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Department of Mechanics, Lanzhou University, Lanzhou 730000, China; 2. ... State Key Laboratory of Frozen Soil Engineering, LIGG, CAS , Lanzhou 730000, ...
V O ~ . 42

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SCIENCE IN CHINA (Series D)

A U ~ U S 1999 ~

Modeling on coupled heat and moisture transfer in freezing soil using mixture theory * MIAOTiande ( $ * f % f l ,

GUOLi

(%

and ZHANG Changqing ( $if

NIUYonghong(q7k&)' )3

( 1 . Department of Mechanics, Lanzhou University, Lanzhou 730000, China; 2 . Southeast University, Nanjing 210096, China;

3 . State Key Laboratory of Frozen Soil Engineering, LIGG, CAS , Lanzhou 730000, China) Received February 8 , 1999

Abstract

A set of perfect constitutive equations including the coupling effects of heat transfer and moisture migration is constructed for freezing soil, after analyzing its thermomechanic properties, in the framework of continuum mechanics and mixture theory. By applying the theory, the influence of void ratio on frost heaving is studied after proposing a criterion for formation of layered ice; the results obtained coincide with experimental data available in the literature. The temperature distribution of freezing soil is analyzed, the controlling equation deduced appears to be a nonlinear Burgers type equation with varying boundaries, which presents a theoretic foundation for studying the nonlinear effects of heatmoisture migration in the freezing process.

Keywords :

freezing soil, heat and moisture transfer, Burgers type equation, continuum mechanics.

Frozen ground (including permafrost and seasonally frozen ground) covers about 60% of the lands on the earth. In these regions, a main engineering calamity is frost heaving or an increase in soil volume that results from freezing of moisture present in soil and moisture drawn in the migration process. It is well known that the principal cause for frost heaving is the migration of moisture toward the freezing front and its freezing, which may result in an increase in volume from tens to hundreds percent, other than the local freezing of the moisture present in soil which can only cause a volume increase of soil of about 9% generally. Therefore, the problem of heat and moisture migration during freezing of soils is of very great importance for understanding the mechanism of frost heaving, and for the study of environment changing in the whole world and analysis of palaeo-climate to which close attention is increasingly paid. The heterogeneity of frozen soil ( including mineral grains, gases, water, vapor and ice) and the possibility of phase transitions make it difficult to study the physical mechanism of moisture and heat migration process in frozen soil. At present, most results in the field are only from trial study, and some existing theoretical formula cannot give a satisfactory explanation to various observed phenomena"' . The mathematical model commonly adopted for calculating the temperature field is still the classical Stefen' s equation'" , in which, however, the coupling effects of moisture and heat cannot be dealt with because of the linearity of the equation. Because freezing soil can be regarded as a kind of mixture with possibility of phase transitions,

* Project

supported by the National Natural Science Foundation of China (Grant No. 19372022) and the State Key Laboratory of

Frozen Soil Engineering (Grant No. 9707)

.

Val. 42

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we try to solve these problems in the theoretical frameworks of continuum mechanics and mixture theory in this paper. A set of proper constitutive equations for frozen soil is constructed referring to frame] applying the constitutive equations, we will discuss works developed recently by de ~ o e r ' ~ Then, two important problems in frozen soil mechanics. Firstly, we will propose a criterion for formation of layered ice in freezing ground, on which the study of influence of soil structure (void ratio) on frost

.

heaving is based. The results coincide well with those from laboratory study available in the literature qualitatively. Then, heat conduction with moisture migration in freezing soil is studied. For the frozen region, the heat conduction equation deduced is the same as the classical heat conduction equation. For the unfrozen region, in which the effect of moisture migration on heat conduction is predominant, the equation deduced is a nonlinear equation of Burgers' type. This equation presents a theoretical starting point for the study of the nonlinear effects, such as ice segregation phenomenon, observed in the freezing process of frozen soil.

1 Fundamental equations In this paper, freezing soil is treated as a kind of porous media consisting of porous solid saturated with liquids and gases, and assumed to satisfy the following two idealizations: ( 1 ) All the three constituents are continuous throughout the configuration, and allowed to occupy common portions in the physical space; (2) The porous solid is elastic and statistically isotropic, and the two fluids are immiscible. Denoting the mass density assigned to each constituent by pa ( a = 1 , 2 , 3 , represents the solid, liquid and gaseous constituents respectively) , we have Pa = naPa~ (1.1) where n, and p , ~are the mass concentration and the intrinsic mass density of the a th constituent respectively. It is evident that n, satisfies the condition 9

If the independent motion function of the ath constituent is supposed to be x, = x, ( X, ,t ) , where X, denotes a material point in the reference configuration, then the velocity and acceleration of Xa are respectively XI,

The material time derivative (

.)A

axll

aZxa = at2

= - and at

-

is defined as follows:

In the mixture theory, each constituent of the mixture should satisfy the balance laws such as conservation of mass, balance of momentum and moment of momentum, and conservation of energy. Moreover, as a whole, the mixture should satisfy the entropy inequality in addition to the balance laws. Considering the characteristics of freezing soil and the constraints imposed by the entropy inequality, the dependent constitutive variables should be chosen as *

*

*

,

.

