Heat and moisture transfer in freezing and thawing soils

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Heat and moisture transfer in freezing and thawing soils

Martynov, G. A.; National Research Council of Canada. Division of Building

Research

In 1950 t h e D i v i s i o n of B u i l d i n g Research i n i t i a t e d a programme of permafrost I n v e s t i g a t i o n s a s p a r t of Its r e s e a r c h on b u i l d i n g p r o b l e n ~ s i n n o r t h e r n Canada. Fundanlental and e n g i n e e r i n g s t u d i e s a r e b e i n g conducted on a c o n t i n u i n g b a s i s t o g a i n a b c t t e r under- starldin?; of and p r o v i d e s o l u t i o n s t o permafrost problerns c o n f r o n t i n g c o n s t r u c t i o n a c t i v i t i e s i n n o r t h e r n Canada.

Because of t h e l o n g h i s t o r y of i n v e s t i g a t i o n s ' a n d c o n s t r u c t i o n i n t h e permafrost r e g i o n of t h e U.S.S.R., p a r t i c u l a r i n t e r e s t i s

b e i n g given t o t h e l a r g e body of Russian l i t e r a t u r e now a v a i l a b l e ' i n t h i s f i e l d . The agency i n t h e U.S.S.R. which i s conducting

r e s e a r c h on permafrost and r e l a t e d phenomenon i s tile V.A. Obruchev I n s t i t u t e of Permafrost S t u d i e s ( I n s t i t u t N e r z l o t o v e d e n i y a ) .

In

1959, t h i s I n s t i t u t e published a two volume work e n t i t l e d ' P r i n c i - p l e s o f ~ e o c r y o l o g y "

-

Volume I d e a l i n g with t h e s c i e n t i f i c a s p e c t s of permafrost and Volume I1 with t h e e n g i n e e r i n g p r o b l e ~ n s a s s o c i a t e d ~ 1 1 t h p e r m a f r o s t . T h i s p u b l i c a t i o n , exceeding 800 pages i n l e n g t h , r e p r e s e n t s t h e most r e c e n t corripilation of Russian e x p e r i e n c e with permafrost.

T h i s work was considered t o be of such g r e a t importance i n t h e p e r m a f r o s t f i e l d t h a t i n v e s t i g a t i o n s were irnrnediately undertaken

as

t o t h e p o s s i b i l i t y of having i t t r a n s l a t e d . Because of i t s g r e a t l e n g t h , i t became e v i d e n t t h a t a coniplete t r a n s l a t i o n would n o t be p o s s i b l e . F o r t u n a t e l y t h e Pernlafrost Subconunlttee of t h e A s s o c i a t e Cornlittee on S o i l and Snow r6lechanics of t h e N a t i o n a l Research Council was a b l e t o a r r a n g e f o r t h e t r a n s l a t i o n of s e l e c t e d c h a p t e r s from Volunle I by v a r i o u s government departments r e p r e s e n t e d on t h e Sub- committee. T h i s t r a n s l a t i o n i s of Chapter V I e n t i t l e d "Heat and

Moisture T r a n s f e r i n F r e e z i n g and Thawing S o i l s " by G.A. Martynov and d e a l s w i t h t h e conductive and c o n v e c t i v e mechanisms of h e a t propaga-

t i o n i n thawed and f r o z e n s o i l s . T h i s c h a p t e r was t r a n s l a t e d f i r s t because t h e mechanisms discussed a r e b a s i c t o t h e formation o f

permafrost and t h e p r o p e r t i e s of f r o z e n s o i l s . The t r a n s l a t i o n of Chapters IV, V I I , I X , X and X I a r e now i n d r a f t form and w i l l be i s s u e d a s s e p a r a t e t r a n s l a t i o n s i n t h e n e a r f u t u r e .

The D i v i s i o n I s g r a t e f u l t o h l r . E.R. Hope of t h e Defence Research Board tiho s p e n t c o n s i d e r a b l e tiirle i n t r a n s l a t i n g t h i s

c h a p t e r i n response t o t h e r e q u e s t of t h e Permafrost Subcommittee.

fo his

t r a n s l a t i o n i s a l s o l i s t e d a s T372R i n t h e Defence Research Board t r a n s l a t i o n s e r i e s . )

Ottawa

March 1963

R.F. Legget D i r e c t o r

Title :

NATIONAL RESEARCH COUNCIL OF CANADA

Technical Translation 1065

Heat and moisture transfer in freezing and.thawing soils

(~eplo- i vlagoperedacha

v

promerzayushchlkh 1 protalvayushchikh grunt akh )

Author : O.A. Martynov

Reference: Principles of geocryology (permafrost studies), Part

I,

General geocryology, Chapter

VI.

Academy of Sciences of the

USSR.

Moscow 1959. p.153-192

(0snov-y geokriologii (merzlotovedeniya), Chast pervaya,

Obshchaya geokriologlya, Glava

VI.

Akademlya Nauk SSSR. Moskva

1959. s .153-192

KEAT AND l4OISTURE TRANSFER I N PFUEZIXG AND THAWING SOILS

In troduc t i o n

The e s t a b l i s h m e n t of t h e thermal regime I n f r o z e n ground i s accompanied by p r o c e s s e s of two t y p e s . The f i r s t type i n c l u d e s p r o c e s s e s of d i r e c t energy and moisture exchange between t h e ground and an e x t e r n a l medium ( t h e atmos- phere, cosmic space, e t c . ) , p r o c e s s e s l o c a l i z e d I n t h e c o n t a c t l a y e r . They i n c l u d e processes of exchange of energy i n thermal form ( h e a t t r a n s p o r t e d by advection o r convection i n t h e atmosphere i s passed through t h e c o n t a c t l a y e r

t o

t h e ground); p r o c e s s e s i n which v a r i o u s forms of energy a r e converted t o thermal energy (conversion of r a d i a n t , chemical, biochemical and o t h e r forms of energy i n t o h e a t ) ; phase conversions of water accompanied by t h e r e l e a s e o r a b s o r p t i o n of h e a t (condensation and evaporation, m e l t i n g o f snow, formation of h o a r f r o s t , and s o f o r t h ) .

Processes of t h e f i r s t type a r e d e a l t with i n d e t a i l i n t h e f o u r t h c h a p t e r of t h e second volume of " P r i n c i p l e s o f Geocryology". Here we s h a l l c o n s i d e r o n l y p r o c e s s e s of t h e second type, those of t h e t r a n s f e r of h e a t coming i n t o a given volume of s o i l o r rock through t h e c o n t a c t l a y e r o r from a d j a c e n t p a r t s of t h e ground.

The p r o c e s s o f h e a t t r a n s m i s s i o n i n d i f f e r e n t k i n d s of ground t a k e s p l a c e mainly by:

(1) A conductive mechanism, i n which t h e t r a n s p o r t of thermal energy i s

n o t accompanied by t r a n s p o r t of substance, and

( 2 ) A convective mechanism, i n which t h e t r a n s p o r t of h e a t i s e f f e c t e d by t h e movement of water and a i r i n t h e pores of t h e s o i l * . Each of t h e s e mechanisms has i t s s p e c i f i c d . i s t i n g u i s h 1 n g f e a t u r e s , a f a c t t h a t makes i t n e c e s s a r y t o c o n s i d e r them s e p a r a t e l y .

Furthermore, f r e e z i n g and thawing s o i l s a r e always s t r u c t u r a l l y inhomo- geneous. Even i n c o n d l t i o n s where l i t h o l o g i c a l composition, moisture c o n t e n t , d e n s i t y and s o f o r t h a r e q u i t e i d e n t i c a l over t h e volume I n q u e s t i o n , they cannot be regarded a s homogeneous, s i n c e i n some p a r t s of t h e ground t h e water has a l r e a d y frozen, while i n o t h e r p a r t s i t has n o t y e t f r o z e n . Depending on t h e phase-composition of t h e water, t h r e e zones may be d i s t i n g u i s h e d i n

f r e e z l n g and thawing s o i l s : thawed s o i l , p h a s e - t r a n s i t i o n a l s o i l and f r o z e n

*

Other mechanisms a r e p o s s i b l e ( f o r i n s t a n c e , r a d i a t i v e t r a n s p o r t of h e a t i n p o r e s ) b u t , a s experience shows, .they a r e of s l i g h t Importance under

s o i l * . A s t h e names theniselves i n d i c a t e , i n t h e m e l t zone t h e thermoactive moisture i s found o n l y i n t h e l i q u i d phase; i n t h e p h a s e - t r a n s i t i o n zone water and i c e may be found i n thcrmodynan~ic e q u i l i b r i w n with each o t h e r , and f i n a l l y , i n t h e frozen zone p r a c t i c a l l y t h e whole of t h e thermoactive moisture is i n t h e i c e phase.

