Page 251 - Introduction to Computational Fluid Dynamics

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PHASE CHANGE
> 1: Although this is a single-phase region, to account for property variation,
we set

C ps
θ pc, j = θ m − ( − 1) 1 − . (7.51)
C pl
Thus, θ pc is specified for the entire > 0 range rather than being restricted to the
0 < < 1 range. Following Voller and Swaminathan [85], computations are car-
2
ried out using x = 0.01 m (or N = 52) and t = 5 so that τ = α s t/ x =
3.96825. Equation 7.48 is solved using TDMA at each time step. It is found that
a maximum of two iterations are required to reduce the residual in the equation to
−5
less than 10 . Figure 7.9(a) shows the time variation of the interface. In this case,
the interface location is identified with = 0 [see definition (7.34)]. The computed
results are compared with the solution obtained by Voller [84] using the heat bal-
ance integral method (HBIM) since exact solution is not available for this highly
nonlinear case. The present computations show some waviness that is also observed
in [85] where computations are carried out using the h = f (T ) relationship rather
than the T = f (h) relationship used here. Figure 7.9(b) shows the temperature
histories at a few values of x. The solutions demonstrate jaggedness (typical of a
highly nonlinear θ– relation) that is also observed by Chiu and Caldwell [6], who
used what is called Broyden’s method.
Finally, we note that the method presented in this section can also be ex-
tended to the case when phase change takes place at a unique temperature, that
is, θ s = θ l = θ m = 0. Because then, f ( ) = 0 (see Equation 7.41) and one can
readily adopt Equation 7.31 to evaluate θ pc . Similarly, the present method can also
be extended to multidimensional phase-change problems. The only care required is
in the evaluation of θ pc because several nodes can undergo phase change simulta-
neously. In Date [12, 13], the necessary considerations and the associated algebra
are explained.

EXERCISES

1. Write a general computer program for solving transcendental equation (7.8)
[63]. Hence, determine the value of C for the two materials and conditions
given in Table 7.2.

2. Modify Equation 7.22 for node j = 2, when the heat transfer coefficient h is
specified at boundary x = 0.
3. Show the validity of Equation 7.23 in a solidification problem.

4. With respect to Figure 7.4, demonstrate the correctness of Equation 7.30 and
hence of Equation 7.31.

5. Show the correctness of Equation 7.51 for > 1.