Page 85 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

THE FINITE ELEMENT METHOD
In the Ritz method, we insert the approximate profile into the governing differential
equation, Equation 3.172, and then the integral of the residual ‘R’ over the domain is
equated to zero to determine the constant B,that is, 77
1 2
d θ(ζ) 2
− µ θ dζ = 0 (3.178)
dζ 2
0
Differentiating Equation 3.177 gives
2
d θ(ζ)
= 2Bθ b (3.179)
dζ 2
Substituting Equation 3.179 into Equation 3.178, we have
1 Bζ 3 1
2 2 2
[2B − µ (1 −{1 − ζ }B)] θ b dζ = 2θ b Bζ − µ θ b ζ − Bζ +
0 3 0
B 2

= 2Bθ b − 1 − B + µ θ b
3
= 0 (3.180)

which gives
µ 2
2
B = (3.181)
µ 2
1 +
3
Substituting Equation 3.181 into 3.177 gives the following solution:
µ 2
θ(ζ) 2 2
= 1 − (1 − ζ ) (3.182)
µ 2
θ b
1 +
3
For the case of a stainless steel fin (k = 16.66 W/m C) of circular cross section with a
◦
diameter of 2 cm and length of 10 cm exposed to a convection environment with h =25
2
2
W/m Cand µ = 3.0and m = 300, the approximate solution is
2◦
θ(ζ) 3 2
= 1 − (1 − ζ ) (3.183)
θ b 4
where the exact solution is
θ(ζ) cosh m(L − x)
= (3.184)
θ b cosh mL
Note that the distance x is taken from the tip of the fin as shown in Figure 3.24. The
comparison between the exact and approximate solutions is given in Figure 3.25. As seen,
the temperatures agree excellently at the base at x = 1 but differ close to the insulated end
at x = 0.