Page 240 - Introduction to Computational Fluid Dynamics

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7.2 1D PROBLEMS FOR PURE SUBSTANCES
state. The overall calculation procedure will be as follows: May 25, 2005 11:14 219
o
o
1. At τ = 0, specify initial condition θ for j = 1to N. Hence, evaluate . Set
j j
o
θ j = θ .
j
2. Choose τ to begin a new step.
3. Solve Equation 7.18 once using the GS method to obtain the l+1 distribution.
j
4. Determine θ l+1 using equations of state (7.15) to (7.17) and return to step 3 to
j
carry out the next iteration.
5. After a few iterations, the change in l+1 between successive iterations will be
j
small and convergence is obtained.
o
6. Set = j and return to step 2 to execute the next time step.
j
Problem 1
To appreciate the nature of the numerical solution, consider a problem with the
following specifications:
3
ρ = 1kg/m , C p = 2.5MJ/kg-K, K = 2W/m-K,
λ = 100 MJ/kg, T m = 0 C, L = 1m,
◦
T (x, 0) = T sup = 2 C, and T w = T (0, t) =−10 C
◦
◦
For this problem St = 0.25 and, as evaluated from Equation 7.8, C = 5.767 ×
−4
10 . A numerical solution is executed with initial conditions θ (τ = 0) = 0.05
and (τ = 0) = 1.05. The boundary condition is θ (X = 0) =−0.25. The time
2
step is determined from τ/ X = 0.2 and the computations are carried out
till τ = 1.6 (or nearly 23 days). Two grid spacings are considered: X = 0.2
(N = 7) and X = 0.0769 (N = 15). At each time step, a converged solution
is obtained in 5–11 iterations. The exact and the numerical solutions for temper-
ature at x = 0.5 m are plotted in Figure 7.3 as a function of time. The figure shows
a wavy temperature history. The waviness, however, decreases with refinement
of the grid size. When X is reduced still further so that N = 51 (say), the re-
sults (not shown) indicate that the exact and the numerical solutions nearly co-
incide. That is, the essentially wavy solution now appears smooth, albeit at the
expense of significantly increased computer time. A few comments are therefore in
order:

1. The numerical procedure is very simple and can be easily extended to multidi-
mensional problems. However, to obtain non-wavy solutions, an extremely fine
mesh size is required. This can be very uneconomical.
2. Why does waviness occur? This can be appreciated from Figure 7.4, where a
phase-change node j is considered. When the interface resides within the control
volume surrounding node j (so that 0 < j < 1),θ j = 0 (see Equation 7.16).