = 2 has a value of 0.91 This value applies with good accuracy for a combustion chamber with L/D = 2.375.

For p CO 2 s m = 0.030 bar m, we read oﬀ from Fig 5.74, an emissivity ε CO 2 = 0.036 2 Then from Figs 5.75 and 5.76, we obtain for p H 2 O s m = 0.060 bar m, ε H 2 O = ε ∗ H 2 O C H 2 O = 0.036 0 ã 1.20 = 0.043 2

At a gas pressure of p = 1.1 bar, which is marginally above 1 bar, pressure corrections for ε CO 2 and ε H 2 O are deemed unnecessary To derive ε G from equation (5.196), a correction factor ∆ε is needed Utilizing the graph from reference [5.59], this correction is determined to be ∆ε = 0.0025, resulting in ε G being calculated as 0.0769.

The determination of the absorptivity a G requires the emissivity at the reduced partial pressures, i.e at p CO 2

For T = T W = 900 K, Fig 5.74 gives ε CO 2 = 0.060, which, with (5.207) gives a CO 2 = 0.10 1 For H 2 O, we ﬁnd ε ∗ H

2 O = 0.22 6 and C H 2 O = 1.20, so a H 2 O = 0.38 8 is yielded from (5.208). With ∆ε = 0.003 from [5.59] we get a G = a CO 2 + a H 2 O − ∆ε = 0.48 6

The gas mixture absorbs the radiation coming from the walls far more strongly than it emits itself radiation in comparison to “black” gas radiation.

The mean heat ﬂux transferred by radiation to the walls of the combustion chamber is found, with the values of ε G and a G , from (5.214), to be ˙ q GW = 0.75 ã 5.67 ã 10 −8 W/(m 2 K 4 )

= 45 1 kW m 2 The heat ﬂow we want to determine follows from (5.215) as

According to (1.64) in section 1.1.6, the heat ﬂux ˙ q GW , corresponds to a heat transfer coeﬃcient of radiation of α rad = q ˙ GW

The heat transfer coefficient for convection is comparable in magnitude to the gas radiation effects, which should not be overlooked in heat transfer calculations for combustion chambers and furnaces.

5.6.5.3 Calculation of the radiative exchange in complicated cases

Exercises

When radiation with a vacuum wavelength of λ = 3.0 µm passes through glass with a refractive index of n = 1.52, the propagation velocity c in glass can be calculated using the formula c = c₀/n, where c₀ is the speed of light in a vacuum Consequently, the wavelength λ in glass is determined by λ_M = λ/n Additionally, the energy of a photon can be calculated using the equation E = hc/λ, where h is Planck's constant and c is the speed of light, for both vacuum and glass conditions.

5.2: A Lambert radiator emits radiation at a certain temperature only in the wavelength interval (λ 1 , λ 2 ), where its spectral intensity

The intensity function is defined as L λ (λ) = L λ (λ m ) − a(λ − λ m )², where L λ (λ m ) equals 72 W/(m²·sr), a is 50 W/(m²·m²·sr), and λ m is 3.5 àm It is determined that L λ is zero for wavelengths λ ≤ λ 1 and λ ≥ λ 2 The values of λ 1 and λ 2 need to be identified Additionally, the intensity L and the emissive power M must be calculated Finally, it is essential to find the fraction of the emissive power that is contained within the solid angle defined by (π/3) ≤ β ≤ (π/2) and 0 ≤ ϕ ≤ (π/4).

5.3: What proportion ∆M of the emissive power M of a Lambert radiator falls in the portion of the hemisphere for which the polar angle is β ≤ 30 ◦ ?

5.4: An opaque body, with the hemispherical total reflectivity r = 0.15, reflects diffusely. Determine the intensity L ref of the reflected radiation and the absorbed radiative power per area for an irradiance E = 800 W/m 2

An enclosure maintains a constant temperature while receiving radiation through a small circular opening with a diameter of 5.60 mm The radiation flow entering the enclosure is measured at 2.35 W To determine the temperature of the enclosure walls, one must analyze the relationship between the incoming radiation and the thermal properties of the enclosure.

5.6: A radiator emits its maximum hemispherical spectral emissive power at λ max = 2.07 àm Estimate its temperature T and its emissive power M (T ), under the assumption that it radiates like a black body.

5.7: What temperature does a black body need to be at, so that a third of its emissive power lies in the visible light region (0.38 àm ≤ λ ≤ 0.78 àm)?

5.8: A diﬀuse radiating oven wall has a temperature T = 500 K and a spectral emissivity approximated by the following function: ε λ =

The oven wall is subjected to radiation from glowing coal, with the spectral irradiance E λ proportional to the hemispherical spectral emissive power M λs (T K) of a black body at a temperature of T K = 2000 K To determine the total emissivity ε of the oven wall, calculations must be performed based on this relationship Additionally, the total absorptivity a of the oven wall for the radiation emitted by the coal needs to be calculated, reflecting its capacity to absorb the emitted energy effectively.

In a scenario where a long cylinder is exposed to radiation from a single, perpendicular direction, its surface acts as a grey radiator with a directional total emissivity of ε(β) = 0.85 cos β To determine the reflected fraction of the incident radiative power, one must analyze the interaction between the incident radiation and the cylinder's emissive properties.

In a scenario where a plate with a hemispherical total absorptivity of 0.36 is uniformly irradiated from both sides, and air at a temperature of 30 °C flows over its surfaces, the plate maintains a steady-state temperature of 75 °C The heat transfer coefficient between the plate and the air is measured at 35 W/m²K A radiation detector indicates that the plate emits a heat flux of 4800 W/m² from both sides To determine the irradiance (E) and the hemispherical total emissivity (ε) of the plate, calculations based on these parameters are necessary.

