THERMAL CONDUCTIVITY OF SOILS—COMPARISON OF EXPERIMENTAL RESULTS AND ESTIMATION METHODS 1 1 Usowicz Boguslaw, 2Usowicz Lukasz Institute of Agrophysics, Polish Academy of Sciences, ul, Doświadczalna 4, 20-290 Lublin 27 e-mail: [email protected] 2 ul. Królowej Jadwigi 6/56, 20-282 Lublin, e-mail: [email protected] Introduction To predict heat transfer in soil under the conditions of steady and non-steady heat flow requires the knowledge of the basic thermal properties of soil (Noborio et al. 1996). While it is possible to determine the heat capacity per unit volume of soil with fairly good accuracy, numerous problems are encountered in the determination of thermal conductivity (Kersten 1949; de Vries, 1963, 1986, 1987; Kohnke and Nakshabandi, 1964; Kasubuchi, 1975; Kossowski, 1977; de Vries and Philip, 1986; Hopmans and Dane, 1986; Abu-Hamdeh, 2001, 2003, Abu-Hamdeh et al., 2001; Bachmann 2001; Gori and Corasaniti, 2003; Nusier and AbuHamdeh, 2003; Tarnawski and Leong, 2000; Usowicz, 1991, 1992). The propose of this study is to presents an evaluation of method for the calculation of the thermal conductivity of soil on the basis of comparison of the thermal conductivity of sand, clay, sandy loam, clay loam, silty loam, silty clay loam and peat, measured and calculated from the statistical-physical model, de Vries model and numerical method. For loam, the evaluation was performed on the basis of comparison of results obtained experimentally, models, Laplace transform, and the method of null-alignment, with and without taking into account the thermal conductivity resulting from water vapour movement due to temperature gradient in the soil. Thermal conductivity One of the components of the heat balance of the active surface is heat flux in the soil G (W m-2). This flux for homogeneous medium can be described from Fourier law: ∂T (1) G = −λ ∂z where λ is thermal conductivity (W m-1 K-1), ∂T / ∂z is temperature gradient (K m-1). In the non-steady state heat flux in the soil is described by equation: ∂T ∂ ⎛ ∂T ⎞ Cv = ⎜λ (2) ⎟ ∂t ∂z ⎝ ∂z ⎠ where Cv (J m-3 K-1) is heat capacity of soil, t (s) is a time. For isotropic medium in one dimension the conductive heat transfer is given by ∂T ∂ 2T =α 2 (3) ∂t ∂z where α is the ratio of thermal conductivity to the heat capacity, called soil thermal diffusivity: α= λ Cv . (4) Method 1 Experimental thermal conductivity used in this paper was published by Kersten (1949), de Vries (1963), Kimball et al. (1976a, 1976b), Asrar and Kanemasu (1983), Sikora and Kossowski (1993) and Ochsner et al. (2001). Thermal properties of soil can be determined by measuring the distribution of temperature in response to heat input to a line sources or were measured using a thermo-time domain reflectometry (Thermo-TDR) probe (Ochsner et al., 001). Thermal conductivity of various soils was measured at different bulk density, water content and temperature of soil. Method 2 De Vries model (de Vries, 1963) based on the application to a granular medium of potential theory. This model considers the soil as a continuous of water for moist and air for dry soil in which granules of soil particles, air or water are dispersed. The thermal conductivity of soil is calculated as the weighted average of the conductivities of various soil components: n λ= ∑k λ x i =1 n i i ∑k x i =1 i i (5) i where n – number of components, λi is the thermal conductivity of each components (Table 1), and xi is the volume friction of each components. Volumes of ki are calculated from: −1 ⎞ ⎤ 1 n ⎡ ⎛λ k i = ∑ ⎢1 + ⎜⎜ i − 1⎟⎟ g i ⎥ (6) 3 i =1 ⎣ ⎝ λ0 ⎠ ⎦ where gi represents the shape factors for i-th components granules considered as ellipsoids. Table 1. Values and expressions for parameters used in calculating the thermal conductivity of soils (T in oC). Reference a Parameters b Expression/value b 9.103 - 0.028 T λq, W m-1 K-1 -1 -1 2 2.93 λm, W m K -1 -1 2 0.251 λo , W m K -1 -1 0.552 + 2.34*10-3 T - 1.1*10-5 T2 1 λw, W m K 1 0.0237 + 0.000064 T λa, W m-1 K-1 -1 -1 2 λapp , W m K λa + hλv 2 h, dimensional exp(ψMw/ρwR(T+273.15)) -1 -1 2 λv, W m K LDaν (dρo/dT) 1 L, J kg-1 2490317 - 2259.4 T Da, m2 s-1 0.0000229*((T+273.15)/273.15)1.75 1 Da, m2 s-1 21.7*10-6 (101.325/P)((T+273.15)/273.15)1.88 2 1 ν, dimensional P/(P-(hρο R (T+273.15)/1000Mw)) -3 -3 10 exp(19.819 - 4975.9/(T+273.15)) 1 ρo kg m -3 -1 1 dρo/dT, kg m K 4975.9 ρo/(T+273.15)2 a 1. Kimball et al. (1976a); 2. De Vries (1963), b ψ − soil water pressure