Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 23
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(3.212) and (3.213) when thesurface heat dissipation is proportional to T m rather than T 4 . Their solution appearsin terms of hypergeometric functions which bear a relationship to the incomplete betafunction. Kraus et al. (2001) provide an extensive collection of graphs to evaluate theperformance of radiating fins of different profiles.3.6.5[212], (52)Longitudinal Convecting–Radiating FinsA finite-difference approach was taken by Nguyen and Aziz (1992) to evaluate theperformance of longitudinal fins (Fig.
3.19) of rectangular, trapezoidal, triangular,and concave parabolic profiles when the fin surface loses heat by simultaneous convection and radiation. For each profile, the performance depends on five parameters,2b/δb , hδb /2k, T∞ /Tb , Ts /Tb , and 2b2 σTb3 /kδb , where Ts is the effective sink temperature for radiation.
A sample result for the fin efficiency is provided in Table 3.11.These results reveal a more general trend—that a convecting–radiating fin has a lowerefficiency than that of a purely convecting fin (2b2 σTb3 /kδb = 0).3.6.6———0.89919pt PgVar———Normal PagePgEnds: TEX[212], (52)Optimum Dimensions of Convecting Fins and SpinesThe classical fin or spine optimization involves finding the profile so that for a prescribed volume, the fin or spine rate of heat transfer is maximized.
Such optimizationsresult in profiles with curved boundaries that are difficult and expensive to fabricate.From a practical point of view, a better approach is to select the profile first and thenfind the optimum dimensions so that for a given profile area or volume, the fin or spinerate of heat transfer is maximized. The results of the latter approach are provided here.For each shape, two sets of expressions for optimum dimensions are given, one set forTABLE 3.11 Efficiency of Longitudinal Convecting–Radiating Fins,T∞ /Tb = Ts /Tb = 0.8, 2hb2 /kδb = 12b2 σTb3 /kδRectangularTrapezoidalδt/δb = 0.25TriangularConcaveParabolic0.000.200.400.600.801.000.69680.46790.36310.30300.26380.23650.69310.46770.36440.30510.26660.23960.68450.46310.36160.30330.26550.23900.62400.42440.33240.28110.24710.2233BOOKCOMP, Inc.
— John Wiley & Sons / Page 212 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 2401 to 2437213EXTENDED SURFACES123456789101112131415161718192021222324252627282930313233343536373839404142434445when the profile area or volume is specified and another set for when the fin or spinerate of heat transfer is specified. Note that qf for fins in the expressions to follow isthe fin rate of heat transfer per unit length L of fin.Rectangular FinWhen the weight or profile area Ap is specified,δopt = 0.9977Ap2 h(3.217)kbopt1/3Ap k= 1.0023h1/3(3.218)and when the fin rate of heat transfer (per unit length) qf is specified,20.6321 qf /(Tb − T∞ )=hk0.7978qf=h(Tb − T∞ )δoptboptTriangular Fin[213], (53)(3.219)Lines: 2437 to 2486(3.220)———Normal Page* PgEnds: EjectWhen the weight or profile area Ap is specified,δb,opt = 1.6710Ap2 hbopt1/3(3.221)kAp k= 1.1969h[213], (53)1/3(3.222)and when the fin rate of heat transfer (per unit length) qf is specified,δb,opt20.8273 qf /(Tb − T∞ )=hkbopt =Concave Parabolic Fin0.8422qfh(Tb − T∞ )(3.223)(3.224)When the weight or profile area Ap is specified,δopt = 2.08011/3(3.225)kbopt = 1.4422Ap2 hAp kh1/3and when the fin rate of heat transfer (per unit length) qf is specified,BOOKCOMP, Inc.
— John Wiley & Sons / Page 213 / 2nd Proofs / Heat Transfer Handbook / Bejan———-5.99965pt PgVar(3.226)214123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERδb,opt =bopt =Cylindrical Spine1hkqf(Tb − T∞ )2(3.227)qfh(Tb − T∞ )(3.228)When the weight or volume V is specified,dopt = 1.5031bopthV 2kV k2= 0.5636h21/5(3.229)1/5(3.230)[214], (54)and when the spine rate of heat transfer qf is specified,dopt = 0.9165hk(Tb − T∞ )21/3qf k= 0.4400 2h (Tb − T∞ )boptConical Spine1/3qf2Lines: 2486 to 2539(3.231)(3.232)When the weight or volume V is specified,db,opt = 1.9536bopt = 1.0008hVk(3.233)1/5(3.234)and when the spine rate of heat transfer qf is specified,db,opt = 1.09881/3qf2boptqf k= 0.7505 2h (Tb − T∞ )2Concave Parabolic Spine(3.235)hk(Tb − T∞ )21/3(3.236)When the weight or volume V is specified,db,opt = 2.0968BOOKCOMP, Inc.
