Page 84 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
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where k is the thermal conductivity, A is the cross-sectional area, h is the heat transfer
coefficient, P is the perimeter and the suffix a represents atmospheric condition.
Simplifying, the governing differential equation becomes
2
d T
kA − hP (T − T a ) = 0 (3.171)
dx 2
with the following boundary conditions:
At x = 0, dT/dx = 0 (tip) and at x = L, T = T b (base)
2 2
2
2
Let (T − T a ) = θ, ζ = x/L, hP/kA = m and m L = µ , then the governing equa-
tion reduces to
2
d θ 2
− µ θ = 0 (3.172)
dζ 2
with the following new boundary conditions:
At ζ = 0, dθ/dζ = 0and at ζ = 1,θ = θ b (3.173)
3.3.1 Ritz method (Heat balance integral method - Goodman’s
method)
An approximate solution of Equation 3.172 along with the appropriate boundary conditions
may be found using the following function:
n

T ≈ T = T(x, a 1 ,a 2 ,... ,a n ) = a i N i (x) (3.174)
i=1
which has one or more unknown parameters a 1 ,a 2 ,... ,a n and functions N i (x) that exactly
satisfy the boundary conditions given by Equation 3.173. The functions N i (x) are referred
to as trial functions, which must be continuous and differentiable up to the highest order
present in the integral form of the governing equation.
The approximations may be carried out using one, two or n terms as follows:

T = a 1 N 1 (x)
T = a 1 N 1 (x) + a 2 N 2 (x) (3.175)
or
n

T = a i N i (x) (3.176)
i=1
When T is substituted into the governing differential equation, it is not satisfied exactly,
leaving a residual ‘R’. The exact solution results when the residual ‘R’ is zero for all
points in the domain. In approximate solution methods, the residual is not in general zero
everywhere in the domain even though it may be zero at some preferred points.
Let us select a profile that satisfies the boundary conditions (Equation 3.173) in the
global sense. By inspection, we find that
θ(ζ) 2
= 1 − (1 − ζ )B (3.177)
θ b
satisfies the boundary conditions, where ‘B’ is an unknown parameter to be determined.