Page 161 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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TRANSIENT HEAT CONDUCTION ANALYSIS
where k x (T ), k y (T ) are k z (T ) are the temperature-dependent thermal conductivities in the
x, y and z directions respectively. The boundary conditions for this type of problem are
T = T b on b (6.8)
and
∂T ∂T ∂T
k x (T ) l + k y (T ) m + k z (T ) n + q + h(T − T a ) = 0on q (6.9)
∂x ∂y ∂z
where, b ∪ q = and b ∩ q = 0. represents the whole boundary. In the above
equation, l, m and n are direction cosines, h is the heat transfer coefficient, T a is the
atmospheric temperature and q is the boundary heat flux. The initial condition for the
problem is
T = T o at t = 0.0 (6.10)
It is now possible to solve the above system, provided that appropriate spatial and
temporal discretizations are available. Before dealing with the temporal discretization, we
introduce in the following subsection, the standard Galerkin weighted residual form for the
transient equations.

6.3.2 The Galerkin method

In this subsection, the application of the Galerkin method for the transient equations sub-
jected to appropriate boundary and initial conditions is addressed. The temperature is
discretized over space as follows:
n

T (x, y, z, t) = N i (x,y,z)T i (t) (6.11)
i=1
where N i are the shape functions, n is the number of nodes in an element, and T i (t) are
the time-dependent nodal temperatures. The Galerkin representation of Equation 6.7 is
∂ ∂T ∂ ∂T ∂ ∂T ∂T

N i k x (T ) + k y (T ) + k z (T ) + G − ρc p d
= 0

∂x ∂x ∂y ∂y ∂z ∂z ∂t
(6.12)
Employing integration by parts on the first three terms of Equation 6.12, we get
∂N i ∂T ∂N i ∂T ∂N i ∂T ∂T

− k x (T ) + k y (T ) + k z (T ) − N i G + N i ρc p d

∂x ∂x ∂y ∂y ∂z ∂z ∂t
∂T ∂T ∂T

+ N i k x (T ) ld q + N i k y (T ) md q + N i k z (T ) nd q = 0 (6.13)
∂x ∂y ∂z
q q q
Note that from Equation 6.9,
∂T ∂T ∂T

N i k x (T ) ld q + N i k y (T ) md q + N i k z (T ) nd q
∂x ∂y ∂z
q q q

=− N i qd q − N i h(T − T a )d q (6.14)
q q