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PergamonInt

0017-9310(94)0028~5

J . Hea t M ass T ramJer. Vol. 38. No. 12. 2291-2303. 1995p.Copyright 10 1995 Elsevier Science Ltd

Printed in Great Britam. All rights reserved0017-9310/95 $9.50+0.00

Evaluation of heat and moisture transferproperties in a frozen-unfrozen water-soilsystem

ASHOK K. SINGHT and D. R. CHAUDHARYThermal Physics Laboratory, Department of Physics, University of Rajasthan, Jaipur 302 004, India

(Received infinalforn~ 10 August 1994)Abstract-An experimental study of heat and mass transfer in a moist unsaturated sand has been carriedout by maintaining the two ends of a sample column at temperatures of -5 and 3OC, respectively.Measurements of temperature have been made as a function of distance from the cold end and time, alongwith effective thermal conductivity and liquid water distribution at the end of the experiment. Comparisonof the measured thermal conductivity values shows close agreement with calculated values from an effectivecontinuous model (Singh et al.). Evaluation of thermal and moisture diffusion coefficients has been donefrom measured steady-state temperature and moisture profiles. A dryout condition has also been determined

from the evaluation of heat and moisture transfer properties.

I. INTRODUCTIONHeat and mass transfer in soils is a complicated prob-lem because of the presence of the solid, liquid andgaseous phases. Dur ing the freezing process the prob-lem is also complicated because the freezing of soilgenerally produces significant changes in its struc-tural, mechanical and thermal properties. The thermalstatus of freezing soils which are in a delicate balanceand disturbance of this region could have seriousengineering and ecological implications. Knowledgeof thermal behaviour is essential in ensuring that sucha disturbance does not occur, and also for the energybalance at the soil surface. The process of simul-taneous heat and mass transfer in the freezing processin soils is of practical interest in many other engin-eering applications. The heat and mass transferproperties of soil systems are also essential for eva-luating the effectiveness of insulatio n of energy storag edevices, bu ildings, and storage of solar and geo-thermal energy in soils. These properties are also use-ful for drying p rocesses, heat transfer calculati onsfrom buried pipelines and underground space util-ization. They include heat dissipation from under-ground electrical pow er cables, heat transfer to andfrom buried heat pump coils, recovery of geothermalenergy and waste heat rejection from power plants.

For schemes of a medium to large scale, buriedpipelines and ground-based reservoirs are appropriatemeans of storage and transportation. Unfortunately,these devices usually need to be surrounded by bulkyand relatively cheap thermal insulation. This suggests

t Present Address : Headquarters, Snow and AvalancheStudy Establishment (SASE), Manali, H. P. 175 131, India.

that sand/soil can act as a natural and robust thermalinsulation. In determining the long-term thermal insu-lation properties of sand/soil, the different properties,i.e. diffusion coefficients, therm al proper ties with dis-tribution of moisture under the effect of a temperaturegradient and dryout possibilities under different con-ditions, have to be known.

Thermal gradients in a moist soil can induce mois-ture movement which, in turn, can result in a decreasein the thermal conductivity of the unfrozen soil andan increase in the thermal conductivity of the freezingside. In particular, the layered structure is associatedwith considerable moisture migration to the freezingfront, as may occur in a frost-susceptible soil. Theyare considered to be coupled processes interacting to-gether. In consequence, changes occur in the proper-ties of both the freezing layer and underlying unfreez-ing zone.

The long range objective of this study is to deter-mine experimentally the effect of moisture migrationon freezing point depression and related phenomenaon the cooling of soil with sufficient precision, and toprovide the basis for the development of mathematicalmodels which take these complications into properaccount. Heat and mass transport properties of moistporous media are basically important in solving suchproblems. However, there are few data and still fewerconvenient means available for the measurement ofthese properties at the moment. An evaluation of theseproperties with the distribution of moisture contentand its effect on the effective therm al conduc tivity(ETC ) and the diffusion coefficients has been made inthe present investigation, along with the conditions forthe existence of a dryout region under experimentalconditions.