A

*

F = {qa, la,q a , T ,~ 2 ~, 3 P2, 9 P3, '2, '3, A}; according to axiom of causality[41,the independent constitutive variables should be

(1.5)

MODELING ON COUPLED HEAT AND MOISTURE TRANSFER

SUPP.

Y = ( 6 1 , A 2 , A 3 , g l , g z , g 3 , C s , p 1 9 p 2 , p a ~ ,x>-

11

(1.6) where 6, , Pa,rl, and qa are respectively the absolute temperature, free energy density, entropy den~ ' 1 ,~ ' 3 - x11I 9 *

,.

+.

sity and heat flux vector of the ath constituent; T* is the Cauchy stress tensor; p a , P, and e, are respectively the mass, momentum and energy supplied to the ath constituent by other constituents per time step ; h is the Lagrangian multiplier; g = grad , gp = gradAp ,Ap = OP - ( ,8 = 2 . 3 ) , and C, is the deformation tensor of the solid constituent, which satisfies the condition det C, = 1

.

Applying the axioms of determinism and equipresence[41, we see that the constitutive equations should be of the form F = F(Y). (1.7) From this equation, we can get the material derivative of free energy of each constituent. Substituting the derivatives into the entropy inequality and considering the independence of the process yields the thermodynamic constraints, and then the dissipation inequality. To study the dissipation effects, we assume that the process of phase transitions is independent of other processes, and take the following definitions : general thermodynamic forces : YD = ( grade1, grad 4, grad A3, x>- X I , x j- x t l , O2 - 81, d3 - 1 ; (1.8) general thermodynamic fluxes :

where ml , m2, m3 are the entroy increments of the three constituents respectively, then the dissipation inequality may be abbreviated as - Y,. J S O . (1.10) ~delen'~ gave ] the general solution of (1.10) for J in the form

J =-

vY,$,

(1.11)

where the dissipation potential $ ( Y " ) > O is a convex function. The dissipation potential $ is subject to the axiom of objectivity, which implies that $ only depends on the invariants of the vectors and tensors in YD If we only consider the linear theory, then only the second order invariants of vectors and tensors in Y , need to be considered.

.

The thermodynamic constraints, independence of the phase transition process and the concrete expressions of ( 1 . I 1 ) form a set of perfect constitutive equations for freezing soil. Since we only discuss the free freezing problem of saturated soil in the paper, that is a case in which n3 = 0,P = 0 and gradn2 = 0 , so only the concrete from of the constitutive equation will be given below. The energy expressions of the solid and liquid constituents are

where ea and 7, ( (Y = 1 ,2) are the internal energy density and heat supply of the solid and liquid constituents, and La is defined by La = grad&.

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The simplified expressions of free energy for the solid and liquid constituents read

a2w ,

where A , = 202 7

A2=-

a2T, a e2

Dissipation function is

where a,, , bz2, c2, are all constants dependent on the configuration temperature. The barycentric velocity of the liquid constituent is

where b2 is the body force density. The thermodynamic constraint is

The momentum supply is

k2 = -

1 9 b ~ ~ ( ~x;) ;-

-

Bczlgrad9 =

- il.

Applying Fourier ' s law, we have q , = KlgradO, q2 = K2grad9,

(1.19) (1.20)

where K1 and K2 are the heat conduction coefficients of the solid and liquid constituents respectively. Letting K, + K2 = K , it can be proved that1) K 6 0. (1.21) We see that the heat conduction coefficient are non-positive, that is, heat can only transfer from regions of higher temperature to those of lower temperature in the mixture.

2

Criterion for formation of layered ice and influence of void ratio on frost heaving

In this paper, we only consider the freezing pattern shown in fig. 1, that is, one dimensional freezing of ground without outside load. Usually, there appear layers of ice lens in the freezing process, which cause frost heaving by repelling the soil around it. To analyze this process, a criterion for formation of layered ice is proposed based on energy equilibrium. Taking a layer of soil of unit thickness at the freezing front, as shown in fig. 2 , we propose that, when QF6 QU, layered ice forms ; otherwise, the bezing front will advance rapidly and layered ice will not appear. Here QF is the heat transferred from frozen region into the freezing front, and Q U stands for the latent heat given out by water migrated from unfrozen region to the freezing front and that conducted by heat flux. They are respectively

.

