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A new model to determine the thermal conductivity of fine-grained soils
International Journal of Heat and Mass Transfer 123 (2018) 407–417
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A new model to determine the thermal conductivity of fine-grained soils Jun Bi a, Mingyi Zhang b,⇑, Wenwu Chen a, Jianguo Lu b, Yuanming Lai b a Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education of China and School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China b State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou 730000, China
a r t i c l e
i n f o
Article history: Received 5 September 2017 Received in revised form 8 February 2018 Accepted 10 February 2018
Keywords: Three-parameter model Thermal conductivity Fine-grained soils Regression analysis Sensitivity analysis
a b s t r a c t In the study, a three-parameter model was presented to calculate the thermal conductivity at full range of degree of saturation (S) for fine-grained soils based on the Fredlund and Xing model and the normalized thermal conductivity method. Three parameters (a, b, c) of the new model are determined by two equations and a measured point at a certain S. Two equations were obtained by the correlations between the parameters a, b, c and the basic properties of 30 Canadian soils by regression analysis. Moreover, the relationship between the thermal conductivity and S at full range of S was defined as the thermal conductivity curve (TCC) and was divided into 3 regions for the improvement of the calculation result. According to the sensitivity analysis, it is found that the calculated TCC is the most reliable when a point is measured in the Region 2. In addition, the new calculation model was also verified by 6 Chinese soils, suggesting the new model could present a good calculation result (R2 = 0.97) for the TCC. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Soil is a multi-phase system consisting of three phases, i.e. soil particle, gas, water [1,2]. Thermal conductivity of soils is one of the most important parameters because of its important role in environment, earth science, and engineering applications [1]. Results from Refs. [3–11] show thermal conductivity of soils is determined by many factors: mineralogical composition, particle size, gradation, packing geometry, dry density, porosity, water content, cementation, temperature, pore size, pore shape, pore orientation, and spatial arrangement of pores, etc. According to Ref. [12], these influencing factors can be classified into three types: (1) compositional factors, including mineralogical composition, particle size, shape, gradation, interparticle physical contact, etc. (2) environmental factors, including water content, density, temperature, etc. and (3) other factors, including properties of soil components, ions, salts, additives, and hysteresis effect, etc. Two methods, i.e. experimental measurement and model calculation, have been put forward to obtain the thermal conductivity of soils. The experimental measurement uses steady state method and transient state method to measure the thermal conductivity of soils [13,14]. However, it is usually time-consuming, expensive, limited, and only available in certain conditions [1,22]. Thus, a considerable number
⇑ Corresponding author. E-mail address: [email protected] (M. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.035 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
of researches have been focused on the development of calculation models for the thermal conductivity [15–22]. Dong et al. [2] suggested the calculation models for the thermal conductivity can be divided into the following three categories: (1) Mixing models (theoretical models), obtained by using the series model and parallel model to connect each component in the cubic cell or representative elementary volume (REV). (2) Mathematical models, derived from the predictive models of the other properties, such as dielectric permittivity, electrical conductivity, and hydraulic conductivity. (3) Empirical models, based on the relationship between the effective thermal conductivity and water content or degree of saturation (S). Mixing models are developed by the simplified derivation process with some assumptions, e.g. uniform particle shape, single particle size, series model and parallel model for heat transfer. However, the formula is very complex and it’s usually difficult to determine the parameters of the mixing models [12]. Mathematical models are developed by the predictive models of the other properties (e.g. dielectric permittivity, electrical conductivity, and hydraulic conductivity), by assuming the similarity between the thermal conductivity and the other soil properties without consideration of the influencing factors, such as particle size, particle shape, packing geometry, mineralogy composition, temperature, stress level and cementation [3,5,23–27]. Empirical models have been widely
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used to fit the experimental data by various mathematical formulas between the thermal conductivity and water content. They have simple formulas and high prediction accuracy [12,19]. Besides, there exists a class of models that considers the effect of microstructural features on the thermal conductivity of soils. Likos [28] obtained the thermal conductivity at full range of S from pore-scale thermal conductivity of a single unit pore by upscaling method. Chen et al. [9–11] proposed a homogenization-based model to calculate the thermal conductivity of unsaturated soils with considerations of the microstructural features of soils, e.g. size, shape, orientation, and spatial arrangement of pores. Compared with empirical models, more factors are considered in the models developed by the upscaling method or the homogenization-based method, leading to the complicate formulas. Therefore, the empirical model was chosen in this study to model the relationship between the thermal conductivity and S at full range of S (thermal conductivity curve, TCC). Moreover, the new model should be constrained by the Wiener bounds and Hashin-Shtrikman bounds (HAS bounds) [8,29]. The Wiener bounds, representing the lower and upper values of thermal conductivities, are usually used to constrain thermal conductivity models for any mixtures [12,29] while the HAS bounds proposed by Hashin and Shtrikman [30] are usually used to constrain thermal conductivity models for isotropic mixtures. Besides, the HAS bounds are narrower than the Wiener bounds [8,29]. The objective of the research is to present a new model for calculation of the thermal conductivity for fine-grained soils at full range of S. Three parameters of the new model were determined by two equations and a measured point at a certain S. Experimental data of 36 soils were collected from the Refs. [16,32,33], of which 30 Canadian soils were used to develop the equations while 6 Chinese soils were used to verify the new model.
