Lecture notes on Heat and mass Transfer

what is radiation in heat and mass transfer, heat and mass transfer fundamentals and applications,advanced heat and mass transfer lecture notes
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Kreith, F.; Boehm, R.F.; et. al. “Heat and Mass Transfer” Mechanical Engineering Handbook Ed. Frank Kreith Boca Raton: CRC Press LLC, 1999 c 1999byCRCPressLLC Heat and Mass Transfer Frank Kreith University of Colorado Robert F. Boehm 4.1 Conduction Heat Transfer................................................4-2 University of Nevada-Las Vegas Introduction • Fourier’s Law • Insulations • The Plane Wall at George D. Raithby Steady State • Long, Cylindrical Systems at Steady State • The Overall Heat Transfer Coefficient • Critical Thickness of University of Waterloo Insulation • Internal Heat Generation• Fins • Transient Systems K. G. T. Hollands • Finite-Difference Analysis of Conduction University of Waterloo 4.2 Convection Heat Transfer ..............................................4-14 N. V. Suryanarayana Natural Convection • Forced Convection — External Flows • Michigan Technological University Forced Convection — Internal Flows 4.3 Radiation ........................................................................4-56 Michael F. Modest Nature of Thermal Radiation • Blackbody Radiation • Pennsylvania State University Radiative Exchange between Opaque Surfaces • Radiative Van P. Carey Exchange within Participating Media University of California at Berkeley 4.4 Phase-Change .................................................................4-82 John C. Chen Boiling and Condensation • Particle Gas Convection • Melting Lehigh University and Freezing 4.5 Heat Exchangers ..........................................................4-118 Noam Lior Compact Heat Exchangers • Shell-and-Tube Heat Exchangers University of Pennsylvania 4.6 Temperature and Heat Transfer Measurements...........4-182 Ram K. Shah Temperature Measurement • Heat Flux • Sensor Environmental Delphi Harrison Thermal Systems Errors • Evaluating the Heat Transfer Coefficient Kenneth J. Bell 4.7 Mass Transfer...............................................................4-206 Oklahoma State University Introduction • Concentrations, Velocities, and Fluxes • Mechanisms of Diffusion • Species Conservation Equation • Robert J. Moffat Diffusion in a Stationary Medium • Diffusion in a Moving Stanford University Medium • Mass Convection Anthony F. Mills 4.8 Applications..................................................................4-240 University of California at Los Angeles Enhancement • Cooling Towers • Heat Pipes • Cooling Electronic Equipment Arthur E. Bergles Rensselaer Polytechnic Institute 4.9 Non-Newtonian Fluids — Heat Transfer ....................4-279 Introduction • Laminar Duct Heat Transfer — Purely Viscous, Larry W. Swanson Time-Independent Non-Newtonian Fluids • Turbulent Duct Heat Transfer Research Institute Flow for Purely Viscous Time-Independent Non-Newtonian Vincent W. Antonetti Fluids • Viscoelastic Fluids • Free Convection Flows and Heat Poughkeepsie, New York Transfer Thomas F. Irvine, Jr. State University of New York, Stony Brook Massimo Capobianchi State University of New York, Stony Brook 4-1 © 1999 by CRC Press LLC 4-2 Section 4 4.1 Conduction Heat Transfer Robert F. Boehm Introduction Conduction heat transfer phenomena are found throughout virtually all of the physical world and the industrial domain. The analytical description of this heat transfer mode is one of the best understood. Some of the bases of understanding of conduction date back to early history. It was recognized that by invoking certain relatively minor simplifications, mathematical solutions resulted directly. Some of these were very easily formulated. What transpired over the years was a very vigorous development of applications to a broad range of processes. Perhaps no single work better summarizes the wealth of these studies than does the book by Carslaw and Jaeger (1959). They gave solutions to a broad range of problems, from topics related to the cooling of the Earth to the current-carrying capacities of wires. The general analyses given there have been applied to a range of modern-day problems, from laser heating to temperature-control systems. Today conduction heat transfer is still an active area of research and application. A great deal of interest has developed in recent years in topics like contact resistance, where a temperature difference develops between two solids that do not have perfect contact with each other. Additional issues of current interest include non-Fourier conduction, where the processes occur so fast that the equation described below does not apply. Also, the problems related to transport in miniaturized systems are garnering a great deal of interest. Increased interest has also been directed to ways of handling composite materials, where the ability to conduct heat is very directional. Much of the work in conduction analysis is now accomplished by use of sophisticated computer codes. These tools have given the heat transfer analyst the capability of solving problems in nonhomo- genous media, with very complicated geometries, and with very involved boundary conditions. It is still important to understand analytical ways of determining the performance of conducting systems. At the minimum these can be used as calibrations for numerical codes. Fourier’s Law The basis of conduction heat transfer is Fourier’s Law. This law involves the idea that the heat flux is proportional to the temperature gradient