1) Guo , L , Constitutive theory of porous media with phase transition and heat-moisture mipation in freezing soil, MS Dissertatwn , Lanzhou University, 1997, 3-28.

SUPP.

MODELING ON COUPLED HEAT AND MOISTURE TRANSFER

13

Ground surface

0

1-

Freezing front

......... . , . . . . Unfrozen .. .. .. . .. .. .. ..region .. .. . . . ...

II

Under-ground water source Fig. 1 .

Fig. 2 .

Sketch of the freezing pattern.

1 = I p2vAn2C 1 + I KAgradO' 1 ,

Heat fluxes at freezing front.

Q, = KoAgrad8- ( ,

(2.1)

Qu

(2.2)

where 8 - and 8 ' are respectively the temperature of frozen and unfrozen regions at the freezing front, KOis the average heat conducting coefficient of soil in the frozen region, A stands for the area of conference cross-section, and C is the latent heat of water.

ae a

Considering that - > 0 in the freezing process, and KO< 0, which is evident from inequality

( 1 .21) , we can express QF concretely as

Then, we go on to consider

QU. Applying (1 .17) , we get

where k is the unit vector in the positive direction of the z axis. Moisture migrates upward in the freezing process, so there should be

ae az

Substituting (2.4) into (2.2) , and considering (2.5) and - > 0 , we get the concrete expression of QU:

Substituting (2.3) and (2.4) into QF< Q U mation of layered ice, it is necessary that

the criterion in the particular case: for for-

For a given temperalure field, the value of the left-hand side of inequality ( 2 . 7 ) decreases as the value of void ratio n 2 increases. This will make it easy for the inequality to hold; in other words,

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when other conditions are the same, it is easier for layered ice to form in soil of higher void ratio, and the corresponding amount of frost-heaving also increases. On the other hand, the cohesive resistance on moisture migration decreases as the void ratio increase and this lowers the dissipation potent 9 . From the expression of $ , eq. (1 . 1 6 ) , we see that the dependence of # on n2 is mainly caused by that of c z l on n , , for a , , describes temperature effects, and the second-order term bz of migration dissipation has little influence since the migration speed x; - x f is small. Hence, the decrease of c2, shows that $ decreases as n2 increases, and this will cause the value of the right-hand side of inequality (2.7) to decrease. This make it more difficult to satisfy the inequality, that is, it is difficult to form layered ice in the freezing process when the void ratio is too great. From the foregoing analysis, we can see that as n2 increases, the chance of formation of layered ice in freezing soil also increases; when n 2 increases beyond a certain limit, the chance will decrease. In other words, there is a critical void ratio n,,, at which it is easiest to form layered ice, and the corresponding harm of frost heaving is the greatest. Xu et al. reported systematic experimental studies on such phenomena. An experiment, with green brick, ceramics, zeolite, gneiss and slate, and an arrangement of void ratio: slate < gneiss < zeolite < green brick < ceramics, gives the final quantity of frost heaving under the same experimental temperature in the following order: slate < gneiss < zeolite < ceramics < green brick. Another experiment reported in the paper was conducted with the following materials : Linxia kaolin, volcanic ash from Japan, clay from Inner Mongolia, Lanzhou loess and Lanzhou sand, with the order according to void ratio: kaolin < volcanic ash < clay < loess < sand. The corresponding order of quantity of frost heaving is sand < clay < loess < kaolin < volcanic ash. This also supports the conclusion: as the void ratio increases, the quantity of frost heaving of volcanic ash is greater than that of kaolin, and the volcanic ash has the greatest quantity of frost heaving, and after that, frost heaving quantity decreases as the void ratio increases. Perhaps, you have seen that the order between clay and loess is different from that of theoretical analysis. This is not surprising since we have assumed in the theoretical analysis that a bigger void ratio corresponds to a less specific surface area. However, this rule is not suitable for the clay from Inner Mongolia, which is well grained. From these two experiments we can see that the procedure can well predict trial results.

3

Heat conduction coupled with moisture migration in freezing soil

The thermodynamic properties of freezing soil are very complex because of its heterogeneity and the possibility of phase transitions under certain conditions. However, since temperature is the predominant factor in the freezing process of soil, it is of great significance to study the temperature field of freezing soil. Concerning mainly with the influence of the temperature field, we need not consider the effects

.

of the stress power ( T" L a ) and kinetic energy

-xl xl 1 (;

when dealing with the energy equations

in the present stage. Considering that there is no phase transitions in the interior of both the frozen and unfrozen regions, and supposing that there is no heat supply, we can combine the energy equations (1 .12) and ( 1 .13) to get the energy equation for the whole soil mass:

SUPP.