5.11: A smooth, polished platinum surface emits radiation with an emissive power of

M = 1.64 kW/m 2 Using the simpliﬁed electromagnetic theory determine its temperature

T , the hemispherical total emissivity ε and the total emissivity ε n in the direction of the surface normal The speciﬁc electrical resistance of platinum may be calculated according to r e = (0.384 6 (T/K) − 6.94) 10 −7 Ωcm

5.12: A long channel has a hemispherical cross section (circle diameter d = 0.40 m). Determine the view factors F 11 , F 12 , F 21 and F 22 , where index 1 indicates the ﬂat surface and index 2 the curved surface.

5.13: A sphere 1 lies on an inﬁnitely large plane 2 How large is the view factor F 12 ?

5.14: An enclosure is formed from three flat surfaces of finite width and infinite length. The three widths are b 1 = 1.0 m, b 2 = 2.5 m and b 3 = 1.8 m Calculate the nine view factors F ij (i, j = 1, 2, 3).

5.15: A very long, cylindrical heating element of diameter d = 25 mm is h = 50 mm away from a reﬂective (adiabatic) wall, Fig 5.80.

Fig 5.80: Cylindrical heating element above a reﬂective wall

In a system where the heating element is at a temperature of T = 700 K and the surroundings are at T_S = 300 K, we consider the heat transfer solely through radiation among black bodies, including the heating element, wall, and surroundings To determine the wall surface temperature as a function of the coordinate x, we calculate T(x) for specific values: at x = 0, x = h, x = 2h, x = 10h, and as x approaches infinity The view factor between an infinitesimal surface strip dA_1 and a ruled or cylindrical surface is also taken into account for accurate radiation heat transfer analysis.

Here, the ruled surface 2 is produced by parallel straight lines of inﬁnite length, perpendicular to the drawing plane in Fig 5.81 The derivation of this equation is available in [5.37], p 197–199.

Fig 5.81: Surface strip dA 1 of inﬁnitesimal width and a ruled surface 2 generated by parallel, inﬁnitely long straight lines perpendicular to the drawing plane

5.16: A long, cylindrical nickel rod (d 1 = 10 mm) is heated electrically and releases a heat ﬂow per length L of ˙ Q 1 /L = 210 W/m The emissivity of nickel can be calculated from ε(T ) = 0.050 + 0.000 10 (T /K)

A nickel rod is encased in a hollow cylinder with a diameter of 25 mm and an emissivity of 0.88 The outer surface of the hollow cylinder is cooled to a temperature of 290 K To determine the surface temperature of the nickel rod, it is essential to consider that heat transfer occurs solely through radiation.

In a rectangular living room measuring 3.5 m in width, 4.8 m in depth, and 2.8 m in height, underfloor heating maintains a constant temperature of 29 °C at the base area The narrow side wall, which is the external wall, features a window with a temperature of 17 °C The remaining walls and ceiling are treated as adiabatic reradiating surfaces, all functioning as grey Lambert radiators with an emissivity of 0.92 The objective is to calculate the heat flow transferred by radiation from the heated floor to the external wall.

To minimize radiative heat transfer between two large, parallel plates, a thin radiation protection shield with differing emissivities on each side is introduced One side has an emissivity ε S < 0.4, while the other has an emissivity of 2.5 ε S To achieve the smallest possible heat flow, the side with lower emissivity (ε S) should face the hotter plate (T1 > T2) This arrangement reduces heat transfer effectively Additionally, this orientation leads to a higher temperature (T S) of the protection shield compared to the alternative configuration.

In a thermal radiation scenario involving two thin radiation protection shields, A and B, positioned between two large parallel plates at temperatures T1 = 750 K and T2 = 290 K, both with an emissivity of ε = 0.82, the temperatures of the shields can be calculated The resulting temperatures TA and TB are determined through radiative heat transfer equations Additionally, the heat flux ˙q exchanged between the plates is calculated, revealing the energy transfer due to radiation When the shields are removed, and plate 1 is subjected to the heat flux ˙q while plate 2 remains at T2 = 290 K, the new temperature T1 of plate 1 can also be evaluated based on the heat transfer principles.

Older houses typically feature walls constructed as two shells, with an air column situated between two brick layers The thermal properties of these walls include a thermal conductivity of λ = 0.95 W/K m, and the thicknesses of the brick walls are δ₁ = 0.24 m and δ₂ = 0.115 m, while the air gap measures δ = 0.060 m The heat transfer coefficients are αᵢ = 7.5 W/m² K for the inner surface and αₐ = 18.0 W/m² K for the outer surface The indoor temperature is maintained at ϑᵢ = 22.0 °C, contrasting with an outdoor temperature of ϑₐ = -5.0 °C Additionally, the thermal conductivity of air is noted as λₐᵢᵣ = 0.0245 W/K m.

In a double shell brick wall with an air gap of width δ, an effective thermal conductivity λ_eff is introduced, accounting for the effects of radiation while neglecting free convection within the air gap To analyze the thermal performance, λ_eff and the heat flux ˙q transferred by the wall must be calculated When the air gap is filled with insulating foam, which has a thermal conductivity λ of 0.040 W/K m, the heat flux ˙q is significantly altered, demonstrating the impact of insulation on thermal transfer efficiency.

In a system where carbon dioxide (CO2) flows through a long, cooled pipe with an internal diameter of 0.10 meters and a velocity of 20 m/s, the mean gas temperature is 1000 °C while the pipe wall temperature is 500 °C The emissivity of the pipe wall is 0.86 To assess the heat transfer mechanisms, it is essential to calculate the convective heat transfer coefficient (α) and the radiative heat transfer coefficient (α_rad) for the gas Relevant property data for CO2 at 1000 °C includes a thermal conductivity (λ) of 0.0855 W/K·m, kinematic viscosity (ν) of 117 × 10^-6 m²/s, and a Prandtl number (Pr) of 0.736.