in any direction n. Thermal conductivity, k, a property of materials that is temperature dependent, is the constant of proportionality. ¶T q =-kA (4.1.1) k ¶n In many systems the area A is a function of the distance in the direction n. One important extension is that this can be combined with the first law of thermodynamics to yield the heat conduction equation. For constant thermal conductivity, this is given as ˙ q 1 ¶T 2 G Ñ T+ = (4.1.2) k a ¶t In this equation, a is the thermal diffusivity and q ˙ is the internal heat generation per unit volume. G Some problems, typically steady-state, one-dimensional formulations where only the heat flux is desired, can be solved simply from Equation (4.1.1). Most conduction analyses are performed with Equation (4.1.2). In the latter, a more general approach, the temperature distribution is found from this equation and appropriate boundary conditions. Then the heat flux, if desired, is found at any location using Equation (4.1.1). Normally, it is the temperature distribution that is of most importance. For example, © 1999 by CRC Press LLC Heat and Mass Transfer 4-3 it may be desirable to know through analysis if a material will reach some critical temperature, like its melting point. Less frequently the heat flux is desired. While there are times when it is simply desired to understand what the temperature response of a structure is, the engineer is often faced with a need to increase or decrease heat transfer to some specific level. Examination of the thermal conductivity of materials gives some insight into the range of possi- bilities that exist through simple conduction. Of the more common engineering materials, pure copper exhibits one of the higher abilities to conduct 2 heat with a thermal conductivity approaching 400 W/m K. Aluminum, also considered to be a good conductor, has a thermal conductivity a little over half that of copper. To increase the heat transfer above values possible through simple conduction, more-involved designs are necessary that incorporate a variety of other heat transfer modes like convection and phase change. Decreasing the heat transfer is accomplished with the use of insulations. A separate discussion of these follows. Insulations Insulations are used to decrease heat flow and to decrease surface temperatures. These materials are found in a variety of forms, typically loose fill, batt, and rigid. Even a gas, like air, can be a good insulator if it can be kept from moving when it is heated or cooled. A vacuum is an excellent insulator. Usually, though, the engineering approach to insulation is the addition of a low-conducting material to the surface. While there are many chemical forms, costs, and maximum operating temperatures of common forms of insulations, it seems that when a higher operating temperature is required, many times the thermal conductivity and cost of the insulation will also be higher. Loose-fill insulations include such materials as milled alumina-silica (maximum operating temperature 2 of 1260°C and thermal conductivities in the range of 0.1 to 0.2 W/m K) and perlite (maximum operating 2 temperature of 980°C and thermal conductivities in the range of 0.05 to 1.5 W/m K). Batt-type insulations include one of the more common types — glass fiber. This type of insulation comes in a variety of densities, which, in turn, have a profound affect on the thermal conductivity. Thermal con- 2 ductivities for glass fiber insulations can range from about 0.03 to 0.06 W/m K. Rigid insulations show a very wide range of forms and performance characteristics. For example, a rigid insulation in foam 2 form, polyurethane, is very lightweight, shows a very low thermal conductivity (about 0.02 W/m K), but has a maximum operating temperature only up to about 120°C. Rigid insulations in refractory form show quite different characteristics. For example, high-alumina brick is quite dense, has a thermal 2 conductivity of about 2 W/m K, but can remain operational to temperatures around 1760°C. Many insulations are characterized in the book edited by Guyer (1989). Often, commercial insulation systems designed for high-temperature operation use a layered approach. Temperature tolerance may be critical. Perhaps a refractory is applied in the highest temperature region, an intermediate-temperature foam insulation is used in the middle section, and a high-performance, low- temperature insulation is used on the outer side near ambient conditions. Analyses can be performed including the effects of temperature variations of thermal conductivity. However, the most frequent approach is to assume that the thermal conductivity is constant at some temperature between the two extremes experienced by the insulation. The Plane Wall at Steady State Consider steady-state heat transfer in a plane wall of thickness L, but of very large extent in both other directions. The wall has temperature T on one side and T on the other. If the thermal conductivity is 1 2 considered to be constant, then Equation (4.1.1) can be integrated directly to give the following result: kA q=- TT (4.1.3) () k 12 L This can be used to determine the steady-state heat transfer through slabs. © 1999 by CRC Press LLC 4-4 Section 4 An electrical circuit analog is widely used in conduction analyses. This is realized by considering the temperature difference to be analogous to a voltage difference, the heat flux to be like current flow, and the remainder of Equation (4.1.3) to be like a thermal resistance. The latter is seen to be L R = (4.1.4) k kA Heat transfer through walls made of layers of