Page 20 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

INTRODUCTION
12
The preceding equations, that is, 1.37 and 1.38 are valid for solving heat conduction
problems in anisotropic materials with a directional variation in the thermal conductivities.
In many situations, however, the thermal conductivity can be taken as a non-directional
property, that is, isotropic. In such materials, the heat conduction equation is written as
(constant thermal conductivity)
2
2
2
∂ T ∂ T ∂ T G 1 ∂T
+ + + = (1.39)
∂x 2 ∂y 2 ∂z 2 k α ∂t
where α = k/ρc p is the thermal diffusivity, which is an important parameter in transient
heat conduction analysis.
If the analysis is restricted only to steady state heat conduction with no heat generation,
the equation is reduced to
2
2
2
∂ T ∂ T ∂ T
+ + = 0 (1.40)
∂x 2 ∂y 2 ∂z 2
For a one-dimensional case, the steady state heat conduction equation is further
reduced to
d dT
k = 0 (1.41)
dx dx
The heat conduction equation for a cylindrical coordinate system is given by
1 ∂ ∂T 1 ∂ ∂T ∂ ∂T ∂T
k r r + k φ + k z + G = ρc p (1.42)
2
r ∂r ∂r r ∂φ ∂φ ∂z ∂z ∂t
where the heat fluxes can be expressed as
∂T
q r =−k r
∂r
k φ ∂T
q φ =−
r ∂φ
∂T
q z =−k z (1.43)
∂z
The heat conduction equation for a spherical coordinate system is given by
1 ∂ 2 ∂T 1 ∂ ∂T 1 ∂ ∂T ∂T
k r r + k φ + k θ sin θ + G = ρc p
2
2
2
r ∂r ∂r r sin θ ∂φ ∂φ r sin θ ∂θ ∂θ ∂t
2
(1.44)
where the heat fluxes can be expressed as
∂T
q r =−k r
∂r
k φ ∂T
q φ =−
r sin θ ∂φ
k θ ∂T
q θ =− (1.45)
r ∂θ
It should be noted that for both cylindrical and spherical coordinate systems,
Equations 1.42 and 1.44 can be derived in a similar fashion as for Cartesian coordinates
by considering the appropriate differential control volumes.