**Table C1. Comparison of closed**

*R*

(1 + *R */ 3)

*K*1 = 1 +

**solution (***G *= 0) and numerical

2

**quadrature (***G *= 0.0001).

*K*2 = *U*(1 + *S*T )

*S*T = 0.144, α = 58.89 m2/yr, ∆*T*1 =

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

*K*3 = α*R*(1 + *R*).

*Freeze*

*Time*

*Time*

*depth*

*(yr)*

*(yr)*

*Percent*

*(m)*

*eq C12*

*eq C17*

*difference*

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

1000

55,867

54,778

1.99

*R*(1 + *R*)

2000

206,996

203,653

1.64

*X*2 =

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.

**Method 2**

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*

2

+ (a1 X - ∆*T*3 )

*T*1 = *T*f + *a*1 X

(C19)

* X*

* X*

(x - *X*)

*x*-*X*

2

[

]

*T*2 = *T*f + *G*(δ + 2 *X *) + 2∆*T*

- (GX + ∆*T *)

(C20)

δ

δ2

* x*

*T*3 = *T*s + ∆*T*1MRX

+ 1

(C21)

*X*d

where

α 21(∆*T *+ *GX *)X

∆*T*1M

*R *= σ d k13 (2 - 1 / *g*) / σ,

*a*1 X =

*g*=

+ 1,

[

]

,

δ *G*(δ + 2 *X *) + 2∆*T*

*g*

[

]

1

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

*M*=

,

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)

*g*

*S*T

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

σ

τ = ∫ K / *Hd*σ

(C23)

0

[(

]

)

*K *= *b*1 + (b2 - *A*σ / 3)β - *M */(6*g*) - *A*σ / 2 + σ d M 2k13 (0.5 / *g *+ 1) g′ / *g*2 - (2 - 1 / *g*) / σ / 3

[

]

()

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

(C24)

32