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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [Fi r s [16 1 Lin e 3.7 3 —— No r m PgE [16 1 CHAPTER 3 Conduction HeatTransfer * A. AZIZ Department of Mechanical Engineering Gonzaga University Spokane, Washington 3.1 Introduction 3.2 Basic equations 3.2.1 Fourier’s law 3.2.2 General heat conduction equations 3.2.3 Boundary and initial conditions 3.3 Special functions 3.3.1 Error functions 3.3.2 Gamma function 3.3.3 Beta functions 3.3.4 Exponential integral function 3.3.5 Bessel functions 3.3.6 Legendre functions 3.4 Steady one-dimensional conduction 3.4.1 Plane wall 3.4.2 Hollow cylinder 3.4.3 Hollow sphere 3.4.4 Thermal resistance 3.4.5 Composite systems Composite plane wall Composite hollow cylinder Composite hollow sphere 3.4.6 Contact conductance 3.4.7 Critical thickness of insulation 3.4.8 Effect of uniform internal energy generation Plane wall Hollow cylinder Solid cylinder Hollow sphere Solid sphere 3.5 More advanced steady one-dimensional conduction 3.5.1 Location-dependent thermal conductivity * The author dedicates this chapter to little Senaan Asil Aziz whose sparkling smile “makes my day.” 161
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162 CONDUCTION HEAT TRANSFER 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [162 Lin e 6.0 p —— Sho r PgE n [162 Plane wall Hollow cylinder 3.5.2 Temperature-dependent thermal conductivity Plane wall Hollow cylinder Hollow sphere 3.5.3 Location-dependent energy generation Plane wall Solid cylinder 3.5.4 Temperature-dependent energy generation Plane wall Solid cylinder Solid sphere 3.5.5 Radiative–convective cooling of solids with uniform energy generation 3.6 Extended surfaces 3.6.1 Longitudinal convecting Fns Rectangular Fn Trapezoidal Fn Triangular Fn Concave parabolic Fn Convex parabolic Fn 3.6.2 Radial convecting Fns Rectangular Fn Triangular Fn Hyperbolic Fn 3.6.3 Convecting spines Cylindrical spine Conical spine Concave parabolic spine Convex parabolic spine 3.6.4 Longitudinal radiating Fns 3.6.5 Longitudinal convecting–radiating Fns 3.6.6 Optimum dimensions of convecting Fns and spines Rectangular Fn Triangular Fn Concave parabolic Fn Cylindrical spine Conical spine Concave parabolic spine Convex parabolic spine 3.7 Two-dimensional steady conduction 3.7.1 Rectangular plate with speciFed boundary temperatures 3.7.2 Solid cylinder with surface convection 3.7.3 Solid hemisphere with speciFed base and surface temperatures 3.7.4 Method of superposition 3.7.5 Conduction of shape factor method 3.7.6 ±inite-difference method
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CONDUCTION HEAT TRANSFER 163 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [16 3 Lin e * 19 . 9 —— Sho * PgE [16 3 Cartesian coordinates Cylindrical coordinates 3.8 Transient conduction 3.8.1 Lumped thermal capacity model Internal energy generation Temperature-dependent speciFc heat Pure radiation cooling Simultaneous convective–radiative cooling Temperature-dependent heat transfer coefFcient Heat capacity of the coolant pool 3.8.2 Semi-inFnite solid model
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