Page 243 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
r momentum component
2
∂u r ∂u r ∂u z 1 ∂p 1 ∂ ∂u r ∂ u r u r 235
+ u r + u z =− + ν r + − (7.207)
∂t ∂r ∂z ρ ∂r r ∂r ∂r ∂z 2 r 2
z momentum component
2
∂u z ∂u z ∂u r 1 ∂p 1 ∂ ∂u z ∂ u z
+ u r + u z =− + ν r + (7.208)
∂t ∂r ∂z ρ ∂z r ∂r ∂r ∂z 2
Energy equation
∂T ∂T ∂T 1 ∂ ∂T ∂ T
2
ρc p + u r + u z = k r + 2 (7.209)
∂t ∂r ∂z r ∂r ∂x r ∂x z
The CBS procedure follows the same steps as for the plane problem. However, the
integration of the matrices will be different as the area of the element will no longer
be two-dimensional. For example, let us consider the diffusion matrix of the momentum
equation. The momentum diffusion matrix for the plane problem is given by Equation
7.148. We can rewrite this as
T T
∂N ∂N ∂N ∂N
] = ν + d
[K me z

∂r ∂r ∂z ∂z
∂N ∂N ∂N ∂N
T T
= ν + 2πrdA (7.210)

∂r ∂r ∂z ∂z
where the radial coordinate r is expressed as
r = N i r i + N j r j + N k r k (7.211)
The formula used in the integration is the same as for any linear triangular element
(Equation 7.110). On applying Equation 7.110,Equation 7.210 becomes

b b i b j b i b k
 2 
2πA i 2
= ν b

K me z (r i + r j + r k ) b j b i j b j b k 
3 2
b k b i b k b j b k
c c i c j c i c k
 2 
2πA i
+ ν (r i + r j + r k ) c j c i c j 2 c j c k  (7.212)

3 2
c k c i c k c j c k
All the other terms of the axisymmetric equations may be discretized in a similar fash-
2
ion. In discretizing the r momentum diffusion terms, the term u r /r can be approximated
by averaging r over an element.
7.14 Summary

In this chapter, we have given a brief overview of convection heat transfer. However, the
subject is vast in extent and it is difficult to cover all aspects within a single chapter.