Page 249 - Introduction to Computational Fluid Dynamics

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1.0

LIQUID

Φ
s
θ
θ = 0 θ l θ m
s
SOLID

Figure 7.8. Schiel’s function.

Thus, solidification commences instantly and calculations can be executed with

θ in = 0.341538, in = 1.10154, =−1,
in
θ w = w =−0.5723,

θ s = 0, s = 0.089,
θ l = 0.226154, l = 1.0,

θ m = 0.258462, m = 1.0.

Figure 7.8 shows the Schiel’s function. We now specify θ pc for the range 0 <
< 1 for which =− .

0 < < s : In this range, θ s = 0 remains stationary. Therefore, we may
employ Equation 7.31.

s < < l : In this range, from Equation 7.46,

− 1 /β
θ pc, j = θ m + (θ l − θ m ) . (7.49)
θ l <θ <θ m : In this range, remains constant at 1. Therefore, the right-hand
side of Equation 7.39 can be equated to zero. Therefore, the solution in discretised
form is
∗ ∗
k θ E + k θ W
e
w
θ pc, j = . (7.50)
k + k w
∗
∗
e