MODELING ON COUPLED HEAT AND MOISTURE TRANSFER

In the frozen region, we have ( 3 . l ) yields

15

=0 ,

substituting ( 1 .14) and ( 1 .18) into

- plAlee;+ divq, = 0.

(3.2)

p,

= 0 , p2

0 and 4,

ae

In the case of infinitesimal deformation, we can take 8; r - and q l = KogradB, then at

If KOis a constant, and B

Bo , then

where Clo= K0/(p,A,8,). We see that eq . ( 3 . 4 ) is just the heat conduction equation for frozen region of the classical heat conduction problem with phase transitions. For the unfrozen region, substituting eqs . ( 1 . 14) , ( 1 . 15 ) , ( 1 18) and ( 1- 1 9 ) into eq . ( 3 . 1 ) yields

.

- pIA188; - pzA,8e;+ Bb,,(x;-

x;)

( x i - x;)

+

Bc,,gradB

In the case of infinitesimal deformation, we can set x', = 0 and omit the second-order terms of speed, then

Taking 8 = Bo , we get

where a = p2A2/(plAl + p2A2), b = K/[(p1Al + p z ~ 2 ) B o ] . If = 0 , then eq . ( 3 . 7 ) is the same, in the form as the classical heat conduction equation for the unfrozen region. However, the term xi cannot be omitted since moisture migration strongly influ] the moisture migration in unfrozen region by the ences frost heaving. ~ u n a r d i n i ' ~approximated change of soil volume to deduce the heat conducting equation. However, the approximation seems to have no physical basis. In fact, since the moisture migration in the unfrozen region is mainly affected by temperature gradient, we can substitute ( 1 .17) into ( 3 . 7 ) and omit the influence of gravity to get

whereZ = - ac2,/bZl = P2A2~21/[b22(P1A1 + p2A2)]. We see that the nonlinearity of the heat conduction equation is induced by moisture migration caused by temperature gradient, and eq. ( 3 . 8 ) is a classical nonlinear equation of Burgers' type. At the freezing front between the frozen and unfrozen regions, there should be a continuous condition of heat conduction :

where h = h ( t ) is the location of the freezing front.

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Expressions ( 3 . 4 ) , ( 3 . 8 ) and ( 3 . 9 ) constitute the proper equations for solving the temperature distribution in freezing soil :

Solving eqs. (3.10) with corresponding initial and boundary conditions, we can determine the temperature field of any given freezing soil. However, it is considerably difficult to get the analytical solutions because eqs . (3.10) are a set of nonlinear partial differential equations with varying boundaries, so that there is no discussion on such equations in the literature up to now. If numerical solutions are wanted, the parameters in these equations should be determined previously. Therefore, we will not solve these equations here. The field eqs. ( 3 . 1 0 ) for the temperature distribution in freezing soil differ from the classical Stefan's equations in that they are nonlinear equations of Burgers' type. The classical Stefan equations are important in both frozen soil mechanics and physico-mathematical problems. However, the nonlinear effects observed in freezing of soils cannot be explained by these equations, for the two basic equations in Stefan problem are both linear. The nonlinear equations of Burgers' type for unfrozen region given here are suitable for explaining the observed nonlinear phenomena[71. Hence, we say that the work gives a theoretical start point for analysing the nonlinear effects of heat and moisture migration in freezing soil. At the same time, the nonlinear equations of Burgers' type with varying boundaries are brand new in physico-mathematics, and provide a new sort of problems for researchers interested in physico-mathematical equations. Acknowledgement

'llanks are due to Profs. Zhu Yuanlin, Xu Xiaozu and Wang Jiacheng for discussions and help.

References 1 2 3 4 5 6 7

Fremond, M . , Mikkola, M . , 'Thermomechanical modelling of freezing soil, in Proceedings of the Sizth International Symposium on Ground Freezing (eds. Y u , X . , Wang, C . ) , Rotterdam: A . A . Balklema, 1991, 17-24. Xu, X. Z. , Wang, J . C. , Zhang, L. X . et al. , M e c h i t r n of Frost and Salt Heaving in Soils (in Chinese), Beijing: Science Press, 1995, 1-126. de Boer, R . , Thermodynamics of phase transitions in porous media, Appl. Mech Rev. , 1995, 48(10) : 613. Eringen, A . C . , Mechanics of Continua, New York : Robert E . Krieger Publishing Company, 1980, 148-189. Edelen , D .G .B . , A nonlinear onsager theory of irreversibility, Int J . Engng . Sci . , 1972, 10(6) : 481 . Lunardini , V .J . , Heat Transfer with Freezing and Thawing, Amsterdam: Elsevier, 1991, 322-363. Wang, M . L . , Nonlinear Developing Equations and Soliton (in Chinese) , Lanzhou: Lanzhou University Press, 1990, 58-87.

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