1. Chapter 2: Heat Conduction EquationYoav PelesDepartment of Mechanical, Aerospace and Nuclear EngineeringRensselaer Polytechnic InstituteCopyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. ObjectivesWhen you finish studying this chapter, you should be able to: Understand multidimensionality and time dependence of heat transfer,and the conditions under which a heat transfer problem can beapproximated as being one-dimensional, Obtain the differential equation of heat conduction in variouscoordinate systems, and simplify it for steady one-dimensional case, Identify the thermal conditions on surfaces, and express themmathematically as boundary and initial conditions, Solve one-dimensional heat conduction problems and obtain thetemperature distributions within a medium and the heat flux, Analyze one-dimensional heat conduction in solids that involve heatgeneration, and Evaluate heat conduction in solids with temperature-dependentthermal conductivity. 3. 2.1. Introduction Although heat transfer and temperature areclosely related, they are of a different nature. Temperature has only magnitude it is a scalar quantity. Heat transfer has direction as well as magnitude it is a vector quantity. We work with a coordinate system and indicatedirection with plus or minus signs. 4. The driving force for any form of heat transfer is thetemperature difference. Three prime coordinate systems: rectangular (T(x, y, z, t)) 5. cylindrical (T(r, , z, t)) spherical (T(r, , , t)). 6. Classification of conduction heat transfer problems: steady versus transient heat transfer, multidimensional heat transfer, heat generation. 7. Steady versus Transient Heat Transfer Steady implies no change with time at any pointwithin the medium Transient implies variation with time or timedependence 8. Multi-dimensional Heat Transfer Heat transfer problems are also classified asbeing: one-dimensional, two dimensional, three-dimensional. In the most general case, heat transfer through amedium is three-dimensional. However, someproblems can be classified as two- or one-dimensional depending on the relativemagnitudes of heat transfer rates in differentdirections and the level of accuracy desired. 9. The rate of heat conduction through a medium ina specified direction (say, in the x-direction) isexpressed by Fouriers law of heat conductionfor one-dimensional heat conduction as:dT Qcond = kA(W) (2-1)dx Heat is conducted in the directionof decreasing temperature, andthus the temperature gradient is negativewhen heat is conducted in the positive x-direction. 10. General Relation for Fouriers Law of Heat Conduction The heat flux vector at a point P on the surface of thefigure must be perpendicular to the surface, and it mustpoint in the direction of decreasing temperature If n is the normal of theisothermal surface at point P,the rate of heat conduction atthat point can be expressed byFouriers law asdTQn = kA(W) (2-2)dn 11. In rectangular coordinates, the heat conductionvector can be expressed in terms of its components asQn = Qx i + Qy j + Qz k(2-3) which can be determined from Fouriers law as T Qx = kAx x T Qy = kAy (2-4) y T Qz = kAz z 12. Heat Generation Examples: electrical energy being converted to heat at a rate of I2R, fuel elements of nuclear reactors, exothermic chemical reactions. Heat generation is a volumetric phenomenon. The rate of heat generation units : W/m3 or Btu/h ft3. 13. The rate of heat generation in a medium may varywith time as well as position within the medium. The total rate of heat generation in a medium ofvolume V can be determined from Egen = egen dV(W)(2-5) V 14. one-dimensional problem ! 15. 