corresponding to Takashi's formulas are derived by using the analytical solution of quasi-
steady problems (Nakano 1994a, Nakano and Primicerio 1995). Comparing the theoretical
formulas with the empirical ones for Kanto loam, Nakano (1996) has shown that M1 is
compatible with the Takashi model. Studying the property of a frozen fringe described by
the Gilpin model (Gilpin 1980), Nakano (1997) has shown that the Gilpin model is essen-
tially one special case of M1 and that it is too restrictive to accurately describe the behavior
of two kinds of porous media studied. Assuming linear temperature profiles in both frozen
and unfrozen parts and neglecting the effect of changing composition in the frozen fringe,
Talamucci has solved the first (Talamucci 1998a) and second (Talamucci 1998b) boundary
value problems of unsteady soil freezing based on M1.
In this work the problem of soil freezing is studied by using M1. We will show that
traveling wave solutions to the problem exist and describe how these solutions have been
used for the empirical verification of M1.
BALANCE EQUATIONS OF MASS AND HEAT
We will consider the one-directional freezing of soils. Let the freezing process advance
from the top down and the coordinate x be positive upward with its origin fixed at some
point in the unfrozen part of the soil. We will treat the soil as a mixture of water in the
liquid phase B1, ice B2 , and soil minerals B3. The bulk density of Bi is denoted by ρi (x,t). If
di is the density of the ith constituent, then the volumetric content θi (x,t) of the ith constit-
uent is given as
θi = ρi/di.
(1)
It is clear that the sum of θi should be unity, namely:
θ1 + θ2 + θ3 = 1.
(2)
We will assume that the density of each constituent remains constant.
We will assume that the unfrozen part of the soil is kept saturated with water at all times
by using an appropriate water supply device. The balance of mass for the ith constituent is
given as (Nakano 1990)
(ρ u ) + λ ,
ρ =-
i = 1, 2, 3
(3)
i
ii
i
t
x
where ui (x,t) is the velocity of the ith constituent, and λi(x, t) the time rate of supply of mass
of the ith constituent per unit volume of the mixture. The summation convention on index
i is not in force here, so that (ρiui) represents only one term. Since none of the constituents is
involved in a chemical reaction, we have
λ1 + λ 2 = 0
λ 3 = 0.
(4)
and
We will assume that the constituents are locally in thermal equilibrium with each other
and that the heat capacity ci of the ith constituent and the latent heat of fusion of water L do
not depend on the temperature T. If k is the thermal conductivity of the mixture, the con-
ductive heat flux q (x,t) in the mixture is assumed to be given as
T
q = -k
.
(5)
x
3
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