Depending on what zone i t t a k e s p l a c e i n , t h e nlechanisnl o f t h e transmls- s i o n process v a r i e s a good d e a l , making i t necessary t o c o n s i d e r each zone s e p a r a t e l y .

When studying t h e t r a n s f e r of h e a t and moisture i n such complex systems as s o i l s we may, g e n e r a l l y speaking, proceed by two r o u t e s : namely, by t h e

I1 _{microscopic" r o u t e , which c o n s i s t s i n a n a l y s i n g t h e mechanism of t h e phenom-}

enon, and by t h e "macroscopic" r o u t e , based on phenomenological d e s c r i p t i o n , wherein we d i s r e g a r d t h e d i s c r e t e s t r u c t u r e of t h e s o i l and c o a s i d e r i t a s a s o l i d body, c h a r a c t e r i z a b l e by averaged, "macroscopic" parameters.

The "macroscopic" approach permits t h e s i m p l e s t d e s c r i p t i o n of t h e

thermal regime of f r e e z i n g and thawing s o i l s , t h a t is, i t p e r m i t s u s t o answer t h e q u e s t i o n most i n t e r e s t i n g f o r geocryolo@;y. Consequently t h e p r e s e n t

c h a p t e r i s mainly devoted t o c o n s i d e r a t i o n of t h e h e a t and moisture conductiv-

i t y p r o c e s s e s from j u s t t h e s e p o i n t s of view. The "macroscopic" approach, however, i s n o t u n i v e r s a l . For i n s t a n c e , I n s t u d y i n g t h e g e n e s i s o f t h e

f r o z e n s t r a t a i t y i e l d s no d e f i n i t e answer, s i n c e t h e c o n d i t i o n s of h e a t exchange between t h e l i t h o s p h e r e and atniosphere i n t h e p a s t a r e , as a r u l e , n o t known. fn t h i s and i n a number of o t h e r c a s e s i t may t u r n o u t t h a t t h e

"microscopic" approach is t h e u s e f u l one, c o n s i s t i n g as it does i n t h e s t u d y of elementary h e a t and moisture t r a n s m i s s i o n e v e n t s . A d e s c r i p t i o n of "micro- s c o p i c " p r o c e s s e s i s given in t h e f i f t h c h a p t e r of t h i s volume. But h e r e we s h a l l b r i n g them up only a s necessary i n t h e course of t h e e x p o s i t i o n .

The Conductive Mechanism of Heat Propagation

k those c a s e s where t h e conductive mechanism of h e a t t r a n s m i s s i o n I s dominant, t h e r e i s no r e a l d i f f e r e n c e between t h e thawed and f r o z e n zones from t h e h e a t t r a n s m i s s i o n p o i n t of view, s i n c e i n e i t h e r zone no water-ice phase conversions a r e p r e s e n t . Consequently both zones a r e h e r e considered t o g e t h e r ; t h e phase t r a n s i t i o n zone i s d e a l t w i t h i n a s e p a r a t e s e c t i o n .

*

N.A. Tsytovich (1954) s u g g e s t s d i s t i n g u i s h i n g f o u r zones: ( 1 ) thawed s o i l ,

( 2 ) a zone of s i g n i f i c a n t phase t r a n s i t i o n s , ( 3 ) a zone i n which phase t r a n s i t i o n s a r e i n s i g n i f i c a n t and ( 4 ) a zone t h a t i s f o r a l l p r a c t i c a l purposes f r o z e n . Yet from t h e thermophysical p o i n t of view t h e r e a r e no d i f f e r e n c e s i n prFnciple between t h e zone of s i g n i f i c a n t phase t r a n s i t i o n s and t h e zone of i n s i g n i f i c a n t phase t r a n s i t i o n s . For t h i s reason they a r e h e r e lumped t o g e t h e r i n a s i n g l e p h a s e - t r a n s i t i o n zone.

Processes of Heat Transfer i n t h e Thawed and Frozen S o i l Zones

Heat flow and thermal c o n d u c t i v i t y . With

a

conductive mechanism of heat t r a n s f e r , t h e h e a t flow is p r o p o r t i o n a l t o t h e temperature g r a d i e n t

v3, t h a t i s :

*

a

a

a

Is the Hamilton o p e r a t o r "nabla" and where t h e w h e r e v = I = + j = + k z

c o e f f i c i e n t of p r o p o r t i o n a l i t y h, t h e s i z e of which depends only on t h e

p r o p e r t i e s of t h e body i n question,

is

c a l l e d t h e c o e f f i c i e n t of thermal con- d u c t i v i t y ;

s ,

and s, a r e p o i n t s l y i n g on

;,

t h e normal t o t h e Isothermal

surf

ace.

Equation ( 6 . 1 ) i s c o r r e c t only f o r continuous homogeneous bodies and, consequently, it may be used f o r d e f i n i n g t h e h e a t flow i n a block of s o i l only under t h e condition t h a t t h e dimensions of t h e l a t t e r a r e s u f f i c i e n t l y g r e a t , t h a t is, when i t i s p o s s i b l e t o disregard t h e complex s t r u c t u r e and t e x t u r e of t h e s o i l . Therefore

a l l

parameters t h a t e n t e r in equation (6.1) r e p r e s e n t q u a n t i t i e s averaged over a g r e a t number of small s t r u c t u r a l f e a t u r e s of t h e ground. The volume over which t h e averaging is c a r r i e d out i s c a l l e d

an

elementary volume. It must be l a r g e enough a s compared with t h e s i z e s of s o i l pores and p a r t i c l e s t o r e n d e r t h e averaged q u a n t i t i e s adequately r e l i a b l e and p r e c i s e . A t the.same time i t should be small enough a s compared with t h e s i z e of t h e whole block of s o i l t o j u s t i f y n e g l e c t i n g t h e v a r i a t i o n of h and 3

w i t h i n the volume, and t o permit t h e t r a n s i t i o n from a d i f f e r e n c e equation t o a d i f f e r e n t i a l equation, with A 3 and A s regarded as i n f i n i t e l y small quanti- t i e s from t h e p h y s i c a l p o i n t of view.

Thus I n (6.1) we must understand A s = s ,

-

s2 a s a d i s t a n c e between c e n t r e s of two neighbouring elementary volumes, and A 3 = 3,

-

3, a s a d i f f e r -

ence of mean temperatures of elementary volumes 3,, 3,. In temperature mea- surements t h e experimenter does not a s a r u l e need t o make any adjustments of t h e q u a n t i t y 3, s i n c e i n p r a c t i c e t h e temperature pick-up

-

t h e thermometer bulb, t h e thermocouple junction, e t c .

-

is of r e l a t i v e l y l a r g e s i z e . Being i n

c o n t a c t with many p a r t i c l e s simultaneously, it a u t o m a t i c a l l y e f f e c t 6 an

averaging of t h e temperature. Nevertheless i n some cases ( f o r i n s t a n c e , when measuring temperatures I n c o a r s e l y "macroskeletal" s o i l s by means of a thermo- couple) t h e r e may be no averaging. Here t h e measured q u a n t i t i e s obviously

with changes of up t o

30$

i n t h i s q u a n t i t y according t o t h e d i r e c t i o n of h e a t flow ( s h c h e r b a n l , 1953). The same phenomenon has been observed by G.V.

Porkhaev in f r o z e n s o i l s too.

h a t has been most s t u d i e d i s t h e dependence of h on t h e u n i t weight y

of t h e s o i l (Chudnovskii, 1948, 1954; Kersten, 1949), t h i s b e i n g an i n t e g r a l c h a r a c t e r i s t i c of t h e t e x t u r e . Unfortunately t h e r e i s no simple r e l a t i o n s h i p between t e x t u r e and u n i t weight, and i t i s o n l y i n an approximate sense t h a t t h e e f f e c t of t e x t u r e on thermal c o n d u c t i v i t y may be judged from h = h ( y ) . We s e e only t h a t under n a t u r a l c o n d i t i o n s t h e volumetric weight of t h e s o i l

changes w i t h i n comparatively small l i m i t s and a s a r u l e t h e change of h due t h e r e t o i s a l s o small. The tendency, however, i s q u i t e c l e a r : t h e l a r g e r t h e q u a n t i t y Y is, t h e b e t t e r t h e c o n t a c t between p a r t i c l e s and t h e l a r g e r t h e q u a n t i t y h ( ~ i g . 1 5 ) .

