Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 16
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— John Wiley & Sons / Page 163 / 2nd Proofs / Heat Transfer Handbook / Bejan[163], (3)Lines: 195 to 286———*19.91pt PgVar———Short Page* PgEnds: PageBreak[163], (3)1641234567891011121314151617181920212223242526272829303132333435363738394041424344453.1CONDUCTION HEAT TRANSFERINTRODUCTIONThis chapter is concerned with the characterization of conduction heat transfer, whichis a mode that pervades a wide range of systems and devices. Unlike convection,which pertains to energy transport due to fluid motion and radiation, which canpropagate in a perfect vacuum, conduction requires the presence of an interveningmedium. At microscopic levels, conduction in stationary fluids is a consequence ofhigher-temperature molecules interacting and exchanging energy with molecules atlower temperatures.
In a nonconducting solid, the transport of energy is exclusivelyvia lattice waves (phonons) induced by atomic motion. If the solid is a conductor, thetransfer of energy is also associated with the translational motion of free electrons.The microscopic approach is of considerable contemporary interest because of itsapplicability to miniaturized systems such as superconducting thin films, microsensors, and micromechanical devices (Duncan and Peterson, 1994; Tien and Chen,1994; Tzou, 1997; Tien et al., 1998). However, for the vast majority of engineering applications, the macroscopic approach based on Fourier’s law is adequate. Thischapter is therefore devoted exclusively to macroscopic heat conduction theory, andthe material contained herein is a unique synopsis of a wealth of information that isavailable in numerous works, such as those of Schneider (1955), Carslaw and Jaeger(1959), Gebhart (1993), Ozisik (1993), Poulikakos (1994), and Jiji (2000).3.23.2.1BASIC EQUATIONS[164], (4)Lines: 286 to 313———-0.65796pt PgVar———Normal PagePgEnds: TEX[164], (4)Fourier’s LawThe basic equation for the analysis of heat conduction is Fourier’s law, which is basedon experimental observations and isqn = −kn∂T∂n(3.1)where the heat flux qn (W/m 2) is the heat transfer rate in the n direction per unit areaperpendicular to the direction of heat flow, kn (W/m · K) is the thermal conductivityin the direction n, and ∂T /∂n (K/m) is the temperature gradient in the directionn.
The thermal conductivity is a thermophysical property of the material, which is,in general, a function of both temperature and location; that is, k = k(T , n). Forisotropic materials, k is the same in all directions, but for anisotropic materialssuch as wood and laminated materials, k is significantly higher along the grain orlamination than perpendicular to it.
Thus for anisotropic materials, k can have astrong directional dependence. Although heat conduction in anisotropic materialsis of current research interest, its further discussion falls outside the scope of thischapter and the interested reader can find a fairly detailed exposition of this topic inOzisik (1993).Because the thermal conductivity depends on the atomic and molecular structureof the material, its value can vary from one material to another by several orders ofBOOKCOMP, Inc.
— John Wiley & Sons / Page 164 / 2nd Proofs / Heat Transfer Handbook / BejanBASIC EQUATIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445165magnitude. The highest values are associated with metals and the lowest values withgases and thermal insulators. Tabulations of thermal conductivity data are given inChapter 2.For three-dimensional conduction in a Cartesian coordinate system, the Fourierlaw of eq. (3.1) can be extended toq = iqx + jqy + kqz(3.2)whereqx = −k∂T∂xqy = −k∂T∂yqz = −k∂T∂z(3.3)and i, j, and k are unit vectors in the x, y, and z coordinate directions, respectively.3.2.2General Heat Conduction EquationsThe general equations of heat conduction in the rectangular, cylindrical, and sphericalcoordinate systems shown in Fig.
3.1 can be derived by performing an energy balance.Cartesian coordinate system:∂∂T∂∂T∂∂T∂Tk+k+k+ q̇ = ρc(3.4)∂x∂x∂y∂y∂z∂z∂tCylindrical coordinate system:1 ∂∂T1 ∂∂T∂∂T∂Tkr+ 2k+k+ q̇ = ρcr ∂r∂rr ∂φ∂φ∂z∂z∂tSpherical coordinate system:1 ∂∂1∂T2 ∂Tkr+kr 2 ∂r∂r∂φr 2 sin2 θ ∂φ1∂T∂T∂+ 2k sin θ+ q̇ = ρcr sin θ ∂θ∂θ∂t(3.5)(3.6)Boundary and Initial ConditionsEach of the general heat conduction equations (3.4)–(3.6) is second order in thespatial coordinates and first order in time. Hence, the solutions require a total of sixBOOKCOMP, Inc.
