ptinciples of heat transfer 2 1684

67706_06_ch06_p350-419.qxd 5/14/10 12:44 PM Page 350 CHAPTER Forced Convection Inside Tubes and Ducts Typical tube bundle of multiple circular tubes and cutaway section of a mini shell-and-tube heat exchanger Source: Courtesy of Exergy, LLC Concepts and Analyses to Be Learned The process of transferring heat by convection when the fluid flow is driven by an applied pressure gradient is referred to as forced convection When this flow is confined in a tube or a duct of any arbitrary geometrical cross section, the growth and development of boundary layers are also confined In such flows, the hydraulic diameter of the duct, rather than its length, is the characteristic length for scaling the boundary layer as well as for dimensionless representation of flow-friction loss and the heat transfer coefficient Convective heat transfer inside tubes and ducts is encountered in numerous applications where heat exchangers, made up of circular tubes as well as a variety of noncircular cross-sectional geometries, are employed A study of this chapter will teach you: • How to express the dimensionless form of the heat transfer coefficient in a duct, and its dependence on flow properties and tube geometry • How to mathematically model forced-convection heat transfer in a long circular tube for laminar fluid flow • How to determine the heat transfer coefficient in ducts of different geometries from different theoretical and/or empirical correlations in both laminar and turbulent flows • How to model and employ the analogy between heat and momentum transfer in turbulent flow • How to evaluate heat transfer coefficients in some examples where enhancement techniques, such as coiled tubes, finned tubes, and twisted-tape inserts, are employed Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 6.1 5/14/10 12:44 PM Page 351 Introduction Heating and cooling of fluids flowing inside conduits are among the most important heat transfer processes in engineering The design and analysis of heat exchangers require a knowledge of the heat transfer coefficient between the wall of the conduit and the fluid flowing inside it The sizes of boilers, economizers, superheaters, and preheaters depend largely on the heat transfer coefficient between the inner surface of the tubes and the fluid Also, in the design of air-conditioning and refrigeration equipment, it is necessary to evaluate heat transfer coefficients for fluids flowing inside ducts Once the heat transfer coefficient for a given geometry and specified flow conditions is known, the rate of heat transfer at the prevailing temperature difference can be calculated from the equation qc = qhc A(Tsurface - Tfluid) (6.1) The same relation also can be used to determine the area required to transfer heat at a specified rate for a given temperature potential But when heat is transferred to a fluid inside a conduit, the fluid temperature varies along the conduit and at any cross section The fluid temperature for flow inside a duct must therefore be defined with care and precision The heat transfer coefficient hqc can be calculated from the Nusselt number hqc DH> k , as shown in Section 4.5 For flow in long tubes or conduits (Fig 6.1a), the significant length in the Nusselt number is the hydraulic diameter, DH, defined as DH = flow cross-sectional area wetted perimeter (6.2) For a circular tube or a pipe, the flow cross-sectional area is pD2> , the wetted perimeter is ␲D, and therefore, the inside diameter of the tube equals the hydraulic Wetted perimeter D2 Flow cross-sectional area (a) D1 (b) FIGURE 6.1 Hydraulic diameter for (a) irregular cross section and (b) annulus 351 Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 352 5/14/10 12:44 PM Page 352 Chapter Forced Convection Inside Tubes and Ducts diameter For an annulus formed between two concentric tubes (Fig 6.1b), we have DH = (p> 4)(D22 - D21) = D2 - D1 p(D1 + D2) (6.3) In engineering practice, the Nusselt number for flow in conduits is usually evaluated from empirical equations based on experimental results The only exception is laminar flow inside circular tubes, selected noncircular cross-sectional ducts, and a few other conduits for which analytical and theoretical solutions are available [13] Some simple examples of laminar-flow heat transfer in circular tubes are dealt with in Section 6.2 From a dimensional analysis, as shown in Section 4.5, the experimental results obtained in forced-convection heat transfer experiments in long ducts and conduits can be correlated by an equation of the form Nu = f(Re)c(Pr) (6.4) where the symbols ␾ and ␺ denote functions of the Reynolds number and Prandtl number, respectively For short ducts, particularly in laminar flow, the right-hand side of Eq (6.4) must be modified by including the aspect ratio x/DH: Nu = f(Re)c(Pr )f a x b DH where f(x> DH) denotes the functional dependence on the aspect ratio 6.1.1 Reference Fluid Temperature The