between Ke and S is defined as the dimensionless thermal conductivity curve (dimensionless TCC).
h ¼ hs C w
1 w c
ln eþð a Þ
9 b > =
lnð1þw=C r Þ C w ¼ 1 lnð1þw 0 =C r Þ
ð1Þ
> ;
where C w is the correction factor; C r is the input value [37]; h is the volumetric water content; hs is the saturated volumetric water content; w is the matrix suction; w0 is the highest suction corresponding to zero water content, taken as 106 kPa [36]; e is the Euler’s number, taken as 2.71828 [38,39]; a, b and c are the curve-fitting parameters.
Ke ¼
8 <0
:
S¼0 1 c
ln eþðaSÞ
b
ð2Þ
S>0
where Ke is the dimensionless Kersten number [15]; S is the degree of saturation (S).
Ke ¼
k kdry ksat kdry
ð3Þ
where k is the thermal conductivity; kdry is the thermal conductivity at dried condition; ksat is the thermal conductivity at saturated condition.
8 < kdry k ¼ kdry þ ðksat kdry Þ :
S¼0 1
ln eþðaSÞ
c
b
S>0
ð4Þ
3. Parameters of the new model 3.1. Effects of the parameters on the dimensionless TCC
2. A new model for TCC Lu and Dong [34] proposed a TCC model based on the similarity between soil water characteristic curve (SWCC) and TCC, and then used the parameters of SWCC model to estimate TCC. He et al. [1] indicated the similar expressions of TCC and SWCC, and proposed a new model for calculation of TCC. Inspired by their work [1,34–36], the Fredlund and Xing model for SWCC (Eq. (1)) [36] was modified to model the relationship between Ke and S (Eq. (2)). Ke (Eq. (3)) is defined as the dimensionless thermal conductivity [31], combination of Eqs. (2) and (3) will lead to Eq. (4). Besides, the relationship
Effect of the parameter a on the dimensionless TCC can be seen in Fig. 1, where a increases from 0.1 to 0.5 when b = 1.5, c = 4. Clearly, the parameter a indicates the critical water content. Critical water content is defined to indicate the water content at which the thermal resistivity increases disproportionately with small reduction in water content [40,41]. Zhang et al. [27] also suggested the thermal resistivity will reach a stable region when the water content increases beyond the critical water content. As the thermal conductivity is in inverse proportion to the thermal resistivity [5], the thermal conductivity will also enter into a stable region when the water content increases beyond the critical water content.
Fig. 1. Relationships between Ke and S when varying the parameter a, b = 1.5, and c = 4.
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Fig. 2. Relationships between Ke and S when varying the parameter b, a = 0.1, and c = 2.
Fig. 3. Relationships between Ke and S when varying the parameter c, a = 0.2, and b = 3.
Effect of the parameter b on the dimensionless TCC is shown in Fig. 2, where b increases from 1.1 to 10 when a = 0.1, c = 2. The parameter b influences the finer content of the soil sample. A small value of b indicates the dimensionless TCC belongs to the coarse-grained soils with sharp increase of Ke at low S (S < 0.1) while a large value of b suggests the dimensionless TCC belongs to the fine-grained soils with small changes in the Ke at low S (S < 0.1) [16,34]. Effect of the parameter c on the dimensionless TCC is shown in Fig. 3, where c increases from 2 to 5 when a = 0.2, b = 3. It is obvious that the parameter c influences the slope of the dimensionless TCC. The slope of the dimensionless TCC increases with the increase of c. 3.2. Methods to determine the parameters 30 fine-grained soils (soil 1 to soil 30) were selected initially from the Refs. [32,33] with P200 (the percentage of soil passing standard sieve No. 200) > 30% [42,43]. The thermal conductivity of each soil sample was measured at full range of S (S = 0.00, 0.10, 0.25, 0.50, 0.70, and 1.00). The 30 soils were used to establish the equations. In addition, 6 Chinese soils (soil 31 to soil 36) from Ref. [16] were used to evaluate the new calculation model. Tarnawski and Leong [32] also collected experimental data from Ref. [16], and used to evaluate the thermal conductivity model for unsaturated soils. The physical properties of these soils are shown in Table 1.
