(1 + *R */ 3)

2

10C, *U *= 1 mm/yr.

Note that if *U *is zero, the phase change interface is

1000

55,867

54,778

1.99

2000

206,996

203,653

1.64

2α*t *.

(C18)

3000

437,935

428,448

2.20

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

2

+ (a1 X - ∆*T*3 )

(C19)

* X*

* X*

(x - *X*)

2

[

]

- (GX + ∆*T *)

(C20)

δ

δ2

* x*

+ 1

(C21)

*X*d

where

α 21(∆*T *+ *GX *)X

∆*T*1M

+ 1,

[

]

,

δ *G*(δ + 2 *X *) + 2∆*T*

[

]

1

∆*T*3 = *T*f - *T*s′(t) = *R*σ∆*T*1M / σ d k13 (2 - 1 / *g*) .

,

1+ *R*

Equation 5 can be used to find a relation between *X *and δ. In nondimensional form this is

2ρ β(g - 1)

β*M*

[

]

- *k*21 σ(β + 2) + 2φ = 21

.

(C22)

Equation 3, the energy integral equation, can now be written nondimensionally as

σ

τ = ∫ K / *Hd*σ

(C23)

0

[(

]

)

[

]

()

+ σ (b2 - *A*σ / 3)β′ + *Mg*′ / 6*g*2 - *A*(β / 3 + 0.5) .

(C24)

32

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