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9.1 Basic Equations 167

9.1.2 Related Equations
The governing differential equations in the preceding
section are derived by using the Fourier’s law. The law relates the
conduction heat flux components with the temperature gradients.
For isotropic material, the conduction heat flux in the x-direction is,
 T
q  k
x
x 
Boundary conditions of heat transfer at the domain
surface may consist of,
(a) Specified temperature,

T  T
s
where T may be constant or varies with x-, y-, z-coordinates and
s
time t.
(b) Specified heat flux,

q   q

s
where q is the specified heat flux which must be in equilibrium
s
with the conduction heat flux q at the surface.
(c) Convection heat transfer,
q  (T  T  )
h
s
where h is the surface convection coefficient and T is the

surrounding medium temperature.
(d) Radiation heat transfer,

q  ( T  4 )T
4
s 
where  is the surface emissivity and  is the Stefan-Boltzmann
constant.
For transient heat transfer, an initial condition is
needed,
Tx y z  T 0 x y z
(, , ,0) ( , , )
where T is the initial temperature of the solid.
0