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Progress in Energy and Combustion Science Vol. 37, No. 2, pp. 204-220, ISSN 0360-1285 Convection and Conduction Heat Transfer 232 Veyret, D. & Tsotridis, G. (2010). Numerical determination of the effective thermal conductivity of fibrous materials. Application to proton exchange membrane fuel cell gas diffusion layers. Journal of Power Sources Vol. 195, No. 5, pp. 1302-1307, ISSN 03787753 Vie, P. J. S. & Kjelstrup, S. (2004). Thermal conductivities from temperature profiles in the polymer electrolyte fuel cell. Electrochimica Acta Vol. 49, No. 7, pp. 1069-1077, ISSN 00134686 Wang, J., Carson, J. K., North, M. F. & Cleland, D. J. (2006). A new approach to modelling the effective thermal conductivity of heterogeneous materials. International Journal of Heat and Mass Transfer Vol. 49, No. 17-18, pp. 3075-3083, ISSN 00179310 Wang, J., Carson, J. K., North, M. F. & Cleland, D. J. (2008). 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Introduction Frozen soils consist of soil solids, ice, unfrozen water, and gas (vapour). The solid particles vary in size and composition and may be composed of one or more minerals or of organic material. Based on particle size, soils are classified into soil types which vary between the many classification systems in use throughout the world. The classification which is most generally used in Russia is that of V.V. Okhotin (Sergeev, 1971), with the basic soil types being sand, sand-silt, silt-clay, and clay which are further subdivided into a large number of subtypes. Soils that have been subject to repeated cycles of freezing and thawing generally have higher silt contents. The bound water is structurally and energetically heterogeneous. Water bonding to the mineral particles is provided predominantly by the active centres on the surface and the exchange cations. The most important active centres for water adsorption in the crystalline lattice of clay minerals are hydroxyl groups and coordinately unsaturated atoms of oxygen, silicon, aluminium and other elements. In quantitative terms, it is an undeniable fact that the pore water freezes over a range of negative temperatures rather than at a single temperature, depending on soil moisture content and solute concentration. This is due to distortion of the bound water structure by the active centres on the particle surfaces and dissolved ions, resulting in a kinetic barrier which makes water crystallization difficult. The phase composition of water (or solution) changes with temperature following the dynamic equilibrium state principle established by Tsytovich (1945) and experimentally confirmed by Nersesova (1953). This principle states that the amount of unfrozen water for a given soil type (non-saline) is a function of the temperature below 0°C and is virtually independent of the total soil moisture content. It is quantitatively described by the equation (Ivanov, 1962): uw 0 2 1 WWA' 1, 1a't b't ⎡ ⎤ =+ − ⎢ ⎥ +Δ+Δ ⎢ ⎥ ⎣ ⎦ (1) Convection and Conduction Heat Transfer 234 where Δt = t – t f ; t f is the initial freezing temperature of water; W 0 is the equilibrium moisture content at t f ; and A’, a’ and b’ are the characteristic soil parameters. For a narrow range of freezing temperatures ( |Δt| ≤ 10°C), Eq. (1) can be simplified by assuming b’ = 0. The thermodynamic instability of the phase composition of water in frozen soils causes their properties to be highly dynamic at subzero temperatures. The presence of unfrozen water below 0°C provides conditions for water migration during freezing. This results in the formation of cryostructures and cryotextures that, in turn, cause the anisotropy of soil thermal and other properties. All cryostructural types can be grouped into three board classes: massive, layered, and reticulate (Everdingen, 2002). Model calculations generally consider heat conduction in frozen soils. It is characterized by an effective value of the heat flux transferred by the solid particles and interstitial medium (ice, water and vapour) and through the contacts. It depends on multiple variables which reflect the origin and history of the soil, including moisture content, temperature, dry density, grain size distribution, mineralogical composition, salinity, structure, and texture. A large number of theoretical models and methods were developed for estimating the