Chapter 2: Heat Conduction Equation

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1 -1 General Relation for Fourier s Law of Heat Conduction - Heat Conduction Equation -3 Boundary Conditions and Initial Conditions

2 -1 General Relation for Fourier s Law of Heat Conduction (1) The rate of heat conduction through a medium in a specified direction (say, in the x-direction) is expressed by Fourier s law of heat conduction for one-dimensional heat conduction as: Q cond dt ka dx (W) Heat is conducted in the direction of decreasing temperature, and thus the temperature gradient is negative when heat is conducted in the positive x-direction. -1

3 -1 General Relation for Fourier s Law of Heat Conduction () The heat flux vector at a point P on the surface of the figure must be perpendicular to the surface, and it must point in the direction of decreasing temperature If n is the normal of the isothermal surface at point P, the rate of heat conduction at that point can be expressed by Fourier s law as Q n dt ka dn (W) -

4 -1 General Relation for Fourier s Law of Heat Conduction (3) In rectangular coordinates, the heat conduction vector can be expressed in terms of its components as Qn Qxi Qy j Qzk which can be determined from Fourier s law as T Qx kax x T Qy kay y T Qz kaz z -3

5 - Heat Conduction Equation (1) Rate of heat conduction at x Rate of heat conduction at x+x Rate of heat generation inside the element - + = Rate of change of the energy content of the element Q x Q Q x xx Q Qx x Q xx x Q x Q x x E gen, element x Q x! x 1 x T ( ka ) x x ( x)... T ( ka ) x x E element t Neglect higher orders -4

6 - Heat Conduction Equation () E E E mc T T cax T T Egen, element egenvelement egen Ax Q element tt t tt t tt t x Qx x t t t e Ax cax gen T T t Dividing by Ax, taking the limit as x 0 and t 0, and from Fourier s law: 1 T T ka egen c A x x t The area A is constant for a plane wall the one dimensional transient heat conduction equation in a plane wall is T T k e c Variable conductivity: gen x x t -5

7 - Heat Conduction Equation (3) Constant conductivity: T e gen 1 T k ; x k t c The one-dimensional conduction equation may be reduces to the following forms under special conditions 1) Steady-state: dt e gen 0 dx ) Transient, no heat generation: k T x 1 T t 3) Steady-state, no heat generation: dt 0 dx -6

8 - Heat Conduction Equation (4) General Heat Conduction Equation Rate of heat conduction at x, y, and z Rate of heat conduction at x+x, y+y, and z+z Rate of heat generation inside the element - + = Rate of change of the energy content of the element Q Q Q x y z ( kt ) e gen Q Q Q, xx yy zz T c t 1) Steady-state: ) Transient, no heat generation: 3) Steady-state, no heat generation: E gen element T T T e gen 0 x y z k E element t T T T 1 T x y z t T T T x y z 0-7

9 - Heat Conduction Equation (5) Cylindrical Coordinates 1 T 1 rk T k T k T e T gen c r r r r z z t -8

10 - Heat Conduction Equation (6) Spherical Coordinates 1 T 1 T 1 kr k k sin T e T gen c r r r r sin r sin t -9

11 -3 Boundary and Initial Conditions (1) Specified Temperature Boundary Condition Specified Heat Flux Boundary Condition Convection Boundary Condition Radiation Boundary Condition Interface Boundary Conditions Generalized Boundary Conditions -10

12 -3 Boundary and Initial Conditions () 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 as T(0, t) = T 1 T(L, t) = T The specified temperatures can be constant, which is the case for steady heat conduction, or may vary with time. -11

13 -3 Boundary and Initial Conditions (3) Specified Heat Flux Boundary Condition The heat flux in the positive x-direction anywhere in the medium, including the boundaries, can be expressed by Fourier s law of heat conduction as q dt k dx Heat flux in the positive x- direction The sign of the specified heat flux is determined by inspection: positive if the heat flux is in the positive direction of the coordinate axis, and negative if it is in the opposite direction. -1

14 -3 Boundary and Initial Conditions (4) Two Special Cases Insulated boundary Thermal symmetry k T (0, t) T (0, t) 0 or 0 x x T L, t x 0-13

15 -3 Boundary and Initial Conditions (5) Convection Boundary Condition Heat conduction at the surface in a selected direction = Heat convection at the surface in the same direction and T(0, t) k h 1 T 1T(0, t) x T ( L, t) k h T ( L, t) T x -14

16 -3 Boundary and Initial Conditions (6) Radiation Boundary Condition Heat conduction at the surface in a selected direction = Radiation exchange at the surface in the same direction and T(0, t) k T T(0, t) x surr,1 T ( L, t) k T( L, t) Tsurr x 4 4, -15

17 -3 Boundary and Initial Conditions (7) Interface Boundary Conditions At the interface the requirements are: (1) two bodies in contact must have the same temperature at the area of contact, () an interface (which is a surface) cannot store any energy, and thus the heat flux on the two sides of an interface must be the same. k T A (x 0, t) = T B (x 0, t) and A T ( x 0, t A ) T B( x 0, t ) kb x x -16

2-1 General Relation for Fourier s Law of Heat Conduction 2-2 Heat Conduction Equation 2-3 Boundary Conditions and Initial Conditions 2-4 Variable Thermal Conductivity 2-5 Steady Heat Conduction in Plane

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