Introduction to tensor notation

In the derivation of balance equations, tensor notation is utilized for clarity and simplicity, focusing on Cartesian coordinates This article highlights the essential aspects of Cartesian tensor notation necessary for deriving balance equations, while extensive literature is available for those seeking more in-depth information For instance, the velocity of a point mass is represented as a vector in a Cartesian coordinate system, expressed through its components: w(w x, w y, w z).

If the unit vector in a cartesian coordinate system is indicated bye x,e y,e z, it holds that w=w x e x+w y e y+w z e z

In tensor notation, the indicesx,y,z are replaced by the indices 1, 2, 3 and we write instead w=w 1 e 1+w 2 e 2+w 3 e 3 3 i=1 w i e i

The velocity vector \( w \) is fully defined by its components \( w_i \) for \( i = 1, 2, 3 \) In tensor notation, it is represented as \( w_i \) Similarly, the position vector \( x(x, y, z) \) is characterized by its components \( x_1 = x \), \( x_2 = y \), and \( x_3 = z \), denoted in tensor notation as \( x_i \) Thus, a vector can be represented using a single index.

Tensors can be classified by their levels, with zero level tensors being scalars that remain unchanged when transitioning between coordinate systems Common examples of scalars include temperature (ϑ), pressure (p), and density, which do not require an index for their identification.

First-level tensors, known as vectors, are represented by a single index, while second-level tensors are defined by two indices An example of a second-level tensor is the stress tensor, which comprises nine components: τ11, τ12, τ13, τ21, and so on, up to τ33 This tensor is typically abbreviated as τji, where both indices j and i can take values of 1, 2, or 3.

In tensor calculations, a free index is defined as an index that appears only once in a term of an equation, allowing it to be replaced by any other index It is essential for all terms in an equation to agree on their free indices For example, the equation \( a_i = c b_i \) (where \( c \) is a constant scalar) indicates that the vectors \( a_i \) and \( b_i \) differ only in magnitude while sharing the same direction, leading to relationships such as \( a_1 = c b_1 \), \( a_2 = c b_2 \), and \( a_3 = c b_3 \) Additionally, the internal product (or scalar product) of two vectors can be expressed in index notation as \( a \cdot b = \sum_{i=1}^{3} a_i b_i \).

In the context of internal products, a "summation convention" has been established where an index appearing twice in a term indicates that it should be summed over that index, known as a bound index This bound index cannot be substituted with another index, and the summation symbol is omitted For example, this convention leads to the expression \( a_i b_i = a_1 b_1 + a_2 b_2 + a_3 b_3 \).

Diﬀerentiation leads to a tensor that is one order higher So the gradient of a scalarp gradp=∇p=e 1

The vector ∂x 3 (A.3) consists of three components denoted as ∂p/∂x i, where i ranges from 1 to 3, and is commonly represented in tensor notation as ∂p/∂x i When differentiating a vector w j, each of its components—w 1, w 2, and w 3—can be differentiated concerning the position coordinates x 1, x 2, and x 3 This process results in the formation of a second-level tensor.

∂x i (i= 1,2,3 ; j= 1,2, 3) , (A.4) that consists of 9 components On the other hand, the divergence of a vector is a scalar, divw=∇ ãw= ∂w i

The formulation of a divergence results in a tensor that is one order lower than the original tensor The Kronecker delta, denoted as δ ij, serves as a key operator, where δ ij equals 1 when i equals j and 0 otherwise This unit tensor property is exemplified by the equation δ ij b j = b i, which can be verified by expanding the equation for each component, confirming that δ 1j b j = b 1, δ 2j b j = b 2, and δ 3j b j = b 3, thereby establishing the relationship δ ij b j = b i.

Relationship between mean and thermodynamic pressure

The mean pressure, defined as ¯p = -1/3δjiτkk, encompasses only normal stresses To connect mean pressure with thermodynamic pressure, we examine a stationary cubic fluid element at temperature T and specific volume v Initially, at time t = 0, the thermodynamic pressure p is present within the element If the external mean pressure ¯p exceeds p, the cube compresses; conversely, if ¯p is less than p, it expands The work done by the external pressure ¯p is represented as -p¯dV, which corresponds to the work during the gas volume change -pdV and the dissipated work Thus, the relationship is expressed as dW = -p¯dV - pdV + dW diss, where the dissipated work is defined as dW diss = - (¯p - p) dV.

According to the second law, a positive outcome is observed since when ¯p > p, the volume change (dV) is negative, while when ¯p < p, dV is positive Additionally, the change in volume (dV) is derived from transport theory, where Z represents volume (V) and z is defined as V/M, resulting in the expression dV/dt.

∂x i dV The dissipated work can also be written as dW diss =−(¯p−p)

The rate of volume change, represented as dV/dt or ∂w_i/∂x_i, is a monotonically decreasing function of overpressure (¯p−p), as illustrated in Fig A2 This indicates that greater overpressure leads to a more rapid reduction in the volume of the cube.

Fig A.1: Interrelation between mean and thermodynamic pressure

Fig A.2: Expansion as a function of the over pressure where the speed of the volume change is not that fast, the curve in Fig A2 may be replaced by a straight line: ¯ p−p=−ζ ∂w i

The factor deﬁned by this, ζ > 0, is the volume viscosity (SI units kg/s m).