different types of materials can be easily found by summing the resistances in series or parallel form, as appropriate. In the design of systems, seldom is a surface temperature specified or known. More often, the surface is in contact with a bulk fluid, whose temperature is known at some distance from the surface. Convection from the surface is then represented by Newton’s law of cooling: q=- h AT T (4.1.5) () cs ¥ This equation can also be represented as a temperature difference divided by a thermal resistance, which 1/ h . is It can be shown that a very low surface resistance, as might be represented by phase change cA phenomena, has the effect of imposing the fluid temperature directly on the surface. Hence, usually a known surface temperature results from a fluid temperature being imposed directly on the surface through a very high heat transfer coefficient. For this reason, in the later results given here, particularly those for transient systems, a convective boundary will be assumed. For steady results this is less important because of the ability to add resistances through the circuit analogy. Long, Cylindrical Systems at Steady State For long (L) annular systems at steady-state conditions with constant thermal conductivities, the following two equations are the appropriate counterparts to Equations (4.1.3) and (4.1.4). The heat transfer can be expressed as 2pLk q = - (4.1.6) TT () k 12 lnrr 21 Here, r and r represent the radii of annular section. A thermal resistance for this case is as shown below. 1 2 lnrr 21 R = (4.1.7) k 2pLk The Overall Heat Transfer Coefficient The overall heat transfer coefficient concept is valuable in several aspects of heat transfer. It involves a modified form of Newton’s law of cooling, as noted above, and it is written as Q=D UAT (4.1.8) U In this formulation is the overall heat transfer coefficient based upon the area A. Because the area for heat transfer in a problem can vary (as with a cylindrical geometry), it is important to note that the U is dependent upon which area is selected. The overall heat transfer coefficient is usually found from UA a combination of thermal resistances. Hence, for a common series-combination-circuit analog, the product is taken as the sum of resistances. © 1999 by CRC Press LLC Heat and Mass Transfer 4-5 11 UA== (4.1.9) n R total R i å i=1 To show an example of the use of this concept, consider Figure 4.1.1. FIGURE 4.1.1. An insulated tube with convective environments on both sides. UA For steady-state conditions, the product remains constant for a given heat transfer and overall temperature difference. This can be written as UA=U A =U A = UA (4.1.10) 11 2 2 3 3 If the inside area, A , is chosen as the basis, the overall heat transfer coefficient can then be expressed as 1 1 U = (4.1.11) 1 r ln rr r lnrr () () r 1 1 21 1 32 1 + + + h k k rh ci,, pipe ins 3 c o Critical Thickness of Insulation Sometimes insulation can cause an increase in heat transfer. This circumstance should be noted in order to apply it when desired and to design around it when an insulating effect is needed. Consider the circumstance shown in Figure 4.1.1. Assume that the temperature is known on the outside of the tube (inside of the insulation). This could be known if the inner heat transfer coefficient is very large and the thermal conductivity of the tube is large. In this case, the inner fluid temperature will be almost the same as the temperature of the inner surface of the insulation. Alternatively, this could be applied to a coating (say an electrical insulation) on the outside of a wire. By forming the expression for the heat transfer in terms of the variables shown in Equation (4.1.11), and examining the change of heat transfer with variations in r (that is, the thickness of insulation) a maximum heat flow can be found. While simple 3 results are given many texts (showing the critical radius as the ratio of the insulation thermal conductivity to the heat transfer coefficient on the outside), Sparrow (1970) has considered a heat transfer coefficient -m n h r that varies as T – T . For this case, it is found that the heat transfer is maximized at c,o 3 3 f,o © 1999 by CRC Press LLC 4-6 Section 4 k ins rr==11 -m +n (4.1.12) ()() 3 crit h co , h By examining the order of magnitudes of m, n, k , and the critical radius is found to be often ins co , on the order of a few millimeters. Hence, additional insulation on small-diameter cylinders such as small- gauge electrical wires could actually increase the heat dissipation. On the other hand, the addition of insulation to large-diameter pipes and ducts will almost always decrease the heat transfer rate. Internal Heat Generation The analysis of temperature distributions and the resulting heat transfer in the presence of volume heat sources is required in some circumstances. These include phenomena such as nuclear fission processes, joule heating, and microwave deposition. Consider first a slab of material 2L thick but otherwise very large, with internal generation. The outside of the slab is kept at temperature T . To find the temperature 1 distribution within the slab, the thermal conductivity is assumed to be constant. Equation (4.1.2) reduces to the following: 2 ˙ q dT G += 0 (4.1.13) 2 dx k Solving this equation by separating variables, integrating twice, and applying boundary conditions gives 2 2 ˙ é ù qL x æ ö G Tx-= T - (4.1.14) () 1 ê ú 1 è ø 2k L ë û A similar type of analysis for a long, cylindrical element of radius r gives 1 2 2 é ù ˙ æ ö qr r G 1 Tr-= T ê- ú (4.1.15) () 1 1 ç ÷ 4k r ê è ø ú 1 ë û Two