PSI - Issue 17

O. Plekhov et al. / Procedia Structural Integrity 17 (2019) 602–609 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

605

4

2. Theoretical model

There are two sources of the cryogenic flow: the moisture gradient and the pressure gradient, which is caused by a drop of the phase transition temperature. The relationship between the pressure and temperature gradients was determined through the application of a standard procedure for the analysis of phase equilibrium. The increments of free energies of moisture (1) and ice (2) are given as

1 f       P  

1 f       T  

2 f       P  

2 f       T  

df

dP

dT

df

dP

dT

.

,

=

+

=

+

(1)

1

2

T

P

T

P

2 , w P p P = + and assume

, i P p P = +

Using the notations 1

0 i dP dT = we can write

  

  

dP

dP

v

i

dp

i

or dPw KdT = .

w

w

1

(2)

1 −  =

= +

dT Tv dT v

dT Tv

2

2

2

where i P is the pressure applied by water to the entire ice phase, w P is the pressure applied by ice to the entire water phase, p is the pressure of water vapor, 1 v , 2 v are the specific volumes of water and ice, K is the coefficient of proportionality equal to 120 m/ С °. The water pressure change leads to the initiation of water flow from unfrozen to frozen area and changes the working regime of the hydro-observation well. To simulate this process we have to use a phase transition model, which take into account kinetic of phase transition and the existence of nonequilibrium moisture content in the soil at negative temperatures. 2.1 A three-phase model of freezing a partially saturated porous medium, taking into account the equilibrium concentration of residual moisture at negative temperatures (model No. 1)

The temperature variation is described by a energy balance law:

T k

p c t  L I 



,

(3)

T −   =

t

c





p

(T) eq I W W t t −  =  ,

W

I





,

(4)

(W) D W =   −

t

t





*

where  is density, p c is heat capacity at constant pressure, k is the coefficient of thermal conductivity, T is the absolute temperature, q is the vector of the heat flow, t is time,   is the divergence operator, I is the ice content, W is the moisture content, ( ) eq W T is the relation determining the equilibrium content of unfrozen moisture in the porous medium at negative temperatures, ( ) D W is the diffusion coefficient, * t is the characteristic time of crystallization. To determine the effective quantities  , p c and k , we use the following relations: ( ) (1 ) w i s a W I n n W I      =  +  + − + − − , (5)