To determine the properties of substances, experimental methods or statistical thermodynamics must be employed, particularly for those with simple molecular structures It is evident that the mean and thermodynamic pressures align precisely only when ζ equals 0 or when the fluid is incompressible, indicated by ∂w i /∂x i = 0.

Navier-Stokes equations for an incompressible ﬂuid of constant viscosity

ible ﬂuid of constant viscosity in cartesian coordinates

The mass force is the acceleration due to gravityk j =g j x 1 =x-direction:

Navier-Stokes equations for an incompressible ﬂuid of constant viscosity

ible ﬂuid of constant viscosity in cylindrical coordinates

The mass force is the acceleration due to gravityk j =g j r-direction:

Entropy balance for mixtures

The Gibbs’ fundamental equation for mixtures, du=T ds−pdv+

M˜ K dξ K or du dt = T ds dt − pdv dt +

M˜ K dξ K dt , (A.14) taking into account dv dt =−1 2 d dt = 1

∂x i + ˙Γ K (A.15) can be rearranged into du dt = Tds dt −p∂w i

Due toà K /M˜ K =h K −T s K this delivers du dt = T ds dt −p∂w i

Putting this into the energy equation (3.81), taking into account the following, yields ˙ q = ˙q i +

We write the following for

∂x i (J S,i − w i s) + ˙σ (A.17) yields the entropy ﬂowJ S,i (SI units W/m 2 K)

K j K,i ∗ s K + w i s (A.18) and the entropy generation ˙σ (SI units W/m 3 K) ˙ σ=−q˙ i

Entropy flow is influenced by heat or material movement, while entropy generation arises from heat flow within a temperature gradient, diffusion driven by mass forces and variations in chemical potential, as well as mechanical dissipation and chemical reactions.

Relationship between partial and speciﬁc enthalpy

When the following summation is written out with the abbreviationδ =∂/∂x i it looks like

K=1 j ∗ K,i = 0, also j N,i ∗ = −j 1,i ∗ −j 2,i ∗ − .−j ∗ N−1,i The sum can also be written as j 1,i ∗ δ(h 1 −h N ) +j 2,i ∗ δ(h 2 −h N ) + j ∗ N−1,i δ(h N −1 −h N )

In the thermodynamics of mixtures [3.1], page 114, it can be shown, that for speciﬁc partial quantities of state the following relationship is valid h A −h N ∂h

On the other hand, the enthalpy of a mixture is given by h=h(T, p, ξ 1 , ξ 2 , ξ N − 1 ) and therefore

So the sum can also run from K = 1 to K = N, instead of from K = 1 to

K=N−1, as the last term withK=N is zero.

Calculation of the constants a n of a Graetz-Nusselt problem (3.246)

Multiplication of (3.247) with ψ m (r + ) (1−r +2 )r + and integration between the limitsr + = 0 andr + = 1 yields

All the integrals on the right hand side disappear whenm=n, yielding (3.248).

To demonstrate the application of the eigenvalues β_m and β_n, we start with the equations d/dr + (r + ψ_m) = -β_m²(1 - r + 2)r + ψ_m and d/dr + (r + ψ_n) = -β_n²(1 - r + 2)r + ψ_n By multiplying the first equation by ψ_n and the second by ψ_m, we derive the equations ψ_n d/dr + (r + ψ_m) = -β_m²(1 - r + 2)r + ψ_mψ_n and ψ_m d/dr + (r + ψ_n) = -β_n²(1 - r + 2)r + ψ_nψ_m These can be reformulated as d/dr + (r + ψ_nψ_m) - r + ψ_nψ_m = -β_m²(1 - r + 2)r + ψ_mψ_n and d/dr + (r + ψ_nψ_m) - r + ψ_nψ_m = -β_n²(1 - r + 2)r + ψ_nψ_m.

Subtraction of both equations and integration between the limits r + = 0 and r + = 1 yields r + (ψ n ψ m −ψ n ψ m ) 33 3 1

The left hand side disappears because of ψ m (r + = 1) = ψ n (r + = 1) = 0 and ψ m (r + = 0) =ψ n (r + = 0) = 0 This means that the integral on the right hand side must also vanish whenn=m.

Table B 1: Properties of air at pressure p = 1 bar ϑ c p β λ ν a P r

Table B 2: Properties of water at pressure p = 1 bar

Table B 3: Properties of water in the saturated state from the triple point to the critical point ϑ p c p c p β β ∆h v

◦ C bar kg/m 3 kJ/kg K 10 −3 /K kJ/kg

Table B 4: Properties of ammonia at pressure p = 1 bar ϑ c p β λ ν a P r

Table B 5: Properties of carbon dioxide at pressure p = 1 bar ϑ c p β λ ν a P r

Table B 6: Properties of nitrogen at pressure p = 1 bar ϑ c p β λ ν a P r

Table B 7: Properties of oxygen at pressure p = 1 bar ϑ c p β λ ν a P r

Table B 8: Properties of helium at pressure p = 1.01325 bar = 1 atm

Table B 9: Diﬀusion coeﬃcients at pressure p = 1.01325 bar = 1 atm a) Gases

The pressure and temperature dependency in the ideal gas state can be estimated from D ∼ T 1.75 /p.

Table B 10: Thermophysical properties of non-metallic solids at 20 ◦ C.

Table B 11: Thermophysical properties of metals and alloys at 20 ◦ C.

Bronze (84 Cu, 9 Zn, 6 Sn, 1 Pb) 8.8 0.377 62 18.7

Table B 12: Emissivities of non-metallic surfaces ε n Total emissivity in the direction of the surface normal, ε hemispherical total emissivity.

Table B 13 presents the emissivities of metal surfaces, highlighting both the total emissivity in the direction of the surface normal (ε) and the hemispherical total emissivity (ε) When a temperature range is specified, the emissivity values can be linearly interpolated between the provided data points.

