Design and Optimization of Thermal Systems Episode 2 Part 6 doc

Acceptable Design of a Thermal System 347 rate of the oil is given as 0.2 kg/s and its inlet temperature as 90 C The water is available at 20 C, but its temperature rise is restricted to 12.5 C because of environmental concerns The outer tube diameter must be less than cm and the inner tube diameter must be greater than 1.5 cm due to constraints arising from space and piping considerations The engine oil must be cooled to a temperature below 50 C Obtain a feasible design if the length of the heat exchanger must not exceed 200 m Redesign the system if the length is restricted to 100 m Even though the fluid properties vary with temperature, take these as constant for simplification, with the specific heat at constant pressure (Cp), viscosity ( ), and thermal conductivity (k) as 2100, 0.03, and 0.15 for the oil, and as 4179, 8.55 10 –4, and 0.613 for water, all in S.I units Solution Several requirements and constraints are given as inequalities Appropriate values may thus be chosen to satisfy these Therefore, the outlet temperature of the oil may be taken as 45 C, the inner tube diameter as cm, and the outer tube diameter as cm These values satisfy the inequalities, but may have to be adjusted if a feasible design is not obtained Total energy lost by the oil Q mhC p,h (Th,i Th,o ) 0.2 2100 ( 90 45) 18.9  kW where the subscript h refers to the hot oil, i to the inlet, and o to the outlet The mass flow rate is represented by m and the temperatures by T Assuming zero heat loss to the ambient, the energy lost by the oil is gained by the water Therefore, mwC p,w (Tw,o Tw,i ) 18, 900 Since the temperature rise is restricted to 12.5 C, mw 18, 900 4179 12.5 0.36  kg/s Let us choose the mass flow rate of water as 0.4 kg/s, which gives the outlet water temperature, which satisfies the given constraint on temperature rise, as Tw,o 20 18, 900 4179 0.4 31.3 C Therefore, the log-mean-temperature-difference, Tm , is Tm (Th,i Tw,o ) (Th,o Tw,i ) ln[(Th,i Tw,o )/(Th,o Tw,i )] 58.7 25 ln(58.7/25) 39.5 C 348 Design and Optimization of Thermal Systems Neglecting the conductive resistances due to tube walls, the overall heat transfer coefficient U is given by (1/hi ) (1/ho ) U where hi and ho are the convective heat transfer coefficients in the inner tube and the outer annulus To determine if the flows are laminar or turbulent, the Reynolds numbers ReD need to be determined For water flow in the inner tube, ReD mw Di 0.4 0.02 8.55 10 2.98 10 Therefore, the flow is turbulent and a correlation such as the Dittus-Boelter equation (Incropera and Dewitt, 2001) may be used to obtain the heat transfer coefficient hi Therefore, NuD 0.023(ReD)0.8 (Pr)0.4 since the Prandtl number Pr hi NuD k Di Cp /k 0.023(29,800)0.8(5.83)0.4 176.8 5.83 for water This gives hi as 176.8 0.613 0.02 5418.9   W/ ( m K ) For flow in the annulus, the hydraulic diameter Dh olds number Re Dh is Do – Di 0.02 m The Reyn- um Dh Re Dh where um is the mean velocity, given by um mh Do Di2 /4 Therefore, Re Dh 4m ( Do Di ) 0.2 ( 0.04 0.02 ) 0.03 141.5 Therefore, the flow in the annulus is laminar For Di /Do 0.5, the Nusselt number for developed annular flow with one surface isothermal and the other insulated may be employed to calculate the heat transfer coefficient at the inner surface of the annulus (Incropera and Dewitt, 1990, Table 8.2) The Nusselt number thus obtained is Nu Dh ho Dh k 5.74 Acceptable Design of a Thermal System 349 Therefore, ho 5.74 0.015 0.02 43.1 W/(m K) and U (1/5418.9) (1/43.1) 42.8 W/(m2 K) Now, the total heat transfer is given by Q UA Tm where A is the heat transfer area, being the inner surface of the annulus Therefore, if L is the length of the heat exchanger, Q U Di L Tm 42.8 0.02 L 39.5 This gives L as L 42.8 18, 900 0.02 39.5 177.9 m This satisfies the given requirement that the length be less than 200 m Therefore, a feasible or acceptable design is obtained Clearly, there are many other acceptable designs because many variables were chosen arbitrarily to satisfy the given constraints and requirements This is typical of most design problems where there is considerable freedom in the choice of design variables, leading to a domain of acceptable designs from which an optimal design may be determined for a given objective function Let us now consider variations in this design to obtain a length less than 100 m It is obvious from the preceding calculations that the heat transfer coefficient in the tube hi is very high and has a small effect on the overall heat transfer coefficient U Therefore, a reduction in Di does not significantly affect U, but it reduces the area A, which leads to an increase in L The outer diameter may be increased up to cm, but this results in a reduction in ho The effect of changing other variables may similarly be considered The best course of action is to reduce Do while keeping Di unchanged If Do is taken as cm, Di /Do 0.667, Re Dh 169.8, and Nu Dh is obtained from Incropera and Dewitt (2001) as 5.45 This gives ho as ho 5.45 0.15 0.01 81.8 W/(m2 K) This gives the value of U as 80.6 W/(m · K), which leads to L 94.5 m Therefore, this is an acceptable design because the length constraint of 100 m is satisfied 350 Design and Optimization of Thermal Systems It is seen here that once an acceptable design is obtained, other designs can easily be generated by varying the design variables In addition, the sensitivity of the results to changes in the variables, as obtained from the simulation or a sensitivity analysis, can be used effectively to obtain acceptable designs for other constraints and requirements Similar design procedures can be used for other heat exchanger designs and with more accurate mathematical models 5.4.5 FLUID FLOW SYSTEMS Fluid flow is an important part of thermal systems because the transport of mass and energy occurs due to the flow of fluids such as refrigerants, combustion products, water, and air Although convective transport is relevant to many different types of thermal systems and has been considered for several applications in the preceding