For t h e most p a r t t h e c o e f f i c i e n t of thermal c o n d u c t i v i t y i s almost i n v a r i a n t with change of temperature, although i n some c a s e s a q u i t e s t r o n g i n c r e a s e of h with i n c r e a s i n g 3 i s p o s s i b l e . To understand why t h i s occurs, i t i s necessary f i r s t t o c o n s i d e r i n somewhat more d e t a i l t h e p h y s i c a l meaning of t h e parameter A.

Heat may be propagated i n s o i l s both through t h e mineral s k e l e t o n by way of conduction, and a l s o through t h e pores, where t h e t r a n s f e r t a k e s p l a c e by conduction, convection and r a d i a t i o n . In measuring t h e q u a n t i t y h* what we determine i s always t h e t o t a l h e a t f l u x , which i s made up of t h e elementary f l u x e s enumerated above. Therefore t h e c o e f f i c i e n t of thermal c o n d u c t i v i t y depends e s s e n t i a l l y on some v i r t u a l o r e f f e c t i v e c h a r a c t e r i s t i c of t h e ground, a c h a r a c t e r i s t i c t h a t lumps t o g e t h e r s e v e r a l d i f f e r e n t mechanisms of h e a t t r a n s m i s s i o n . The d e r i v a t i o n of such an e f f e c t i v e c h a r a c t e r i s t i c is, however, p o s s i b l e only when t h e a c t i v i t y of each mechanism i s c o n t r o l l e d by one and t h e same cause ( f o r I n s t a n c e , by t h e temperature g r a d i e n t ) . Consequently pro- c e s s e s such a s t h e conductive t r a n s f e r of h e a t , in which $c

-

v3, and t h e con- v e c t i v e t r a n s f e r of h e a t by m i g r a t i n g moisture, i n which t h e h e a t flow i s pro- p o r t i o n a l t o t h e moisture g r a d i e n t

$,

-

m, cannot be j o i n t l y expressed by any s i n g l e c o e f f i c i e n t of t h e type of A. In f a c t , f o r one and t h e same value of conductive h e a t flow t h e convective flow may have any value, and conse- q u e n t l y t h e t o t a l flow

5

$c

+

5

_cv a l s o may t a k e any value. Furthermore t h e d i r e c t i o n of t h e temperature g r a d i e n t v 3 i s q u i t e a r b i t r a r y with r e s p e c t t o t h e moisture g r a d i e n t

vw.

Consequently t h e , t o t a l h e a t f l u x

5

d e f i n e d a s t h e

*

3

v e c t o r sun of Qc and Qcv depends on t h e angle between

v3

and o w . Since t h e r e

*

On t h i s q u e s t i o n t h e r e is a d e t a i l e d review a r t i c l e by A . F . Chudnovsldi

i s no simple r e l a t i o n s h i p between

$

and 03 o r

o w ,

i t i s inlpossible t o d e r i v e any c o e f f i c i e n t of p r o p o r t i o n a l i t y between them. The same i s t r u e a l s o i n the case of f i l t r a t i o n .

In c o n t r a s t t o migration and f i l t r a t i o n , which represent t h e i n t e r - p o r e movement of moisture, t h e transmission of h e a t by pore processes (namely conduction Qc*, convection QcvX and r a d i a t i o n Qr*) may indeed b e handled by a s i n g l e e f f e c t i v e parameter. According t o M.A. Mikheev (1949) and A.F.

Chudnovskii (1954) we have:

where A 3 = 3,

-

3, i s the temperature d.ifference between opposite walls of a pore, Ax i s the mean pore diameter, and A and a a r e constants. Hence t h e t o t a l flux through a pore is:

where A,, t h e e f f e c t i v e c o e f f i c i e n t of thermal conductivity of a porous substance, i s

Since the thermal conductivity c o e f f i c i e n t h of the s o i l i s some f u n c t i o n of t h e thermal conductivity c o e f f i c i e n t AM of t h e mineral skeleton and the e f f e c t i v e thermal conductivity c o e f f i c i e n t Ax f o r a porous substance, then h

A 3

t o o must depend on 3 and

E.

Experiment shows, however, t h a t i n thawed and

A 3

frozen s o i l s

ax

p r a c t i c a l l y does n o t a f f e c t the q u a n t i t y A, though t h i s was t o be expected s i n c e the second term i n ( 6 . 2 ) i s small.

The temperature dependence of h manifests i t s e l f somewhat more c l e a r l y , but i t too i s r a t h e r f e e b l e (with a temperature i n c r e a s e of about 50°C, t h e c o e f f i c i e n t of thermal conductivity i n c r e a s e s by 10

-

2 6 ; Chudnovskii, 1954). The explanation of t h i s i s t h a t under ordinary conditions the f r a c t i o n of h e a t transported by convection and r a d i a t i o n within t h e pores, t h e amount of which u s u a l l y v a r i e s s t r o n g l y with temperature, i s i n s i g n i f i c a n t l y small ( ~ i g . 1 6 ) .

O n t h e o t h e r hand the thermal conductivity c o e f f i c i e n t of t h e mineral skeleton and i n t e r s t i t i a l matter (water, a i r ) i s almost independent of temperature ( f o r instance, with a temperature i n c r e a s e from 0 t o 20°C t h e thermal conductivity of water i n c r e a s e s by j u s t 2 . s ) .

The f r a c t i o n of h e a t transported by convection and r a d i a t i o n i n t h e pores Increases r a p i d l y with increase of the s o i l pore diameter above a few

m i l l i m e t r e s ( ~ i r p i c h e v e t a l . , 1940; ~lullokandov, 1947), with i n c r e a s e of t h e mean temperature of t h e ground above 30

-

5oeC ( ~ i g . 1 7 ) , and w i t h i n c r e a s e

of t h e temperature g r a d i e n t above 1 de@;/cm approximately. Thus f o r i n s t a n c e i n macroskeletal s o i l s i t i s no longer p o s s i b l e t o n e g l e c t t h e dependence of t h e thermal c o n d u c t i v i t y c o e f f i c i e n t on temperature.

Heat accumulation and h e a t c a p a c i t y . A most important s o i l c h a r a c t e r i s - t i c , d e f i n i n g i t s a b i l i t y t o accumulate h e a t , i s t h e c o e f f i c i e n t of h e a t

c a p a c i t y . A s i s w e l l known, t h e s p e c i f i c h e a t c a p a c i t y of any substance ( s e e f o r i n s t a n c e Epshtein, 1948) i s defined by t h e formula:'

where m i s t h e t o t a l mass of t h e substance, q i s t h e q u a n t i t y of h e a t communi- c a t e d t h e r e t o , and 3 i s t h e temperature. Since thawed s o i l c o n s i s t s of t h e mineral s k e l e t o n , of a i r , and of bound and f r e e water, then m = InM

+

mA

+

msw

+

'%we

Let q with a s u b s c r i p t denote t h e q u a n t i t y of h e a t communicated t o u n i t mass of each component. Then

Taking i n t o account t h a t m does n o t depend on temperature we get, a f t e r s u b s t i t u t i n g ( 6 . 4 ) in ( 6 . 3 ) :

d q ~

where cN =

T ,

cA = dqA and s o f o r t h a r e t h e s p e c i f i c h e a t c a p a c i t i e s of each s o i l component. For f u l l y f r o z e n s o i l we must add t o t h i s e x p r e s s i o n y e t a n o t h e r component r e p r e s e n t i n g i c e .

Since s p e c i f i c h e a t c a p a c i t y i s t h e h e a t c a p a c i t y p e r u n i t mass, i t does n o t ( o t h e r c o n d i t i o n s b e i n g equal), depend on t h e u n i t weight and t h e t e x t u r e of t h e s o i l . This p e r n i t s a l l conclusions reached by i n v e s t i g a t i n g t h e h e a t c a p a c i t y of l a b o r a t o r y samples t o be extended almost u n c o n d i t i o n a l l y t o

s o i l s under n a t u r a l c o n d i t i o n s . A s i s seen from formula ( 6 . 5 ' ) t h e s p e c i f i c h e a t c a p a c i t y of a s o i l i s determined by t h e h e a t c a p a c i t i e s of t h e s o i l

cM, cA, CgW, Cml and by t h e i r masses.