convection heat transfer coefficient used to build the Nusselt number for heat transfer to a fluid flowing in a conduit is defined by Eq (6.1) The numerical value of hc, as mentioned previously, depends on the choice of the reference temperature in the fluid For flow over a plane surface, the temperature of the fluid far away from the heat source is generally uniform, and its value is a natural choice for the fluid temperature in Eq (6.1) In heat transfer to or from a fluid flowing in a conduit, the temperature of the fluid does not level out but varies both along the direction of mass flow and in the direction of heat flow At a given cross section of the conduit, the temperature of the fluid at the center could be selected as the reference temperature in Eq (6.1) However, the center temperature is difficult to measure in practice; furthermore, it is not a measure of the change in internal energy of all the fluid flowing in the conduit It is therefore a common practice, and one we shall follow here, to use the average fluid bulk temperature, Tb, as the reference fluid temperature in Eq (6.1) The average fluid temperature at a station of the conduit is often called the mixing-cup temperature because it is the temperature which the fluid passing a cross-sectional area of the conduit during a given time internal would assume if the fluid were collected and mixed in a cup Use of the fluid bulk temperature as the reference temperature in Eq (6.1) allows us to make heat balances readily, because in the steady state, the difference Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 5/14/10 12:44 PM Page 353 6.1 Introduction 353 in average bulk temperature between two sections of a conduit is a direct measure of the rate of heat transfer: # (6.5) qc = mcp ¢Tb where qc ϭ rate of heat transfer to fluid, W # m ϭ flow rate, kg/s cp ϭ specific heat at constant pressure, kJ/kg K ¢Tb ϭ difference in average fluid bulk temperature between cross sections in question, K or °C The problems associated with variations of the bulk temperature in the direction of flow will be considered in detail in Chapter 8, where the analysis of heat exchangers is taken up For preliminary calculations, it is common practice to use the bulk temperature halfway between the inlet and the outlet section of a duct as the reference temperature in Eq (6.1) This procedure is satisfactory when the wall heat flux of the duct is constant but may require some modification when the heat is transferrred between two fluids separated by a wall, as, for example, in a heat exchanger where one fluid flows inside a pipe while another passes over the outside of the pipe Although this type of problem is of considerable practical importance, it will not concern us in this chapter, where the emphasis is placed on the evaluation of convection heat transfer coefficients, which can be determined in a given flow system when the pertinent bulk and wall temperatures are specified 6.1.2 Effect of Reynolds Number on Heat Transfer and Pressure Drop in Fully Established Flow For a given fluid, the Nusselt number depends primarily on the flow conditions, which can be characterized by the Reynolds number, Re For flow in long conduits, the characteristic length in the Reynolds number, as in the Nusselt number, is the hydraulic diameter, and the velocity to be used is the average over the flow crosssectional area, Uq , or ReDH = UqDHr UqDH = v m (6.6) In long ducts, where the entrance effects are not important, the flow is laminar when the Reynolds number is below about 2100 In the range of Reynolds numbers between 2100 and 10,000, a transition from laminar to turbulent flow takes place The flow in this regime is called transitional At a Reynolds number of about 10,000, the flow becomes fully turbulent In laminar flow through a duct, just as in laminar flow over a plate, there is no mixing of warmer and colder fluid particles by eddy motion, and the heat transfer takes place solely by conduction Since all fluids with the exception of liquid metals have small thermal conductivities, the heat transfer coefficients in laminar flow are relatively small In transitional flow, a certain amount of mixing occurs through eddies that carry warmer fluid into cooler regions and vice versa Since the mixing Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 12:44 PM Page 354 Chapter Forced Convection Inside Tubes and Ducts 200 100 50 hc D k NuD ∝ ReD0.8 20 NuD = 354 5/14/10 10 Laminar Transitional Turbulent 5.0 NuD ∝ ReD0.3 2.0 1.0 100 200 500 1000 2000 5000 10,000 20,000 50,000 ReD = U∞ D/v FIGURE 6.2 Nusselt number versus Reynolds number for air flowing in a long heated pipe at uniform wall temperature motion, even if it is only on a small scale, accelerates the transfer of heat considerably, a marked increase in the heat transfer coefficient occurs above ReDH = 2100 (it should be noted, however, that this change, or transition, can generally occur over a range of