sections, fluid flow systems such as the pipe networks shown in Figure 1.20 are also important in many practical circumstances Figure 5.31 shows a couple of fluid systems that involve a piping network as well as pumps to Storage tank Cleaning plant Pump Lake (a) Reservoir B Reservoir A Pump (b) FIGURE 5.31 Two fluid flow systems, involving flow in pipes and pumps Acceptable Design of a Thermal System 351 move the fluid The design of such fluid flow systems generally involves the following two considerations: Selection of fluid flow equipment Design of the piping system for the flow The first consideration is directed at the selection of equipment such as pumps, fans, blowers, compressors, valves, and storage vessels Obviously, the design of these components may also be undertaken, depending on the application and the scope of the overall effort The second aspect relates to pressure or head losses in pipes due to friction, bends, pipe fittings, joints, valves, etc., and the appropriate overall pressure difference needed to maintain a given flow rate Selection of Equipment In selecting appropriate equipment for a given application, it is necessary to know the desired flow rate, pressure head needed, and the fluid involved Constraints on dimensions, speed, weight, etc., are also important considerations The requirements and constraints are then matched with the specifications of the available hardware and a selection is made based on cost and characteristics of the equipment A brief description of some of this equipment is given in the following for the sake of completeness A pump is a device used to move fluid by drawing the fluid into itself and then forcing it out through an exhaust port It may be used to move liquids in pipelines, to lift water from a water processing plant to a storage tank high above the city, to empty a container, or to put an oil under pressure as in a hydraulic brake system Many different types of pumps are available, often being classified as reciprocating, rotary, or centrifugal Sketches of these three types of pumps are shown in Figure 5.32 In a reciprocating pump, an inlet valve, which opens at appropriate points during the motion of a piston, allows the lower pressure fluid to flow into a chamber Then the back-and-forth movement of the piston is employed to push the fluid through an outlet valve In a rotary pump, the rotating elements contain fluid that is physically pushed out Both of these types of pumps employ fixed movements of the fluid and are thus positive displacement devices The centrifugal Rotary Reciprocating Centrifugal FIGURE 5.32 Different types of pumps (Adapted from Boehm, 1987.) Design and Optimization of Thermal Systems p 352 f p m m FIGURE 5.33 Typical curve representing the characteristics of a pump pump raises the pressure by imparting kinetic energy to the fluid The fluid picks up velocity as it flows in the pump and, as it exits, a pressure rise is generated due to the centrifugal force Further subdivision of centrifugal pumps as axial, radial, and mixed flow pumps may be made according to the direction of fluid flow through these with respect to the axis of rotation Several other types of pumps are available for fluid flow systems The characteristics of a pump may be written in terms of the pressure difference p generated by it and the mass flow rate m of a given fluid Figure 5.33 shows a typical characteristic curve, indicating the decrease in the pressure head due to friction and other losses as the flow rate increases Curve fitting may be used to obtain a correlating equation, which represents such data on pump characteristics This equation may be of the form f ( p, m ) or, for example, p A0 A1m A2 m (5.25) where A0, A1, and A2 are constants A0 gives the highest pressure head generated, which arises under no-flow conditions Similar correlations may be derived for different types of pumps if experimental data on their characteristics are given Therefore, the constants, such as the A’s in Equation (5.25), characterize a particular pump for a given fluid The specifications of a pump may then be given in terms of the pressure head generated for a specified flow rate of a particular fluid, the maximum flow rate that can be delivered, or the maximum pressure that can be generated The requirements for an application can then be employed to select a pump for the purpose See Pollak (1980), Warring (1984), and Boehm (1987) for further details A fan is also a device used to move fluids, though the pressure head is generally quite small The flow is generated by producing a low compression ratio, as in ventilation Blowers are fans that operate with most of the resistance to flow downstream of the fan Exhausters are also fans that operate with most of the resistance upstream of the fan Three main types of fans are usually defined These are axial, propeller, and centrifugal, with the first two employing the angle of attack of the rotating blade to move the fluid The housing plays an important role in controlling the flow rate in axial fans, whereas propeller fans are not good Acceptable Design of a Thermal System 353 Backward inclined (shown at right) Backward curved Airfoil Radial Rotation direction Single blade Radial tip or forward curved Forward curved Backwardly curved blades Radial tip blades Forwardly curved blades FIGURE 5.34 Different blade profiles in fans (Adapted from Boehm, 1987.) for controlling the flow In centrifugal fans, the centrifugal force acting at the perimeter of the fan results in a pressure rise as the gas leaves the fan The blade profiles, some of which are shown in Figure 5.34 from Boehm (1987), affect the performance of the fan significantly The characteristics may again be given in terms of the pressure head p and mass flow rate m The diameter D and the revolutions per minute (RPM) N may also be included to obtain an equation of the general form f ( p, m, D, N ) or, for example, p Am a D b N c (5.26) where A, a, b, and c are constants For further details on fans and their characteristics, see Thompson and Trickler (1983) and Avallone and