Considering temperature dependence of thermo-physical properties of sandy soils

Aleksey V.Malyshev ,Anatoly M.Timofeev

Institute of Physical-Technical Problems of the North.V.P.Larionov,Siberian Branch of the Russian Academy of Sciences,Yakutsk 677000,Russia

1 Introduction

The knowledge of thermophysical properties of sandy soils subject to oil pollution and similar contamination is essential in applied geoecology,specifically to predict the temperature regime for containment of contaminated sites and further recovery.Thermophysical properties of soils are determined experimentally.However,the experimental determination of thermophysical properties such as thermal conductivity of,for example,coarse rocks containing large inclusions is more difficult.The only viable solution is to calculate heat conductivity based on the generalized conductivity theory.The heat conductivity of oil-contaminated sandy soils can be determined by analysis of multicomponent dispersion systems calculations (Stepanov and Timofeev,1994;2003),with varying degrees of contamination and humidity.

In this paper we calculate the heat conductivity of river sand contaminated with oil products according to the model of polydispersive systems (Stepanov and Timofeev,1994),using data of particle size distribution,coordination number (which indicates the number of contacts per particle),porosity,density,volume concentrations of gas-air mixture,oil,water,ice and sand particles.The heat capacity of the contaminated sandy soil was estimated by the formula proposed by Malyshev and Timofeev (2006) utilizing experimental data on the specific heat and density of the components comprising the contaminated soil,such as oil,water,ice,the mixture of air and gas hydrocarbons and the skeleton of the soil.Earlier,we had calculated the thermophysical properties of river sand contaminated with oil in two scenarios of pollution,depending on concentration and order of soil contamination with oil(Malyshev and Timofeev,2008).In this paper we continue our studies within the framework of the presented calculation models,but now take into account the temperature dependence of the amount of unfrozen water.

2 The objects of study

River sand with a bulk density of 1,560 kg/m3was taken as our object of modeling thermophysical properties of contaminated sandy soil.Distilled water with a density of 998.9 kg/m3and diesel fuel of 813 kg/m3was selected as the liquid phase.The sand moisture content was determined by the ratio of the water mass to the mass of dry sand,W=Pw/P.The concentration of oil product in the sand was set with a ratio by analogy with humidity,z=Po/P.The moisture content in the sand was taken as equal to 7.5%,and the concentration of the diesel fuel was 10.5%.To calculate the thermal conductivity of such a multicomponent dispersion system,it was necessary to know the thermophysical properties of the components;some of the data were taken from reference materials (Table 1).

Table 1 Thermophysical properties of polluted sand components

The calculation of the thermophysical properties of the contaminated sand was performed by taking into account the phase transition of pore moisture in two different pollution scenarios:in the first case (Figure 1a),an oil product was introduced into wet sand,and in the second case (Figure 1b),dry sand was polluted by the oil product and then moistened with water.

The location of water and oil in the second pollution scenario can be presented as follows.First,oil was introduced into dry river sand,the sand particles were moistened with it,and then the water was introduced into the polluted sand.The water was mostly adsorbed into areas that were not affected by oil product,that is,into the pores that were free from oil products.In some areas,water with better moisturizing ability could partially occupy space in the pores which was previously occupied by the oil product.It was assumed that the oil product did not completely soak the mineral particles,but only stayed in the vicinity.At temperatures below 0 °C all the water except unfrozen water freezes thus turns into ice.In the second scenario,pollution,despite the fact that the particle is soaked with oil in some places the water is still in contact with mineral particle,so at temperature below 0 °C during ice formation relations and a certain amount of unfrozen water,this fact,as we will see below confirmed by experiment.

Figure 1 Model location of components in the dispersion medium in both pollution scenarios

3 Research methods

To calculate the porosity of sand,upon which the coordination number depends,it was necessary to know the density of the mineral particles.By use of a densimeter,this was found to be 2,750 kg/m3.Porosity,defined by the ratioΠ=1-γ/ρ,amounted to 0.43.We used a method of continuous heat input (Efimov,1986)to determine the composition of the pore moisture phase.The obtained results are given in table 2 and figure 2.

Table 2 Temperature dependence of the amount of unfrozen water amount in the sand

Figure 2 Amount of unfrozen water in the sand

4 Heat capacity

The heat capacity of the moistened sand at low temperatures was an effective value,since it was characterized by the heat of the water–ice phase transition.To determine the volumetric heat capacity of the sand polluted in the two scenarios,we used the concept that each component of a dispersion system is additive and contributes to the total heat capacity of that system(Gavriliev,1970).In our case,the value of the volumetric heat capacity was determined by the expression:

wherecsc,co,cw,andciare the specific heat capacities of the soil skeleton,oil,water,and ice,respectively(units:J/(kg·K));Lis the heat of the phase transition(units:J/kg);andWufis the amount of unfrozen water(units:kg/kg).Cγis a volumetric heat capacity of polluted soil (units:J/(m3·K));Wis a total moisture of soil(units:kg/kg);zis a weight concentration of impurities(units:kg/kg);Tis an independent variable temperature of soil (units:K);γis a volumetric weight of the skeleton (units:kg/m3).