thermal conductivity of various particulate materials. However, most of them do not address the structural transformations and their validity is limited to a narrow range of material's density. In permafrost investigations, it is essential that properties of snow, soils and rocks be studied in relation to the history of sediment formation through geologic time. Therefore, a universal theoretical model with changing particle shapes was proposed by the present author to describe the processes of rock formation, snow compaction and glacierization with account for diagenetic and post-diagenetic structural modifications, as well the processes of rock weathering and soil formation. A detailed description of the model was given in earlier publications (Gavril’ev, 1992, Gavriliev, 1996, Gavriliev, 1998). Since then, the model has been amended and improved. We therefore find it necessary to present a brief description of the geometric models and the final predictive equations. 2. Theoretical model accounting for structural transformations of sediments 2.1 Soils and sedimentary rocks A model for estimating the thermal conductivity of soils and sedimentary rocks should take into account the changes in particle shape over the entire range of porosity from 0 to 1 in order to consider the entire cycle of sediment changes since its deposition. In developing such a model, it should be kept in mind that mineral rock particles undergo some kind of plastic deformation through geologic time, gradually filling the entire space. Particles bind together at the contacts (“the contact spot”) and rigid crystal bindings develop between the particles. Following the real picture of rock weathering and particle shape changes through diagenesis, the author has proposed a model, which presents the solid component in a cubic cell by three intersecting ellipsoids of revolution (Fig. 1) (Gavril’ev, 1992). In this scheme, depending on the semi-axes ratio of the ellipsoids a/R, the porosity of the system varies from 0 to 1 and the particle attains a variety of shapes, such as cubical, faceted, spherical, worn, and cruciate. This logically represents the real changes in particle shape through the sedimentary history, i.e., the key requirement to the model - adequate representation of the real system – is met. In this scheme, the particles always maintain contacts with each other and the system remains stable and isotropic. The coordinate Analytical Methods for Estimating Thermal Conductivity of Multi-Component Natural Systems in Permafrost Areas 235 number is constant and equal to 6; the relation between the thermal conductivity and porosity is realized by changing the particle shape at various size ratios of the ellipsoids of revolution. At a/R ≥ 1, a contact spot appears automatically in the model, which represents rigid bonding between the particles that provides hard, monolithic rock structure (Gavriliev, 1996). Fig. 1. Particle shapes in the soil thermal conductivity model at different semi-axes ratios of ellipsoids δ = a/R: 1 – faceted ( δ > 1); 2 – spherical (δ = 1); 3 – worn (δ < 1); 4 – cruciate (δ < 1) All calculations are made in terms of the parameter δ = a/R, which is a unique function of the porosity m 2 (dry density γ s ): mod sc λ =λ +ϕ , (2) where λ mod is the resulting thermal conductivity of the model and ϕ sc is the correction for heat transfer across the contact spot, W/(m•K): mod ad 2 2 1.3 11sinm, 1 0.5 0.26 ⎡ ⎤ ⎛⎞ λ=λ + − π ⎢ ⎥ ⎜⎟ ⎜⎟ +ϑ− ϑ ⎢ ⎥ ⎝⎠ ⎣ ⎦ (3) where 21 ;0 1;ϑ=λ λ ≤ϑ≤ λ ad is the thermal conductivity of the system where the elementary cell is divided by adiabatic planes; the subscripts “1” and “2” refer to the particle and the fill (air, water and ice), respectively. The thermal conductivity of the model, λ ad , is given by the following equations: at δ ≤ 1 () () () 2 ad 2111 4 1 1 11 1 H arcsin X 1 1 ln 2KK1K K 4arcsin /X 11 1ln11 XK X 21 ⎡ ⎛⎞ λ πδ π = −−δ+δ δ+ −δ− − ⎢ ⎜⎟ ⎜⎟ λδδ−δ ⎢ ⎝⎠ ⎣ ⎤ ⎡⎤ ⎛⎞ δδ−πδ −−δ + − ×− ⎥ ⎢⎥ ⎜⎟ δπ−δ ⎢⎥ ⎥ ⎝⎠ ⎣⎦ ⎦ , (4) Convection and Conduction Heat Transfer 236 at δ ≥ 1 22 1 ad 2 211 1 1 K 11 1H ln1 42K K XX K arcsin1/X XK 4 ⎛⎞ δ λ ππ δ ⎛⎞ = −−δ + δ− × − + + ⎜⎟ ⎜⎟ λ δ ⎝⎠ ⎝⎠ δ π ⎛⎞ +δ − ⎜⎟ −δ ⎝⎠ , (5) where δ = a/R; 2 X1 ;=+δ 1 11 1 Hln1K1; 2K K ⎛⎞ π =−+ ⎜⎟ ⎝⎠ 2 1 1 K1 . λ =− λ The correction factor ϕ sc is given by 22 2 cc 11 sc 22 22 2 2 1 1 c 1 22 rr 1K 11 ln 2K 2R R K r 1K 1 R ⎡ ⎤ ⎢ ⎥ ⎛⎞ πλ δ ϑ ϑ − ⎢ ⎥ ⎜⎟ ϕ= + − − + ⎢ ⎥ ⎜⎟ δδ ⎝⎠ ⎢ ⎥ −− ⎢ ⎥ δ ⎣ ⎦ , (6) where r c is the radius of the contact spot between the particles. It is assumed in Eq. (6) that the spot contact between particles is formed of the same material as the particle by its flattening at high pressure or by its squeezing (solution and crystallization) due to selective growth of cement in sandstones (quartz cement grows on quartz particles and feldspar on feldspar particles). In a general case however, the contact spot may consist of a foreign material resulting, for example, from precipitation of salts from solution at the particle contacts. In this case, the correction factor ϕ sc is given by () () ( ) 222 1c 2c 3 2 sc 22 2 12 12 22 3 2 c 21 1K 1ra 1K 1ra a ln ln 1K 1K 2R K K 11ra KK ⎡ −− −− π λ λ ⎢ ϕ= − + ⎢ −− ⎣ ⎤ ⎛⎞ λ λ +− −− ⎥ ⎜⎟ ⎥ ⎝⎠ ⎦ , (7) where K 2 = 1 - λ 3 /λ 1 ; λ 1 , λ 2 and λ 3 are the thermal conductivities of the solid, medium and contact spot (contact cement), respectively. The relative size of the contact spot is expressed in terms of the system’s porosity as: () c 3 2 r 1.69 1 . R61m π =− − (8) The soil porosity m 2 or the volume fraction of the mineral particle m 1 is a unique function of the parameter δ and is given by the following equations: at δ ≤ 1 2 2 1 2 1111 m1 3, 6X X ⎡ ⎤ πδ −δ −δ +δ ⎛⎞ =−+ −+ ⎢ ⎥ ⎜⎟ δδ ⎝⎠ ⎢ ⎥ ⎣ ⎦ (9) Analytical Methods for Estimating Thermal Conductivity of Multi-Component Natural Systems in Permafrost Areas 237 at δ ≥ 1 ( ) 2 2 2 1 2 2 12 111 1 m1 4arcsin. 6X X X 16X 2 δ+δ ⎛⎞ ⎛⎞ ⎛⎞ πδ δ− =−+ + ×δ −π ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ δδ +δ ⎝⎠ ⎝⎠ ⎝⎠ (10) The increase in the volume fraction of the solids due to the contact spot is expressed by 22 2 2 cc c sc 22 22 rr r m211. 4 RR R ⎡ ⎤ ⎛⎞ ⎛⎞ π ⎢ ⎥ ⎜⎟ =−δ− −− ⎜⎟ ⎜⎟ ⎜⎟ ⎢ ⎥ δ ⎝⎠ ⎝⎠ ⎣ ⎦ (11) The dry density of the soil is ( ) d1scs mm , γ =+ ρ (12) where ρ s is the solids unit weight. The above equations can be used to calculate the thermal conductivity of soils and sedimentary rocks in the saturated frozen and unfrozen states, as well as in the air-dry state in relation to the porosity m 2 and the thermal conductivity λ 1 of the solid particles (a two- component system). The predictions obtained are presented as nomograms in Fig. 2. It should be noted that in this case, the porosity m 2 refers to the entire volume fraction of the soil or rock which is completely filled either with ice, water, or air. This porosity is related to the volume fraction m s and dry density γ s by 2s1scss m1m1mm 1 . = −=−− =−γρ (13) The model assumes that the material consists of mineral particles of the same composition. However, naturally occurring soils always contain particles of various compositions and they can be treated in modelling as multi-component heterogeneous systems with a statistical particle distribution. In computations based on the universal model, the average thermal conductivity of soil solid particles may be used, which is approximately estimated in terms of the thermal conductivity and volume fraction of constituent minerals according to the equation (Gavriliev, 1989): n 1jj n j j1 j j1 1 0.5 m , m = = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ λ= λ + ⎢ ⎥ ⎢ ⎥ λ ⎢ ⎥ ⎣ ⎦ ∑ ∑ (14) where λ j and m j are the thermal conductivity and volume fraction of the j-th mineral of the soil, respectively. This equation can also be used for calculating the thermal conductivity of rocks characterized by the plane contacts between mineral aggregates. 2.2 Snow In snowpack, the structural changes of ice crystals occur continuously throughout the winter. The thermodynamic processes in snowpack result in a multi-branch openwork structure of Convection and Conduction Heat Transfer 238 contacting ice crystals with shapes that continuously change throughout the period of snow existence. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Porosity (m2) Thermal conductivity (λ), W/(m•K) (a) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Porosity (m2) Thermal conductivity (λ), W/(m•K) (b) [...]