Baumeister (1987) A compressor is a machine that increases the pressure of a gas or vapor by reducing the fluid specific volume as it passes through the equipment Compressors are used in a wide variety of applications such as cleaning, pneumatic tools, paint spraying, refrigeration, and tire inflating Again, there are several types of compressors such as reciprocating, rotary, centrifugal, jet, or axial flow, depending on the mechanical means used to compress the fluid The thermodynamics of compressors are given in most textbooks on thermodynamics, such as Van Wylen et al (1994), Howell and Buckius (1992), and Cengel and Boles (2002) The energy needed for an actual compressor is compared against ideal isothermal 354 Design and Optimization of Thermal Systems Floating roof Fixed (conical) roof Gas holder Bin Cylindrical (bullet) tank Open yard (pile) Spherical FIGURE 5.35 Common types of storage vessels (Adapted from Boehm, 1987.) or adiabatic processes to yield the compression efficiency The characteristics of a compressor may be given in terms of the flow rate and the pressure generated for a given fluid such as air or a refrigerant like ammonia For further details on different types of compressors and their characteristics, see Gulf (1979), Bloch et al (1982), and Brown (1986) Other fluid-flow equipment may similarly be considered and selected for different applications Storage vessels are commonly used to provide a buffer between the supply and the demand An example of this is the household hotwater storage tank, which is used to meet the demand for hot water when the outflow exceeds the inflow, as in the morning Similarly, storage of thermal energy in solar energy utilization is an essential part of the overall system Figure 5.35 shows a few common types of storage vessels from Boehm (1987) Valves are important ingredients in the successful design and operation of several thermal systems Many types of valves, such as globe, gate, butterfly, and ball, shown in Figure 5.36, are available to provide shut-off, open-flow, or throttling The main consideration in throttling valves is the pressure drop as a function of the flow rate for a given fluid The characteristics may therefore be given as f ( p, m ) Check valves are used to obtain flow in one direction only The flow rate is then given as a function of the opening size Extensive information is available in the literature on fluid-flow equipment, particularly on the different types of devices and their basic characteristics The manufacturers of these devices generally give the specifications in terms of the flow rates and the pressures, along with limitations on their use with respect to the fluid, temperatures, and environmental conditions The characteristic curves obtained from prototype testing are also available in most cases From this information and the needs of the given application, the appropriate equipment may be Acceptable Design of a Thermal System 355 (a) (b) (c) (d) FIGURE 5.36 Schematic diagrams for (a) globe, (b) gate, (c) butterfly, and (d) ball valves (Adapted from Boehm, 1987.) selected However, it must be reiterated that the design of these equipment may also be undertaken if the needs of the project require it Piping Systems The design of piping networks and systems is based on flow rates and the pressure head needed to generate the desired flows Pressure losses occur in pipe flows due to friction, couplings, pipe fittings, bends, joints, elbows, etc., and these must be taken into account to determine the total pressure head needed The flow rate in a circular tube of diameter D yields the Reynolds number Re as Re VD/ m / D , where is the fluid density, V is the average velocity in the tube, m is the mass flow rate, and is the fluid dynamic viscosity Similarly, the Reynolds number may be defined and calculated for noncircular tubes, annuli, and channels (Incropera and Dewitt, 2001; Shames, 1992) The Reynolds number determines whether the flow is laminar or turbulent (see Example 5.7) Empirical results are generally used to obtain the friction coefficient f as a function of Re and the pipe surface roughness e, which depends on the material and affects the head loss due to friction The basic concepts involved are given in most books on fluid mechanics Only a brief discussion is given here for completeness The pressure drop p due to friction in a pipe of constant diameter D over a length L is given by the expression p f L V2 D (5.27) 356 Design and Optimization of Thermal Systems where f may be obtained for laminar and turbulent flow from, respectively, f 64 Re f 0.5 and 2.0 log e/ D 3.7 2.51 Re f 0.5 (5.28) Here, the second equation is the widely used Colebrook formula for f, and log represents the logarithm to base 10 Iteration is used to solve for the root f in this equation, using the techniques discussed in Chapter Other equations and experimental results presented in graphical form, known as Moody’s chart, are available for determining the friction factor f (Fox and McDonald, 2003) For noncircular channels, the hydraulic diameter Dh, defined earlier, is used instead of D The flow through a variety of fittings, bends, abrupt changes in area, joints, etc., also gives rise to head losses, mainly due to flow separation These losses are usually known as minor losses and are expressed as ( p)minor K V2 (5.29) where the loss coefficient K is determined experimentally for each circumstance and the total pressure loss is obtained by summing the different losses Extensive information on experimentally determined coefficients for pipe entrances, contractions, bends, valves, fittings, etc., is available in the literature (Fox and McDonald, 2003; Janna, 1993) and may be used for calculating the pressure drop in a piping system The modified Bernoulli equation is generally used to include the effects of friction and minor losses and may be written as p1 V12 gz1 p2 V22 gz2 fL V Dh K V2 (5.30) where the subscripts and refer to two locations in the flow, g is the magnitude of gravitational acceleration, z is the vertical location with respect to a chosen