The h e a t c a p a c i t y of t h e mineral s k e l e t o n , cM, f o r a l l s o i l s i s i n t h e mean 20.20 kcal/kg.deg. With i n c r e a s i n g temperature i t s value r i s e s somewhat. For i n s t a n c e i n q u a r t z sand a t -9.5OC we have cM = 0.16 kcal/k.g.deg,

and a t +65OC we have cM = 0.19 kcal/kg.deg (Kersten, 1949). The q u a n t i t y c M

v a r i a t i o n i s within small l i m i t s (from 0.17 kcal/kg-deg f o r sands t o 0.22 kcal/kg-deg f o r clays a t room temperature; Kersten, 1949). According t o the q u i t e p r e c i s e d a t a of S.M. Skuratov (1951), t h e heat capacity of the bound water and t h a t of the f r e e water a r e f o r p r a c t i c a l purposes equal, t h a t i s ,

cBW

=

cFW cW kcal/kg-deg. I n s o i l s , t h e r e f o r e , when c a l c u l a t i n g t h e heat capacity i t i s permissible t o make no d i s t i n c t i o n between bound and f r e e water. Although the s p e c i f i c heat capacity of a i r i s q u i t e high ( c A = 0.24 kcal/kg- deg), i t s absolute value i n s o i l s i s s o small t h a t i n equation ( 6 . 5 ' ) t h e term

m ~ may always be neglected. C ~ I n view of a l l the above, we may r e w r i t e

equation (6.51) i n the following form:

where

%

=

msW

+

mFW. For f u l l y frozen s o i l i n which t h e r e can e x i s t only i c e

and firmly bound water, we g e t i n place of ( 6 . 5 ) t h e expression:

where t h e q u a n t i t y of i c e in t h e s o i l i s mI = n-jq

-

%W. The heat capacity

C~ of i c e i s 0.5 kcal/kg-deg ( ~ o r c e ~ , 1940).

I f we put wtot f o r

mw

m~

% + m w

and f o r

-

m~ (where wtot means t o t a l

moisture and imax means maximum i c e c o n t e n t ) , we o b t a i n f o r thawed s o i l :

and f o r frozen s o i l

It must be emphasized t h a t the s p e c i f i c heat capacity a s defined by equations (6.5) and (6.6) i s an averaged c h a r a c t e r i s t i c of t h e medium.

The equation of thermal conductivity and t h e l i m i t s of i t s a p p l i c a b i l i t y . S e t t i n g up t h e heat balance equation f o r an elementary volume, with ( 6 . 1 ) and

( h e r e t i s tirne, yo = mP4 i s t h e u n i t weight of t h e s o i l s k e l e t o n , V t h e s o i l volun~e, and t h e r e s t of t h e n o t a t i o n as above)*. I f t h e ground i s s o uniform t h a t we may n e g l e c t any v a r i a t i o n of h over the volume, then, t a k i n g i n t o account t h a t f o r p r a c t i c a l l y a l l s o i l s except macroskeletal s o i l s t h e r e i s no dependence of h on 3 , we may r e w r i t e equation ( 6 . 7 ) i n t h e form:

a

a

where

v2

=

-

=

-

=

-

a 2

i s t h e Laplace o p e r a t o r and a =

-

h =

-

i s t h e

ax2 av2 az2 Y,C YC

c o e f f i c i e n t of d i f f u s i v i t y . A t t h e p r e s e n t time it i s by means of t h i s equa- t i o n t h a t a l l c a l c u l a t i o n s of t h e temperature f i e l d i n t h e thawed and f r o z e n zones of f r e e z i n g and thawing s o i l s a r e c a r r i e d o u t , given i n t h e c a s e of

a

conductive mechanism of h e a t t r a n s f e r .

Equation ( 6 . 7 ) i s a d i f f e r e n t i a l equation, and t h e r e f o r e c o r r e c t o n l y f o r elementary volumes of s o i l . Therefore t h e p o s s i b i l i t y of d e l i m i t i n g an e l e - mentary volume i n t h e s o i l block, w i t h dimensions much g r e a t e r t h a n t h o s e of t h e i n d i v i d u a l s o i l p a r t i c l e s and a t t h e same time much small t h a n t h e dimen- s i o n s of t h e whole block, i s a necessary c o n d i t i o n f o r i t s a p p l i c a b i l i t y .

Equation ( 6 . 1 ) , on which ( 6 . 7 ) i s based, d e s c r i b e s o n l y t h o s e p r o c e s s e s of h e a t t r a n s f e r t h a t a r e due t o t h e e x i s t e n c e of

a

temperature g r a d i e n t . Consequently i t i s s u f f i c i e n t l y p r e c i s e o n l y i n c a s e s where i t is p o s s i b l e t o n e g l e c t pore convection due t o migration o r f i l t r a t i o n of water.

Equations ( 6 . 5 ) and ( 6 . 6 ) , which a l s o form t h e b a s i s f o r ( 6 . 7 ) , do n o t t a k e i n t o account t h e h e a t involved i n i n t r a - p o r e evaporation and condensation. This, however, does n o t i n p r a c t i c e a f f e c t t h e p r e c i s i o n of ( 6 . 7 ) , because i n thawed and f r o z e n s o i l s t h e h e a t expended in phase conversions between water and vapour i s only

a

few r n i l l i o n t h s of t h e t o t a l q u a n t i t y of h e a t going t o warm t h e s o i l .

Furthermore, equation ( 6 . 7 ) does n o t t a k e i n t o account t h e e v o l u t i o n of h e a t r e s u l t i n g from chemical r e a c t i o n s i n t h e s o i l , s i n c e t h e i n t e n s i t y of t h e s e i s , as a r u l e , i n s i g n i f i c a n t l y small.

Heat T r a n s f e r Processes i n t h e Phase T r a n s i t i o n Zone

Conditions of phase e q u i l i b r i u m of water i n f r o z e n s o i l s . The p r o c e s s e s of water f r e e z i n g and i c e thawing i n t h e s o i l e x e r t a v e r y s t r o n g i n f l u e n c e on

t h e s o i l ' s thermal regime, t h e main r e a s o n b e i n g t h a t a g r e a t d e a l of h e a t Is expended i n t h e s e p r o c e s s e s . Moreover, t h e conversion of w a t e r I n t o i c e , con- s t i t u t i n g a change i n t h e conlposition of t h e s o i l , i s accompanied by t h e f o r - mation of a f r o s t t e x t u r e and consequently by a r e c o n s t i t u t i o n of t h e e n t i r e system. The r e c o n s t i t u t i o n b r i n g s about change i n t h e thermophysical charac- t e r i s t i c s of t h e c o n s t i t u e n t s , which I n c r e a s e s s t i l l more t h e e f f e c t of t h e phase conversions on t h e thermal regime of t h e s o i l . T h e r e f o r e i t i s neces- s a r y , b e f o r e proceeding d i r e c t l y t o t h e p r o c e s s e s of h e a t t r a n s f e r i n t h e phase t r a n s i t i o n zone, t o r e c a l l c e r t a i n d a t a on t h e phase composition of f r o z e n s o i l s ( t h i s q u e s t i o n i s d e a l t w i t h i n more d e t a i l in Capter

v ) .

Numerous r e s e a r c h e s have e s t a b l i s h e d t h a t a t b e l o w - f r e e z i n g t e m p e r a t u r e s i n n e a r l y a l l s o i l s , w i t h t h e e x c e p t i o n of g r a v e l s and c o a r s e sands, some p a r t of t h e w a t e r c o n t e n t may s t i l l be i n an unfrozen c o n d i t i o n (Jung, 1932;

Fedosov, 1942,; Tsytovich, 1945; G o l ' d s h t e i n , 1948; Nersesova, 1950, 1951; G r i g o r t e v a , 1957; a n d o t h e r s ) . This i s e x p l a i n e d by I s u r f a c e 1 f o r c e s , t h a t

is, by t h e same f o r c e s t h a t cause m i g r a t i o n of t h e w a t e r .

It i s obvious t h a t t h e amounts of i c e wI and of unfrozen water wU

t o g e t h e r d e f i n e t h e m o i s t u r e c o n t e n t of t h e s o i l : w = w I

+

w

_u

(w

_{u = q}

mu,

where

mu =

%

-

mI i s t h e mass of t h e unfrozen w a t e r ) . It i s s u f f i c i e n t to--know what s o r t of e f f e c t i s e x e r t e d by each of t h e s e f a c t o r s on t h e amount o f any one phase, i n o r d e r t o have a f u l l r e p r e s e n t a t i o n of t h e phase c o n s t i t u t i o n of t h e w a t e r i n t h e f r o z e n s o i l ( t h e q u a n t i t y w m y be e a s i l y determined by

o r d i n a r y methods). It i s most convenient t o o p e r a t e w i t h t h e q u a n t i t y of un-

f r o z e n water, wU. Let us enumerate t h e b a s i c f a c t o r s a f f e c t i n g i t .