— John Wiley & Sons / Page 165 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 313 to 372———-1.30785pt PgVar———Normal PagePgEnds: TEX[165], (5)In eqs. (3.4)–(3.6), q̇ is the volumetric energy addition (W/m 3), ρ the density ofthe material (kg/m 3), and c the specific heat (J/kg · K) of the material. The generalheat conduction equation can also be expressed in a general curvilinear coordinatesystem (Section 1.2.4). Ozisik (1993) gives the heat conduction equations in prolatespheroidal and oblate spheroidal coordinate systems.3.2.3[165], (5)166123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERqz ⫹zdzqy ⫹q ⫹r sin dyxdzq ⫹dr dqxqx ⫹qyddydxqzrdydxxT(r,,)drqr ⫹ydrq[166], (6)qz(a)qz ⫹(c)dzLines: 372 to 379———r dqr0.951pt PgVarq ⫹ddzzxqrT(r,,z)qr ⫹drdryqz(b)Figure 3.1 Differential control volumes in (a) Cartesian, (b) cylindrical, and (c) sphericalcoordinates.boundary conditions (two for each spatial coordinate) and one initial condition.
Theinitial condition prescribes the temperature in the body at time t = 0. The threetypes of boundary conditions commonly encountered are that of constant surfacetemperature (the boundary condition of the first kind), constant surface heat flux (theboundary condition of the second kind), and a prescribed relationship between thesurface heat flux and the surface temperature (the convective or boundary conditionof the third kind). The precise mathematical form of the boundary conditions dependson the specific problem.For example, consider one-dimensional transient condition in a semi-infinite solidthat is subject to heating at x = 0. Depending on the characterization of the heating,the boundary condition at x = 0 may take one of three forms. For constant surfacetemperature,BOOKCOMP, Inc. — John Wiley & Sons / Page 166 / 2nd Proofs / Heat Transfer Handbook / Bejan———Long Page* PgEnds: Eject[166], (6)SPECIAL FUNCTIONS123456789101112131415161718192021222324252627282930313233343536373839404142434445T (0, t) = Ts167(3.7)For constant surface heat flux,∂T (0, t)= qs∂x(3.8)∂T (0, t)= h [T∞ − T (0, t)]∂x(3.9)−kand for convection at x = 0,−kwhere in eq.
(3.9), h(W/m 2 · K) is the convective heat transfer coefficient and T∞ isthe temperature of the hot fluid in contact with the surface at x = 0.Besides the foregoing boundary conditions of eqs. (3.7)–(3.9), other types ofboundary conditions may arise in heat conduction analysis. These include boundary conditions at the interface of two different materials in perfect thermal contact,boundary conditions at the interface between solid and liquid phases in a freezingor melting process, and boundary conditions at a surface losing (or gaining) heatsimultaneously by convection and radiation.
Additional details pertaining to theseboundary conditions are provided elsewhere in the chapter.3.3Lines: 379 to 443———0.24222pt PgVar———Long Page* PgEnds: EjectSPECIAL FUNCTIONSA number of special mathematical functions frequently arise in heat conduction analysis. These cannot be computed readily using a scientific calculator. In this sectionwe provide a modest introduction to these functions and their properties. The functions include error functions, gamma functions, beta functions, exponential integralfunctions, Bessel functions, and Legendre polynomials.3.3.1[167], (7)Error FunctionsThe error function with argument (x) is defined as x22erf(x) = √e−t dtπ 0(3.10)where t is a dummy variable.
The error function is an odd function, so thaterf(−x) = −erf(x)(3.11)In addition,erf(0) = 0anderf(∞) = 1The complementary error function with argument (x) is defined as ∞22erfc(x) = 1 − erf(x) = √e−t dtπ xBOOKCOMP, Inc. — John Wiley & Sons / Page 167 / 2nd Proofs / Heat Transfer Handbook / Bejan(3.12)(3.13)[167], (7)168123456789101112131415161718192021222324252627282930313233343536373839404142434445CONDUCTION HEAT TRANSFERThe derivatives of the error function can be obtained by repeated differentiationsof eq. (3.10):d22erf(x) = √ e−xdxπandd242erf(x) = − √ xe−x2dxπThe repeated integrals of the complementary error function are defined by ∞i n erfc(x) =i n−1 erfc(t) dt(n = 1, 2, 3, .
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