ground level, and the summations indicate head losses due to friction, bends, etc Therefore, this equation may be applied to compute the pressure head needed for a given flow in a chosen piping system, including the effects of gravity involved in raising or lowering the fluid Many examples of the application of this equation and of the calculations for head losses are given in most textbooks on fluid mechanics Example 5.8 A water distribution system consisting of two centrifugal pumps in two parallel flow channels, as shown in Figure 5.37, is to be designed The total mass flow rate m is the sum of the flow rates m1 and m2 in the two paths Therefore, m m1 m2 (a) Acceptable Design of a Thermal System 357 m m2 Storage tank Pump m He m1 Pump Cleaning facility FIGURE 5.37 Fluid flow system considered in Example 5.8 Also, the characteristics of the two pumps are given in terms of the pressure difference P and flow rates as P P1 A( m1 )2.75 and P P2 B( m2 )2.75 (b) where the pressures are in kPa and flow rates in kg/s Here, P1 and P2 are the maximum pressure heads generated, for no-flow conditions, and A, B are constants Curve fitting has been used to derive these equations from experimental data, as outlined in the preceding section and in Chapter The energy balance, considering elevation change He and friction losses, is obtained from Bernoulli’s equation as P (c) H C (m )2 The initial design values of P1, P2, A, B, and C are given as 500, 700, 7, 22, and 4.75, respectively In addition, the pressure head H due to elevation is fixed and given as 140 kPa The required total flow rate is 6.5 kg/s Determine if this initial design is satisfactory If not, vary the design variables from their initial values up to 35% of the initial values to obtain an acceptable design From these results, determine conditions under which maximum flow rate is obtained Solution The mathematical model has been derived using the basic ideas presented in the preceding section The fixed quantity is the elevation pressure head H, and the requirement is that the flow must be greater than 6.5 kg/s The design variables that refer to the pump are P1, P2, A, and B, while C refers to the friction and other losses in the pipes The constraints are placed on all the design variables as 35% of the initial values We need to simulate this system by solving the given set of nonlinear algebraic equations to determine the total flow rate Although all the equations may be used directly in the simulation, the problem may be simplified by eliminating m and P to obtain the following two equations for m1 and m2: F ( m1, m2 ) H C ( m1 m2 )2 P1 A( m1 )2.75 (d) G ( m1, m2 ) H C ( m1 m2 )2 P2 B( m2 )2.75 (e) 358 Design and Optimization of Thermal Systems The Newton-Raphson method may be used conveniently to solve this system of two nonlinear equations, as discussed in Chapter The resulting values of m1 and m2 would then yield the total flow rate from Equation (a) The pressure head P, if needed, may be calculated from the other equations If the values of m1 and m2 after the ith iteration are m1,i and m2,i , the values for the next iteration are given by m1,i m1,i m1,i m2,i F m2 i G m2 m2,i m2,i m2,i Fi (g1) i m2,i Gi (g2) (f) where F m1 G m1 m1,i i m1,i i Here, the derivatives and functions are evaluated for the ith iteration and the equations are solved for m1,i and m2,i , as given in Chapter The derivatives are easily obtained mathematically from the expressions for F and G Equation (f) then yields the new values for m1 and m2 Using the approach just outlined, the individual as well as the total flow rates are computed A convergence criterion of 10 –4 is applied to the sum of the squares of the functions F and G, i.e., the iteration is terminated if |F2 G 2| 10 –4 It is ensured that the results are negligibly affected by a further reduction in the convergence parameter Starting with initial, guessed values of 1.0 for both m1 and m2 , convergence is achieved in seven iterations, with m1 3.277 and m2 2.826 The pressure head P is obtained as 316.932 and the total flow rate m as 6.103, which is less than the required value of 6.5 Therefore, the given initial design is not acceptable and the different design variables are changed over the given ranges to obtain higher flow rates Figure 5.38 and Figure 5.39 show the computed results in terms of the total flow rate m over the range of variation of the design variables In each case, one design variable is changed while the others are held constant at the base or initial values It is easy to see that the required total flow rate m is 6.5 or more if C is at its lowest value of 3.09 It exceeds 6.5 also if P1 is greater than 605 In both cases, the other variables are at the base values For changes in other design variables, the flow rate m is less than 6.5 Therefore, a range of acceptable designs is obtained Here, the design variables were changed one at a time in order to follow the basic trends and minimize the change needed in the initial design Clearly, these results indicate that the highest total flow rate is obtained with the largest allowable values of P1 and P2, which represent the largest no-flow pressures generated by the pumps, and the smallest values of A, B, and C, which indicate Total flow rate m(kg/s) Acceptable Design of a Thermal System 6.8 6.7 6.6 6.5 6.4 6.3 6.2 6.1 6.0 5.9 5.8 5.7 5.6 5.5 5.4 5.3 5.2 5.1 300 359 Flow rate vs P1 400 500 P1 600 700 Flow rate vs P2 6.5 6.4 Total flow rate m(kg/s) 6.3 6.2 6.1 6.0 5.9 5.8 5.7 5.6 5.5 400 600 800 1000 P2 FIGURE 5.38 Effect of P1 and P2 on the total flow rate m in Example 5.8 the smallest head losses This result is physically expected If the costs of making these changes in the design variables are also considered, an optimal design may be sought that minimizes the cost while meeting the given requirements and constraints This aspect is considered in detail in later chapters 360 Design and Optimization of Thermal Systems Flow rate vs A 6.5 Total flow rate m(kg/s) 6.4 6.3 6.2 6.1 6.0 5.9 5.8 10 A (a) Flow rate vs B Total flow rate m(kg/s) 6.5 6.4 6.3 6.2 6.1 6.0 5.9 14 16 