Reynolds number, 2000 ReDH 5000) This change can be seen in Fig 6.2, where experimentally measured values of the average Nusselt number for atmospheric air flowing through a long heated tube are plotted as a function of the Reynolds number Since the Prandtl number for air does not vary appreciably, Eq (6.4) reduces to Nu = f(ReDH), and the curve drawn through the experimental points shows the dependence of Nu on the flow conditions We note that in the laminar regime, the Nusselt number remains small, increasing from about 3.5 at ReDH = 300 to 5.0 at ReDH = 2100 Above a Reynolds number of 2100, the Nusselt number begins to increase rapidly until the Reynolds number reaches about 8000 As the Reynolds number is further increased, the Nusselt number continues to increase, but at a slower rate A qualitative explanation for this behavior can be given by observing the fluid flow field shown schematically in Fig 6.3 At Reynolds numbers above 8000, the flow inside the conduit is fully turbulent except for a very thin layer of fluid adjacent to the wall In this layer, turbulent eddies are damped out as a result of the viscous forces that predominate near the surface, and therefore heat flows through this layer mainly by conduction.* The edge of this sublayer is indicated by a dashed line *Although some studies [1] have shown that turbulent transport also exists to some extent near the wall, especially when the Prandtl number is larger than 5, the layer near the wall is commonly referred to as the “viscous sublayer.” Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 5/14/10 12:44 PM Page 355 6.1 Introduction 355 Edge of viscous sublayer Edge of buffer or transitional layer Turbulent core FIGURE 6.3 Flow structure for a fluid in turbulent flow through a pipe in Fig 6.3 The flow beyond it is turbulent, and the circular arrows in the turbulentflow regime represent the eddies that sweep the edge of the layer, probably penetrate it, and carry along with them fluid at the temperature prevailing there The eddies mix the warmer and cooler fluids so effectively that heat is transferred very rapidly between the edge of the viscous sublayer and the turbulent bulk of the fluid It is thus apparent that except for fluids of high thermal conductivity (e.g., liquid metals), the thermal resistance of the sublayer controls the rate of heat transfer, and most of the temperature drop between the bulk of the fluid and the surface of the conduit occurs in this layer The turbulent portion of the flow field, on the other hand, offers little resistance to the flow of heat The only effective method of increasing the heat transfer coefficient is therefore to decrease the thermal resistance of the sublayer This can be accomplished by increasing the turbulence in the main stream so that the turbulent eddies can penetrate deeper into the layer An increase in turbulence, however, is accompanied by large energy losses that increase the frictional pressure drop in the conduit In the design and selection of industrial heat exchangers, where not only the initial cost but also the operating expenses must be considered, the pressure drop is an important factor An increase in the flow velocity yields higher heat transfer coefficients, which, in accordance with Eq (6.1), decrease the size and consequently the initial cost of the equipment for a specified heat transfer rate At the same time, however, the pumping cost increases The optimum design therefore requires a compromise between the initial and operating costs In practice, it has been found that increases in pumping costs and operating expenses often outweigh the saving in the initial cost of heat transfer equipment under continuous operating conditions As a result, the velocities used in a majority of commercial heat exchange equipment are relatively low, corresponding to Reynolds numbers of no more than 50,000 Laminar flow is usually avoided in heat exchange equipment because of the low heat transfer coefficients obtained However, in the chemical industry, where very viscous liquids must frequently be handled, laminar flow sometimes cannot be avoided without producing undesirably large pressure losses It was shown in Section 4.12 that, for turbulent flow of liquids and gases over a flat plate, the Nusselt number is proportional to the Reynolds number raised to the 0.8 power Since in turbulent forced convection the viscous sublayer generally controls the rate of heat flow irrespective of the geometry of the system, it is not surprising that for turbulent forced convection in conduits the Nusselt number is related to the Reynolds number by the same type of power law For the case of air flowing in a pipe, this relation is illustrated in the graph of Fig 6.2 Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 356 5/14/10 12:44 PM Page 356 Chapter Forced Convection Inside Tubes and Ducts 6.1.3 Effect of Prandtl Number The Prandtl number Pr is a function of the fluid properties