Page 124 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

STEADY STATE HEAT CONDUCTION IN ONE DIMENSION
116
T w
r i
r o

h, T a

Figure 4.10 Radial heat conduction in an infinitely long cylinder

where T w is the inside wall temperature, T o is the outside wall temperature, k is the
thermal conductivity, h is the heat transfer coefficient at the outside surface and T a is the
atmospheric temperature.
Integrating Equation 4.57, we obtain

(4.59)
kT = C 1 ln r + C 2
Subjecting the above equation to the boundary conditions of Equation 4.58 results in
C 1 =−hr o (T o − T a ) and C 2 = kT w − C 1 ln r i (4.60)

Substituting the constants and rearranging Equation 4.59, we obtain the exact solution as
(T − T w ) hr o r i
= ln (4.61)
(T o − T a ) k r o
With the use of the finite element method and assuming a linear variation of temperature,
the resulting stiffness matrix is given by

T T
[K] = [B] [D][B]d
+ h[N] [N]dA s

A s
1
 
r o
1 1 N i
−
 l 
=  k − 2πr dr + h N i N j dA s
1 l l N j

r i A s
l
2πk (r i + r j ) 1 −1 00
= + 2πr o h (4.62)
l 2 −1 1 01
per unit length of a cylinder. In the above equation, the variation of r is expressed as
r = N i r i + N j r j . The surface area per unit length is A s = 2πr o . The loading vector is

T 0
{f}= hT a [N] dA s = hT a 2πr o (4.63)
1
A s