1. The q u a n t i t y of unfrozen w a t e r i s p r i m a r i l y determined by t h e tempera- t u r e o f t h e s o i l ( ~ u n g , 1932; Tsytovich, 1945; Nersesova, 1950, 1351;

G r i g o r l e v a , 1957). F i g u r e 18 shows curves of = wU(3) f o r a few of t h e most t y p i c a l l i t h o l o g i c a l s p e c i e s . Fron t h e s e d a t a we s e e t h a t i n a r e g i o n where t h e temperatures a r e s u f f i c i e n t l y low t h e amount of unfrozen w a t e r remains p r a c t i c a l l y c o n s t a n t and e q u a l t o t h e amount of f i r m l y bound water, wmIq.

In

d i f f e r e n t s o i l s t h e q u a n t i t y wFBbJ d i f f e r s : i n sands i t i s z e r o , w h i l e i n c l a y s i t may be 1 0

-

20% and h i g h e r . With i n c r e a s i n g temperature t h e unfrozen w a t e r c o n t e n t wU i n c r e a s e s , w i t h t h e s t e e p e s t change of wU t a k i n g p l a c e a t a temperature n e a r t o O°C.

2. The q u a n t i t y of unfrozen w a t e r depends on t h e d i s p e r s i o n of t h e s o i l (Nersesova, 1950, 1951); w i t h i d e n t i a l m o i s t u r e c o n t e n t s , t h e s o i l d i s p e r s i o n i s t h e g r e a t e r t h e more unfrozen water i t c o n t a i n s ( F i g . 18).

3 .

The q u a n t i t y o f unfrozen w a t e r depends on t h e m i n e r a l o g i c a l composi- t i o n of t h e s o i l and p a r t i c u l a r l y on i t s exchange complex

rigorle leva,

1957; Nersesova, 1957).

4. The q u a n t i t y of unfrozen wator depends on t h e c o n c e n t r a t i o n of water- s o l u b l e substances i n t h e s o i l , w i t h wU b e i n g g r e a t e r t h e h i g h e r t h e concen- t r a t i o n of t h e s o i l s o l u t i o n .

5. The q u a n t i t y of unfrozen w a t e r i s p r a c t i c a l l y independent o f t h e t o t a l s o i l moisture IP = wU

+

wI, provided o n l y t h a t wI 2 0 ( ~ e d o s o v , 1942 ;

G o l l d s h t e i n , 1948). On t h e o t h e r hand, t h e q u a n t i t y of i c e WI v e r y s t r o n g l y depends on w .

6 . The q u a n t i t y of unfrozen w a t e r depends o n l y t o an i n s i g n i f i c a n t degree on t h e t e x t u r e of t h e s o i l , a f a c t which, i n p a r t i c u l a r , has served a s a b a s i s f o r t h e s o - c a l l e d " r a t i n g method" o f d e t e r m i n i n g t h e i c e c o n t e n t ( ~ e r s e s o v a , 1954 [ s i c ] ) .

7 . The q u a n t i t y of unfrozen w a t e r i s v e r y l i t t l e dependent on e x t e r n a l p r e s s u r e a p p l i e d t o t h e s o i l . This was e x p e r i m e n t a l l y e s t a b l i s h e d by

I.V.

Boiko ( 1 9 5 6 ) ~ who found t h a t when t h e p r e s s u r e was i n c r e a s e d by one atmosphere t h e c r y s t a l l i z a t i o n t e m p e r a t u r e o f t h e i c e i n t h e s o i l was reduced by A 3 =

0.0055°C. (We rernerrlber t h a t f o r p u r e i c e A 3 = 0.0075 dedatrr,. )

8.

The q u a n t i t y of unfrozen w a t e r depends on t h e course t a k e n by t h e p r o c e s s ( ~ e d o s o v , 1942; Nersesova, 1950, 1951). I f a s o i l sample i s brought t o a g i v e n temperature by g r a d u a l c o o l i n g ( f r e e z i n g c y c l e ) , t h e q u a n t i t y of unfrozen water i n i t i s g r e a t e r t h a n i n t h e c a s e when i t r e a c h e s t h e same s t a t e by a g r a d u a l i n c r e a s e of temperature from a lower s t a r t i n g p o i n t (thaw- i n g c y c l e ) . The h y s t e r e s i s loop i s as a r u l e small.

9 . The q u a n t i t y o f unfrozen wateP i s independent o f t i m e . T h i s s t a t e m e n t i s n o t c o r r e c t , however, f o r a l l t h e unfrozen w a t e r p r e s e n t under t h e given c i r c u n s t a n c e s , but f o r o n l y t h a t p a r t of i t which can e x i s t i n s t a b l e therrno- dynamic e q u i l i b r i u m w i t h t h e i c e . The q u a n t i t y o f supercooled w a t e r i n t h e s o i l may change very s t r o n g l y w i t h time. It i s important t o emphasize t h a t a l l t h e above d a t a c h a r a c t e r i z e o n l y c o n d i t i o n s o f thermodynamic e q u i l i b r i u m of t h e w a t e r . On t h e b a s i s of t h e f a c t s h e r e s e t f o r t h , we can d r a w some c o n c l u s i o n s t h a t a r e v e r y i m p o r t a n t f o r what f o l l o w s . In t h e f i r s t p l a c e , s i n c e t h e e q u i l i b r i u m amount of unfrozen w a t e r i n a s o i l of g i v e n n i n e r a l o g i c a l c o n s t i t u t i o n i s p r a c t i c a l l y independent of t h e n o i s t u r e , t e x t u r e , e x t e r n a l p r e s s u r e and time, and s i n c e t h e h y s t e r e s i s loop

i s g e n e r a l l y s n i l l , t h e n t h e q u a n t i t y o f unfrozen w a t e r i s i n t h e f i r s t

approximation i n a s i n g l e - v a l u e d f u n c t i o n o f t h e temperature. T h i s , however, i s c o r r e c t o n l y f o r t h e c a s e when t h e f r e e z i n g p r o c e s s t a k e s p l a c e s l o w l y enough, t h a t Is, when i t i s q u a s i - s t a t i o n a r y . In r a p i d f r e e z i n g t h e wU =

v+,(3) curve i s no l o n g e r deteriiiined by t h c m i n e r a l o g i c a l composition o f t h e s o i l and i t s temperature, s i n c e s u p e r c o o l i n g of t h e w a t e r i s p o s s i b l e . It

must be enlphasized t h a t t h e q u a n t i t y of i c e I n t h e s o i l i s never c h a r a c t e r i s - t i c f o r t h e l a t t e r , s i n c e wI = w

-

wU, and t h e t o t a l moisture c o n t e n t w may have any value.

I n t h e second p l a c e , t h e r e e x i s t s i n s o i l s

a

type of water t h a t i s n o t converted t o i c e even a t very low temperatures. It i s known a s t h e f i r m l y bound water (wFBW). A l l t h e r e s t of t h e water i n t h e s o i l i s thermoactive and

may f r e e z e under a p p r o p r i a t e c o n d i t i o n s . The amount of t h e l a t t e r i s wa = w

-

W ~ ~ '

I n t h e t h i r d p l a c e , t h e s p e c i f i c c h a r a c t e r of t h e r e l a t i o n s h i p between q u a n t i t y of unfrozen water and temperature makei i t p o s s i b l e t o d i s t i n g u i s h a t l e a s t t h r e e zones i n t h e s o i l , according t o a temperature c r i t e r i o n , namely:

( a ) The zone of low temperature (3 i 0, ), in which wU r. wm z const, and where f o r a l l p r a c t i c a l purposes water-ice phase t r a n s i t i o n s do n o t occur. This i s c a l l e d t h e f r o z e n zone [ f r o s t zone].

( b ) The zone of temperatures below f r e e z i n g but r e l a t i v e l y n e a r O°C, in which i n t e n s i v e change i n t h e phase composition of t h e water occurs. On t h e one s i d e i t i s bounded by t h e 0, isotherm, and on t h e o t h e r by 0. It must be emphasized t h a t whereas t h e q u a n t i t y O, i s t o

a

c o n s i d e r a b l e e x t e n t a r b i t r a r y , because t h e q u a n t i t y wU of unfrozen water tends a s y m p t o t i c a l l y toward i t s

l i m i t i n g value wFBii, t h e q u a n t i t y O, on t h e o t h e r hand, may be q u i t e p r e c i s e l y d e f i n e d , s i n c e with i n c r e a s i n g temperature t h e wU

-

w U ( 0 ) curve rises s h a r p l y upward. In p r a c t i c e , some p a r t of t h e water in s o i l s w d l l f r e e z e i h a sudden

jump a t 3 = O. This zone i s c a l l e d t h e phase t r a n s i t i o n zone.

( c ) The zone of temperatures h i g h e r t h a n O, in which wU(3) = w and wI =

0. It i s obvious t h a t w i t h 3 > O no water-Ice phase conVersions occur. This

i s c a l l e d t h e thawed zone.

