This paper gives an exact solution of heat transfer equations in freezing soil with constant freezing speed accompanying uniform suction flow to the freezing front, as a boundary value problem. Temperatures in frozen and unfrozen soil, θ
1 and θ
2 respectively, are :
θ
1=κ
1/k
1 (k
2/κ
2U+ν/
U+υ
hθ
∞+γ
wLwu
fU+υ
w/
U+υ
h) {1-exp (-
U+υ
h/κ
1ζ)} for ζ<0 and
θ
2=θ
∞ {1-exp (-
U+ν/κ
2ζ)}, for ζ>0,
where, ζis variable of moving coordinate system with the same speed of the freezing front (at the freezing front ζ=0),
U the constant advancing speed of freezing front, υ
wthe constant suction speed of water from unfrozen soil to the freezing front, υ
h the heaving speed of frozen soil,
υ
h = (1+Γ) (υ
w+n
f1/1+Γ
U),
where Γ is the ratio of volume increase of water when freezing (≅ 0.09), n
f the volumetric content of freewater in the vicinity of the freezing front, κ
1, κ
2 the thermal diffusivity of frozen and unfrozen soil respectively, k
1, k
2 the thermal conductivity of frozen and unfrozen soil respectively; νthe parameter defined as
ν=
Cwγ
w/
Csγ
sυ
wwhere γ
w the weight of unit volume of pore water, γ
s the weight of unit volume of unfrozen soil,
Cw the specific heat of pore water, and
Cs the specific heat of unfrozen soil;
Lw the latent heat of pore water in freezing, and θ
∞ the initial temperature of unfrozen soil.
In these equations the value of suction speed of pore water υ
w can be taken independently to the freezing speed
U. However authors have shown previously (Takashi
et al., 1974) that υ
w is a function of
U;
υ
w=
U/1+Γσ
0/σ (1+√
U0/
U) -n
fΓ/1+Γ
U,
where, σ the effective stress in soil under freezing, σ
0,
U0 the characteristic constants of soil.
Therefore if the last equation is applicable to all kind of soils it would be said that the present problem was solved completely.
View full abstract