head, kPa; Mw – molecular weight of water, 0.018 kg mol–1; ρw – density of liquid water, 1.0 Mg m–3; R – universal gas constant, 8.3143 J mol–1; h – relative humidity; L – latent heat of vaporisation; Da – diffusion coefficient for water vapour in air; ν – mass flow factor; ρo – saturated vapour density; P – barometric pressure, kPa, thermal conductivity of: quartz, λq, other minerals, λm, organic matter, λo, water or salt solution, λw, and air, λa. In a soil not saturated with water, and with the presence of a temperature gradient, a part of the energy flowing through the soil is transmitted by water vapour. Water vapour flow through soil causes an increase in the thermal conductivity of the soil. The thermal conductivity component related to water vapour and soil air is referred to by de Vries  by the general term compound thermal conductivity and expressed as a sum of the thermal conductivity of air λa and the thermal conductivity of water vapour λv: λapp = λa + λv (7) The thermal conductivity of water vapour λv can be calculated from the equation formulated by Kirscher and Rohnalter (after de Vries, 1963), or from the equation modified by Philip and de Vries (de Vries, 1987) (Table 1). Relative humidity h can be calculated from the Kelvin equation (Table 1) or from an empirical relation (Kimball et al. 1976a), De Vries considers two ranges of water moisture in which relative humidity equals one, or varies according to the Kelvin equation. Within the range between field water capacity and full saturation with water, the relative humidity of the soil air equals one. Below the field water capacity the relative humidity can be calculated from the Kelvin formula, or from an empirical (Kimball et al. 1976a) propose another division of the moisture range into two intervals. The first of these is from zero to a moisture value corresponding to the wilting point, and the second - from the wilting point to soil saturation with water. Method 3 The null-alignment method (Kimball and Jackson, 1975, Pikul and Allmaras, 1984) is based on measurements of moisture and temperatures over a small increments of time and depth and initial estimation of the thermal conductivity at a given reference level. The thermal conductivity of soil at the references depth, i = r was calculated as: Gi λi = . (8) ⎛ ∂T ⎞ ⎜ ⎟ ⎝ ∂z ⎠ i Soil heat flux density, G, for all depths above references depth was determined from: z ⎛ ∂T ⎞ G (z ) = G(z r ) − ∫ ⎜ Cv ⎟dz ∂t ⎠ zr ⎝ (9) where Cv is heat capacity. In the null- alignment method, soil thermal conductivity was calculated by means of the calorimetric method (9) and temperature gradient method, the calculations being started from the point where zero gradients occurred. Method 4 The Laplace transform method (Asrar and Kanemasu, 1983) based on the numerical integration of Laplace transform of heat conduction equation in soil (3): d 2 L[T ( z , t )] s − L[T (z , t )] = 0 (10) dt α where the Laplace time transform is given by: ∞ L[T ( z , t )] = ∫ T ( z , t ) exp(− st )dt , (11) 0 and where s ≥ 5.0 / t max is the Laplace transform parameter, tmax is the maximum duration of the experiment. The transform is used to determine the thermal diffusivity of soil, while the volumetric heat capacity was measured or calculated (18). Thermal conductivity was determined from the relation between thermal diffusivity and heat capacity λ = α ⋅ Cv . (12) Method 5 The statistical-physical model (Usowicz 1991, 1992) is based on terms of heat resistance (Ohm’s law and Fourier’s law), two laws of Kirchhoff and polynomial distribution (Eadie et al., 1989). The volumetric unit of soil in the model (fig. 1a) consists of solid particles, water and air, is treated as a system made up of the elementary geometric figures, in this case spheres, that form overlapping layers (fig. 1b). c) b) a) Fig. 1. Schematic diagram of the statistical model construction: a) unit volume of soil, b) the system of spheres that form overlapping layers, c) parallel connection of resistors in the layers and series between layers. It is assumed that connections between layers of the spheres and the layer between neighbouring spheres will be represented by the serial and parallel connections of thermal resistors, respectively (fig. 1c). Comparison of resultant resistance of the system, with consideration of all possible configurations of particle connections together with a mean thermal resistance of given unit soil volume, allows