... to understand heat transfer processes and to analyze experimental data on thermal properties of soils and rocks from a common point of view 254 Convection and Conduction Heat Transfer Considering the history of sediment formation over a geologic time, a universal model with changing particle shapes is proposed which describes the processes of rock formation from sediments, snow compaction and glacierization... (October-December, 195 9), рр 9- 49 Tsyganov, M ( 195 8) Soil Science (in Russian) Selkhozgiz, Moscow, USSR Tsytovich, N ( 194 5) On the Theory of the Equilibrium State of Water in Frozen Soils (in Russian) Izvestia AN SSSR, Seria Geograficheskaya i Geofizicheskaya, Vol 9, No 56 (November-December, 194 5), pp 493 -502 Vargaftik, N (Ed.) ( 196 3) Handbook of Thermophysical Properties of Gases and Fluids (in Russian),... Heat transport Thermal transport modes in bio-thermal systems involve three typical modes: conduction, convection and radiation Limitation and restrictions of therapeutic temperatures on heating (or freezing) subjects are first, to remove tumorous tissues and at the same time without damaging the normal tissues Table 1 shows significance of thermal transport modes in 258 Convection and Conduction Heat. .. Thermal Engineering and Sciences for Cold Regions, pp 516-521, Ottawa, Canada, May 19- 22, 199 6 Gavrilyev, R ( 199 6b) Calculation of Thermophysical Properties of Large-fragmented Rocks at the Non-uniform Composition of Gruss-crushed Stone and Gravel-pebble Inclusions Proceedings of International Symposium on Cold Regions Engineering, pp 186-1 89, Harbin, China, September 11-14, 199 6 Gavrilyev, R (2001)... Moscow, USSR 256 Convection and Conduction Heat Transfer Sergeev, E., Golodkovskaya, G., Osipov, V & Trofimov, V ( 197 1) Soil and Rock Engineering (in Russian), Izdatelstvo MGU, Moscow, USSR Smyslov, A., Moiseenko, U & Chadovich, T ( 197 9) Thermal Regime and Radioactivity of the Earth (in Russian), Nedra, Leningrad, USSR Sulakvelidze, G.K & Okudzhava, A.M ( 195 9) Snow and Its Properties (in Russian), Transactions... Khusid, B ( 198 1) Thermal Conductivity of Materials with Shell-Covered Dispersed Fillers (in Russian) Vestsi AN BSSR Seryia fizika-matematicheskikh navuk, No.1 (January-March 198 1), pp 91 -95 , ISSN 0374-4760 Birch, F., Schairer, J & Spicer, H (Eds.) ( 194 2) Handbook of Physical Constants, Spec Pap 36, Geological Society of America, Boulder, Colorado, USA Carslaw, H & Jaeger, J ( 195 9) Conduction of Heat in... (hyperthermia or thermal therapy) and much higher temperatures (thermal ablation treatments) than previous one Heating in the biothermal systems involves two primary heat transport modes: thermal conduction and convection Extreme complicating living vasculatures and organs make heating the target volume and raising temperature to therapeutic temperature at the target volume difficult and, thus, a challenging... in Solids (2nd edition), Oxford University Press, ISBN 0- 19- 85-3303 -9, Oxford, UK Clark, S (Ed.) ( 196 6) Handbook of Physical Constants (rev ed.), Geological Society of America, ISBN 081371 097 9, New York, USA Everdingen, R van (Ed.) ( 199 8, revised January, 2002) Multi-Language Glossary of Permafrost and Related Ground-Ice Terms, National Snow and Ice Data Center/World Data Center for Glaciology, Boulder,... Engineering and Sciences for Cold Regions, pp 1 39- 143, ISBN 89- 952282-5-3, Seoul, Korea, July 12-14, 2001 Ivanov, N & Gavriliev, R ( 196 5) Thermal Properties of Frozen Soils (in Russian), Nauka, Moscow, USSR Ivanov, N ( 196 2) Heat Exchange in Permafrost (in Russian), Nauka, Moscow, USSR Kobranova, V ( 196 2) Physical Properties of Rocks (in Russian), Gostopttekhizdat, Moscow, USSR Kokshenov, B ( 195 7) Determining... estimated considering a three-component shell system (mineral particle + unfrozen water + ice) as shown in Fig 8 Mineral particles in this scheme are assumed to be spherical in shape 248 Convection and Conduction Heat Transfer Fig 7 Thermal conductivity vs saturation moisture content for alluvial sediments in frozen state: 1 - sand; 2 – sand-silt; 3 – silt-clay; 4 - experimental curves; 5 – predicted . consider heat conduction in frozen soils. It is characterized by an effective value of the heat flux transferred by the solid particles and interstitial medium (ice, water and vapour) and through. (Gavril’ev, 199 2, Gavriliev, 199 6, Gavriliev, 199 8). Since then, the model has been amended and improved. We therefore find it necessary to present a brief description of the geometric models and the. of elementary functions (Carslaw & Jaeger, 195 9). Let us consider two examples. Convection and Conduction Heat Transfer 242 1. The particles have a shape of an oblate ellipsoid of