Table C1. Comparison of closed
(1 + R / 3)
K1 = 1 +
solution (G = 0) and numerical
quadrature (G = 0.0001).
K2 = U(1 + ST )
ST = 0.144, α = 58.89 m2/yr, ∆T1 =
10C, U = 1 mm/yr.
K3 = αR(1 + R).
Note that if U is zero, the phase change interface is
R(1 + R)
1 + R / 2(1 + R / 3)
This is identical to the well known Stefan solution given in Lunardini (1991). We may compare the closed
form solution (for which G = 0) with the numerical quadrature of eq C12 by letting G be very small. Table
C1 shows that the results are quite good even for very long freeze times.
We can examine the same problem with a different approximation method by referring to Figure 13. For
region 3, a quasi-steady approach will be used, leading to a linear temperature profile. The basic equations
for heterogenetic growth are valid except that the surface temperature will be replaced by a transient func-
tion Ts′(t) . Equations 15 are valid but the temperature profiles are changed as follows. Quadratic tempera-
ture profiles in regions 1 and 2 and a linear temperature in region 3 that satisfy the boundary conditions are
x - X
x - X
+ (a1 X - ∆T3 )
T1 = Tf + a1 X
(x - X)
T2 = Tf + G(δ + 2 X ) + 2∆T
- (GX + ∆T )
T3 = Ts + ∆T1MRX
α 21(∆T + GX )X
R = σ d k13 (2 - 1 / g) / σ,
a1 X =
δ G(δ + 2 X ) + 2∆T
∆T3 = Tf - Ts′(t) = Rσ∆T1M / σ d k13 (2 - 1 / g) .
Equation 5 can be used to find a relation between X and δ. In nondimensional form this is
2ρ β(g - 1)
- k21 σ(β + 2) + 2φ = 21
Equation 3, the energy integral equation, can now be written nondimensionally as
τ = ∫ K / Hdσ
K = b1 + (b2 - Aσ / 3)β - M /(6g) - Aσ / 2 + σ d M 2k13 (0.5 / g + 1) g′ / g2 - (2 - 1 / g) / σ / 3
+ σ (b2 - Aσ / 3)β′ + Mg′ / 6g2 - A(β / 3 + 0.5) .