additional cases will be given. Both involve the situation when the heat generation rate is dependent upon the local temperature in a linear way (defined by a slope b), according to the following relationship: qq ˙˙=+ 1 bT-T (4.1.16) () GG,oo For a plane wall of 2L thickness and a temperature of T specified on each surface 1 Tx ()-+ T 1 b cos mx o (4.1.17) = TT-+ 1 b cos mL 1 o For a similar situation in a long cylinder with a temperature of T specified on the outside radius r 1 1 Tr ()-+ T 1 b Jr () m o o = (4.1.18) TT-+ 1 b Jr m () 1 o o 1 © 1999 by CRC Press LLC Heat and Mass Transfer 4-7 In Equation (4.1.18), the J is the typical notation for the Bessel function. Variations of this function are o tabulated in Abramowitz and Stegun (1964). In both cases the following holds: ˙ bq Go , m º k Fins Fins are widely used to enhance the heat transfer (usually convective, but it could also be radiative) from a surface. This is particularly true when the surface is in contact with a gas. Fins are used on air- cooled engines, electronic cooling forms, as well as for a number of other applications. Since the heat transfer coefficient tends to be low in gas convection, area is added in the form of fins to the surface to decrease the convective thermal resistance. The simplest fins to analyze, and which are usually found in practice, can be assumed to be one- dimensional and constant in cross section. In simple terms, to be one-dimensional, the fins have to be long compared with a transverse dimension. Three cases are normally considered for analysis, and these are shown in Figure 4.1.2. They are the insulated tip, the infinitely long fin, and the convecting tip fin. FIGURE 4.1.2. Three typical cases for one-dimensional, constant-cross-section fins are shown. For Case, I, the solution to the governing equation and the application of the boundary conditions of the known temperature at the base and the insulated tip yields coshmL-x () Case I: q== q (4.1.19) b cosh mL For the infinitely long case, the following simple form results: -mx = Case II: q(xe )q (4.1.20) b The final case yields the following result: mL cosh m L - x + Bisinhm L - x ( ) () Case III: = q(x)q (4.1.21) b mL cosh mL + Bisinh mL where Bi ºh Lk c © 1999 by CRC Press LLC 4-8 Section 4 In all three of the cases given, the following definitions apply: hP 2 c qºTx-T ,qºT x= 0-º T , and m ( ) ( ) ¥¥ b kA Here A is the cross section of the fin parallel to the wall; P is the perimeter around that area. To find the heat removed in any of these cases, the temperature distribution is used in Fourier’s law, Equation (4.1.1). For most fins that truly fit the one-dimensional assumption (i.e., long compared with their transverse dimensions), all three equations will yield results that do not differ widely. Two performance indicators are found in the fin literature. The fin efficiency is defined as the ratio of the actual heat transfer to the heat transfer from an ideal fin. q actual hº (4.1.22) q ideal The ideal heat transfer is found from convective gain or loss from an area the same size as the fin surface area, all at a temperature T . Fin efficiency is normally used to tabulate heat transfer results for various b types of fins, including ones with nonconstant area or which do not meet the one-dimensional assumption. An example of the former can be developed from a result given by Arpaci (1966). Consider a straight fin of triangular profile, as shown in Figure 4.1.3. The solution is found in terms of modified Bessel functions of the first kind. Tabulations are given in Abramowitz and Stegun (1964). 12 ˜ I 2mL () 1 h= (4.1.23) 12 12 m˜˜ L I 2mL () o Here, ˜ º 2 m h L kb. c The fin effectiveness, e, is defined as the heat transfer from the fin compared with the bare-surface transfer through the same base area. FIGURE 4.1.3. Two examples of fins with a cross-sectional area that varies with distance from the base. (a) Straight triangular fin. (b) Annular fin of constant thickness. © 1999 by CRC Press LLCHeat and Mass Transfer 4-9 q q f actual e= = (4.1.24) q h AT -T () bare base cb ¥ Carslaw and Jaeger (1959) give an expression for the effectiveness of a fin of constant thickness around ˜ (mº 2h kb). a tube (see Figure 4.1.3) This is given as c ˜ ˜ ˜ ˜ Imr Kmr -Kmr Imr ( ) ( ) ( ) ( ) 2 1 2 1 1 1 2 1 1 e (4.1.25) = ˜ ˜˜ ˜˜ mbImm rK r +K mrImr ( ) ( ) ( ) ( ) oo 1 1 2 1 1 2 Here the notations I and K denote Bessel functions that are given in Abramowitz and Stegun (1964). Fin effectiveness can be used as one indication whether or not fins should be added. A rule of thumb indicates that if the effectiveness is less than about three, fins should not be added to the surface. Transient Systems Negligible Internal Resistance Consider the transient cooling or heating of a body with surface area A and volume V. This is taking h place by convection through a heat transfer coefficient to an ambient temperature of T . Assume the c ¥ thermal resistance to conduction inside the body is significantly less than the thermal resistance to convection (as represented by Newton’s law of cooling) on the surface of the body. This ratio is denoted by the Biot number, Bi. R h(V A) k c Bi== (4.1.26) R k c The temperature (which will be uniform throughout the body at any time for this situation) response with time for this system is given by the following relationship. Note that the shape of the body is not important — only the ratio of its volume to its area matters. T(t) -T -r h At Vc ¥ c (4.1.27) = e TT - o ¥ Typically, this will hold for the Biot number being less than (about) 0.1. Bodies with