18 20 22 B (b) 24 26 5.4 5.8 28 Flow rate vs C 6.6 Total flow rate m(kg/s) 6.5 6.4 6.3 6.2 6.1 6.0 5.9 5.8 5.7 3.0 3.4 3.8 4.2 5.0 4.6 6.2 6.6 C (c) FIGURE 5.39 Effect of the parameters A, B, and C on the total flow rate m in Example 5.8 Acceptable Design of a Thermal System 361 5.4.6 OTHER AREAS In the preceding sections, we considered several different areas of practical application in which thermal systems are of particular interest The main concerns that arise in the design of the system were outlined A few examples were also given to illustrate the use of the design procedures presented in earlier chapters However, it is not possible to consider every type of thermal system that arises in engineering practice Similarly, even for the few areas considered in detail in the preceding sections, only a few salient features and examples could be discussed However, these examples and the accompanying discussions serve to indicate the basic nature of the design process to obtain an acceptable thermal system for a given application Starting with the formulation of the design problem and the conceptual design, the detailed, quantitative design process is illustrated, employing different strategies for converging to an acceptable design The different steps involved in the design process were discussed in earlier chapters and the coupling of all these aspects is illustrated here The main considerations presented here are expected to apply to other types of thermal systems and to different problems Some of these areas and problems are considered again in later chapters with respect to optimization 5.4.7 DESIGN OF COMPONENTS VERSUS DESIGN OF SYSTEMS Throughout this chapter, we have focused on thermal systems, ranging from small systems consisting of only two or three parts to large systems consisting of many parts that interact with each other Similarly, in earlier chapters, the treatment and discussion have been largely directed at systems Not much has been said about the design of components, even though each system obviously consists of a number of components and the design of the system is often closely coupled with that of the components In addition, in many cases, the components themselves consist of separate parts and may be considered as subsystems or systems for design purposes Heat transfer and flow equipment such as heat exchangers, pumps, blowers, fans, and compressors are examples of items that are generally treated as components even though these may involve interacting and coupled parts Many of these components have been considered in preceding sections as parts of a larger system such as an air-conditioning system or a water-flow system Therefore, it is worthwhile to clarify the design of a system as compared to that for a component A component is basically an independent item that is often available, readymade, over wide ranges of specifications The characteristics of each component, such as a pump or a blower, are also available from the manufacturer, who is obviously involved in the design and production of such components Therefore, the design of the components precedes that of the system However, we have employed the availability of standard items and their characteristics to design systems that are obtained by combining different components to obtain a desired thermal process It is largely a question of focus and interest, these being on the 362 Design and Optimization of Thermal Systems overall system in this book rather than on individual components Some of the discussion in Section 5.4.4 and Section 5.4.5 has been directed toward components and their selection, with a few relevant references being given for additional information The design process outlined for systems can often be employed for components that involve a number of interacting parts, such as heat transfer and flow equipment For others, such as pipes, sensors, heaters, and valves, that not contain interacting parts, modeling and simulation can again form the basis for design, but this problem has not been considered here The components are largely treated as available items whose characteristics are employed in the modeling and simulation of the complete system and which can be selected for the overall design process 5.5 ADDITIONAL CONSIDERATIONS FOR LARGE PRACTICAL SYSTEMS In all our discussions of the design of thermal systems, we have focused on the thermal aspects arising from heat and mass transfer, fluid flow, and thermodynamics This is obviously because of the types of systems that are of particular interest to us in this book Thermal aspects are the dominant mechanisms in the processes and systems under consideration and, therefore, the design is largely based on these However, the successful design and implementation of practical systems generally involve several additional considerations that must be included if the system is to perform satisfactorily Some of these additional considerations have been mentioned earlier and may be listed here as Safety Control of the system Environmental impact Structural integrity and mechanical strength Selection and availability of materials Costs involved Availability of facilities and utilities Regulation, legal issues Safety is a very important consideration and is generally addressed by providing sensors that monitor the levels of temperature, pressure, concentration, and other physical quantities that may affect the safety of material and personnel In general, unsafe levels of such variables are given, and the system is turned down or turned off if these are exceeded An alarm may also sound to alert the operator In many cases, components of the system cannot be turned on if the specified conditions are not met For instance, the heaters in a boiler may be set so that they can be turned on only if the water level is adequate Many such safety features are usually built into the design to avoid damage to the system as well as to the user Acceptable Design of a Thermal System 363 Control of the system is one of the most important technical aspects that must be included in the design for