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2298 A. K. SINGH and D. R. CHAUDHARY

NOMENCLATURE

;constant

;Tvolume fraction

slope of density temperature curve non-dimensional steady-stateD diffusion coefficient temperatureh latent heat of evaporation non-dimensional steady-state moistureJ moisture flux Frn phase dispersionK moisture diffusion coefficient 4 ratio of energy stored in vapourLe Lewis number enthalpy to that stored as internalM moisture content by weight at energy.

saturationm moisture content by weight at Subscripts

unsaturation a air4 power per unit length supplied to the coldprobe heater Zl dryT temperature [Clt timeW moisture contentx coordinate (distance).

Greek symbols;

thermal diffusivitynew thermal diffusivity

/I thermal conductivityP densityE porosityV amass condensation/evaporation per

unit volumew mass content of water vapour in

saturated air at 760 mm of Hg

e effectiveECM effective continuous m ediag gasH hoti initialj 1 for frozen and 2 for unfrozenma moist airS saturationS solidS.S. steady-stateV vapourW moisture (water)I9 thermalBV thermal vapourf3W thermal moisture.

2. MEASUREMENT AND ANALYSISTemperature and water content profiles in a vertical

soil column, with the cold side temperature below thefreezing point of water, and subjected to temperaturegradient, were measured as a function of time anddistance from the cold end. The moisture content pro-file was measured at the steady state. Th e soil samplewas dune sand of particle size 150-1 77 pm , porosity0.42 in dry state and density 1620 kg m-. The samplehad a predetermined moisture content of 14.47% andwas in a PVC pipe of length 27.0 cm and diameter15.0 cm. The moisture content was measured by agravimetric method using a precise electric balance :accurate to 0.1 mg. This oven drying technique is themost widely used method for measuring soil moisturecontent and is the standard for the calibration of allother soil moisture determination methods [l]. Theerror associated with this method can be up to 10%[2] and consists of drying a soil sample in an oven at105C until a constant weight is obtained.

The experimental arrangement is similar to thatillustrated in Fig. 1. Controlled liquids of constanttemperatures, i.e. -5C and 30C were circulatedthrough hollow copper circulation plates placed atboth the ends of the soil column. Thirteen thermal

5

6

7

Fig. 1. Schematic diagram of experimental arrangement.

conductivity probes were placed a t intervals of 2.0 cmthroughout the length of the column. Temperaturerecording was achieved throughout the length ofcolumn by an inbuilt copper

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Evaluation of heat and moisture transfer properties 2299

2, =: q In !Z0NT*--T,) t1where Tz and T, are two temperatures at times t2an dt,, respectively. By knowing the value of the slopebetween In(t), the temperature rise, and q, power perunit length supplied to the probe heater, one can deter-mine the thermal conductivity I,. At the end of theexperiment, the probes were removed and a smallamount of the sample was extracted from the vicinityof the probe position with the help of hollow brasstubes. The determination of the water content wasachieved by weighing a sample specimen with a preciseelectrical balance before and after drying in an oven at105C. The calculation of the ETC w as accomplishedusing temperature-dependent expressions [4].

The moisture concentration ( W) and temperature(T) in an unsaturated porous medium has beendescribed as [5. 61:

2: = D,V2 T+ KV W (2)and

2 = &VT, (3 )where

BI

= , (1 + (~,b,h,/(~~),)(D,/ac,))(1 +%hh,l(Pc),)

The term fl, is a rather new diffusivity term which isdependent on the thermal diffusivity of the material(a,), the latent h eat of evaporation (h,), the slope ofdensity temperature curve (b,), the volumetric heatcapacity of the material (PC),, and it reveals thecharacter of the system. A brief account of thedevelopment of equation (3) is given below.

At any point within the system, the rate of changein vapour density (E&) equals the diffusion rate tothat point minus/plus the condensation/evaporation(v), i.e.

(34Here subscript j is 1 for the frozen region and j is 2for the unfrozen region. The change in internal energyat any point is expressed as

%Pd,T,ir = 1, $ + ,v,. (3b)The empirical relation for vapour density with tem-perature can be written as [7]

P, = a,+b,T,. (3c)Using equation (:~a), (3b) with (3~) one can obtainequation (3).

For one-dimensional flow of heat the moisture con-centration flux from equation (2) is

where Do = DBw Dev an d K = K,+ K, are thermaland moistu re diffu sion coefficients, respectively.Under the condition of steady-state flow, the net fluxof moisture must be zero, i.e.