alone It has been defined as the ratio of the kinematic viscosity of the fluid to the thermal diffusivity of the fluid: Pr = cpm n = a k The kinematic viscosity v, or m> r, is often referred to as the molecular diffusivity of momentum because it is a measure of the rate of momentum transfer between the molecules The thermal diffusivity of a fluid, k> cpr, is often called the molecular diffusivity of heat It is a measure of the ratio of the heat transmission and energy storage capacities of the molecules The Prandtl number relates the temperature distribution to the velocity distribution, as shown in Section 4.5 for flow over a flat plate For flow in a pipe, just as over a flat plate, the velocity and temperature profiles are similar for fluids having a Prandtl number of unity When the Prandtl number is smaller, the temperature gradient near a surface is less steep than the velocity gradient, and for fluids whose Prandtl number is larger than one, the temperature gradient is steeper than the velocity gradient The effect of Prandtl number on the temperature gradient in turbulent flow at a given Reynolds number in tubes is illustrated schematically in Fig 6.4, where temperature profiles at different Prandtl numbers are shown at ReD = 10,000 These curves reveal that, at a specified Reynolds number, the temperature gradient at the wall is steeper in a fluid having a large Prandtl number than in a fluid having a small Prandtl number Consequently, at a given Reynolds number, fluids with larger Prandtl numbers have larger Nusselt numbers Liquid metals generally have a high thermal conductivity and a small specific heat; their Prandtl numbers are therefore small, ranging from 0.005 to 0.01 The Prandtl numbers of gases range from 0.6 to 1.0 Most oils, on the other hand, have large Prandtl numbers, some up to 5000 or more, because their viscosity is large at low temperatures and their thermal conductivity is small 6.1.4 Entrance Effects In addition to the Reynolds number and the Prandtl number, several other factors can influence heat transfer by forced convection in a duct For example, when the conduit is short, entrance effects are important As a fluid enters a duct with a uniform velocity, the fluid immediately adjacent to the tube wall is brought to rest For a short distance from the entrance, a laminar boundary layer is formed along the tube wall If the turbulence in the entering fluid stream is high, the boundary layer will quickly become turbulent Irrespective of whether the boundary layer remains laminar or becomes turbulent, it will increase in thickness until it fills the entire duct From this point on, the velocity profile across the duct remains essentially unchanged Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 5/14/10 12:44 PM Page 357 6.1 Introduction 357 Viscous layer Buffer layer 0.1 =1 Pr u(r) umax 0.6 0.0 01 TS – T TS – Tcenter 0.8 10 0.0 10 1.0 0.4 ReD = 10,000 0.2 0.2 0.4 0.6 0.8 1.0 y r0 FIGURE 6.4 Effect of Prandtl number on temperature profile for turbulent flow in a long pipe (y is the distance from the tube wall and r0 is the inner pipe radius) Source: Courtesy of R C Martinelli, “Heat Transfer to Molten Metals”, Trans ASME, Vol 69, 1947, p 947 Reprinted by permission of The American Society of Mechanical Engineers International The development of the thermal boundary layer in a fluid that is heated or cooled in a duct is qualitatively similar to that of the hydrodynamic boundary layer At the entrance, the temperature is generally uniform transversely, but as the fluid flows along the duct, the heated or cooled layer increases in thickness until heat is transferred to or from the fluid in the center of the duct Beyond this point, the temperature profile remains essentially constant if the velocity profile is fully established The final shapes of the velocity and temperature profiles depend on whether the fully developed flow is laminar or turbulent Figures 6.5 on the next page and Figure 6.6 on page 359 qualitatively illustrate the growth of the boundary layers as well as the variations in the local convection heat transfer coefficient near the entrance of a tube for laminar and turbulent conditions, respectively Inspection of these figures shows that the convection heat transfer coefficient varies considerably near the entrance If the entrance is square-edged, as in most heat exchangers, the initial development of the hydrodynamic and thermal boundary layers along the walls of the tube is quite similar to that along a flat plane Consequently, Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 358 5/14/10 12:44 PM Page 358 Chapter Forced Convection Inside Tubes and Ducts u/U∞ u/U∞ u/U∞ x Velocity profile δ –hydrodynamic boundary layer Ts T/Tb T/Tb T/Tb Temperature profile for fluid being cooled (Ts = 0) δr – thermal boundary layer Ts hcx hc∞ 1.0 x/D FIGURE 6.5 Velocity distribution, temperature profiles, and variation of the local heat transfer coefficient near