Aluminium, polished 20 0.045 rough 75 0.055 0.07 rolled smooth 170 0.039 0.049 commercial foil 100 0.09 oxidised at 600 ◦ C 200 600 0.11 0.19 strongly oxidised 100 500 0,32 0.31

Lead, not oxidised 127 227 0.06 0.08 grey oxidised 20 0.28

Chromium, polished 150 0.058 oxidised by red heat 400 800 0.11 0.32

Iron, polished − 73 727 0.04 0.19 0.06 0.25 oxidised − 73 727 0.32 0.60 polished with emery 25 0.24 electrolytically polished 150 0.128 0.158 casting skin 100 0.80 rusted 25 0.61 very rusted 20 0.85

Copper, polished 327 727 0.012 0.019 oxidised 130 0.76 0.725 highly oxidised 25 0.78

DIN 1.4301=AISI 304 polished 50 200 0.111 0.132 sand blasted, R a = 2.1 àm − 50 200 0.446 0.488

1.3: By diﬀerentiation of ˙ q = − λ(ϑ) dϑ/dx follows dλ dϑ = − λ d 2 ϑ/ dx 2 ( dϑ/ dx) 2 < 0 λ decreases with rising temperature.

1.6: a) ϑ max = ϑ 0 along the hypotenuse y = x; ϑ min = ϑ 0 − ϑ 1 at x = l, y = 0 ϑ 1 = ϑ max − ϑ min b) gradϑ = 2ϑ 1 l

1.13: Q/L ˙ = 514 W/m; ϑ 3 = 193 ◦ C lies below the permitted value of 250 ◦ C.

1.14: ϑ F = 17.0 ◦ C; t ∗ = 3.16 h = 3 h 10 min These values are valid under the assumption that one of the circular ends is adiabatic.

; ϑ + 2 (z) = C 1 ϑ + 1 (z) + ε 1 − 1 ε 1 = ε 1 (N 1 , C 1 ) is to be calculated according to Table 1.4.

1.16: Countercurrent: kA = 423 W/K; Cross-ﬂow with one tube row: kA = 461 W/K, Cross countercurrent ﬂow with two tube rows as in Fig 1.59: kA = 433 W/K.

1.17: Under the assumptions mentioned ˙ N A = u j A 2πrL = − D∂c A /∂r2πrL = const and with that d ˙ N A dr = 0 = ∂

Integration between the limits c A (r 0 ) = c AW and c A (r 0 +δ) = c Aδ yields the concentration proﬁle c A − c AW c Aδ − c AW = ln r/r 0 ln [(r 0 + δ)/r 0 ] From this, by diﬀerentiation and introduction into the equation for ˙ N A the expression given is found.

In a steady-state solution, the diffusion flow is determined using the equation where the molar flux of component A is 3.848 × 10⁻⁶ kmol/m²·s, resulting in a mass flux of 1.773 × 10⁻⁴ kg/(m²·s) Over a time interval dt, an enthalpy amount equivalent to ˙mA A dt evaporates This analysis incorporates the variable y(t) for the concentration, leading to the relationship dt = -L.

1 ln(p B2 /p B1 ) dy = B [y 2 − y(t)] dy Integration and putting in the numerical values gives t = 20.57 h.

1.19: Equimolar counterdiﬀusion prevails in the tubes It therefore follows from (1.176): ˙ N A =

− cDA∂˜ x A /∂y As ˙ N A is constant, a linear concentration drop over the length L of the tubes develops It then holds that

The mole fraction of ammonia in the pipes is given as ˜x Aa = 1, while in the air it is ˜x Ae = 0 The ammonia loss rate is calculated to be ˙N A = 3.99 × 10^−13 kmol/s, resulting in a mass flow rate of ˙M A = ˜M A N ˙A = 6.78 × 10^−12 kg/s The total ammonia flow through the pipes is ˙M = 1.91 × 10^−3 kg/s, corresponding to ˙N = 1.123 × 10^−4 kmol/s Additionally, the amount of air that enters the ammonia system is also measured.

N ˙ B = − N ˙ A = − 3.99 ã 10 −13 kmol/s, ˙ M B = ˜ M B N ˙ B = 1.16 ã 10 −11 kg/s The mole fraction of air in the pipes is extremely small, namely ˜ x B = | N ˙ B | / N ˙ = 3.55 ã 10 −9

1.20: Unidirectional mass transfer prevails From (1.195), we have ˙ m A = ˜ M A n ˙ A = ˜ M A p

In order to use (1.195a), the moisture content is required

1.21: From (1.210) and (1.203), using steam tables, the solution is found by trial and error to be ϑ I = 2.56 ◦ C It is p s (2.56 ◦ C) = 7.346 mbar.

1.22: We will assume a small blowing rate The lowest temperature is the wet bulb temperature ϑ I In (1.211), the factor is

M ˜ A c pA = 1.097 ã 10 −3 mol K/J This allows (1.211) to be written as follows, as X Aδ = 0,

Using a steam table, the wet bulb temperature is determined to be 65.1 °C through trial and error, effectively satisfying the required conditions Additionally, the water fed to the chamber is calculated to be 0.0227 kg/m²s.

1.23: The amount of benzene transferred is ∆ ˙ N B = 11.88 kmol/h The molar ratio of the gas in cross section e is ˜ Y e = 2.5 ã 10 −3 According to (1.224), ˜ X o = 0.216.

