Page 248 - Introduction to Computational Fluid Dynamics

P1: IBE
0521853265c07
7.3 1D PROBLEMS FOR IMPURE SUBSTANCES
and CB908/Date 0 521 85326 5 May 25, 2005 11:14 227
= 1 for θ l <θ <θ m , (7.44)
where θ s <θ l <θ m , and the values of these temperatures and n (0.2 to 0.5) are
known. In another form, known as Schiel’s equation, the relationship is given by

θ = θ s for 0 < < s , (7.45)

− β

θ − θ m
= for s < < 1, (7.46)
θ l − θ m
= 1 for θ l <θ <θ m , (7.47)

where β = (1 − γ ) −1 and γ is the partition coefficient. The values of γ , φ s , and θ l
are known.
The discretised version of Equation 7.39 will read as

∗ ∗ ∗ ∗ ∗
ρ X k + k k k
P e w e w
+ P = E + W
τ X X X
∗ ∗
k k

+ e ( − ) + w ( − )
P
W
E
P
X X
∗
ρ X
P
o
+ . (7.48)
P
τ
The trick now is to correctly interpret function f ( ) so as to calculate θ pc since
(see Equation 7.20) can be easily calculated from . This will enable calculation

of (see Equation 7.25).
Problem 2
To illustrate the procedure, consider a specific case of Al–4.5% Cu alloy for which
the data are as follows and Schiel’s equation is used:
K s = 200 W/m-K, K l = 90 W/m-K,
C ps = 900 J/kg-K, C pl = 1,100 J/kg-K,
3
ρ s = ρ l = 2,800kg/m ,
5
λ = 3.9 × 10 J/kg, L = 0.5m,
and
T s = 821 K, T l = 919 K, T m = 933 K.

The initial state is superheated T in = 969 K and T w (x = 0) = 573 K, with γ =
0.14 (or β = 1.163).