Do CdW/d.4-= _~K (dT/dx) . (5)From equation (5) the evaluation of the ratio (Do/K)can be obtained under the condition of one-dimen-sional heat flow from the steady-state measurementsof temperature and moisture profiles. T he thermaldiffusion coefficient from the definition itself can beexpressed as

D &D&f,0Pd satThe air density (p,) can be expressed as

P* = 1.2929(T$;313).

The vapour diffusivity through the porous material(D,), is equal to approximately one-fifth of thediffusion coefficient of water vapour through air,which is expressed as

D,=2.29*iO-5[T~~:;13]is [ms_].

The mass content of water vapo ur (0) in saturatedhumid air under 760 mm of Hg with variation oftemperature was derived by Yu [8] and is expressedasdw-I)= 1.8041 x IO6 exp (g)dr [760-0.378 xexp(g)12(T+227.02) [K-l].

The parameter g is(7 )

g(T) = 18.304- Ty;;;402.Here one obtains the concept of dryout in a moistporous medium when the moisture profile reaches itssteady state [9] by writing equations (2) and (3) indimensionless form and using Greens theorem understeady-state conditions. The non-dimensional steady-state temperature (rjT) and moisture (I,!I~) profiles arerelated by (1/M - rjT+ (rjT), irrespective of the ther-mal boundary conditions with impermeable boun-daries. One then has(*M)ss = +TwL - WJ = -(h)ss. + ). (74From this, one defines the wet region ( W > 0) as

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2300 A. K. SINGH and D. R. CHAUDHARY

(*TL < g$W...). (7b)H

Using the Laplace transform technique, the solutionat steady state is

($T)s.s. = 1 --X with ((IcITks > = l/2 (8 )(rikk = -(1-X)+D,VT/Kand the wet region is given by equation (7b)

(10)

1 KW ,4 = 2D,VT < X. (11)When the left-hand side of equation (11) is set equalto zero, one obtains the upper bound of W, for whichdryout is possible.

(12)The effective thermal conductivity (ETC ) has been

calculated using the expressions developed by Singhet al. [4]. This model has been tested for different typesof porous materials saturated with various liquids andfor materials having a wide variation in porosity. TheETC of moist soil is expressed as

The volume fraction of moisture in the pore spacewill be

Here [, = tis - 0.5, where rjjs, [,, I, and I,,, denote the Evolum e fraction of the solid phase, the solid phase edispersion, the therm al con ductivity of solid phase and the thermal conductivity of the effective con-tinuous medium, which is expressed for a moistporous medium as

where r/k,,,= (m/M)tia. Here m and M represent thevarying moisture content and the moisture content atsaturation by weight percent, respectively. In the caseof a frozen system, 1, will be replaced by the thermalconduct ivity of ice (A,,,).

3. RESULTS AND DISCUSSIONThe results of the study are presented in Figs. 2-6.

In Fig. 2 the transient temperature measurements arepresented. This figure presents temperature recordingsfor various times after the temperature at the upperend of the column is lowered suddenly to -5C(below the freezing point of water) and the bottomend of the column is at 30C which is approximatelyroom temperature. From Fig. 2 it can be observedthat the soil temperature decreases and approachesa linear distribution, because of the initial uniformmoisture content, and then approaches a steady-statenon-linear profile. It might be due to the fact that the

at 15.0 cm

at 9.0 cm

QO 10.0 20.0 30.0 co.0 540 60.0 70.0 (Time (how)

L O

nECM = 1.092(1 1 )Ias f (14) Fig. 2. Variation of non-dimensional temperature [(T- fc)/The thermal conductivity of moist air (I,,,,) is ex- (TH- Tc)] with non-dimensional distance (.X/L) from coldpressed as (for t,k,,. volume fraction of moisture in end (L = 27.0 em, T, = - 5C and T H = 30C).pore space, 0 < I+ @: < 0.4)

A,,,, = I, 1+3 .844 S&J). (15)

When rj,,,, lies between 0.4 and 1.0, the thermal con-ductivity of moist air becomes

3,,, = 1, (16)

The thermal conductivities of water (,I,) and air (1,)are expressed as

i, =0.55+2.34x 10m3 T-l.1 x lO-5 Tand QO 0.2 0.b 0.6 0.6Non-dimonsionol dis tance

1, = 0.0237+6.41 x lop5 T. Fig. 3. Variation of temperature with time at specific dis-tances (cm).

