Page 243 - Introduction to Computational Fluid Dynamics

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222
PHASE CHANGE
introduction of the variable yields two further advantages: May 25, 2005 11:14

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1. The terms containing and can be treated as sources. Thus, at the current
iteration level, Equation 7.22 can be solved by TDMA. This can achieve con-
siderable economy in computer time. For example, for the problem considered
in the previous subsection, with N = 51, the TDMA solution turns out to be
nearly 2.5 times faster than the GS solution.
2. It is easy to recognize that at each time step, when a converged solution is
obtained, X i (τ) can be estimated from the simple formula
N−1

X i = (1 + ) X. (7.23)
j
j=2
This is because (1 + ) represents the solid fraction for each node j. Again,

j
the bookkeeping requirement is eliminated.
Although useful for obtaining faster solutions on fine grids, the introduction

of the variable does not eliminate the problem of wavy temperature histories
on coarse grids. This is because the replacement indicated in Equation 7.19 still
renders θ = 0 at the phase-change node (0 < j < 1). In the next subsection, it
will be shown that accurate solutions can be obtained even on coarser grids while
still employing the TDMA procedure. Thus, we seek an economic solution that
combines the beneficial effects of computations at fewer nodes with the speed of
the line-by-line procedure.

7.2.4 Accurate Solutions on a Coarse Grid
To prevent θ from remaining stationary at zero at the phase-change node,
Equation 7.19 is rewritten as

θ = + , (7.24)
where

= + θ pc , (7.25)
with, θ pc denoting the nodal value of θ at the phase-change node 0 < j < 1. Note
that θ pc = 0 at all single phase nodes. Making these substitutions in Equation 7.18
leads to

τ l+1 τ l+1 l+1
1 + 2 j = j+1 + j−1
X 2 X 2
τ
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+ ( j+1 − 2 + j−1 ) + . (7.26)
j
j
X 2