Page 244 - Introduction to Computational Fluid Dynamics

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7.2 1D PROBLEMS FOR PURE SUBSTANCES
This equation is the same as Equation 7.22 except that is replaced by and the 11:14 223

latter will again lag behind by one iteration. Equation 7.24 is therefore amenable
to solution by TDMA.

To make further progress, a procedure for evaluating or, in effect, θ pc must be

set out since can be evaluated from its definition (7.20). Thus, consider Figure 7.4
again and define
X i = X i − X j , (7.27)
where X i is the location of the interface where θ is truly zero and X j is the coordinate
of node j. At the time instant considered in the figure, therefore, X i is positive
and we may evaluate θ pc, j by linear interpolation as

X i
θ pc, j = θ j−1 . (7.28)
X i + X
At another earlier time instant, X i may be negative (X i < X j ) and we may write

| X i |
θ pc, j = θ j+1 . (7.29)
| X i |+ X
Note, however, that for both positive or negative values of X i

X i = X i − X j = (0.5 − pc, j ) X = (0.5 + ) X (7.30)
pc, j
since pc, j =− at the phase-change node. Equations 7.28 and 7.29 therefore
pc, j
can be generalised to read as

F
θ pc, j = (A +|A|)θ j−1 − (A −|A|)θ j+1 , (7.31)
2
where
0.5 +
pc, j
A = (7.32)
|0.5 + |+ 1
pc, j
and
(1 + )

j j
F =− . (7.33)
(1 − j ) j
In these equations, F = 0 at the single phase nodes (rendering θ pc = 0) but
F = 1 at the phase-change node as desired. Thus, the phase-change node tempera-
ture can be evaluated without bookkeeping. Therefore, in step 4 of our calculation

procedure, is also evaluated without bookkeeping.
j
Problem 1 of Section 7.2.2 is now solved again for the coarse grid with N = 7
(or X = 0.2) for St = 0.25, and the predicted temperature history is shown in
Figure 7.5. Now, even the coarse grid solution is nearly accurate. In the same figure,
−3
−3
computations for St = 1(C = 1.075 × 10 ) and St = 3(C = 1.6 × 10 ) are
also shown and the grids used are indicated in the figure. Again, smooth histories