Chapter 2 HEAT CONDUCTION EQUATION

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1 Heat Transfer: A Practical Approach Second Edition Yunus A. Cengel McGraw-Hill, 2002 Chapter 2 HEAT CONDUCTION EQUATION Universitry of Technology Materials Engineering Department MaE216: Heat Transfer and Fluid

2 aluate heat conduction in solids with temperature-dependent rmal conductivity. jectives derstand multidimensionality and time dependence of heat transfer, d the conditions under which a heat transfer problem can be proximated as being one-dimensional. tain the differential equation of heat conduction in various ordinate systems, and simplify it for steady one-dimensional case. ntify the thermal conditions on surfaces, and express them thematically as boundary and initial conditions. lve one-dimensional heat conduction problems and obtain the perature distributions within a medium and the heat flux. alyze one-dimensional heat conduction in solids that involve heat neration.

3 TRODUCTION lthough heat transfer and temperature are closely related, they are of a fferent nature. mperature has only magnitude. It is a scalar quantity. eat transfer has direction as well as magnitude. It is a vector quantity. e work with a coordinate system and indicate direction with plus or minus gns.

4 he driving force for any form of heat transfer is the temperature fference. he larger the temperature difference, the larger the rate of heat ansfer. hree prime coordinate systems: rectangular T(x, y, z, t) cylindrical T(r,, z, t) spherical T(r,,, t).

5 ady versus Transient Heat Transfer teady implies no change ith time at any point within e medium ansient implies variation ith time or time pendence the special case of riation with time but not ith position, the mperature of the medium anges uniformly with e. Such heat transfer stems are called lumped stems.

6 ltidimensional Heat Transfer eat transfer problems are also classified as being: one-dimensional two dimensional three-dimensional the most general case, heat transfer through a medium is threeimensional. However, some problems can be classified as two- or ne-dimensional depending on the relative magnitudes of heat ansfer rates in different directions and the level of accuracy desired. ne-dimensional if the temperature in the medium varies in one irection only and thus heat is transferred in one direction, and the ariation of temperature and thus heat transfer in other directions are egligible or zero. wo-dimensional if the temperature in a medium, in some cases, aries mainly in two primary directions, and the variation of mperature in the third direction (and thus heat transfer in that irection) is negligible.

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8 e rate of heat conduction through a medium in a specified direction y, in the x-direction) is expressed by Fourier s law of heat nduction for one-dimensional heat conduction as: at is conducted in the direction decreasing temperature, and s the temperature gradient is gative when heat is conducted the positive x -direction.

9 he heat flux vector at a point P on e surface of the figure must be rpendicular to the surface, and it ust point in the direction of creasing temperature n is the normal of the isothermal rface at point P, the rate of heat nduction at that point can be pressed by Fourier s law as

10 amples: electrical energy being converted to heat at a rate of I 2 R, fuel elements of nuclear reactors, exothermic chemical reactions. at generation is a volumetric phenomenon. e rate of heat generation units : W/m 3 or Btu/h ft 3. e rate of heat generation in a medium may vary with time as well as sition within the medium. Heat Generation

11 E-DIMENSIONAL HEAT CONDUCTION UATION ider heat conduction through a large plane wall such as the wall of a, the glass of a single pane window, the metal plate at the bottom of ssing iron, a cast-iron steam pipe, a cylindrical nuclear fuel element, ctrical resistance wire, the wall of a spherical container, or a ical metal ball that is being quenched or tempered. conduction in these and many other geometries can be ximated as being one-dimensional since heat conduction through geometries is dominant in one direction and negligible in other ions. we develop the onedimensional heat conduction equation in gular, cylindrical, and spherical coordinates.

12 (2-6) Heat Conduction Equation in a Large Plane Wall

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14 Heat Conduction Equation in a Long Cylinder

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16 Conduction Equation Sphere

17 bined One-Dimensional Heat Conduction ation xamination of the one-dimensional transient heat conduction tions for the plane wall, cylinder, and sphere reveals that all equations can be expressed in a compact form as n = 0 for a plane wall n = 1 for a cylinder n = 2 for a sphere e case of a plane wall, it is customary to replace the variable x. equation can be simplified for steady-state or no heat

18 NERAL HEAT CONDUCTION EQUATION e last section we considered one-dimensional heat conduction assumed heat conduction in other directions to be negligible. heat transfer problems encountered in practice can be oximated as being one-dimensional, and we mostly deal with problems in this text. ever, this is not always the case, and sometimes we need to ider heat transfer in other directions as well. ch cases heat conduction is said to be multidimensional, and s section we develop the governing differential equation in systems in rectangular, cylindrical, and spherical coordinate ms.