estimating thermal conductivity of soil λ (W m–1 K–1) according to the equation (Usowicz, 1991, 1992): 4π (13) λ= L u∑ j =1 P ( x1 j ,..., x kj ) x1 j λ 1 (T )r1 + ... + x kj λ k (T )rk where u is the number parallel connections of soil particles treated as thermal resistors, L is the number of all possible combinations of particle configuration, x1, x2 ,..., xk – a number of particles of individual particles of a soil with thermal conductivity λ1, λ2 ,..., λk and particle radii r1, r2 ,..., rk, k where ∑ x ij = u , j=1,2,...,L, P(xij) – probability of occurrence of a given soil particle i =1 configuration calculated from the polynomial distribution: u! x x P ( x1 j ,..., x kj ) = ! f 1 1 j ... f k kj . (14) ! x1 j ...x kj The condition: ∑ P(X = x )= 1 must also be fulfilled. The probability of selecting a given L j =1 j soil constituent (particle) fi, i = s, c, g, in a single trial was determined based on fundamental physical soil properties. In this case f s , f c , and f g are the content of individual minerals and organic matter – f s = 1 − φ , liquid – f c = θ v and air – f g = φ − θ v in a unit of volume, φ – soil porosity. Parameters of the model were defined earlier based on empirical data (Usowicz, 1992; 1995). Degrees of freedom u characterizing the number of parallel combinations of thermal resistors as a function of saturation of soil with water (θv/φ ) are presented on Figure 2, where θv(m3 m–3) is the soil water content, φ(m3 m–3) is the porosity. Equivalent radius of the spheres rk (m) for all soil components were defined according to the equation (16): 14 12 10 u 8 6 4 2 0 0 0.2 0.4 0.6 Water saturation (θv /φ) 0.8 rk = 0.036 f o + 0.044 1 (15) Fig. 2. Number of parallel connections of resistors as a function of soil water saturation. where: fo(m3 m–3) is the volumetric organic matter content. The radius increased with increasing soil organic matter content, with maximum up to 0.08. Sharp transition of the value u with water saturation of soil (θv/φ) resulting in sharp increase in calculated values of soil thermal conductivity and thus in greater evaluation error of the thermal conductivity. To avoid such transition and to reduce the error a procedure was suggested that allow determining the thermal conductivity in the range of any saturation using a linear interpolation of the conductivity in the range (Usowicz and Usowicz, 2004). In this procedure, the thermal conductivity of the medium was determined from general equation of the thermal conductivity (1) for two successive values of u and u+1 (Fig. 2) and corresponding water contents of the medium, θv(u), θv(u+1) and then thermal conductivity for the searched water content of the medium, θv from the following equation: λ =λ (u ) + θ v − θ v (u ) (λ (u + 1) − λ (u )) . θ v (u + 1) − θ v (u ) (16) In data pertaining to a specific soil, five main components were distinguished, of the following values of thermal conductivity: quartz, λq, other minerals, λm, organic matter, λo, water or salt solution, λw, and air, λa (Usowicz and Usowicz, 2004). The values of coefficients of thermal conductivity and their relation to temperature, pressure, and soil water potential are presented in Table 1. For a soil non-saturated with water, with high salt concentration ((Kimball et al. 1976a; Noborio and McInnes, 1993) and with high soil temperature gradient in the soil the coefficient of air thermal conductivity is replaced by apparent thermal conductivity of soil air, λapp, composed of the air thermal conductivity, λa, and the thermal conductivity of water vapour, λv, (de Vries, 1963; Usowicz, 1993, 1995). Method 6 Numerical method (Sikora and Kossowski, 1993) based on solution of the one dimensional of heat conduction equation with it finite-difference form: Ti , j − Ti , j −1 Ti −1, j −1 − 2Ti , j −1 + Ti +1, j −1 =α (17) ∆t (∆z )2 where T is soil temperature, i and j indices of depth z (cm) and time t(s). Numerical method was used to calculated thermal diffusivity of soil. In this case the thermal diffusivity was 2 assumed to be constant through out the soil and stability criterion α ⋅ (∆z ) ≤ 0.5 is ensured (Sikora, Kossowski 1993). Volumetric heat capacity of soil was calculated from (19), thermal conductivity from (12). Heat capacity Volumetric heat capacity Cv (MJ m–3 K–1) was calculated using empirical formulae proposed by Vries de (1963): C v = (2.0 x s + 2.51xo + 4.19 x w ) ⋅ 10 6 (18) or by Sikora and Kossowski (1993): C