PSI - Issue 17

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

606

5

(

)

c W c I c =  + 

n c n W I c

(1 ) + − + − −

,

(6)

p

, p w

, p i

, p s

, p a

(

)

k  + − + I

n W I k

+ − −

,

(7)

k W k = 

n k

(1 )

a

w

i

s

I W n +  ,

(8)

, p a c is the air temperature at constant pressure, a k is the coefficient of thermal

where

a  is the air density,

conductivity of air. The approximation of the function

( ) ( T T T T T T a T T − + − − − − ) c c

( ) eq W T is given as 0 W is the initial moisture content of the medium, c T is the temperature of crystallization, min T is the minimal temperature of the examined region, 0 a is the material parameter. The distinguishing feature of this model is the presence of two additional equations, which allow us to determine the fraction of the pore space occupied by ice, liquid and air. As a result, there is no need to calculate two sets of constants in the frozen and unfrozen states. The parameters of the medium continuously change according to the current state of the pore space. 2.2 A two-phase model of freezing a partially saturated porous medium, taking into account the equilibrium concentration of residual moisture at negative temperatures (model No. 2) ( ) ( ) min 0 min 0 ( ) W T W = eq c c , where

In the case of a two-phase ice-water mixture, model No. 1 can be simplified. The equation (2) takes the form:

(T) eq W W W t t −  = − 

, I n W = − .

(9)

*

Density, heat capacity at constant pressure and thermal conductivity coefficient of the system under consideration can be written as:

W I    = 

(1 ) n

,

(10)

+  + −

s 

w

i

c W c I c =  + 

, (1 ) p s n c

+ −

,

(11)

p

, p w

, p i

k  + − + I

k W k = 

n k

(1 )

.

(12)

w

i

s

3. Results of the numerical simulations

Let us develop a solution to the problem of phase transition in a porous saturated soil in the framework of two models described above. The initial and boundary conditions with respect to temperature are written as ( ) 0 0 T t T = = , 2  is the boundary of the freeze well, 2 T is the temperature at the external wall of the well, 1 T is the temperature at the boundary of the examined region, 0 T is the initial temperature of the examined region. If needed, the following initial and boundary conditions for moisture and ice contents are specified for the model under consideration ( ) 0 0 t W W = = , ( ) 0 0 t I = = , 1 0 W n  =   , where 0 W is the initial moisture content of the medium. Table 1 presents the soil parameter data obtained from the results of surveying the Petrikov mine rocks (Belorussia) j j T T  = , 1, 2 j = , where 1  is the boundary of the examined region,