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22 lindrical Coordinates ations between the coordinates of a point in rectangular cylindrical coordinate systems:

23 herical Coordinates ations between the coordinates of a point in rectangular spherical coordinate systems:

24 UNDARY AND INITIAL CONDITIONS scription of a heat transfer problem in a medium is not complete without a full tion of the thermal conditions at the bounding surfaces of the medium. dary conditions: The mathematical expressions of the thermal conditions at the ries. perature at any the wall at a d time depends ondition of the ry at the ng of the heat tion process. condition, which lly specified at 0, is called the ondition, which thematical ion for the

25 Boundary Conditions Specified Temperature Boundary Condition Specified Heat Flux Boundary Condition Convection Boundary Condition Radiation Boundary Condition Interface Boundary Conditions Generalized Boundary Conditions

26 ecified Temperature Boundary Condition emperature of an exposed surface sually be measured directly and. fore, one of the easiest ways to fy the thermal conditions on a surface pecify the temperature. ne-dimensional heat transfer through ne wall of thickness L, for example, pecified temperature boundary itions can be expressed as re T 1 and T 2 are the specified eratures at surfaces at x = 0 and, respectively.

27 ecified Heat Flux Boundary Condition at flux in the positive x-direction anywhere in the, including the boundaries, can be expressed by plate of thickness L subjected to heat 50 W/m 2 into the medium from both for example, the specified heat flux ary conditions can be expressed as

28 cial Case: Insulated Boundary ll-insulated surface can be modeled surface with a specified heat flux of. Then the boundary condition on a ctly insulated surface (at x = 0, for ple) can be expressed as n insulated surface, the first ative of temperature with respect e space variable (the temperature ient) in the direction normal to the lated surface is zero.

29 her Special Case: Thermal Symmetry heat transfer problems possess thermal try as a result of the symmetry in imposed l conditions. ample, the two surfaces of a large hot plate ness L suspended vertically in air is ted to the same thermal conditions, and thus perature distribution in one half of the plate ame as that in the other half., the heat transfer problem in this plate ses thermal symmetry about the center t x = L/2. ore, the center plane can be viewed as an ed surface, and the thermal condition at this f symmetry can be expressed as

30 nvection Boundary Condition -dimensional heat transfer in the x-direction te of thickness L, the convection boundary ns on both surfaces:

31 diation Boundary Condition tion boundary condition on a surface: e-dimensional heat transfer in the ction in a plate of thickness L, the ion boundary conditions on both es can be expressed as

32 erface Boundary Conditions oundary conditions at an interface sed on the requirements that o bodies in contact must have the temperature at the area of contact interface (which is a surface) t store any energy, and thus the lux on the two sides of an interface e the same. oundary conditions at the interface bodies A and B in perfect contact at can be expressed as

33 eneralized Boundary Conditions eneral, however, a surface may involve convection, ation, and specified heat flux simultaneously. boundary condition in such cases is again obtained a surface energy balance, expressed as

34 UTION OF STEADY ONE-DIMENSIONAL T CONDUCTION PROBLEMS section we will solve a wide range of heat tion problems in rectangular, cylindrical, herical geometries. l limit our attention to problems that result ary differential equations such as the one-dimensional heat conduction s. We will also assume constant thermal tivity. lution procedure for solving heat ction problems can be summarized as ulate the problem by obtaining the ble differential equation in its simplest nd specifying the boundary conditions, tain the general solution of the differential n, and

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43 T GENERATION IN A SOLID ractical heat transfer applications the conversion of some form of energy rmal energy in the medium. ediums are said to involve internal heat tion, which manifests itself as a rise in ature throughout the medium. examples of heat generation are ance heating in wires, ermic chemical reactions in a solid, and ar reactions in nuclear fuel rods electrical, chemical, and nuclear s are converted to heat, respectively. eneration in an electrical wire of outer r o and length L can be expressed as

44 ntities of major interest in a medium with neration are the surface temperature T s maximum temperature T max that occurs edium in steady operation.

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46 RIABLE THERMAL CONDUCTIVITY, k(t) When the variation of thermal conductivity with temperature in a specified temperature interval is large, it may be necessary to account for this variation to minimize the error. When the variation of thermal conductivity with temperature k(t) is known, the average value of the thermal conductivity in the temperature range between T 1 and T 2 can be determined from

47 ariation in thermal conductivity of a material with rature in the temperature range of interest can often be ximated as a linear function and expressed as perature coefficient rmal conductivity. verage value of thermal conductivity temperature range T 1 to T 2 in this can be determined from verage thermal conductivity in this is equal to the thermal conductivity at the average temperature.

48 ummary Introduction Steady versus Transient Heat Transfer Multidimensional Heat Transfer Heat Generation One-Dimensional Heat Conduction Equation Heat Conduction Equation in a Large Plane Wall Heat Conduction Equation in a Long Cylinder Heat Conduction Equation in a Sphere Combined One-Dimensional Heat Conduction Equation General Heat Conduction Equation Rectangular Coordinates Cylindrical Coordinates Spherical Coordinates Boundary and Initial Conditions Solution of Steady One-Dimensional Heat Conduction Problems