Page 262 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
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8.4 Non-isothermal Flows
Several examples of porous medium flow problems are non-isothermal in nature. The main
focus in this case will be to demonstrate non-isothermal flow through a porous medium. As
mentioned previously, an energy equation needs to be solved, in addition to the momentum
and pressure equations if the flow is non-isothermal. For steady state problems, if no
coupling exists between the momentum and energy equation, the temperature field can be
established after calculation of the velocity fields. The temporal discretization of the energy
equation can be written in a similar form to the momentum equation and is given as
2 n+θ 2
n+1
n
T − T ∂T n+θ 1 k ∗ ∂ T

σ =− u i + (8.66)
t ∂x i ReP r ∂x 2
i
where θ 1 and θ 2 have the same meaning as previously discussed in Section 8.3. The variable
involved in this case is temperature and can be spatially approximated as
T = [N]{T} (8.67)
The weak form of the energy equation can be written (assuming θ 1 and θ 2 are both equal
to zero) as
∂T k t ∂[N] ∂T
n ∗ T n
T
T
σ[N] T d
=− t [N] u i d
− d
+ b.t.

∂x i ReP r
∂x i ∂x i
(8.68)
where
T = T n+1 − T n (8.69)
The substitution of Equation 8.67 into Equation 8.68 yields the final global matrix form of
the energy equation, that is,
n
σ[M p ]{ T}=− t [C p ]{T}+ [K T ]{T}− {f 4 } (8.70)
where the elemental matrices are
b b i b j b i b k c c i c j c i c k
 2   2 
k ∗ i 2 k ∗ i 2
[K Te ] =  b j b i b j b j b k  +  c j c i c j c j c k  (8.71)
4AReP r 2 4AReP r 2
b k b i b k b j b c k c i c k c j c
k k
and the forcing vector is
 n

b i T i + b j T j + b k T k
1
{f 4 }=  b i T i + b j T j + b k T k  n 1
4A ReP r
0
 n

1 c i T i + c j T j + c k T k
+  c i T i + c j T j + c k T k  n 2 (8.72)
4A ReP r
0
It should be noted that both the flux and convective heat transfer boundary conditions are
treated by using the boundary integral, as discussed in the previous chapter. At higher
Reynolds numbers convection stabilization of Equation 8.70 is essential. This can be
achieved by introducing characteristic Galerkin method (Chapter 7).