Heat of c r y s t a l l i z a t i o n . Change i n t h e temperature of c r y s t a l l i z a t i o n of any substance i s n e c e s s a r i l y accompanied by change i n i t s h e a t of c r y s t a l l i z a - t i o n . Pure d i s t i l l e d water f r e e z e s a t O°C w i t h a h e a t l i b e r a t i o n of qo =

79.67 c a l o r i e s p e r gram

a and book

of Chemistry and Physics, 1957), but t h e moisture in t h e s o i l , f r e e z i n g a t a temperature 0 2 Of 2 O,, l i b e r a t e s

a

q u a n t i t y of h e a t t h a t i s l e s s by Aqo = f ( a f ) c a l o r i e s ( s e e , f o r i n s t a n c e , Epshtein, 1948), with

Here Aqo = qo(oOC)

-

q 0 ( a f ) > 0; To = 273OK i s t h e temperature of c r y s t a l l i z a - t i o n of water i n t h e volume phase (measured i n t h e a b s o l u t e s c a l e of tempera- t u r e s ) ; AT =

To

-

Tf = -3 > f 0, where Of i s t h e temperature of c r y s t a l l i z a t i o n

of t h e thermoactlve water i n t h e C e l s i u s s c a l e (af c o O C ) ; Ac = c u

-

c I i s

t h e change of h c a t c a p a c i t y o f w a t e r i n i t s conversion t o i c e ( c U and c I a r e nieasured a t c o n s t a n t e x t e r n a l p r e s s u r e P ) ; Av = vU

-

v I i s t h e change of

s p e c i f i c volume of water i n t r a n s i t i o n from one phase t o t h e o t h e r . The above formula i s v a l i d o n l y i n t h e c a s e where t h e bound water, by f r e e z i n g , p a s s e s o u t of t h e s p h e r e o f i n f l u e n c e of t h e s u r f a c e f o r c e s e x e r t e d by t h e m i n e r a l s k e l e t o n of t h e s o i l , and forms " f r e e " i c e . We know t h a t i n t h e f r e e z i n g o f f i n e l y d i s p e r s e s o i l s ( t h e o n l y ones i n which a c o n s i d e r a b l e displacement o f t h e f r e e z i n g temperature o c c u r s ) t h e bulk o f t h e w a t e r c r y s t a l l i z e s i n t h e form of "macroscopic" i n t e r l a y e r s o f i c e [ i c e l e n s e s ] , and t h u s o b v i o u s l y cannot be w i t h i n t h e s p h e r e o f a c t i o n o f t h e s u r f a c e f o r c e s . X-ray s t u d i e s c a r r i e d o u t by T.P. Kostetskaya and G.A. Martynov (1951) have shown t h a t even when t h e r e i s a formation o f "microscopic" c r y s t a l s of ice-cement t h e l a t t e r a r e o u t o f d i r e c t c o n t a c t w i t h t h e s u r f a c e and consequently t h e y a l s o c o n e t i - t u t e " f r e e " i c e . Thus i t is c l e a r t h a t formula ( 6 . 8 ) can be used t o c a l c u l a t e t h e q u a n t i t y Aqo.

I n o r d e r t o e s t i m a t e t h e s i z e o f Aqo l e t u s suppose t h a t t h e h e a t c a p a c i t y and s p e c i f i c volume o f t h e bound w a t e r do n o t d i f f e r from t h o s e of t h e f r e e water. T h i s may be c o n s i d e r e d as an e x p e r i m e n t a l l y e s t a b l i s h e d f a c t a s f a r as t h e h e a t c a p a c i t y i s concerned ( v . s u p r a ) , b u t f o r t h e s p e c i f i c volume we have a s y e t no d a t a . But i f t h e term i n Av i n e q u a t i o n ( 6 . 8 ) i s t a k e n t o be o n l y a s m a l l increment, we may assume t h a t t h e c a l c u l a t i o n i s s u f - f i c i e n t l y p r e c i s e . Using t h e d a t a g i v e n in t h e Handbook o f Chemistry and Physics (1951), we f i n d t h a t Aqo 2. -0.76 J f .

Since in t h e m a J o r i t y of s o i l s t h e phase t r a n s i t i o n r e g i o n o f tempera- t u r e s does n o t extend below

-3

o r -5OC, t h e change of t h e h e a t o f c r y s t a l l i z a - t i o n i n t h i s temperature i n t e r v a l i s s o s m a l l t h a t i n p r a c t i c e i t may always be n e g l e c t e d . T h e r e f o r e in what f o l l o w s we s h a l l everywhere suppose t h a t b 0 ( J f )

=

qo

=

80 c a v e .

Heat accumulation and h e a t c a p a c i t y . I n c r e a s i n g t h e t e m p e r a t u r e of a s u b s t a n c e r a i s e s i t s i n t e r n a l energy by i n c r e a s i n g t h e speed o f t h e r m a l motion of i t s molecules.

In

t h e absence of phase c o n v e r s i o n s t h e s p e c i f i c e f f e c t

( t h a t i s , t h e e f f e c t p e r u n i t Inass and p e r OC) o f t h i s change o f i n t e r n a l energy i s d e s c r i b e d by t h e h e a t c a p a c i t y of t h e s u b s t a n c e . When phase conver- s i o n s a r e involved, t h e r e l i k e w i s e t a k e s p l a c e a n i n c r e a s e i n t h e molecular thermal v e l o c i t y , b u t now a p a r t of t h e energy i s expended i n b r e a k i n g down t h e c r y s t a l l a t t i c e o f t h e body. If we l e a v e a s i d e t h e q u a l i t a t i v e changes and c o n s i d e r o n l y t h e energy s i d e of t h e process,, t h e n t h e d i f f e r e n c e between h e a t o f c r y s t a l l i z a t i o n and h e a t c a p a c i t y d i s a p p e a r s . Accordingly t h e l i b e r a - t i o n of h e a t i n phase t r a n s i t i o n s c a n be t a k e n i l l t o account by t h e use of an e f f e c t i v e h e a t c a p a c i t y .

The fundamental equation ( 6 . 3 ) d e f i n i n g s p e c i f i c h e a t c a p a c i t y a l s o remains t r u e f o r s o i l s a t phase convcrsion t c n ~ p e r a t u r e s . The same i s t r u e a l s o of expression (6.4), which can be r e w r i t t e n i n t h e forrn:

w i t h mk, h e r e being expressed a s

%

+

rnI and t h e a i r being taken a s having z e r o mass. But i n c o n t r a s t t o ( 6 . 4 ) i t i s h e r e no longer p o s s i b l e t o regard t h e q u a n t i t y of i c e and unfrozen water I n t h e s o i l a s independent of tempera- t u r e . Therefore upon s u b s t i t u t i n g ( 6 . 9 ) i n

(6.3)

i n s t e a d of

( 6 . 5 )

we g e t :

The l a s t term i n s i d e t h e moulded b r a c k e t s d e f i n e s t h e q u a n t i t y of h e a t t h a t must be removed from dmU grams of water i n o r d e r t o c o n v e r t t h i s q u a n t i t y of water i n t o i c e . Obviously i t i s e q u a l t o qodmU.

We move mM o u t s i d e t h e moulded b r a c k e t s and put:

m u ( 3

where wU(3) =

-

i s

t h e q u a n t i t y of unfrozen water a t temperature 3. Then:

m~

This q u a n t i t y i s c a l l e d t h e e f f e c t i v e h e a t c a p a c i t y of t h e s o i l ( c e f f ) . It is a more g e n e r a l c h a r a c t e r i s t i c o f t h e system t h a n t h e t r u e s p e c i f i c h e a t c a p a c i t y

s i n c e ceff t a k e s i n t o account t h e h e a t of phase conversions, a s w e l l as every- t h i n g e l s e

.

Since w = ~ ~ ( 3 )

+

~ ( ~ ( 3 ) we have

v a r i a b l e q u a n t i t y . But a t no vcry g r e a t v a l u e s of t h e moisture c o n t e n t , t h e niaximum r e l a t i v e change of h e a t c a p a c i t y 13 :

-

-

C(0)

-

~ ( 0 , )

(4)

-

-

"

0 . 1 t o 0.2. max ~ ( 0 , )

Consequently t h e dependence of h e a t c a p a c i t y on tenlperature may i n many c a s e s be n e g l e c t e d .