2.2 One-Dimensional Heat Conduction Equation - (i) Plane WallRate of heat Rate of heat Rate of heat Rate of change ofconduction - conduction + generation inside = the energy contentat x at x+xthe element of the elementEelementQx Qx +x + Egen,element =t (2-6) 16. EelementQx Qx+x + Egen,element =(2-6)t The change in the energy content and the rate of heatgeneration can be expressed asEelement = Et +t Et = mc (Tt +t Tt ) = cAx (Tt +t Tt ) (2-7)Egen,element = egenVelement = egen Ax (2-8) Substituting into Eq. 26, we get Tt +t Tt (2-9) Qx Qx+x +egen Ax = cAx t Dividing by Ax, taking the limit as x 0 and t 0,and from Fouriers law: 1 TT kA + egen = c(2-11) A x x t 17. The area A is constant for a plane wall the one dimensionaltransient heat conduction equation in a plane wall is TTVariable conductivity:k + egen = c (2-13) x x t 2T egen 1 TkConstant conductivity:+ = ; =(2-14) x 2k t c The one-dimensional conduction equation may be reduces to the following forms under special conditionsd 2T egen1) Steady-state: 2 +=0(2-15)dx k 2T 1 T2) Transient, no heat generation:=(2-16)x 2 t d 2T3) Steady-state, no heat generation:2=0(2-17) dx 18. One-Dimensional Heat Conduction Equation (ii) Long Cylinder Rate of heat Rate of heat Rate of heat Rate of change of conduction - conduction + generation inside = the energy content at r at r+rthe element of the element EelementQr Qr +r + Egen,element= t (2-18) 19. EelementQr Qr +r + Egen,element = (2-18) t The change in the energy content and the rate of heatgeneration can be expressed asEelement = Et +t Et = mc (Tt +t Tt ) = cAr (Tt +t Tt ) (2-19)Egen,element = egenVelement = egen Ar (2-20) Substituting into Eq. 218, we getTt +t Tt(2-21) Qr Qr +r +egen Ar = cArt Dividing by Ar, taking the limit as r 0 and t 0,and from Fouriers law: 1 T T kA + egen = c (2-23) A r rt 20. Noting that the area varies with the independent variable raccording to A=2rL, the one dimensional transient heatconduction equation in a plane wall becomes1 T T(2-25)Variable conductivity: rk + egen = cr r rt1 T egen 1 TConstant conductivity: r += (2-26)r r r k tThe one-dimensional conduction equation may be reducesto the following forms under special conditions 1 d dT egen 1) Steady-state: r+ = 0 (2-27) r dr dr k1 T 1 T 2) Transient, no heat generation: r =(2-28)r r r t d dT 3) Steady-state, no heat generation: r =0(2-29) dr dr 21. One-Dimensional Heat Conduction Equation- (iii) Sphere1 2 T T r k + egen = c(2-30) Variable conductivity: r r 2r t 1 2 T egen 1 T Constant conductivity: r += (2-31) r r r2 k t 22. One-dimensional steady-stateheat conduction equation:d 2T =0dx 2 23. 2.3 General Heat Conduction EquationRate of heat Rate of heatRate of heat Rate of change conduction- conduction + generation of the energyat x, y, and zat x+x, y+y,inside the= content of the and z+zelementelementEelementQx + Qy + QzQx +x Qy +y Qz +z + Egen ,element = (2-36)t 24. Repeating the mathematical approach used for the one-dimensional heat conduction the three-dimensional heatconduction equation is determined to beTwo-dimensional 2T 2T 2T egen 1 T Constant conductivity:+ 2 + 2 + = (2-39)x 2yzk tThree-dimensional 2T 2T 2T egen + 2 + 2 += 0 (2-40) 1) Steady-state: x 2yzk 2T 2T 2T 1 T 2) Transient, no heat generation: x 2 + y 2 + z 2 = t(2-41) 2T 2T 2T 3) Steady-state, no heat generation: 2 + 2 + 2 = 0 (2-42) x y z 25. Cylindrical Coordinates1 T 1 T TT rk + 2 k + z k z + egen = c(2-43)r r r r t 26. Spherical Coordinates1 2 T 1 T 1 T T kr + 2 2 k + 2 k sin + egen = cr 2 r r r sin r sin t (2-44) 27. 2.4 Boundary and Initial Conditions (i) Specified Temperature Boundary Condition (ii) Specified Heat Flux Boundary Condition (iii) Convection Boundary Condition (iv) Radiation Boundary Condition (v) Interface Boundary Conditions (vi) Generalized Boundary Conditions 28. (i) Specified Temperature Boundary Condition For one-dimensional heat transfer through a plane wall of thickness L, for example, the specified temperature boundary conditions can be expressed asT(0, t) = T1 (2-46)T(L, t) = T2 The specified temperatures can be constant, which is the case for steady heat conduction, or may vary with time. 29. (ii) Specified Heat Flux Boundary ConditionThe heat flux in the positive x-direction anywhere in the medium,including the boundaries, can beexpressed by Fouriers law of heatconduction as dTHeat flux inq = k= (2-47) dxthe positive x-directionThe sign of the specified heat flux is determined byinspection: positive if the heat flux is in the positivedirection of the coordinate axis, and negative if it is inthe opposite direction. 