the inlet of a tube for air being cooled in laminar flow (surface temperature Ts uniform) the heat transfer coefficient is largest near the entrance and decreases along the duct until both the velocity and the temperature profiles for the fully developed flow have been established If the pipe Reynolds number for the fully developed flow UqDr> m is below 2100, the entrance effects may be appreciable for a length as much as 100 hydraulic diameters from the entrance For laminar flow in a circular tube, the hydraulic entry length at which the velocity profile approaches its fully developed shape can be obtained from the relation [3] a xfully developed D b lam = 0.05ReD (6.7) whereas the distance from the inlet at which the temperature profile approaches its fully developed shape is given by the relation [4] a xfully developed D b lam,T = 0.05ReD Pr (6.8) In turbulent flow, conditions are essentially independent of Prandtl numbers, and for average pipe velocities corresponding to turbulent-flow Reynolds numbers, entrance effects disappear about 10 or 20 diameters from the inlet Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_06_ch06_p350-419.qxd 5/14/10 12:44 PM Page 359 6.1 Introduction q q q q q q 359 Growth of boundary layers Variation of velocity distribution hcx hc∞ Turbulent flow behavior Laminar flow behavior Laminar boundary layer Turbulent boundary layer Fully established velocity distribution x/D FIGURE 6.6 Velocity distribution and variation of local heat transfer coefficient near the entrance of a uniformly heated tube for a fluid in turbulent flow 6.1.5 Variation of Physical Properties Another factor that can influence the heat transfer and friction considerably is the variation of physical properties with temperature When a fluid flowing in a duct is heated or cooled, its temperature and consequently its physical properties vary along the duct as well as over any given cross section For liquids, only the temperature dependence of the viscosity is of major importance For gases, on the other hand, the temperature effect on the physical properties is more complicated than for liquids because the thermal conductivity and the density, in addition to the viscosity, vary significantly with temperature In either case, the numerical value of the Reynolds number depends on the location at which the properties are evaluated It is believed that the Reynolds number based on the average bulk temperature is the significant parameter to describe the flow conditions However, considerable success in the empirical correlation of experimental heat transfer data Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_13_app3_pA50-A55.qxd A52 5/14/10 10:28 AM Page A52 Appendix Output: I A B 1.00 Ϫ0.60 0.90 C D T 0.00 0.1666 0.0999 Ϫ0.50 Ϫ0.30 0.2022 0.1112 0.80 Ϫ0.40 Ϫ0.20 0.2177 0.1444 1.10 Ϫ0.70 Ϫ0.70 0.5155 0.1999 0.95 Ϫ0.60 Ϫ0.50 0.5906 0.2779 0.85 Ϫ0.40 Ϫ0.10 0.5489 0.3778 1.15 Ϫ0.60 Ϫ0.30 1.0750 0.5001 0.70 Ϫ0.40 Ϫ0.20 0.8755 0.6443 0.75 Ϫ0.80 Ϫ0.10 1.4728 0.8111 10 1.20 0.00 Ϫ0.50 1.6056 1.0000 (b) Computer Program in Cϩϩ This program first defines and sets up the matrix coefficients, then calls a subroutine “tridiag” to perform the actual matrix inversion, or solution The tridiag subroutine can be incorporated into any computer simulation program written in Cϩϩ that requires the solution of a tridiagonal system of equations /*C++ program for solving a given Tridiagonal Matrix using the Thomas algorithm*/ /*The size of the Tridiagonal Matrix in this example is taken to be 10*/ /*Including the necessary header files*/ #include #include #include #include #include using namespace std; /*Defining a function which takes the diagonal, super-diagonal and sub-diagonal elements of the Tridiagonal Matrix along with the right hand array elements and the size of the matrix to solve the matrix*/ /*The Tridiagonal Matrix is of the general form*/ /* | A(1) |-C(2) | | | | | */ -B(1) A(2) -B(2) -C(i) A(i) -C(N-1) -B(i) A(N-1) -B(N-1) -C(N) A(N) | | | | | | | |T(1) | |D(1) | |T(2) | |D(2) | | | | | |T(i) | = |D(i) | | | | | |T(N-1)| |D(N-1)| |T(N) | |D(N) | Copyright 2011 Cengage Learning, Inc All Rights Reserved May not be copied, scanned, or duplicated, in whole or in part 67706_13_app3_pA50-A55.qxd 5/14/10 10:28 AM Page A53 Tridiagonal Matrix Computer Programs A53 /*N is the size of the matrix*/ void tridiag(int m, double W[10], double X[10], double Y[10], double Z[10]) { /*W, X, Y and Z are the diagonal, super-diagonal, sub-diagonal and right hand array elements*/ /*m is the size of the Tridiagonal Matrix*/ double P[10]={0}; double Q[10]={0}; double T[10]={0}; /*P and Q are the Recursion Variables*/ /*T is the temperature variable or the Solution Array*/ /* Calculate the initial values of the Recursion Variables*/ P[0]=X[0]/W[0]; Q[0]=Z[0]/W[0]; /*Calculate the subsequent values of the Recursion Variables*/ for(int i=1;i>=0;j— —) { T[j]=(P[j]*T[j+1])+Q[j]; } /*Display the Solution Array*/ for(int i=0;i