successful operation of the system If a system is designed for a particular temperature at the surface or in the fluid, a control scheme is needed to ensure that these are maintained at these values, within an allowable tolerance Sensors are used to monitor the appropriate physical quantity, and the control scheme makes the appropriate correction, as needed Similarly, the flow rate of a fluid or a given material must be maintained at the design value in order to obtain the required product quality or production rate A control strategy may be employed to preserve these within acceptable levels Automation also depends strongly on the control scheme used Many different control strategies are available and may be used effectively to ensure satisfactory performance of the system These include on/off arrangements, which are very commonly used in thermal systems, proportional control, and integral control, among others Control systems constitute an important area and, though beyond the scope of this book, generally form an important ingredient in the final implementation of the thermal system, as outlined in Chapter Environmental impact is generally an important consideration in system design today because of increasing concerns with pollution, depletion of the ozone layer, greenhouse effect, global warming, and solid waste disposal Therefore, the impact of the designed system on the environment has to be evaluated for successful implementation In particular, the types and amounts of pollutants discharged into the atmosphere or into bodies of water need to be estimated The amount of solid waste generated and the procedures to dispose of these must be determined Similarly, if gases such as carbon dioxide or sulfur dioxide are being generated, it may be necessary to develop means to convert these into harmless byproducts One of the most important technical, though not necessarily thermal, considerations involved in the design of a system is that pertaining to the mechanical strength of the system It is crucial that the structural integrity of the system be maintained and that the various elements that constitute the system not fail under the temperatures, pressures, loads, and other forces acting on these Therefore, the mechanical strength of the various materials must be considered in terms of stresses in different parts of the system, ensuring that these not fail by fracture or excessive deformation Even if a particular item satisfies the requirements and constraints imposed by the thermal aspects, it must still meet the strength requirements However, in most cases, the constraints due to strength considerations may be translated into the appropriate limitations on temperature, pressure, speed, weight, etc This is particularly true of thermal stresses that arise due to temperature gradients in the material Thus, excessive thermal stresses can be expressed in terms of a constraint on the temperature gradient or difference across a given system part Similarly, considerations like wear, fatigue, and buckling are generally taken care of by limiting the speed, duration of daily usage, total time for which the system is employed, temperatures, pressures, and so on Consequently, many of the constraints considered in this book may be the outcome of mechanical strength and structural integrity considerations 364 Design and Optimization of Thermal Systems The selection of materials is another important consideration, as discussed in Chapter With the development of new materials such as composites, ceramics, alloys, and different types of polymers, the choice of materials has expanded substantially in recent years The design would therefore be influenced by the availability of appropriate materials In most cases, effort is made to choose the most suitable material with respect to the cost and the desired thermal properties Economic considerations are always critical to the success of the design effort because the viability of a project depends strongly on the overall financial return Therefore, it is important to evaluate the system with respect to the costs incurred for the hardware as well as for the operation of the system The productivity in terms of the output can then be considered along with the price to determine the rate of return on the investment Some of these aspects are presented in the next chapter The availability of appropriate facilities and utilities like power and water is an important consideration in the overall design process Additional issues like governmental regulations and legal matters also have to be satisfied for a successful system The following example presents the design of a practical thermal system Example 5.9 Discuss the modeling, simulation, and design of the batch-annealing furnace, shown in Figure 5.40, which is used for the annealing of steel sheets rolled up in the form of annular cylindrical coils Solution The problem considered is an actual industrial system that is used in the steel industry Annealing is employed for relieving the stresses in the material, which has undergone a rolling process such as that shown in Figure 1.10(d) during manufacture The annealing process restores the ductility in the material for further machining and forming operations As shown in Figure 5.40, the steel sheets are rolled into the form of cylindrical coils and stacked vertically with convector plates, which aid the protective inert gas flow, at both ends of each coil A stainless steel cover encloses the coils and an inert environment is maintained between the coils and the cover by the flow of inert gases These gases, which include nitrogen and helium, are driven by a fan as shown The region between the cover and the furnace walls contains flue gases, which are generally obtained from the blast furnace of the steel plant These gases contain various combustion products, such as carbon dioxide and moisture, due to combustion occurring at the two burners These burners are located circumferentially and the flow enters tangentially causing swirl in the flow The dimensions in Figure 5.40 are shown in terms of symbols for generality Typically, the height H6 is approximately 4.0 m and diameter Db is approximately 1.8 m Different types and sizes of furnaces are used For further details on this system, see Harvey (1977) and Jaluria (1984) The basic thermal process involves heating of the coils to the annealing temperature of approximately 723 C; maintaining the temperature at this value for a given time known as the soaking period so that this temperature level is attained Acceptable Design of a Thermal System 365 Flue gas H5 H4 H3 d4 s2 Di Furnace gas Do + 2s s1 Do H Convector plate Inert gas circulation Furnace gases Fire clay and insulation d3 r z H6 Protection hood d2 Steel coils Burner II H2 Dc d1 Do Burner I z1 H1 Fan Db FIGURE 5.40 A batch-annealing furnace for cylindrical coils of steel sheets (Adapted from Jaluria, 1984.) 