Briefly, the solving process for the model parameters can be summarized as follows: (1) Thermal conductivity at full range of S and the basic soil properties from Refs. [32,33] were initially imported to Microsoft Excel 2003 with P200 > 30% [42,43]. (2) From the above selection, the thermal conductivities of 30 soils were curve-fitted with the new model (Eq. (4)) by MATLAB in order to obtain 30 groups of fitted parameters (a, b, c). (3) After step (2), regression analysis was used to establish the correlations between the fitted parameters a, b, c and the basic soil properties, such as clay content, silt content, sand content, thermal conductivity of soil particles, thermal conductivities of soils at dried and saturated conditions, quartz content, particle density, and porosity. (4) Once the correlations were obtained, the parameters a, b, c can be expressed as functions of the basic soil properties in terms of the maximum correlation coefficients. 3.3. Analysis of parameters Table 2 shows the parameters b and c are poorly correlated with the physical properties, and all the correlation coefficients between the parameters and the soil properties range from 0.71 to 0.77.
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Table 1 Physical properties of soils used for establishing and evaluating the new model. Soil No.
Soil name
Particle size distribution (PSD) Clay
Silt
Sand
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
a
0.10 0.05 0.05 0.06 0.12 0.08 0.09 0.15 0.17 0.10 0.10 0.33 0.08 0.18 0.07 0.14 0.14 0.24 0.21 0.26 0.06 0.15 0.05 0.10 0.42 0.42 0.30 0.41 0.33 0.10 0.22 0.19 0.27 0.32 0.30 0.09
0.57 0.34 0.37 0.38 0.67 0.42 0.39 0.82 0.83 0.66 0.64 0.67 0.56 0.75 0.37 0.54 0.69 0.55 0.76 0.74 0.27 0.83 0.28 0.52 0.58 0.58 0.70 0.59 0.67 0.58 0.51 0.70 0.54 0.60 0.38 0.41
0.32 0.61 0.57 0.56 0.22 0.50 0.51 0.03 0.00 0.24 0.26 0.00 0.37 0.07 0.56 0.32 0.17 0.22 0.03 0.00 0.67 0.02 0.68 0.38 0.00 0.00 0.00 0.00 0.00 0.32 0.27 0.11 0.19 0.08 0.32 0.50
NS-1 a NS-2 a NS-3 a NS-6 a NS-7 a PE-1 a PE-2 a NB-1 a NB-2 a NB-3 a NB-4 a NB-5 a ON-1 a ON-2 a ON-5 a ON-7 a MN-1 a MN-2 a MN-3 a SK-1 a SK-2 a SK-3 a SK-5 a AB-1 a BC-1 a BC-2 a BC-3 a BC-4 a BC-5 a BC-6 b #5 b #6 b #7 b #8 b #9 b #11
ks (W m1 K1)
kdry (W m1 K1)
ksat (W m1 K1)
q quartz content
qs particle density (kg m3)
n porosity
4.23 4.99 5.08 5.28 4.24 4.75 4.50 4.05 3.75 3.20 3.22 4.09 3.29 2.26 3.30 2.97 4.01 5.31 2.63 4.45 4.07 2.91 4.57 3.80 2.43 2.40 2.88 2.16 2.32 3.62 – – – – – –
0.18 0.25 0.29 0.21 0.18 0.26 0.29 0.18 0.16 0.13 0.15 0.20 0.26 0.20 0.25 0.24 0.19 0.26 0.15 0.28 0.23 0.19 0.23 0.17 0.21 0.20 0.20 0.19 0.19 0.19 – – – – – –
1.46 1.93 2.17 1.76 1.40 1.92 1.94 1.46 1.35 1.14 1.31 1.46 1.60 1.16 1.74 1.46 1.42 2.19 1.05 1.97 1.73 1.27 1.84 1.39 1.20 1.21 1.30 1.12 1.14 1.43 – – – – – –
0.51 0.61 0.63 0.65 0.34 0.66 0.58 0.57 0.56 0.55 0.60 0.39 0.28 0.17 0.36 0.25 0.38 0.20 0.21 0.48 0.61 0.37 0.63 0.55 0.21 0.19 0.27 0.17 0.17 0.37 0.47 0.62 0.34 0.36 0.41 0.57
2708 2711 2680 2684 2781 2636 2663 2590 2540 2569 2588 2707 2704 2758 2754 2760 2685 2788 2739 2693 2703 2702 2677 2640 2740 2718 2713 2782 2775 2757 – – – – – –
0.55 0.45 0.40 0.51 0.57 0.44 0.42 0.54 0.56 0.62 0.54 0.54 0.43 0.51 0.38 0.45 0.55 0.41 0.63 0.41 0.45 0.53 0.45 0.55 0.51 0.50 0.51 0.52 0.53 0.52 0.51 0.52 0.52 0.52 0.52 0.49
Note: a Data from Tarnawski et al. [32,33] were used to establish the new model. b Data from Lu et al. [16] and Tarnawski and Leong [32] were used to evaluate the new model.