The e f f e c t i v e h e a t c a p a c i t y c e f f i s u s u a l l y many tlrnes g r e a t e r t h a n t h e t r u e h e a t c a p a c i t y c, s i n c e t h e i c e c o n t e n t i n c r e a s e s q u i t e s h a r p l y with f a l l - i n g temperature. The value of ceff i n t h i s r e g i o n i s t h e r e f o r e determined mainly by t h e qov term, t h e s i z e of which depends on t h e moisture c o n t e n t and on t h e shape of t h e curve f o r unfrozen water c o n t e n t w U = wU(3). Because of t h e i n t e r m i t t e n t c h a r a c t e r of t h i s curve, v and consequently ceff tend t o i n f i n i t y f o r 3 = O, t h u s l o s i n g t h e i r p h y s i c a l meaning. I n t h i s case i t i s expedient t o allow f o r t h e h e a t of c r y s t a l l i z a t i o n n o t by means of an e f f e c - t i v e h e a t c a p a c i t y but i n some o t h e r way ( v . i n f r a ) . But i f v ( 3 ) has a f i n i t e value, t h e n t h e u s e of ceff i s f u l l y j u s t i f i e d .

Heat flow and thermal c o n d u c t i v i t y . The b a s i c law of t h e conductive mechanism of thermal t r a n s m i s s i o n

i s a l s o v a l i d f o r t h e phase t r a n s i t i o n zone. But h e r e , i n c o n t r a s t t o t h e o t h e r zones, t h e c o e f f i c i e n t of thermal c o n d u c t i v i t y A depends on temperature even i n t h e case of f i n e l y d i s p e r s e s o i l s . In f a c t i f t h e s o i l i s a t phase t r a n s i t i o n temperatures and i f t h e temperature i s non-uniform w i t h volume, t h e n a t d i f f e r e n t p o i n t s i n t h e s o i l t h e r e w i l l be d i f f e r e n t amounts of i c e

wI and d i f f e r e n t amounts of unfrozen water wU, even under t h e c o n d i t i o n t h a t t h e t o t a l moisture c o n t e n t w = w I ( 3 )

+

w U ( 3 ) i s c o n s t a n t . The consequence o f t h i s i s t h a t t h e composition and s t r u c t u r e of t h e s o i l a r e f u n c t i o n s of t h e temperature. Accordingly t h e q u a n t i t y A a l s o w i l l v a r y w i t h temperature.

Taking i n t o account t h e d i r e c t connection between t h e phase composition of t h e s o i l water and thermal c o n d u c t i v i t y , we may w r i t e ( ~ a r t y n o v , 1957):

Here hT and hF a r e t h e thermal c o n d u c t i v i t y c o e f f i c i e n t s of thawed and f ' u l l y f r o z e n s o i l , r e s p e c t i v e l y . Thus t h e l i m i t s of v a r i a t i o n of h i n t h e phase t r a n s i t i o n zone a r e determined by t h e r a t i o while t h e dependence

of A on temperature i s d e f i n e d by t h e c h a r a c t e r of t h e unfrozen water curve

w,(a)

= , w, + wa(s)

-

In t h e mean f o r a l l a o i l s t h e r a t i o

Ap/hT

l i e s i n t h e i n t e r v a l 1.1 t o

1.3,

v a r y i n g a l i t t l e a c c o r d i n g t o t h e volumetric weight y , t h e t e x t u r e , t h e moisture c o n t e n t w, t h e mineral composition, d i s p e r s i o n and o t h e r f a c t o r s . With I n c r e a s i n g p o r o s i t y t h e f r a c t i o n of h e a t p a s s i n g by way of a i r ( i n t h e pores ) is i n c r e a s e d , with

a

corresponding d e c r e a s e i n t h e v a r i a t i o n of t h e thermal c o n d u c t i v i t y c o e f f i c i e n t due t o phase conversions of t h e o t h e r s o i l component, t h e water. T y p i c a l curves f o r t h e r e l a t i o n s h i p of t o 'y a r e shown I n F i g . 19 ( t h e graph i s c a l c u l a t e d from t h e d a t a of Kersten, 1949).

I n v e r y f r i a b l e s o i l s i n which t h e amount of c o n t a c t i s s m a l l and which a r e under an e x t e r n a l p r e s s u r e n o t exceeding 25 g/cm2, t h e r a t i o

hp/hT

may even be l e s s t h a n u n i t y , a f a c t t h a t i s e x p l a i n e d by c r y s t a l s of i c e growing and

moving t h e s o i l p a r t i c l e s a p a r t , t h a t is, by a n i n c r e a s e of p o r o s i t y ( ~ e s k o w , 1947

1

The e f f e c t of t e x t u r e on t h e q u a n t i t y hp/hT i s a t p r e s e n t completely un-

I n v e s t i g a t e d . I n d i r e c t l y one may judge from t h e r e l a t i o n s h i p between and u n i t weight t h a t t e x t u r e does p l a y a r o l e , s i n c e t h e u n i t weight

i s

r e l a t e d t o t h e make-up of t h e s o i l . A s we have a l r e a d y noted, however, t h i s l a t t e r

r e l a t i o n s h i p i s n o t simple.

The q u a n t i t y A d $ depends on t h e s o i l moisture c o n t e n t ( ~ i g . 2 0 ) : t h e g r e a t e r t h e moisture c o n t e n t , t h e g r e a t e r t h e amount of h e a t t h a t i s t r a n s m i t - t e d d i r e c t l y by way of t h e water, and t h e s t r o n g e r w i l l be t h e c h v g e i n A as

a r e s u l t of t h e water f r e e z i n g In t h e s o i l p o r e s . And s i n c e t h e thermal con- d u c t i v i t y of i c e i s h i g h e r t h a n t h e thermal c o n d u c t i v i t y of water, t h e r a t i o

must i n c r e a s e with i n c r e a s i n g moisture c o n t e n t . However, a t s n l a l l

v a l u e s of w t h e o p p o s i t e p i c t u r e i s observed. The e x p l a n a t i o n of t h i s is, i n a l l p r o b a b i l i t y , t h a t t h e growing c r y s t a l s of i c e a t t r a c t water t o themselves and d e s t r o y t h e water f i l m between t h e s o i l p a r t i c l e s and t h u s worsen t h e c o n d i t i o n s f o r h e a t t r a n s f e r from one p a r t i c l e t o a n o t h e r .

The d i r e c t e f f e c t of t h e s o i l t s m i n e r a l o g i c a l composition and d i s p e r s i o n on t h e r a t i o must be very s m a l l . But t h e t e x t u r e and t h e amount of thermoactive moisture w i l l depend on t h e s e f a c t o r s , and t h e r e f o r e s o w i l l

A@T. For i n s t a n c e , i n t h e c a s e of q u a r t z sand we have AF = 1 . 6 AT f o r w

-

212 and y = 1.78 g/cm2, while i n t h e c a s e of a heavy c l a y s o l l we have A F z

$

f o r w = 212: and y = 1.95 d o n 2 ( ~ h i m a n o v s k l l and Shlmanovskaya, 1952).

The d i s p e r s i o n and m i n e r a l o g i c a l composition of t h e s o i l mainly a f f e c t n o t t h e r a t i o hFjhT b u t t h e c h a r a c t e r of t h e w U = wU(3) curve and t h u s t h e dependence of A on temperature. The more d i s p e r s e d t h e s o l l , t h e l a r g e r t h e phase t r a n - s i t i o n r e g i o n and t h e l e s s s t e e p l y 'Its thermal c o n d u c t i v i t y w i l l change.

Whereas in sand the change of h is a sudden jump, in heavy clay h may still be changing even at a temperature below -2OC. Even in clays, however, the

greater part of the water crystallizes at the temperature of inception of freezing ( ~ l g . 18) and, consequently, at this same temperature the principal change in h will occur. Since the said change is small (about 20

-

30$ In the mean), it is nearly always possible in thermophyslcal calculations to regard h

as changing discontinuously at the incipient freezing temperature O of water, and thereafter remaining constant.

The absolute value of h, as may be seen from formula (6.13) Is, mainly determined by the thawed soil thermal conductivity

AT.

The equation of thermal conductivity and the limits of its applicability. The thermal conductivity equation for the phase transition zone is derived in exactly the same way as for thawed soils. The sole difference is in the fact that the heat coming into the elementary volume goes not only to heat the soil but also into phase transitions. Therefore instead of the true heat capacity we must

In

the thermal conductivity equation put the effective heat capacity 'eff.

