Page 63 - Introduction to Computational Fluid Dynamics

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Table 2.5: Coefficients in the discretised 1D HEAT CONDUCTION
equation – Problem 2.
i 2 3 4 5 6
AW i 0 4.5 4.5 4.5 4.5
AE i 4.5 4.5 4.5 4.5 0
2025.6 0.6 0.6 0.6 0.6
Su i
9.024 0.024 0.024 0.024 0.024
Sp i
used. Greater accuracy can be obtained with finer grids; however, this will require
more computational effort.
From the converged solution, the fin heat loss is estimated as Q loss = AW 2 ×
(T 1 − T 2 ) = 9(225 − 222.42) = 23.26 W. This also compares favourably with the
exact solution already mentioned.
Table 2.7 shows the execution of the same problem using TDMA. The table
shows values of A i and B i derived fromTable2.5and Equations2.74 and 2.75. Since
these are constants, solution is now obtained in only one iteration. Also, the initial
guess becomes irrelevant. The estimated heat loss is Q loss = 9(225 − 222.45) =
22.967 W.
Thus, compared to GS, the TDMA procedure is considerably faster. Experience
shows that this conclusion is valid even in nonlinear problems. For this reason, the
TDMA is the most preferred solution procedure in generalised codes.

Problem 3 – Annular Composite Fin
Consider an annular fin put on a tube (of outer radius r 1 = 1.25 cm), as shown
in Figure 2.11. The fin is made from two materials: The inner material has radius
r 2 = 2.5 cm and conductivity k 2 = 200 W/m-K and the outer material extends
to radius r 3 = 3.75 cm and has conductivity k 3 = 40 W/m-K. The fin thickness
◦
t = 1 mm. The tube wall (and hence the fin base) temperature is T 0 = 200 C. The

Table 2.6: Solution by Gauss–Seidel method – Problem 2.

l FCMX 0 cm 0.2 cm 0.6 cm 1.0 cm 1.4 cm 1.8 cm 2.0 cm
0 225 223 219 215 211 207 205
1 0.01 225 222.65 218.31 214.15 210.08 209.1 209.1
2 0.0034 225 222.42 217.77 213.44 210.77 209.78 209.78
3 0.0021 225 222.24 217.32 213.54 211.16 210.18 210.18
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
22 0.00012 225 222.41 218.28 215.22 213.19 212.19 212.19
23 0.00011 225 222.41 218.30 215.24 213.21 212.21 212.21
24 0.000092 225 222.42 218.31 215.25 213.23 212.23 212.23
Exact − 225 222.58 218.52 215.51 213.49 212.49 212.37