v = (c s + 4.19 ⋅ 10 3 θ w )ρ (19) 3 –3 where: xs , xo , xw (m m ) are volumetric contributions of mineral and organic components and water, respectively, cs – specific heat of soil solid (713 J kg–1 K–1), θw – soil water content (kg kg–1), ρ (Mg m–3) – soil bulk density. Validation of methods The agreement between predicted and measured results was determined with a mean square error (σb) and relative maximum error (ηb): n σb = ∑( f i =1 mi − f ci ) 2 , (20) k where: fmi is the measured value, fci is the calculated value, k = n – 1 if n < 30 and k = n, n > 30, n – number of data. The relative maximum error was calculated using the following equation: ⎧ f − f ci ⎫ ⋅ 100%⎬ . η b = max ⎨ mi (21) i =1, 2 ,L, n f mi ⎩ ⎭ Also linear regression equations of the thermal conductivity and determination coefficient R2 were developed. Results and discussion The comparisons of the thermal conductivity values of various soils (Fairbanks sand, Healy clay, Felin silty loam, Fairbanks peat and Avondale loam), measured and calculated from the models, are presented in Table 2 and Figure 3. Also comparison results obtained from de Vries model and from the null-alignment method, statistical-physical model and numerical method are presented in Figure 4. The data in Figure 3a indicate that the results obtained from the de Vries model and the statistical-physical models are on a similar level of agreement with the measured values. Regression coefficients were close to unity, however permanent factors in the equation were close to zero. Determination coefficients of the linear regression were high 0.987 and 0.985 respectively. Mean square errors σ (W m–1 K–1) and relative maximum errors η (%) ranged from 0.057 to 0.123 (W m–1 K–1) and from 12 to 38.3% for Stat model and ranged from 0.017 to 0.127 (W m–1 K–1) and from 22 to 70.7% for de Vries model. In the case of data in Figure 3b the calculated and measured thermal conductivity generally agreed wall, R2 = 0.872 for Stat model and 0.938 for de Vries model. Calculated values from de Vries model are notably higher then measured values at low thermal conductivity. In the case of Stat model calculated values are notably lower then measured at low thermal conductivity. Mean square errors and relative maximum errors were 0.238 (W m–1 K–1) and 58.3% for Stat model and 0.149 (W m–1 K–1) and 62.5% for de Vries model. The results presented in Table 2, without taking into account the effect of water vapour, indicate that the statistical-physical model gives the best approximation of the calculated values to the measured ones (Usowicz, 1995). The other methods also provide fairly good approximations. However, their mean square error is nearly twice as high, or even higher than in the case of the statistical-physical model. When the effect of water vapour is taken into account, the de Vries model significantly increases thermal conductivity while the statisticalphysical model only slightly, though the thermal conductivity due to water vapour was calculated identically for both the models (Kimball et al. 1976a). The high contribution of thermal conductivity due to water vapour in the overall thermal conductivity of soil in the de Vries model was caused, most probably, by poor selection of the shape coefficient in the function of moisture. And the question of how significantly the shape coefficient affects the calculated values of thermal conductivity of soil is discussed in the papers by Kimball et al. (1976a,b). Table 2. Average soil temperature, moisture content, heat capacity, and thermal conductivity for 0 to 10 cm of Avondale loam at an average bulk density of 1.4 Mg m–3, quartz fraction of 0.373 g g–1 and particle density of 2.71 Mg m–3 (Usowicz, 1995) Date Thermal conductivity (W m–1 K–1) Average Moisture Heat temp.