In

view of this we may immediately write (~artynov, 1957)":

The equation of thermal conductivity

(6.1.4)

may be given a more conven- ient form for calculation by differentiating and then dividing the left-hand

and

right-hand sides by yOzeff:

As a rule, the second term

In

the right-hand side of (6.14') plays only a small role. For instance, if in (6.14') we substitute data obtained with the electrometric equipment of the Podmoskovski station of the Obruchev Perma- frost Institute (Martynov, Shimanovskii, 1957), we find that the second compo- nent is about 150 times smaller than the first. Under these conditions the second component, when no particularly high precision is required, may be omitted, whereupon (6.141) takes the form:

*

The first attempt to derive the thermal conduction equation for the phase transition zone was made by A.G. Kolesnikov (1952). However, he overlooked a mistake in his calculations.

where

Is t h e e f f e c t i v e temperature c o n d u c t i v i t y coef f i c i e n t

.

Equation ( 6 . 1 4 ) Is a f u r t h e r g e n e r a l i z a t i o n of e q u a t i o n

( 6 . 7 ) .

It i s v a l i d f o r s o i l s i n which w a t e r - i c e phase t r a n s i t i o n s a r e t a k i n g p l a c e . Consequently i t holds o n l y i n t h e absence of pore convection. Furthermore, s i n c e t h e e f f e c t i v e thermal c a p a c i t y c e f f e n t e r s i n t o (6.14) i t may be used o n l y i n those r e g i o n s of t h e phase t r a n s i t i o n zone where v ( 3 ) does n o t tend t o i n f i n i t y

.

The l a t t e r l i m i t a t i o n , however, i s n o t fundamental, s i n c e by simple mathematical t r a n s f o r m a t i o n s , without t h e i n t r o d u c t i o n of any new p h y s i c a l concepts, e q u a t i o n ( 6 . 1 4 ) may be g e n e r a l i z e d f o r t h e case when t h e r e a r e i n t h e f r e e z i n g and thawing s o i l s s u r f a c e s of d i s c o n t i n u i t y of c e f f . Thus e q u a t i o n ( 6 . 1 4 ) e s s e n t i a l l y c o n t a i n s t h e system of e q u a t i o n s o f t h e S t e f a n problem, on t h e b a s i s of which a l l thermophysical c a l c u l a t i o n s a r e today conducted.

The dependence o f ceff on temperature means t h a t e q u a t i o n ( 6 . 1 4 ) belongs t o t h e c l a s s of n o n - l i n e a r e q u a t i o n s of mathematical p h y s i c s . Methods f o r s o l v i n g t h i s t y p e of e q u a t i o n a r e c o n ~ p l i c a t e d , and s o f a r a r e almost complete- l y undeveloped. In c a s e s where no p a r t i c u l a r l y g r e a t p r e c i s i o n i s r e q u i r e d t h e c a l c u l a t i o n s may be c a r r i e d o u t on t h e b a s i s of t h e l i n e a r i z e d form o f e q u a t i o n ( 6 . 1 5 ) : where

-

A* a:ff

-

= cons t

.

ro%f f The p o s s i b i l i t y of r e p l a c i n g t h e n o n - l i n e a r e q u a t i o n ( 6 . 1 5 ) w i t h t h e l i n e a r e q u a t i o n ( 6 . 1 6 ) Is t h e b a s i c r e a s o n f o r d i s t i n g u i s h i n g

a

phase t r a n s i - t i o n zone i n f r o z e n s o i l . A c t u a l l y t h e r e i s no s h a r p boundary between t h e phase t r a n s i t i o n zone and t h e f r o z e n s o i l zone, and t h e r e f o r e t h e p r o c e s s e s of h e a t propagation i n them could be d e s c r i b e d by a s i n g l e e q u a t i o n o f t h e type of ( 6 . 1 5 ) , b u t under t h e c o n d i t i o n t h a t t h e c o e f f i c i e n t s X and c e f f t h e r e i n w i l l be c o n s i d e r a b l y dependent on temperature. This, however, w i l l markedly complicate a l l c o ~ n p u t a t i o n s . It i s found more e x p e d i e n t t o d i v i d e t h e f r o z e n ground i n t o two zones, w i t h i n each o f which X and c e f f may be t a k e n as con- s t a n t s . Generally speaking, i t would be p o s s i b l e t o d i s t i n g u i s h s e v e r a l zones, each with c o n s t a n t X and c e f f , and i n t h e l i m i t t h i s would p e r m l t e q u a t i o n (6.15) t o be solved p r e c i s e l y ( ~ o l e s n l k o v and Martynov, 1953). But conlparison of a c t u a l d a t a with t h e r e s u l t s of computation on t h e b a s i s of

equation ( 6 . 1 5 ' ) shows t h a t t h e r e

is

l i t t l e p r a c t i c a l n e c e s s i t y f o r such refinement, s i n c e t h e d i v i s i o n i n t o j u s t two zones w i l l ensure a r e l a t i v e e r r o r of c a l c u l a t i o n l e s s than

5%.

Processes of Heat T r a n s f e r i n t h e Case of Simultaneous Existence of t h e Three Zones

System of equations of t h e generalized S t e f a n problem and t h e l i m i t s of i t s a p p l i c a b i l i t y . A temperature c r i t e r i o n has been e s t a b l i s h e d a s b a s i s f o r t h e d i v i s i o n of f r e e z i n g s o i l s i n t o zones. This i s r e f l e c t e d i n t h e f a c t t h a t t h e zone boundaries a r e t h e 0 and 0, isotherms. Since h e a t exchange between zones t a k e s p l a c e only a t t h e i r boundaries, while w i t h i n t h e zones t h e mechan- i s m of h e a t propagation remains j u s t t h e same a s i f no o t h e r zones were pre- s e n t , we can immediately w r i t e down t h e thermal c o n d u c t i v i t y equation system:"

Here t h e s u b s c r i p t Index T i n d i c a t e s t h a t t h e q u a n t i t y i n q u e s t i o n r e f e r s t o t h e thawed zone; t h e index P, t o t h e phase t r a n s i t i o n zone; t h e index F, t o t h e frozen s o i l zone.

The s p a t i a l p o s i t i o n of t h e 0 and 0, isotherms does n o t remain c o n s t a n t , s i n c e t h e temperature f i e l d In t h e s o i l v a r i e s . Therefore i f we p u t h f o r t h e h e i g h t (z-coord.inate) o f t h e O isotherm and h , f o r t h a t of t h e 0, isotherm, then, g e n e r a l l y speaking, h and h, w i l l be f u n c t i o n s of t h e time t . I n view of t h i s we may r e w r i t e system ( 6 . 1 7 ' ) in t h e form:

- -

where cp = ceff i s t h e e f f e c t i v e thermal c a p a c i t y

-

s e e ( 6 . 1 1 ) ~ and Ap

-

X

i s

t h e thermal c o n d u c t i v i t y of t h e s o i l i n t h e phase t r a n s i t i o n zone, a s defined by equation (6.13 )

.

The mutual thermal e f f e c t of t h e zones one on t h e o t h e r i m p l l e s t h a t a t t h e mobile boundaries h ( t ) and h , ( t ) t h e temperature f i e l d c o n t i n u i t y

c o n d i t ions

and a l s o t h e energy conservation c o n d i t i o n s must be met. In o r d e r t o express t h e l a t t e r i n a n a l y t i c form, we s h a l l suppose t h a t in passage from t h e thawed zone t o t h e phase t r a n s i t i o n zone t h e change AwU of t h e q u a n t i t y of unfrozen water

wU

w i l l n o t be discontinuous b u t gradual, extending over a c e r t a i n small r e g i o n i h ( t )

+

~ , h ( t )

-

E ] , in such a way t h a t f o r h

-

e and temperature

~ ( h

-

& , t )

-

8

+

1A13

we have

wU

-

w, and f o r h

+

8 and temperature 3 ( h

+

8 , t ) -

0

-

f ~ 3 we have

wU

= w

-

AwU. Since 3(h, t ) = 0, we have

which upon s u b s t i t u t i o n in (6.14) gives:

I n t e g r a t i n g (6.20) from h

+

c t o h

-

e and u s i n g t h e theorem of t h e mean, we g e t t h i s i n t o t h e form:

where h

+

E 1

E

,> h

-

E ; t h e meaning of

2)

_h+e

i s t h e q u a n t i t y of h e a t flowing o u t t o t h e s u r f a c e

z

= h ( t ) from he phase t r a n s i t i o n zone, and t h e meaning of

(

b

$)

h-e i s t h e q u a n t i t y of h e a t flowing from t h e s u r f a c e h

-

E

i n t o t h e thawed zone. Because of t h e c o n t i n u i t y of t h e temperature f i e l d , we have A 5 4 0 when E -+ 0. Meanwhile, however, t h e q u a n t i t y

a

dWu w i l l remain

f i n i t e and equal t o AwU, s i n c e wU(0) breaks c o n t i n u i t y a t 5 = 0 . Taking t h i s i n t o account, we f i n a l l y o b t a i n :