30. Two Special Cases Insulated Boundary Insulated boundaryThermal symmetryT (0, t ) T (0, t )k=0 or=0x xT ( L / 2, t ) (2-49) = 0 (2-50) x 31. (iii) Convection Boundary Condition Heat conduction Heat convectionat the surface in a =at the surface inselected directionthe same direction 32. T (0, t )k= h1 [T1 T (0, t ) ](2-51a) xand T ( L, t )k = h2 [T ( L, t ) T 2 ] (2-51b) x 33. (iv) Radiation Boundary ConditionHeat conduction at the Radiation exchange atsurface in a selected=the surface indirection the same direction 34. T (0, t ) k= 1 Tsurr ,1 T (0, t ) 4 4 (2-52a)xandT ( L, t ) k = 2 T ( L, t ) 4 Tsurr ,2 4 (2-52b)x 35. (v) Interface Boundary ConditionsAt the interface the requirements are:(1) the same temperature at the area of contact,(2) the heat flux on the two sides of an interface must be the same.TA(x0, t) = TB(x0, t) (2-53)and TA ( x0 , t ) TB ( x0 , t )k A= kB(2-54)x x 36. (vi) Generalized Boundary ConditionsIn general a surface may involve convection, radiation,and specified heat flux simultaneously. The boundarycondition in such cases is again obtained from a surfaceenergy balance, expressed as Heat transfer Heat transfer to the surface =from the surfacein all modes in all modes 37. 2.5 Solution of Steady One-DimensionalHeat Conduction Problems 38. (a)OK ! 39. ! 40. 2.6 Heat Generation () in Solids 41. The quantities of major interest in a mediumwith heat generation are(i) the surface temperature Ts(ii) the maximum temperature Tmaxthat occurs in the medium in steady operation. 42. Heat Generation in Solids - (i) The Surface TemperatureRate ofRate ofheat transfer = energy generation (2-63) from the solidwithin the solidFor uniform heat generation within the mediumQ = egenV (W) (2-64)The heat transfer rate by convection can also beexpressed from Newtons law of cooling as- Q = hAs (Ts T )(W) (2-65)egenVTs = T + (2-66)hAs 43. (a) For a large plane wall of thickness 2L (As=2Awalland V=2LAwall)egen L Ts , plane wall = T + (2-67) h(b) For a long solid cylinder of radius r0 (As=2r0Land V=r02L)egen r0 Ts ,cylinder = T +(2-68)2h(c) For a solid sphere of radius r0 (As=4r02 and V=4/3r03)egen r0 Ts , sphere = T + (2-69) 3h 44. Heat Generation in Solids (ii)The maximum Temperature in a Cylinder (the Centerline)The heat generated within an innercylinder must be equal to the heatconducted through its outer surface. dTkAr= egenVr(2-70) drSubstituting these expressions into the above equationand separating the variables, we get( ) dT egenk ( 2 rL )= egen r L dT = 2 rdr dr 2kIntegrating from r =0 where T(0) =T0 to r=ro egen r02Tmax,cylinder = T0 Ts = (2-71) 4k 45. Variable Thermal Conductivity, k(T) - The thermal conductivity of amaterial, in general, varies withtemperature. An average value for thethermal conductivity iscommonly used when thevariation is mild. This is also common practicefor other temperature-dependent properties such asthe density and specific heat. 46. Variable Thermal Conductivity for One- Dimensional CasesWhen the variation of thermal conductivity with temperatureis known, the average value of the thermal conductivity in thetemperature range between T1 and T2 can be determined fromT2 kave =T1 k (T )dT(2-75) T2 T1The variation in thermal conductivity of a materialwith can often be approximated as a linear functionand expressed ask (T ) = k0 (1 + T )(2-79) the temperature coefficient of thermal conductivity. 47. Variable Thermal Conductivity For a plane wall thetemperature varies linearlyduring steady one-dimensional heat conductionwhen the thermal conductivityis constant. This is no longer the casewhen the thermal conductivitychanges with temperature(even linearly).