* content* capacity*. Methods °C 17 May 1973 11 July 1973 20 Sept 1973 21 Sept. 1973 2 Oct. 1973 7 Dec. 1973 25.2 32.3 26.9 26.9 27.2 11.6 3 –3 m m 0.259 0.282 0.276 0.266 0.225 0.261 –3 –1 MJ m K 1 2 2# 3 4 5 5# 2.34 2.43 2.40 2.36 2.20 2.34 1.13 1.21 1.55 1.13 1.04 1.15 1.17 1.20 1.34 1.63 0.96 1.09 1.17 1.19 1.18 1.34 1.59 1.13 1.14 1.17 1.19 1.15 1.26 1.55 1.05 1.01 1.16 1.17 1.00 1.00 1.51 0.88 1.01 1.12 1.14 1.14 1.17 1.46 1.17 1.09 1.16 1.17 0.456 0.131 0.094 0.057 0.067 σb (W m–1 K–1) 0.114 ηb (%) 13.6 51 20 12.2 12 14 Data: 2,2#, 3 from Kimball et al. (1976b) and *, 1,4 from Asrar and Kanemasu (1983). Method: 1 – Measured in the laboratory (Asrar and Kanemasu, 1983); 2 – de Vries model without vapour and 2# – de Vries model with vapour (Kimball et a.l. (1976b); 3 – Null-alignment (Kimball et al. (1976b)); 4 – Laplace transform (Asrar and Kanemasu, (1983)) 5 – statistical-physical model without vapour, 5# – statistical-physical model with vapour. As follows from the comparisons and analyses, the values of thermal conductivity of soil obtained from the models (Tables 2 and Fig. 3a), without taking into account the effect of water vapour, from the method with Laplace transform, and from the null-alignment method, are in fairly good agreement over a wide range of moisture and soil density values. Greater differences were observed at very low moisture values. If no high temperature gradients occur in the soil and it can be assumed that the effect of water vapour on the overall effect of conductivity is negligible, the methods presented can be applied alternatively for the determination of thermal conductivity. As it was to be expected, the values of thermal conductivity of soil obtained from the null-alignment method and from the method with Laplace transform are somewhat underestimated with relation to values calculated from the statistical-physical model and considerably underestimated with relation to values obtained from the de Vries model with the conductivity due to the effect of water vapour taken into account (Table 2). This situation is caused by the fact that a part of the energy is transmitted through the latent heat which is hidden in water vapour. The null-alignment method and the method with Laplace transform do not fully provide for that fact in their calculations. It is known, however, that when water vapour flows through the soil some of the latent energy is released during vapour condensation, which is manifested in an increase in the temperature of the soil layer in which the vapour condensed. That part of the latent heat which passes through the object freely has a decreasing effect on the total thermal conductivity of soil. And thus, with high temperature gradients and high water vapour flow rates it is recommended to use the statistical-physical model and the de Vries 2.5 K-1) 2 1.5 1 de Vries - y = 0.963x + 0.061 2 R = 0.987 0.5 Stat - y = 0.979x + 0.028 R2 = 0.985 0 0 deVries -1 de Vries Stat Linear (de Vries ) Linear (Stat) Modeled thermal conductivity (W m -1 -1 Calculated conductivity (W m K ) model, with special emphasis on correct weight determination. The application of the nullalignment method and the method with Laplace transform is not recommended in this case. 2.5 b) a) 3 y = 0.798x + 0.232 1 2 -1 -1 Measured conductivity (W m K ) Stat 2.0 R2 = 0.938 y = 1.150x - 0.186 R2 = 0.872 1.5 1.0 deVries Stat 1:1 Linear (deVries) Linear (Stat) 0.5 0.0 3 0.0 0.5 1.0 1.5 2.0 2.5 Measured thermal conductivity (W m -1 K-1) Fig. 3. Comparisons of the thermal conductivity values of various soils measured and calculated from the de Vries and statistical-physical (Stat) models. Measured and calculated from the de Vries model data presented in figure (a) and (b) were published by de Vries (1963) and Ochsner et al. (2001). Thermal conductivity, others models 3 Stat - y = 1.011x - 0.030 Null - y = 0.814x + 0.095 2 2 R = 0.989 R = 0.792 2.5 Numerical - y = 0.938x - 0.063 2 R = 0.272 2 1.5 1 Null-alignment Numerical method Model Stat Model Stat Null-alignment Numerical method 0.5 0 0 0.5 1 1.5 2 2.5 3 Thermal conductivity, de Vries model Fig 4. Comparisons of thermal conductivity obtained from de Vries model and from the nullalignment method, numerical method and statistical-physical model (Stat). Predicted data from de Vries model were published in papers: de Vries (1963), Sikora and Kossowski (1993); Null-alignment data, Asrar and Kanemasu (1983) and Sikora and Kossowski (1993); Numerical data, Sikora and Kossowski (1993). Generally, similar values of thermal conductivity were obtained from de Vries model and from Stat model (Fig. 4). Calculated thermal conductivity from the de Vries model and the statistical-physical models were higher then those determined by numerical method and nullalignment method. Conclusion The methods presented can be used alternatively for the determination of thermal conductivity if no high temperature gradients occur in the soil and if it can be assumed that the effect of water vapour on the overall effect of conductivity is slight. 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