The specific heat capacity of the sand included in expression(1)was determined by the method of continuous heat input (Efimov,1986),and its value was 782 J/(kg·K).The heat of the phase transition was taken as equal to the heat of crystallization of the free water,which was 334,000 J/kg.

5 The model for calculating the heat conductivity of polluted soil

In the first stage of the calculation,the multicomponent system containing water,ice,air,oil,and mineral particles was reduced a two-component mixture with different values of the volume concentration.The volumetric concentrations of air and oil werema+mo=1.

The volumetric concentration of the oil product in the air-oil product system was:

whereΠ,ρ,ρo,ρw,z,andWare,respectively,porosity,density of particles,oil,water,weight concentration of impurities,and moisture content of the sand.

The air in this case was considered as a matrix,and the oil product as an inclusion.Therefore,to calculate the effective conductivity of the mixtures with closed inclusions,this formula (Odelevsky,1951)was applied:

whereλaois the air-oil product system coefficient of heat conductivity,andλa,λoare the values of the heat conductivity of air and oil,respectively,mais a volume fraction of air (units:dimensionless),vaois a ratio heat conduct of oil product to heat conduct of air (units:dimensionless).

In the second step,heat conductivity of a rigid particle–water film–ice-adjacent binary mixture(air-oil product) system was calculated according to the known model of a granular system.Figure 3 shows the averaged element of the soil particles with a film of unfrozen water,ice,and the mixture (air-oil product).The averaged element was divided into adiabatic rings,and there were thermal resistances in each section of the element.In the case of frozen sandy soil,the derivation of the expressions for thermal resistances of the averaged element’s sections (taking into account the variable thickness of water cuffs and ice) was taken from Malyshev and Timofeev (2008).

Figure 3 Averaged element of the soil particles and its equivalent electrical circuit corresponding to thermal resistances of the averaged element’s sections

In wet dispersed material,calculation of thermal conductivity is done by taking into account the thickness of a water film:

whereNkis the coordination number indicating the number of contacts per one particle,this depends on the porosity;ris the radius of the particle (units:m);Δris the thickness of the layer of water;W0is the moisture content of the soil in a thawed state,in a frozen state it is equal to the amount of unfrozen water.

In the case of the frozen state,we used the thermal conductivity of iceλi,instead of the thermal conductivity of waterλw;also,the thickness of the water film was set equal to the thickness of the ice,that is,the difference between the thickness of the water film of the melt sample atW0=W-W1moisture and the unfrozen water film thickness of corresponding moisture atW0=Wuf+W1,whereW1is the ice content (units:kg/kg).It was necessary to consider the expansion of water in the process of freezing:

where Δriis the thickness of the ice layer (units:m);Δrmis the thickness of the cuff water (units:m);ρiis a density of the ice (units:kg/m3).

The heat resistances of the discrete sections were defined by integrals:

whereR1is the thermal resistance the spherical parts of a segment of a particle starting from the horizontal dotted lines and part of the part water cuff (units:(m·K)/W);λcis a heat conduct of the particle (units:W/(m·K));ymis the dimensionless value equal to ratio of the radius of the water cuff to the radius of the particle (units:dimensionless);υwis the dimensionless value equal to the ratio of the heat conductivity of the water (λw) to heat conductivity of the particle (λc)(units:dimensionless);yis a dimensionless variable of integration coordinate (units:dimensionless).

whereR2is a thermal resistance the spherical parts of a segment of a particle starting from the horizontal dotted lines,part water cuff and part ice layer (units:(m·K)/W);yiis the dimensionless value is equal to ratio of the radius of the ice layer to the radius of the particle (units:dimensionless);υiis the dimensionless value equal to the ratio of the heat conductivity of the ice (λi) to heat conductivity of the particle(λc) (units:dimensionless).

whereR3is a thermal resistance of the particles to the horizontal dotted lines in figure 3 (units:(m·K)/W).

whereR4is a thermal resistance of the mixture of air and oil product (units:(m·K)/W);the value ofAis introduced for the compactness of the expression(10).υaois a ratio heat conduct of oil product to heat conduct of air (units:dimensionless).

whereR5is the thermal resistance the spherical parts of a segment of a particle starting from the horizontal dotted lines,part water cuff,part ice layer and part mixture air-oil product (units:(m·K)/W).

whereR6is a thermal resistance adjacent to the particle part of the water cuff (units:(m·K)/W);λwis a heat conduct of the water cuff (units:W/(m·K)).

whereR7is a thermal resistance adjacent to the water cuff part of the ice layer (units:(m·K)/W);λiis a heat conduct of the ice layer (units:W/(m·K)).

whereR8is a thermal resistance part of the water cuff,part of the ice layer and part of the mixture air-oil product (units:(m·K)/W).

whereR9is a thermal resistance part of ice layer and part of the mixture air-oil product (units:(m·K)/W).