Significant Internal Resistance When a body is being heated or cooled transiently in a convective environment, but the internal thermal resistance of the body cannot be neglected, the analysis becomes more complicated. Only simple geometries (a symmetrical plane wall, a long cylinder, a composite of geometric intersections of these geometries, or a sphere) with an imposed step change in ambient temperature are addressed here. The first geometry considered is a large slab of minor dimension 2L. If the temperature is initially uniform at T , and at time 0+ it begins convecting through a heat transfer coefficient to a fluid at T , the o ¥ temperature response is given by ¥ æ ö sin l L 22 n q = 2 exp -lL Fo coslx (4.1.28) ( ) () ç ÷ nn å lL + sin lLcoslL è ø n n n n=1 and the l are the roots of the transcendental equation: l L tan l L = Bi. The following definitions hold: n n n © 1999 by CRC Press LLC4-10 Section 4 hL TT - at c ¥ Bi º Foº qº 2 k L TT - o ¥ The second geometry considered is a very long cylinder of diameter 2R. The temperature response for this situation is ¥ 22 exp -ll R FoJ r ( ) () n on q 2Bi (4.1.29) = å 22 2 ll R + BiJ R () ( ) n on n=1 Now the l are the roots of l R J (l R) + Bi J (l R) = 0, and n n 1 n o n hR TT - at c ¥ Bi = Fo= q= 2 k R TT - o ¥ The common definition of Bessel’s functions applies here. For the similar situation involving a solid sphere, the following holds: ¥ sin lR -lRcoslR sin l r () () () n n n 22 n q 2 exp -l R (4.1.30) = Fo () n å lR - sin lR coslR l r ( ) () n n n n n=1 and the l are found as the roots of l R cos l R = (1 – Bi) sin l R. Otherwise, the same definitions as n n n n were given for the cylinder hold. Solids that can be envisioned as the geometric intersection of the simple shapes described above can be analyzed with a simple product of the individual-shape solutions. For these cases, the solution is found as the product of the dimensionless temperature functions for each of the simple shapes with appropriate distance variables taken in each solution. This is illustrated as the right-hand diagram in Figure 4.1.4. For example, a very long rod of rectangular cross section can be seen as the intersection of two large plates. A short cylinder represents the intersection of an infinitely long cylinder and a plate. The temperature at any location within the short cylinder is = (4.1.31) q q q 22 RL , Rod Infinite 2R Rod 2L Plate Details of the formulation and solution of the partial differential equations in heat conduction are found in the text by Arpaci (1966). Finite-Difference Analysis of Conduction Today, numerical solution of conduction problems is the most-used analysis approach. Two general techniques are applied for this: those based upon finite-difference ideas and those based upon finite- element concepts. General numerical formulations are introduced in other sections of this book. In this section, a special, physical formulation of the finite-difference equations to conduction phenomena is briefly outlined. Attention is drawn to a one-dimensional slab (very large in two directions compared with the thickness). The slab is divided across the thickness into smaller subslabs, and this is shown in Figure 4.1.5. All subslabs are thickness Dx except for the two boundaries where the thickness is Dx/2. A characteristic temperature for each slab is assumed to be represented by the temperature at the slab center. Of course, this assumption becomes more accurate as the size of the slab becomes smaller. With © 1999 by CRC Press LLCHeat and Mass Transfer 4-11 FIGURE 4.1.4. Three types of bodies that can be analyzed with results given in this section. (a) Large plane wall of 2L thickness; (b) long cylinder with 2R diameter; (c) composite intersection. FIGURE 4.1.5. A one-dimensional finite differencing of a slab with a general interior node and one surface node detailed. the two boundary-node centers located exactly on the boundary, a total of n nodes are used (n – 2 full nodes and one half node on each of the two boundaries). In the analysis, a general interior node i (this applies to all nodes 2 through n – 1) is considered for an overall energy balance. Conduction from node i – 1 and from node i + 1 as well as any heat generation present is assumed to be energy per unit time flowing into the node. This is then equated to the time rate of change of energy within the node. A backward difference on the time derivative is applied here, and the notation T ; T (t + Dt) is used. The balance gives the following on a per-unit-area basis: ¢ i i TT ¢ - ¢ TT ¢ - ¢ TT ¢ - ii -11i + i ii qx ˙ rxc + +=DD (4.1.32) Gi , p DD xk xk Dt - + In this equation different thermal conductivities have been used to allow for possible variations in properties throughout the solid. The analysis of the boundary nodes will depend upon the nature of the conditions there. For the purposes of illustration, convection will be assumed to be occurring off of the boundary at node 1. A balance similar to Equation (4.1.32) but now for node 1 gives the following: © 1999 by CRC Press LLC4-12 Section 4 TT ¢ - ¢ TT ¢ - ¢ TT ¢ - DD xx ¥ 12 1 11 + += q ˙ r c (4.1.33) Gp ,1 12 h Dxk 2 Dt c + After all n equations are written, it can be seen that there are n unknowns represented in these equations: the temperature at all nodes. If one or both of the boundary conditions are in terms of a specified temperature, this will decrease the number of equations and unknowns by one or two, respec- tively. To determine the temperature as a function of time, the time step is arbitrarily set, and all the temperatures are