— John Wiley & Sons / Page 214 / 2nd Proofs / Heat Transfer Handbook / BejanhV 2k———Normal PagePgEnds: TEX[214], (54)2 1/5V k2h2———14.10623pt PgVar1/5(3.237)TWO-DIMENSIONAL STEADY CONDUCTION123456789101112131415161718192021222324252627282930313233343536373839404142434445boptV k2= 1.4481h22151/5(3.238)and when the spine rate of heat transfer qf is specified,qf2db,opt = 1.1746bopt1/3hk(Tb − T∞ )21/3qf k= 1.0838 2h (Tb − T∞ )2(3.239)(3.240)When the weight or volume V is specified,Convex Parabolic Spine1/5hV 2= 1.7980k 2 1/5Vk= 0.7877h2[215], (55)db,optbopt(3.241)Lines: 2539 to 2574(3.242)———Normal PagePgEnds: TEXand when the spine rate of heat transfer qf is specified,db,opt = 1.0262boptqf21/3hk(Tb − T∞ )21/3qf k= 0.5951 2h (Tb − T∞ )2(3.243)[215], (55)(3.244)The material presented here is but a small fraction of the large body of literatureon the subject of optimum shapes of extended surfaces.
The reader should consultAziz (1992) for a comprehensive compilation of results for the optimum dimensionsof convecting extended surfaces. Another article by Aziz and Kraus (1996) providessimilar coverage for radiating and convecting–radiating extended surfaces. Both articles contain a number of examples illustrating the design calculations, and both aresummarized in Kraus et al. (2001).3.7 TWO-DIMENSIONAL STEADY CONDUCTIONThe temperature field in a two-dimensional steady-state configuration is controlled bya second-order partial differential equation whose solution must satisfy four boundaryconditions. The analysis is quite complex, and consequently, exact analytical solutions are limited to simple geometries such as a rectangular plate, a cylinder, and asphere under highly restrictive boundary conditions.
Problems that involve complexgeometries and more realistic boundary conditions can only be solved by using anBOOKCOMP, Inc. — John Wiley & Sons / Page 215 / 2nd Proofs / Heat Transfer Handbook / Bejan———-1.31589pt PgVar216123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERapproximate technique or a numerical method. Approximate techniques that are employed include the integral method, the method of scale analysis, and the method ofconduction shape factors.
The two most popular numerical techniques are the finitedifference and finite-element methods. There are numerous sources for informationon approximate and numerical techniques, some of which are Bejan (1993), Ozisik(1993, 1994), Comini et al. (1994), and Jaluria and Torrance (1986). In the following section we provide an example of an exact solution, a table of conduction shapefactors, and a brief discussion of the finite-difference method and its application totwo-dimensional conduction in a square plate and a solid cylinder.3.7.1Rectangular Plate with Specified Boundary TemperaturesFigure 3.26 shows a rectangular plate with three sides maintained at a constanttemperature T1 , while the fourth side is maintained at another constant temperature,T2 (T2 = T1 ).
Definingθ=T − T1T2 − T 1the governing two-dimensional temperature distribution becomes∂ 2θ∂ 2θ+ 2 =02∂x∂y(3.245)[216], (56)Lines: 2574 to 2632———3.08197pt PgVar———Normal Page(3.246) * PgEnds: Ejectwith the boundary conditions[216], (56)θ(0,y) = 1(3.247a)θ(x,0) = 0(3.247b)θ(L,y) = 0(3.247c)θ(x,H ) = 0(3.247d)Use of the separation of variables method gives the solution for θ as∞4 sinh [(2n + 1)π(L − x)/H ] sin [(2n + 1)πy/H ]θ=π n=0sinh [(2n + 1)πL/H ]2n + 1(3.248)Using eq.
(3.248), Bejan (1993) developed a network of isotherms and heat flux lines,which is shown in Fig. 3.27 for H /L = 2 (a rectangular plate) and for H /L = 1 (asquare plate).The heat flow into the plate from a hot left face is given by∞q81= k(T2 − T1 )Wπ(2n+1)tanh+ 1)πL/H ][(2nn=0BOOKCOMP, Inc. — John Wiley & Sons / Page 216 / 2nd Proofs / Heat Transfer Handbook / Bejan(3.249)217TWO-DIMENSIONAL STEADY CONDUCTION123456789101112131415161718192021222324252627282930313233343536373839404142434445[217], (57)Figure 3.26 Two-dimensional steady conduction in a rectangular plate.=0H=0H———yyLines: 2632 to 26480.97401pt PgVarIsothermHeatfluxline = 1 0.8 0.60.4 0.200H =2__Lx0=0L = 1 0.8 0.600.4[217], (57)0.20=00H =1__LxLFigure 3.27 Isotherms and heat flux lines in a rectangular plate and a square plate.
(FromBejan, 1993.)where W is the plate dimension in the z direction. Solutions for the heat flux andconvective boundary conditions are given in Ozisik (1993) and Poulikakos (1994).3.7.2Solid Cylinder with Surface ConvectionFigure 3.28 illustrates a solid cylinder of radius r0 and length L in which conductionoccurs in both radial and axial directions. The face at z = 0 is maintained at a constantBOOKCOMP, Inc. — John Wiley & Sons / Page 217 / 2nd Proofs / Heat Transfer Handbook / Bejan———Normal PagePgEnds: TEX218123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERFigure 3.28 Radial and axial conduction in a hollow cylinder.[218], (58)temperature T1 , while both the lateral surface and the face at z = L lose heat byconvection to the environment at T∞ via the heat transfer coefficient h.
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