Chapter 2: Heat conduction and mass diﬀusion

2.2: At x = 0, ϑ(x) has a horizontal tangent The tangent at x = δ intersects the horizontal ϑ = ϑ F at point R with the abscissa x R = δ + λ/α =

1 + Bi −1 δ = 1.667 δ The tangent to the ﬂuid temperature plot at x = δ intersects the line ϑ = ϑ F at x F = δ + λ F /α =

2.3: a) The plate heats up, because ∂ 2 ϑ + /∂x +2 > 0. b) x + T = 1/2; (∂ϑ/∂t) max = 4.40 K/s. c) x + min = 0.3778. d) B(t) = 0.850 exp

2.5: a) ϑ 0 = 55.39 ◦ C, ϑ L = 37.39 ◦ C. b) ˙ Q 0 = 3.204 W, ˙ Q L = 0.0622 W. c) The results do not diﬀer within the numbers given The simple calculation with the replacement bolts of length L C is very exact.

2.7: a) 87 ﬁns/m This means the heat ﬂow increases by a factor 6.019 b) 1.40.

2.10: Surface temperature: 73.1 ◦ C; at 10 cm depth: 27.2 ◦ C.

2.11: Surface temperature: 45.8 ◦ C; at 10 cm depth: 24.3 ◦ C.

2.12: a) The amplitude at a depth of 1 m is only 3.7 ã 10 −4 K. b) Highest temperature 18.6 ◦ C on 2nd October, lowest temperature 1.4 ◦ C on 2nd April.

2.13: a) α = 19.3 W/m 2 K b) The temperatures of the insulated surface are ϑ(t 1 ) = 146.4 ◦ C and ϑ(t 2 ) = 112.3 ◦ C.

2 (1 + Bi ∗ ) = 0.4137 is satisfied Temperature profile at time t ∗ = 15 min: ϑ 15 1 = 80.0 ◦ C; ϑ 15 2 = 63.9 ◦ C; ϑ 15 3 = 50.7 ◦ C; ϑ 15 4 = 40.8 ◦ C; ϑ 15 5 = 34.9 ◦ C; ϑ 15 6 = 30.8 ◦ C. b) Steady-state temperature profile (t → ∞ ): ϑ ∞ 1 = 80.0 ◦ C; ϑ ∞ 2 = 71.6 ◦ C; ϑ ∞ 3 = 64.1 ◦ C; ϑ ∞ 4 = 57.4 ◦ C; ϑ ∞ 5 = 51.3 ◦ C; ϑ ∞ 6 = 45.7 ◦ C.

The first term of the series yields t + D = 1.039, resulting in t = 14.1 days Subsequent terms in the series are negligible, confirming that only the first term is necessary for accurate calculations.

In section 2.22, diffusion is analyzed in relation to the x- and y-coordinates The heat conduction problem discussed in section 2.3.5 reveals that for a block with dimensions 2X and 2Y, the temperature profile is defined by the equation (2.191), which illustrates the relationship between the temperature at the surface and the internal temperature.

For the diﬀusion problem, in the centre of the rod x = y = 0 and for βX/D = β Y /D → ∞ it correspondingly holds that ξ + A = ξ A (x = y = 0) − ξ AS ξ Aα − ξ AS = c + Pl

Under the assumption still to be checked, i.e that the ﬁrst term of the series from the solution of the previous exercise is satisfactory, follows ξ A + =

Which gives t + D = 0.569 and t = 7.7 days It is easy to prove that the remaining series terms are actually negligible in comparison with the ﬁrst term.

2.23: a) We have ˙ M La = 0.7 kg/s, ˙ M Ga = 6.0 kg/s and ˙ M W = 0.532 kg/s. b) The required time is obtained from (2.389) In which c Aα = 0 and so c Am /c AS = ξ L Aa /ξ A0 = 0.6

It then follows from (2.390) that

The solution is found by trial and error to be t + D = 0.0485 The ﬁrst three terms of the series suﬃce They give t = 72.7 s. c) L = tw = 7.27 m.

2.24: We can approximately say that Bi D → ∞ In addition to this the surface of the spheres are immediately completely immersed in the water, ξ A0 = 1 Therefore c A0 = ξ A0 / M ˜ A = 55.5 kmol/m 3

− i 2 π 2 t + D follows, with c Aα = 0: c Am /c A0 = 0.6927 and c Am = 38.42 kmol/m 3 Each sphere takes in 0.241 g of water.

Chapter 3: Convective heat and mass transfer Single phase

3.1: From − q ˙ j = λ ji ∂ϑ/∂x i follows, under consideration from λ 12 = λ 21 ,

In steady heat conduction, we generally have ∂ q ˙ j /∂x j = 0 as the plate is thin in the x 2 -direction, then ∂ϑ/∂x 2 = 0 and the diﬀerential equation for steady heat transfer looks like

3.2: For the model (index M) and the original (index O) it has to hold that

N u = f (Re, P r) a) It has to be P r M = P r O With P r O = 4.5, the associated temperature is T = 311 K. b) We have (w M ) 1 = 0.0355 m/s and (w M ) 2 = 0.355 m/s. c) Because N u M = N u O or (α M d M )/λ M = (α O d O ) /λ O , α O = 484 W/m 2 K.

3.3: For the model (subscript M) and the original (subscript O), according to (3.333), it holds that

N u m ∼ Gr 1/4 with N u m = α m L/λ and Gr = β ∞ (∆ϑ) g L 3 /ν 2 This then gives α mO α mM =

3.4: Firstly, a dimension matrix is set up L indicates the dimension of a length:

L = dimL ; correspondingly t = dim t, T = dim ϑ, M = dimM The dimension matrix looks like

To determine the rank \( r \) of a matrix, we utilize equivalent transformations, which involve creating linear combinations of its rows or columns This process continues until the diagonal of a sub-matrix consists solely of ones, while the adjacent diagonals are filled with zeros The rows are labeled as \( Z_1, Z_2, \) and so forth, leading to the formation of a new matrix represented by dashes For instance, we have \( Z_4 = -Z_4 \) and \( Z_3 = Z_3 + Z_4 \).