found by simultaneous solution at t = 0 + Dt. The time is then advanced by Dt and the temperatures are then found again by simultaneous solution. The finite difference approach just outlined using the backward difference for the time derivative is termed the implicit technique, and it results in an n ´ n system of linear simultaneous equations. If the forward difference is used for the time derivative, then only one unknown will exist in each equation. This gives rise to what is called an explicit or “marching” solution. While this type of system is more straightforward to solve because it deals with only one equation at a time with one unknown, a stability criterion must be considered which limits the time step relative to the distance step. Two- and three-dimensional problems are handled in conceptually the same manner. One-dimensional heat fluxes between adjoining nodes are again considered. Now there are contributions from each of the dimensions represented. Details are outlined in the book by Jaluria and Torrance (1986). Defining Terms Biot number: Ratio of the internal (conductive) resistance to the external (convective) resistance from a solid exchanging heat with a fluid. Fin: Additions of material to a surface to increase area and thus decrease the external thermal resistance from convecting and/or radiating solids. Fin effectiveness: Ratio of the actual heat transfer from a fin to the heat transfer from the same cross- sectional area of the wall without the fin. Fin efficiency: Ratio of the actual heat transfer from a fin to the heat transfer from a fin with the same geometry but completely at the base temperature. Fourier’s law: The fundamental law of heat conduction. Relates the local temperature gradient to the local heat flux, both in the same direction. Heat conduction equation: A partial differential equation in temperature, spatial variables, time, and properties that, when solved with appropriate boundary and initial conditions, describes the variation of temperature in a conducting medium. Overall heat transfer coefficient: The analogous quantity to the heat transfer coefficient found in convection (Newton’s law of cooling) that represents the overall combination of several thermal resistances, both conductive and convective. Thermal conductivity: The property of a material that relates a temperature gradient to a heat flux. Dependent upon temperature. References Abramowitz, M. and Stegun, I. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55. Arpaci, V. 1966. Conduction Heat Transfer, Addison-Wesley, Reading, MA. Carslaw, H.S. and Jaeger, J.C. 1959. Conduction of Heat in Solids, 2nd ed., Oxford University Press, London. Guyer, E., Ed. 1989. Thermal insulations, in Handbook of Applied Thermal Design, McGraw-Hill, New York, Part 3. Jaluria, Y. and Torrance, K. 1986. Computational Heat Transfer, Hemisphere Publishing, New York. Sparrow, E. 1970. Reexamination and correction of the critical radius for radial heat conduction, AIChE J. 16, 1, 149. © 1999 by CRC Press LLCHeat and Mass Transfer 4-13 Further Information The references listed above will give the reader an excellent introduction to analytical formulation and solution (Arpaci), material properties (Guyer), and numerical formulation and solution (Jaluria and Torrance). Current developments in conduction heat transfer appear in several publications, including The Journal of Heat Transfer, The International Journal of Heat and Mass Transfer, and Numerical Heat Transfer. © 1999 by CRC Press LLC4-14 Section 4 4.2 Convection Heat Transfer Natural Convection George D. Raithby and K.G. Terry Hollands Introduction Natural convection heat transfer occurs when the convective fluid motion is induced by density differences that are themselves caused by the heating. An example is shown in Figure 4.2.1(A), where a body at surface temperature T transfers heat at a rate q to ambient fluid at temperature T T . s ¥ s FIGURE 4.2.1 (A) Nomenclature for external heat transfer. (A) General sketch; (B) is for a tilted flat plate, and (C) defines the lengths cal for horizontal surfaces. In this section, correlations for the average Nusselt number are provided from which the heat transfer rate q from surface area A can be estimated. The Nusselt number is defined as s hL qL c Nu== (4.2.1) k A DTk s where DT = T – T is the temperature difference driving the heat transfer. A dimensional analysis leads s ¥ to the following functional relation: Nu = f Ra, Pr, geometric shape, boundary conditions (4.2.2) ( ) For given thermal boundary conditions (e.g., isothermal wall and uniform T ), and for a given geometry ¥ (e.g., a cube), Equation (4.2.2) states that Nu depends only on the Rayleigh number, Ra, and Prandtl number, Pr. The length scales that appear in Nu and Ra are defined, for each geometry considered, in a separate figure. The fluid properties are generally evaluated at T , the average of the wall and ambient f temperatures. The exception is that b, the temperature coefficient of volume expansion, is evaluated at T for external natural convection(Figures 4.2.1 to 4.2.3) in a gaseous medium. ¥ The functional dependence on Pr is approximately independent of the geometry, and the following Pr-dependent function will be useful for laminar heat transfer (Churchill and Usagi, 1972): 49 9 16 C= 0.671 1 + 0.492 Pr (4.2.3) () () l V H C C and are functions that will be useful for turbulent heat transfer: t t © 1999 by CRC Press LLCHeat and Mass Transfer 4-15 FIGURE 4.2.2 Nomenclature for heat transfer from planar surfaces of different shapes. FIGURE 4.2.3 Definitions for computing heat transfer from a long circular cylinder (A), from the lateral surface of a vertical circular cylinder (B), from a sphere (C), and from a compound body (D). 