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166 Chapter 9 Heat Transfer Analysis

9.1 Basic Equations

9.1.1 Differential Equation
The conservation of energy at any location in an iso-
tropic three-dimensional solid is described by the differential
equation,

 T    T     T     T   
c     k  k      k   Q   0
t   x   x   y   y   z     z  
  
where  is mass density of the solid material, c is the specific
heat, k is the thermal conductivity coefficient, Q is the internal
heat generation rate per unit volume, and T is the temperature that
varies with the coordinates ,x ,y z and time t .
For steady-state heat transfer, the differential equation
above becomes,
  k T   +   k   T   +   k T   + Q  0
x    x   y    y    z     z   

If heat transfer occurs only in the two-dimensional -xy plane with
constant thermal conductivity coefficient k , the differential equa-
tion reduces to,
 2 T +  2 T   Q
x  2 y  2 k

which is in form of the Poisson’s equation. In addition, if there is
no internal heat generation, the governing differential equation
reduces further to,
 2 T +  2 T  0
x  2 y  2
which is called the Laplace’s equation.

Even though the Laplace’s equation above looks very
simple, its exact solution (, )Tx y is still difficult to derive especial-
ly when the problem geometry is complicated.