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Evaluation of heat and moisture transfer properties 2301

Temperature %-5.0 0.0 5.0 10.0 15.0 zao 25.0 300

13.0 14.0 VS.0 16.0 17.0 18.0 19.0 20.0Mois ture content by weight percent

Fig. 4. Variation of effective thermal conductivity (ETC)with moisture content and distribution of steady-state tem-perature and moisture with axial distance from cold end(cm).

effective thermal conductivity of moist porousmaterials prim arily depends upon moisture contentand composition [10, 111. Also if the effective thermalconductivity of the moist material is to remainuniform, then a linear temperature distribution wouldbe expected. As th.e heat and moisture migration is acoupled process, therefore the steady-state tem-perature profile will be non-linear, because ofm oistureredistribution resulting from the applied temperaturedifference. Since the slope of the temp erature profiledecreases with increase in axial distance from the coldend, this profile show s the character istics of thematerial, whose E.TC decreases with x/L. Transienttemperature mea,surements reveal that the tem-perature adjusts rapidly compared with the moistureand that a quasi-steady state exists during the time ofmoisture redistribution.

In Fig. 3 the variation of temperature with time atspecific distances is presented. This figure presents theeffectiveness of time-depen dent cooling of the mate rialwith the axial distance from the cold end. In Fig. 4

I I I I I I , I I ,14.0 15.0 16.0 120 18.0 19.0 :Moisture uwltent by weight percent 20.0

Fig. 5. Variation of the ratio of thermal and moisturediffusion coefficients with moisture content.

10-8 10-6I.- . . D,oo K

Fig.l-:ll;lo-g zao

Mois ture content b y weight percent6. Variation of thermal and moisture diffusion

coefficients with moisture content.

the steady-state temperature and moisture profiles areshown as a function of axial distance from the coldend. These steady-state temperature and moistureprofiles have been used for the evaluation of diffusioncoefficients. Also on this figure are plotted the mea-sured and calculated values of the effective thermalconductivity. The Singh et al. model [4] was used tocalculate the ETC values. A reasonable agreement hasbeen found between the measured and calculated ETCvalues. A sharp increase in the ETC has been foundat the interface that m ight be due to the freezing ofsoil. At the interface, the moistu re content within thesoil freezes, which increases the ETC of soil becausefreezing water has a thermal conductivity which isfour times greater than that of free water. In the cal-culations, the thermal conductivity of ice was takento be 2.20 W mm K-.

An analysis of the proposed diffusivity term /?, andsome points of interest are discussed here. The newdiffusivity term is

pI

= ~, (1+ 4,lLeJ(1+4,,)

where Le, = a,,lD, represents the Lewis number, whichis of basic importance in determining the behaviourof the system. The term 4, which also affects themagnitude of the difference between a, and p,, is theratio of energy stored in the vapour enthalpy to thatstored in the internal energy

E,b,h,4J PC), .The vapour density temperature slopes for the frozenand unfrozen regions are b, = 1.94 x lop4 andb2 = 6.18 x 10e4, respectively. The magnitude of L e,presents the existence of evapor ation or condensatio n.When L e, > 1 evaporation will occur in the system,which w ill increase the thermal diffusivity of the

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2302 A. K. SINGH and D. R. CHAUDHARYsystem. The ratio of the energy flux in the vapourenthalpy to that in the internal energy is