The total resistance of the circuit,and thus the thermal conductivity of the whole system,was found on the basis of the equivalent electric circuit shown in figure 3:

whereRis a total thermal resistance of the multicomponents dispersion system (units:(m·K)/W).

The calculation of the heat conductivity in the second pollution scenario took into account the temperature dependence of unfrozen water (Figure 1).In the first stage of the calculation,the heat conductivity of a binary mixture of interlinked components (unfrozen water and air) was determined.The volume fraction of water and air was determined from the ratiosmw′=mw/(mw+ma) andma′=1-mw′,wheremaandmware volume fraction of air and unfrozen water,respectively,defined asma=Π-mw-zγ/ρoandmw=W0γ/ρw.Calculations were made for models with penetrating components of the air and water,and the heat conductivity was equal to (Dulnev and Zarichnak,1974):

Cwwas determined by the solution of the cubic equation:

The dimensionless parameterCwis equal to the ratio of the rod thickness to its length.The rod stands for the unfrozen water.In a frozen condition,Cwis variable and depends on the temperature.

In the second stage of the calculation,the conductivity of a binary mixture of interlinked components (ice and system air-unfrozen water) was determined.The calculation was similar according to the formula(17).

whereλiis a heat conductivity of the ice (units:W/(m·K));λawis a heat conductivity of the mixture air–unfrozen water system (units:W/(m·K));mawis a relative volume fraction air–unfrozen water system(units:dimensionless);miis a volume fraction ice;mi´is a relative volume fraction ice;υawiis a ratio heat conductivity of the mixture air–unfrozen water system to heat conductivity of ice (units:dimensionless);Wis a total moisture content of the soil (units:kg/kg);Wufis the amount of unfrozen water (units:kg/kg);γis a volumetric weight of the sceleton (units:kg/m3);ρiis a density of ice.

In the third stage,the heat conductivity of the binary mixturesλawiand heat conductivityλowas calculated according to formula(3).

whereλois a heat conductivity of the oil product(units:W/(m·K));λawiis a heat conductivity of the mixture air–unfrozen water–ice system (units:W/(m·K));mawiis a relative volume fraction air–unfrozen water–ice system (units:dimensionless);υawiois a ratio heat conductivity of the mixture air–unfrozen water–ice system to heat conductivity of oil product (units:dimensionless).

Finally,in the fourth and final stage,the calculation was made according to formula(3)and a air-unfrozen water-ice-oil product system with a heat conductivity ofλawiowas considered as a matrix,and an particlewith a heat conductivity ofλcwas an inclusion.

whereλcis a heat conductivity of the particle (units:W/(m·K));λawiois a heat conductivity of the mixture air–unfrozen water-ice-oil product system (units:W/(m·K));mawiois a volume fraction air–unfrozen water–ice–oil product system equal a porosity (units:dimensionless);υawiocis a ratio heat conductivity of the mixture air–unfrozen water-ice-oil product system to heat conductivity of particle (units:dimensionless).

The results of the temperature dependence calculation of the contaminated sand thermal properties are given in table 3.

Table 3 Temperature dependence of thermal properties sand in both pollution scenarios

6 Conclusion

The calculation of heat conductivity in the first scenario of diesel pollution showed that this characteristic was increased by 5% in the frozen state and by 9%in the thawed state,compared to the uncontaminated sand.In contrast,in the second pollution scenario the heat conductivity was decreased by 11% in the frozen state and by 17% in the thawed state compared to uncontaminated sand.Thus,the overall impact of pollution on volumetric heat capacity of sandy soil appeared to increase this characteristic,and this increase was apparently proportional to the increase of pollution.At pollution concentration of about 10% and humidity of 7.7% the increase in volumetric heat capacity amounted to 18%.However,the increase of volumetric heat capacity was not dependent on the pollution scenario (the different methods by which pollution was introduced).This indicates that the most significant impact of oil pollution on the thermophysical properties of sandy soil was to change the heat conductivity and volumetric heat capacity of the sandy soil.By determining these characteristics of the temperature dependencies,we can predict the temperature field of contaminated sandy soil with various structures and densities.

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