− Z 2 + 3Z 4 The new matrix looks like

It already contains only ones in the front main diagonal By a further transformation

The left-hand sub-matrix features ones along the main diagonal, while the neighboring diagonals consist solely of zeros The matrix's rank is r = 4, indicating that there are four linearly independent rows, with no additional rows convertible through equivalent transformations Following Buckingham's theorem, the number of π-quantities is calculated as m = n − r, where n represents the original variables, which in this instance is n = 7 Thus, m = 7 − 4 = 3 π-quantities are derived from the matrix, specifically: π 1 = νL −1 w m −1 −0 λ −0 = ν w m L = 1/Re, π 2 = cL 1 w 1 m 1 λ −1 = c λ w m L = w m L a = ReP r, and π 3 = α m L 1 w m 0 0 λ −1 = α m L λ = N u.

By equivalent transformations, new rows Z i are obtained from the original rows Z i (i =

1, 2, 3) The following equivalent transformation of the rows Z 1 = Z 1 + Z 2 + 3Z 3 , Z 2 =

The rank of the matrix is r = 3 This gives m = n − r = 5 − 3 = 2 The dimensionless quantities are π 1 = W d 0 w 0 −1 L = W

L π 2 = gd 1 w −2 0 L = gd w 2 b) We have π 2 = f(π 1 ) or gd/w 2 = f ( W / L ).

3.6: Introducing the velocity proﬁle in the integral condition that follows from (3.165) for the momentum d dx

2 The integral has the value 1 − π/4 With that, after integration δ = π

= 4.795 x Re −1/2 x This result diﬀers from (3.170) by the fact that in place of the factor 4.64 in (3.170), the factor 4.795 appears.

To determine if the flow is turbulent, the Reynolds number is calculated using the formula Re = w ∞ L/ν at the end of the plate Given that viscosity is only available at a pressure of 0.1 MPa, we assume η = ν = const for a constant temperature This leads to the relationship p 1 /(RT 1 )ν 1 = p 2 /(RT 2 )ν 2, allowing us to express ν 2 as ν 1 p 1 /p 2 under constant temperature conditions With ν 1 (p 1 , ϑ m ) calculated as 30.84 × 10 −6 m²/s at p 1 = 0.1 MPa and a mean temperature of ϑ m = 162.5 °C, we find ν 2 to be 30.84 × 10 −5 m²/s, which is essential for determining the associated Reynolds number.

Re = 3.243 ã 10 4 The ﬂow is laminar to the end of the plate The mean Nusselt number is, see also Example 3.8, N u m = 105.5 This yields α m = 3.84 W/m 2 K and ˙ q = 1056 W/m 2

The Reynolds number at the lake's end is Re = 2.606 × 10^6, indicating that the flow starts as laminar and transitions to turbulent flow after a distance of x_cr = 3.84 m The Sherwood number, derived from equation (3.208) by substituting the Nusselt number, yields Sh_m,lam = 904.1 for laminar flow and Sh_m,turb = 3592 for turbulent flow.

We also have p A0 = p Ws and therefore ˙ m A = β m p Ws

3.9: M ˙ = V ˙ = 0.25 kg/s Further, the energy balance holds

3.10: The solar energy caught by the reﬂector is transferred to the absorber tube ˙ q S sL = ˙ q d o πL/2

The heat ﬂux absorbed by the absorber tube is therefore ˙ q = 1.567 ã 10 4 W/m 2 It serves to heat the water: ˙ q d o πL/2 = ˙ M c p (ϑ e − ϑ i ) with ˙ M = w m d 2 i π/4 = 8.468 ã 10 −2 kg/s.

L = 2 ˙ M c p (ϑ e − ϑ i ) ˙ q d o π = 13.3 m The wall temperature ϑ 0 at the outlet follows from ˙ q = α (ϑ 0 − ϑ F ) = α (ϑ 0 − ϑ e ) as ϑ 0 = ˙ q/α + ϑ e = 139.4 ◦ C.

3.11: The heat losses are yielded from the energy balance as ˙ Q = ˙ M c p (ϑ i − ϑ e ) = 11.97 kW. The heat ﬂux transferred at the end of the tube is ˙ q = k(ϑ e − ϑ 0 ) with 1 k = 1 α + 1 α e

Here, α is the heat transfer coeﬃcient of the superheated steam on the inner tube wall.

We set d i ≈ d o here For the calculation of α, the Reynolds number has to be found ﬁrst

The ﬂow is turbulent In addition L/d > 100 We obtain from (3.262) N u m = 1145, α m = 1267 W/m 2 K and ˙ q = 1779 W/m 2 Furthermore, it holds that ˙ q = α m (ϑ e − ϑ 0 ) and therefore ϑ 0 = ϑ e − q/α ˙ m = 118.6 ◦ C.

3.12: The speciﬁc surface area of the particle from (3.269) is a P = 6(1 − ε)/d = 180 m 2 /m 3 The arrangement factor from (3.271) is f ε = 1.9, the Reynolds number

Re = w m d εν = 6.098 ã 10 3 The Nusselt number is calculated according to section 3.7.4, No 5, from

N u 2 m,lam + N u 2 m,turb with N u m,lam = 46.47 and N u m,turb = 35.57 to be N u m = 60.52 This yields α m = 79.28 W/m 2 K The total particle surface area is, according to (3.267), nA P = a P V = a P A 0 H =

117 m 2 a) With that ˙ Q = α m nA P (ϑ A − ϑ 0 ) = 171601 W ≈ 172 kW. b) The amount of water evaporated due to the heat fed is

M ˙ W = ˙ Q/∆h v = 7 ã 10 −2 kg/s The amount of water evaporated due to the partial pressure drop

R L T (X WS − X ) is around two orders of magnitude smaller, and can be neglected This can be checked using the mass transfer coeﬃcients and speciﬁc humidity X = 0.622 p WS /(p/ϕ − p WS ) from section 3.7.4, No 5.