In Fig. 5 variation of the ratio of thermal and mois-ture diffusion coefficient (Do/K) with moisture contentis presented. The ratio of Do/K is a function of bothcomposition and soil moisture content. V ariation ofthe ratio can also be seen by combining the above twowith moisture content individually. These results alsoverify that the ratio should not be more than lo-, asproposed by Luikov [121 from the theoretical analysis.It can be concluded from the graph that the ratiobecomes greater with the increase of moisture contentand thereafter decreases. The ratio is small for a lowmoisture content. As the moisture content increases,the ratio also increases up to a certain value of mois-ture content. which depends on soil type. Furtherincrease in water content results in a decrease in theratio of the diffusion coefficients. The decrease in D,/Kat higher water content is mainly because of thedecrease in porosity of soil, which reduces the amountof vapour transfer. The experimental data presentedby Luikov [12] indicated that D,,/K is approximatelyzero when moisture content reaches saturation. In thedry region of the soil, the ratio should also decreaseto a small value (i.e. zero), since the mass transfer ofwater vapour is not possible due to the absence ofliquid water. Figure 4 shows a small dip in the mois-ture content distribution between 0.7 and 0.85 (x/L).This dip affects the dW/dx values in the region andhence causes the aforementioned D,/K valley, whichmay be caused by measurement error. The ratio ofthermal and moisture diffusion coefficients has beenfound to be in the range of lo--lo-C-. Based ontheoretical analysis and experimental data, Luikovhas proposed that, for the coupled heat and masstransfer process in porous media, the ratio of diffusioncoefficients should be less than 1 O x lo-C.

The variation of individual thermal and moisturediffusion co efficients with moist ure content has beenplotted in Fig. 6. The moisture diffusion coefficientexhibits a broad minimum , because it is inversely pro-portional to Do/K an d D,, varies monotonically withW/W,. For the sand sample, the thermal diffusioncoefficient has been found to be in the range IO- -lo- m2 s-C and the moisture diffusion coefficienthas been found to be in the range of 10-8-10P6 ms-(lo--lo- for unfrozen soil and lo--lop6 forfrozen soil). The order of these values is in agreementwith results of Wang and Yu [131.

The analysis of the dryout condition presents amethod for determining the existence of a dryoutregion by knowing only the boundary conditions andthe ratio of diffusion coefficients. It has been foundthat, for the present therm al boundary conditions,dryout is possible only when the initial moisture con-tent is less than 3.0%. Thus the existence of a dryoutregion will depend upon the initial moisture content.

The extent of the dryout region increases with adecrease in initial moisture content. Dryout starts ata time when the temperature field has already reacheda steady state. It can be observed that the moisturecontent decreases near the hot surface of the slab andincreases near the cold surface. After a certain time,the moisture content at the hot surface reaches thevalue zero. At that moment, dryout starts and thedryout region expands into the interior of the slab, anasymptotic steady state is finally reached in whichthe moisture content increases linearly w ith increasingdepth of the slab. The depth to which dryo ut pen-etrates increases with larger values of thermal/massdiffusion parameters. The total moisture content inthe slab must rem ain unchanged. As such the localmoisture content value at the cold side increases withincrease in value of the mass diffusion parameter.

4. CONCLUSIONRatio s of diffusion coefficients, Do/K, were exper-

imentally determined by measuring the one-dimen-sional temperature and moisture distribution in thetemperature range of -5-30C. The ratio of Do/Kw as found to be function of the moisture content. Theratio w as found to be small for low moisture contentvalues, increases with moisture content and thereafterdecreases. The ratio Do/K was found to be in the rangeof 10-3~1 0-C. For the soil used in this study, themoistu re diffusion coefficient was estimated to be intherangeof 10-8~10-6m2s- (10~8-10-7forunfrozenregion and 10-7-10m6 for frozen region).

The transient thermal response of the sand is muchquicker than its moisture response, the thermalresponse soon settles down to being quasi-steady. Thesteady-state moisture and temperature profiles werefound to be functions of initial moisture. In general,an increase in initial moisture content leads to uniformmoisture distribution, thus a uniform thermal con-ductivity and linear temperature profile. Since bothDo and Kare physical properties, they should be inde-pendent of time. Theref ore, the diffusion coefficientdetermined at any specific instant (e.g. quasi-steadystate) must be applicable through the process.

The results show that the effective thermal con-ductivity model is adequate for a system of porousmedia under different conditions. This analysis estab-lishes a method of predicting the existence of a dryoutregion for different boundary conditions using thediffusion coefficients.

The moisture diffusion coefficient K is very sensitiveto moisture content. Small errors in the measurementof the moisture content give rise to large errors in theK value.

Acknowledgemenrs-The authors are thankful to Dr R.Singh and Mr A. K. Shrotriya for discussion and suggestionsduring preparation of the manuscript. Financial assistances

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from DNES, New Delhi and TWAS, Trieste (Italy) are grate-fully acknowledged

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