366 Design and Optimization of Thermal Systems Soaking Initial cooling Final cooling (furnace walls removed) Temperature Heating Time FIGURE 5.41 Typical temperature cycle of the annealing process (Adapted from Jaluria, 1984.) at all points in the coil and the internal stresses are relieved; initial slow cooling to allow the microstructure to settle down; and, finally, rapid cooling with the furnace walls removed The typical temperature cycle undergone by the material at a point in the coil is shown in Figure 5.41 The numerical simulation of the system must, therefore, include the heating, soaking, and cooling processes and determine the temperatures at various locations in the system as functions of time This is a fairly involved problem and requires a transient, distributed model to obtain the inputs needed for design However, many simplifications can be employed to reduce the complexity of the problem and make it amenable to the numerical simulation procedures discussed in Chapter First, we break the system down in terms of the following components or parts: Coils Convector plates Inert gases Protective cover Furnace or flue gases Furnace walls The temperature in each component varies, in general, with the height z, taken from the base of the bottom coil, the radial distance r, taken from the axis of the coils, the circumferential location , and time Mass, momentum, and energy balances lead to the governing equations for the different parts, and all these equations are coupled to each other through the boundary conditions With respect to model development, it is first noted that axisymmetry may be assumed due to the cylindrical configuration of the system and the anticipated Acceptable Design of a Thermal System 367 circumferential symmetry in the energy exchange mechanisms This simplifies the problem to an axisymmetric transient circumstance The coil is essentially a hollow cylinder with inner radius Ri and outer radius Ro There are usually gaps, filled with inert gases, that exist between the different layers of the coil As a result, the thermal conductivity in the radial direction kr is generally much smaller than that in the axial direction kz The governing energy equation for the steel coils is m Cm Tm T rkr m r r r z kz Tm z where the subscript m indicates coil material and the other symbols have their usual connotations The conductivities and other properties may be obtained from the available literature on this problem All the properties depend on temperature The initial and boundary conditions for the preceding equation are obtained from the initial temperature Tr , heat transfer with the cover and inert gases at the outer surface of the coils, and convective heat transfer with the inert gases at the inner surface of the coils Similarly, simplifying approximations are made for other components, shown schematically in Figure 5.42 Using the techniques discussed in Chapter 3, the temperature in the convector plate Tp is assumed to vary only with radial position and time, since the Biot number based on its thickness is small For the cover, temperature variation across its thickness is neglected because of its small thickness and high conductivity Thus, the temperature Tc of the cover varies with z and The Convector plate Tp(r, τ) Tm(r, z, τ) Tc(z, τ) Tg(z, τ) r Tw(r, z, τ) Tg(z, τ) Wall Wall Tf (z, τ) Inert Tc(z, τ) Coils z Tg(z, τ) Tf (z, τ) Tw(r, z, τ) Cover Subscripts: c : Protective cover f : Flue gases g : Inert gases m : Coil material p : Convector plate w : Furnace walls FIGURE 5.42 Components of the system for developing a mathematical model 368 Design and Optimization of Thermal Systems furnace wall is treated as an axisymmetric conduction problem, yielding the energy equation as w Cw Tw kw T r w r r r z Tw z where the subscript w denotes the wall For the gases, radial temperature uniformity is assumed because of turbulent mixing and only the variations with height and time are considered The resulting equation for the inert gases is of the form g (C p ) g UA dTg dz Phi (Tm Tg ) at r Di / Pho [(Tm Tg ) (Tc Tg )] at r Do / where the subscripts g and c refer to the gas and the cover, U is the average velocity, A is the cross-sectional area, P is the perimeter for heat transfer, subscripts i and o refer to inside and outside the core region in the center, and D is the diameter Similarly, the appropriate equations are written for the cover, flue gases, and convector plates As discussed in detail in Chapter 4, all the components of the system are studied individually, with constant, specified boundary conditions to decouple them from each other These uncoupled problems allow one to validate the mathematical model and the corresponding numerical scheme for each component It was found that the various approximations made for the model are valid and that the gases and the cover have a very fast transient response The coils are the slowest in response and the largest time step can be employed for simulating these The various numerical schemes discussed in Chapter can be used for the numerical simulation of the different components Explicit methods are particularly useful because of the variable properties However, implicit methods can also be used for better numerical stability The mathematical