3.13: The height H mf of a ﬂuidised bed at the ﬂuidisation point follows from the condition of constant sand mass

The total pressure drop in a fluidized bed is approximately given by ∆p = S (1 − ε mf )gH, where ε mf represents the minimum fluidization void fraction This relationship indicates that the pressure drop remains nearly constant due to the constancy of (1 − ε mf )H Experimental validation supports this finding, and at the fluidization point, (1 − ε S )H 0 can be used to represent the quiescent sand layer, resulting in a pressure drop value of approximately ∆p ∼ = 7848.

The pressure at the blower outlet, denoted as p2, is determined by the pressure at the inlet of the fluidized bed, p1, plus the pressure drop, Δp This results in a total pressure of p2 = p1 + Δp, which equals 107,848.

Pa The mean pressure of the air in the ﬂuidised bed is mG = p m /RT = 0.322 kg/m 3 The

ﬂuidisation velocity follows from (3.275) This includes the Archimedes number formed with the mean density mG

According to (3.275) it is Re mf = 0.310; and w mf = 0.094 m/s The actual velocity is w m = 10w mf = 0.94 m/s The mass ﬂow rate of the air at the inlet is

The required blower power is

M c ˙ p (ϑ 1 − ϑ 1 ) P when ϑ 1 is the ﬁnal temperature of the compression It follows from this, that ϑ 1 = 299.5K = 26.4 ◦ C The heat ﬂow fed in is ˙ Q = ˙ M L c pmG (ϑ 2 − ϑ 1 ) = 2126 kW.

3.14: The density of the air over the ground is

= 1.2084 kg/m 3 , that of the waste gases is

R G T G1 = 1.1946 kg/m 3 G1 < A1 ; the exhaust gases can rise They would no longer rise if G1 ≥ A1 or

= 438.1K = 164.9 ◦ C , if the exhaust gas temperature was to lie below 165 ◦ C.

It holds for the air that dp = − g dx and so v dp = − g dx With v = R A T A /p follows dp p = − g

R A T A dx Through integration the barometric height formula is obtained p 2 = p 1 exp

The air pressure at 100m is p 2 = 0.09882 MPa The density of the air at 100 m height is

R A T A = 1.194 kg/m 3 The density of the exhaust gases at 100m follows from

G1 = 1.184 kg/m 3 The exhaust gases are lighter than air at 100m height, they can rise further.

Gr = β ∞ (ϑ 0 − ϑ ∞ )gL 3 ν 2 = 4.49 ã 10 9 and Ra = GrP r = 3.20 ã 10 9 Then, from (3.331) the mean Nusselt number for free ﬂow is found to be N u mF = 168.8 Furthermore

Giving N u m,lam = 142.7 and N u m,turb = 203.99 The mean Nusselt number for forced convection is obtained as

As free and forced ﬂow are directed against each other, the minus sign in (3.337) holds,

N u m = 219.7 and α m = 5.97 W/m 2 K The two sides of the plate release the heat ﬂow

3.16: In this section 3.9.3, page 385, No 5 is used We have

At the end of the plate x 1 = 0.4 m, Ra x1 = 1.84 ã 10 9 The ﬂow at the end of the plate is just about still laminar It is

It further follows from ˙ q = α(ϑ 0 − ϑ ∞ ) : ϑ 0 = q ˙ α + ϑ ∞ = 7.649 W m 1/5 x 1/5 + 283.15 K The wall temperature increases with x 1/5 , and at the end of the plate x 1 = 0.4 m is ϑ 0 = 289.5 K = 16.4 ◦ C

The cooling process is influenced by heat transfer in free flow, with specific equations applicable for different cylinder orientations For vertical cylinders, the relevant equation is (3.331), while for horizontal cylinders, reference should be made to equation No 3 in section 3.9.3, page 385 The Rayleigh number for vertical cylinders (Ra ver) is based on the cylinder's height, whereas the Rayleigh number for horizontal cylinders (Ra hor) is determined by the cylinder's diameter.

Ra ver = β ∞ (ϑ o − ϑ ∞ )gL 3 ν 2 P r = 8.81 ã 10 6 and Ra hor = 5.64 ã 10 5

The calculations reveal that the Nusselt number for vertical flow (N u mver) is 30.14, resulting in a heat transfer coefficient (α mver) of 4.72 W/m²K In contrast, the Nusselt number for horizontal flow (N u mhor) is 12.40, leading to a higher heat transfer coefficient (α mhor) of 4.86 W/m²K, indicating that α mhor exceeds α mver Additionally, heat is dissipated through the ends of the horizontal cylinder, allowing it to cool more efficiently when positioned horizontally.

Tiêu đề	Heat and Mass Transfer
Tác giả	Hans Dieter Baehr, Karl Stephan
Người hướng dẫn	Dr.-Ing. E. h. Dr.-Ing. Hans Dieter Baehr, Dr.-Ing. E. h. mult. Dr.-Ing. Karl Stephan
Trường học	University of Hannover, University of Stuttgart
Chuyên ngành	Thermodynamics
Thể loại	book
Năm xuất bản	2006
Thành phố	Berlin

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Số trang	705
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