04 . 2 V 0.22 0.81 C=+ 01.. 3Pr 1 061Pr (4.2.4) () t 1 + 0.0107Pr H æ ö C = 01 . 4 (4.2.5) t è ø 10 + .01Pr The superscripts V and H refer to the vertical and horizontal surface orientation. The Nusselt numbers for fully laminar and fully turbulent heat transfer are denoted by Nu and Nu , , t respectively. Once obtained, these are blended (Churchill and Usagi, 1972) as follows to obtain the equation for Nu: 1 m m m Nu = Nu + Nu (4.2.6) () () () l t The blending parameter m depends on the body shape and orientation. T The equation for Nu in this section is usually expressed in terms of Nu , the Nusselt number that , T would be valid if the thermal boundary layer were thin. The difference between Nu and Nu accounts l for the effect of the large boundary layer thicknesses encountered in natural convection. It is assumed that the wall temperature of a body exceeds the ambient fluid temperature (T T ). s ¥ For T T the same correlations apply with (T – T ) replacing (T – T ) for a geometry that is rotated s ¥ ¥ s s ¥ © 1999 by CRC Press LLC4-16 Section 4 180° relative to the gravitational vector; for example, the correlations for a horizontal heated upward- facing flat plate applies to a cooled downward-facing flat plate of the same planform. Correlations for External Natural Convection This section deals with problems where the body shapes in Figures 4.2.1 to 4.2.3 are heated while immersed in a quiescent fluid. Different cases are enumerated below. 1. Isothermal Vertical (f = 0) Flat Plate, Figure 4.2.1B. For heat transfer from a vertical plate 12 (Figure 4.2.1B), for 1 Ra 10 , 20 . T 14 Nu == C Ra Nu (4.2.7) ll T ln 1 +2.0 Nu () V 13 9 Nu =C Ra 1+1.4´ 10 Pr Ra () tt V C C and are given by Equations (4.2.3) and (4.2.4). Nu is obtained by substituting Equation l t (4.2.7) expressions for Nu and Nu into Equation (4.2.6) with m = 6. l t 2. Vertical Flat Plate with Uniform Heat Flux, Figure 4.2.1B. If the plate surface has a constant (known) heat flux, rather than being isothermal, the objective is to calculate the average temper- 5 DT, ature difference, between the plate and fluid. For this situation, and for 15 Ra 10 , 15 3 4 14 1.83 T V Nu =G Ra Nu = Nu =C Ra (4.2.8a) () ( ) ( ) ll tt T ln 1 +1.83 Nu () 15 6 æ Pr ö G = (4.2.8b) ç ÷ l è ø 5 49+ P r + 10Pr V C Ra is defined in Figure 4.2.1B and is given by Equation (4.2.4). Find Nu by inserting these t G expressions for Nu and Nu into Equation (4.2.6) with m = 6. The expression is due to Fujii l t l and Fujii (1976). 3. Horizontal Upward-Facing (f = 90°) Plates, Figure 4.2.1C. For horizontal isothermal surfaces of various platforms, correlations are given in terms of a lengthscale L (Goldstein et al., 1973), defined in Figure 4.2.1C. For Ra ³ 1, 20 . T 14 H 13 Nu == 0.835C Ra Nu Nu =C Ra (4.2.9) ll tt T ln 1 +1.4 Nu () Nu is obtained by substituting Nu and Nu from Equation 4.2.9 into Equation 4.2.6 with m = 10. l t DT. For non-isothermal surfaces, replace DT by 4. Horizontal Downward-Facing (f = –90°) Plates, Figure 4.2.1C. For horizontal downward-facing plates of various planforms, the main buoyancy force is into the plate so that only a very weak force drives the fluid along the plate; for this reason, only laminar flows have been measured. For 10 this case, the following equation applies for Ra 10 , Pr ³ 0.7: 0.527 2.45 T 15 Nu == H Ra H Nu = (4.2.10) ll 29 T 9 10 ln 1 +2.45 Nu () 1+ 1.9 Pr () H fits the analysis of Fujii et al. (1973). l © 1999 by CRC Press LLCHeat and Mass Transfer 4-17 5. Inclined Plates, Downward Facing (–90° £ f £ 0), Figure 4.2.1B. First calculate q from Case 1 with g replaced by g cos f; then calculate q from Case 4 (horizontal plate) with g replaced by g sin (–f), and use the maximum of these two values of q. 6. Inclined Plates, Upward Facing (0 £ f £ 90), Figure 4.2.1B. First calculate q from Case 1 with g replaced by g cos f; then calculate q from Case 3 with g replaced by g sin f, and use the maximum of these two values of q. 7. Vertical and Tilted Isothermal Plates of Various Planform, Figure 4.2.2. The line of constant c in Figure 4.2.2 is the line of steepest ascent on the plate. Provided all such lines intersect the T plate edges just twice, as shown in the figure, the thin-layer (Nu ) heat transfer can be found by subdividing the body into strips of width Dc, calculating the heat transfer from each strip, and adding. For laminar flow from an isothermal vertical plate, this results in 14 W æ ö L T 14 34 Nu =º CC Ra C Sdc (4.2.11) ç ÷ 1 l 1 ò è A 0 ø Symbols are defined in Figure 4.2.2, along with L and calculated C values for some plate shapes. 