models and the numerical schemes for individual components are validated by considering the physical trends obtained from the simulation, eliminating the dependence on numerical parameters like grid size and time step, and comparing the results for a few idealized cases with analytical results These individual numerical models are then coupled with each other by using the actual boundary conditions, arising from heat transfer between different components The overall system is then simulated In practical systems, the process is controlled by monitoring a thermocouple in contact with the base of the bottom coil This is known as the control thermocouple and it is important to use numerical simulation to obtain the temperature cycle measured by this thermocouple Using typical values for the operating conditions, such as initial temperature, flow rates of the gases, and composition of the flue gases, the temperature variation with time was computed at various locations in existing furnaces These results were compared with measurements Figure 5.43 and Figure 5.44 show the comparison between the numerical results and experimental data, indicating fairly good agreement The operating conditions were varied and it was confirmed that the behavior of the system follows expected trends Large variations in the governing Acceptable Design of a Thermal System 369 0.85 Temperature ( ) 0.65 Numerical Experimental 0.45 0.25 0.05 10 20 30 40 50 Time ( × 8) FIGURE 5.43 Comparison between the numerical simulation results and the temperature measurements of the control thermocouple (Adapted from Jaluria, 1984.) parameters and operating conditions were also tried to determine safe levels of operation and to generate system characteristics Therefore, the annealing furnace is satisfactorily simulated numerically For further details, consult the references cited here The results generated by the simulation may be used for the design and optimization of the system Different materials, coil sizes, and heat treatment applications 0.85 Numerical Experimental Temperature ( ) 0.65 0.45 Middle coil 0.25 0.05 Top coil 10 20 30 40 50 Time ( × 8) FIGURE 5.44 Comparison between the numerical simulation results and the temperature measurements in the steel coils (Adapted from Jaluria, 1984.) 370 Design and Optimization of Thermal Systems require different designs, in terms of configuration, dimensions, and heating/cooling arrangement Different design strategies may be used to obtain an acceptable or optimal design The results obtained may also be employed for modification of existing systems to improve the performance In the example considered here, considerable improvement in the process and the product was achieved simply by controlling the flow rate m of the flue gases entering the furnace This controls the heat input since the heat released by combustion at the burners depends on the flow rate The control thermocouple is set to follow a given temperature cycle and this, in turn, controls the gas flow rate at the burners The temperature variation in the system is a strong function of the heat input and, therefore, the desired temperature cycle for heat treatment is obtained in the coils The overall result of this effort was a more uniform annealing than that obtained earlier, and a consequent reduction in material wastage Other Large Systems The preceding example presents a typical practical thermal system: a large one with many interacting parts and components and different flows and thermal transport processes The sketch shown in Figure 5.40 is a schematic of the much more complicated industrial system, but it presents the main features of the thermal system under consideration Additional subsystems that are involved due to safety, control, material loading, furnace top removal, etc., are important and must be included in the actual thermal system However, these are usually brought in after the essential thermal design of the system has been concluded Similarly many other thermal systems considered in the earlier chapters are large systems, even though the schematic diagrams shown indicate the main features of the system in a relatively simplified manner An example of such a large system is the Czochralski crystal-growing process, shown schematically in Figure 5.45 (also see P2.1) A photograph of an industrial facility for the same Pulling Seed Crystal Melt Heater Crucible FIGURE 5.45 Schematic of the Czochralski crystal-growing process Acceptable Design of a Thermal System 371 process is shown in Figure 5.46, indicating the complexity of the system and the inclusion of many auxiliary arrangements for heating, feeding, control, safety, and other practical issues However, the sketch shows the main parts of the system and can be used to develop the mathematical model, using simplifications, approximations, and idealizations, and to simulate the system for wide ranges of the design variables and operating conditions From these results, an acceptable design that meets the given requirements and constraints may be obtained Optimization of the system, as well as of the operating conditions, may also be undertaken, following the determination of the domain of acceptable designs Another large system is the heat rejection system that was outlined in Example 5.6 The design problem involves the cooling water body, which may FIGURE 5.46 An industrial facility for the Czochralski crystal-growing process (From Ferrofluids Corp.) ... 6. 0 5.9 14 16 18 20 22 B (b) 24 26 5.4 5.8 28 Flow rate vs C 6. 6 Total flow rate m(kg/s) 6. 5 6. 4 6. 3 6 .2 6. 1 6. 0 5.9 5.8 5.7 3.0 3.4 3.8 4 .2 5.0 4 .6 6 .2 6. 6 C (c) FIGURE 5.39 Effect of the parameters... chapters  360 Design and Optimization of Thermal Systems Flow rate vs A 6. 5 Total flow rate m(kg/s) 6. 4 6. 3 6 .2 6. 1 6. 0 5.9 5.8 10 A (a) Flow rate vs B Total flow rate m(kg/s) 6. 5 6. 4 6. 3 6 .2 6. 1 6. 0... pumps, and the smallest values of A, B, and C, which indicate Total flow rate m(kg/s) Acceptable Design of a Thermal System 6. 8 6. 7 6. 6 6. 5 6. 4 6. 3 6 .2 6. 1 6. 0 5.9 5.8 5.7 5 .6 5.5 5.4 5.3 5 .2 5.1