1 If the plate is vertical, follow the procedure in Case 1 above (isothermal vertical flat plate) except T replace the expression for Nu in Equation (4.2.7) by Equation (4.2.11). If the plate is tilted, follow the procedure described in Case 5 or 6 (as appropriate) but again use Equation (4.2.11) T for Nu in Equation (4.2.7) 8. Horizontal Cylinders, Figure 4.2.3A. For a long, horizontal circular cylinder use the following expressions for Nu and Nu : l t 2 f T 14 13 Nu == 0.772C Ra Nu Nu =C Ra (4.2.12) ll tt T 12 + f Nu () –2 –10 –2 C is given in the table below. For Ra 10 , f = 0.8 can be used, but for 10 Ra 10 use t T 0.16 f = 1 – 0.13/(Nu ) . To find Nu, the values of Nu and Nu from Equation (4.2.12) are substituted l t into Equation (4.2.6) with m = 15 (Clemes et al., 1994). C for Various Shapes and Prandtl Numbers t Pr® 0.01 0.022 0.10 0.71 2.0 6.0 50 100 2000 Horizontal cylinder 0.077 0.81 0.90 0.103 0.108 0.109 0.100 0.097 0.088 Spheres 0.074 0.078 0.088 0.104 0.110 0.111 0.101 0.97 0.086 9. Vertical Cylinders (f = 90°), Figure 4.2.3B. For high Ra values and large diameter, the heat T transfer from a vertical cylinder approaches that for a vertical flat plate. Let the Nu and Nu l T equations for a vertical flat plate of height L, Equation (4.2.7), be rewritten here as Nu and p Nu , respectively. At smaller Ra and diameter, transverse curvature plays a role which is accounted p for in the following equations: 09 . xNu 2LD p Nu = x = (4.2.13) l T ln 1+0.9x Nu () p These equations are valid for purely laminar flow. To obtain Nu, blend Equation (4.2.13) for Nu l with Equation (4.2.7) for Nu using Equation (4.2.6) with m = 10. t © 1999 by CRC Press LLC4-18 Section 4 10. Spheres, Figure 4.2.3C. For spheres use Equation (4.2.6), with m = 6, and with 14 13 Nu =+ 2 0.878 CC Ra and Nu = Ra (4.2.14) ll tt C The table above contains values. t 11. Combined Shapes, Figure 4.2.3D. For combined shapes, such as the cylinder in Figure 4.2.3D with spherical end caps, calculate the heat transfer from the cylinder of length L (Case 8), the heat transfer from a sphere of diameter D (Case 10) and add to obtain the total transfer. Other shapes can be treated in a similar manner. Correlations for Open Cavities Examples of this class of problem are shown in Figure 4.2.4. Walls partially enclose a fluid region (cavity) where boundary openings permit fluid to enter and leave. Upstream from its point of entry, the fluid is at the ambient temperature, T . Since access of the ambient fluid to the heated surfaces is ¥ restricted, some of the heated surface is starved of cool ambient to which heat can be transferred. As the sizes of the boundary openings are increased, the previous class of problems is approached; for example, when the plate spacing in Figure 4.2.4A (Case 12) becomes very large, the heat transfer from each vertical surface is given by Case 1. FIGURE 4.2.4 Nomenclature for various open-cavity problems. 12. Isothermal Vertical Channels, Figure 4.2.4A and B. Figure 4.2.4A shows an open cavity bounded by vertical walls and open at the top and bottom. The large opposing plates are isothermal, at DT temperatures T and T , respectively, and the spacing between these plates is small. is the 1 2 average temperature difference between the plates and T , as shown in Figure 4.2.4A, but T and ¥ 1 T must not straddle T . For this case 2 ¥ 1 m m æ ö æ ö m Ra 14 5 Nu = + CC Ra Ra 10 £ (4.2.15) () ç ç ÷ 1 l ÷ è f Re ø è ø © 1999 by CRC Press LLCHeat and Mass Transfer 4-19 where f Re is the product of friction factor and Reynolds number for fully developed flow through, and C is a constant that accounts for the augmentation of heat transfer, relative to a vertical flat 1 plate (Case 1), due to the chimney effect. The fRe factor accounts for the cross-sectional shape (Elenbaas, 1942a). Symbols are defined in Figure 4.2.4A and B; in the Nu equation, q is the total heat transferred to the ambient fluid from all heated surfaces. For the parallel plate channel shown in Figure 4.2.4(A), use f Re = 24, m = –1.9, and for gases C » 1.2. It should be noted, however, that C must approach 1.0 as Pr increases or as the plate 1 1 spacing increases. For channels of circular cross section (Figure 4.2.4B) fRe - 16, m = –1.03, and for gases C » 1.17. For other cross-sectional shapes like the square (fRe = 14.23), hexagonal 1 (fRe = 15.05), or equilateral triangle (fRe = 13.3), use Equation (4.2.15) with the appropriate fRe, and with m = –1.5, and C » 1.2 for gases. 1 The heat transfer per unit cross-sectional area, q/A , for a given channel length H and temperature c difference, passes through a maximum at approximately Ra , where max 43 æ ö f ReCC 1 l Ra = (4.2.16) max ç ÷ 1 m 2 è ø Ra provides the value of hydraulic radius r = 2A /P at this maximum. max c 13. Isothermal Triangular Fins, Figure 4.2.4C. For a large array of triangular fins (Karagiozis et al., 5 1994) in air, for 0.4 Ra 5 ´ 10 -13 3 é ù 3.26 14 æ ö 5 Nu = C Ra 1 + 04 . Ra 5 10 ´ (4.2.17) ê ú l 0.21 è ø Ra ë û In this equation, b is the average fin spacing (Figure 4.2.4C), defined such that bL is the cross- sectional flow area between two adjacent fin surfaces up to the plane of the fin tips. For Ra 0.4, Equation (4.2.17) underestimates the convective heat transfer. When such fins are mounted horizontally (vertical baseplate, but the fin tips are horizontal), there is a substantial reduction of the convective heat transfer (Karagiozis et al., 1994). 14. U-Channel Fins, Figure 4.2.4C. For the fins most often used as heat sinks, there is uncertainty about the heat transfer at low Ra. By using a conservative approximation applying for Ra 100 (that underestimates the real heat transfer), the following equation may be used: -05 . -2 é ù -2 Ra æ ö Nu = CC Ra + (4.2.18) ê () ú 1 l è ø 24 ë û For air C depends on aspect ratio of the fin as follows (Karagiozis, 1991): 1 é H ù æ ö C=+1 ,. 1 16 (4.2.19) 1 ê ú è ø b ë û min Equation (4.2.18) agrees well with measurements for Ra 200, but for smaller Ra it falls well below data because the leading term does not account for heat transfer from the fin edges and for three-dimensional conduction from the entire array. 15. Circular Fins on a Horizontal Tube, Figure 4.24D. For heat transfer from an array of circular 4 fins (Edwards and Chaddock, 1963), for H/D = 1.94, 5 Ra 10 , and for air, i © 1999 by CRC Press LLC

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