Table 2 Coefficients of correlation R between the fitted parameters of the new model and the basic soil properties.
a b c Clay content Silt content Sand content ks kdry ksat q
qs n
a
b
c
Clay content
Silt content
Sand content
ks
kdry
ksat
q
qs
n
1.00 0.87 0.84 0.77 0.49 0.71 0.47 0.27 0.46 0.60 0.20 0.32
1.00 0.65 0.56 0.34 0.51 0.28 0.08 0.22 0.42 0.13 0.10
1.00 0.75 0.42 0.65 0.58 0.42 0.60 0.51 0.10 0.47
1.00 0.44 0.79 0.57 0.18 0.47 0.70 0.36 0.20
1.00 0.90 0.48 0.58 0.63 0.43 0.12 0.62
1.00 0.61 0.48 0.66 0.63 0.09 0.52
1.00 0.48 0.86 0.70 0.23 0.40
1.00 0.83 0.14 0.27 0.94
1.00 0.52 0.03 0.80
1.00 0.71 0.14
1.00 0.23
1.00
Nevertheless, both of them are strongly correlated with the parameter a, and the correlation coefficients reach 0.87 and 0.84, respectively. The exponential functions are attempted to model the relationship between b and a and between c and a (Eqs. (5), (6)). Figs. 4 and 5 show that the parameter b decreases with the increase of the parameter a while the parameter c increases with the increase of the parameter a. Both coefficients of determination (R2) are larger than 0.72, indicating good performance of the exponential functions.
b ¼ 2:59 þ 3:06a0:40
ð5Þ
c ¼ 1:83 þ 7:71a2:58
ð6Þ
All the three parameters can be expressed as functions of the parameter a. If a was obtained, the other two parameters can be calculated by Eqs. (5), (6), respectively. In order to obtain a, three thermal conductivities (kdry ; kunsaturate ; ksat ) were measured at dried, unsaturated, and saturated conditions, respectively. Then Ke at
J. Bi et al. / International Journal of Heat and Mass Transfer 123 (2018) 407–417
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Fig. 4. Relationship between the parameter a and the parameter b.
Fig. 5. Relationship between the parameter a and the parameter c.
unsaturated condition was obtained when kunsaturate was normalized by kdry and ksat . After normalization, all the parameters were put into Eq. (2), and the dimensionless TCC was forced to pass through or come close to the Ke at unsaturated condition, which yielded the parameter a. Obviously, kunsaturate has significant influences on the calculation results. Fig. 6 shows the dimensionless TCCs are completely different based on the five kunsaturate (at S = 0.05, 0.25, 0.50, 0.75, 0.95). Therefore, it’s necessary to determine which region of the TCC (or dimensionless TCC) provides the most reliable calculation result. Lu et al. [16] indicated the TCC of fine-grained soils can be characterized by three regions in terms of slope of each region, suggesting S = 0.13 and 0.30 are the dividing points. Actually, the dividing points are not constant and they will change with soil texture [2]. Therefore, a new three-region was proposed to divide the TCC based on the study of Lu et al. [16], as shown in Fig. 7. Region 1 shows the thermal conductivity seldom changes with the increase of water content. When the water content increases in Region 2, it leads to the significant increase of the thermal conductivity until it reaches the border between Region 2 and Region 3. After that, the water content has hardly effect on the thermal conductivity in Region 3. The border between Region 1 and Region 2 can be
approximated by the residual water content (Sr) [34] while the border between Region 2 and Region 3 can be approximated by the critical water content (Sc), showing the thermal conductivity (thermal resistivity) will reach a stable region when the water content increases beyond a certain water content [27,40,41]. In order to determine which region gave the most reliable calculation result, sensitivity analysis was performed to research the effects of the measured points on the calculation results of the dimensionless TCCs. The fitted dimensionless TCC was used because of the variability of the measured thermal conductivity. Five values of S (S = 0.05, 0.25, 0.50, 0.75, 0.95) were selected to calculate the dimensionless TCCs by the previous calculation process. Among the five points, 0.05 is in Region 1, 0.50 is in Region 2, and 0.95 is in Region 3. Moreover, R2 was recommended to judge the calculation results of the proposed method for the dimensionless TCCs. Table 3 shows values of coefficient of determination (R2) for the five points, which indicates the measured point at the S of 0.50 gives the best calculation result for the dimensionless TCC. It also reveals that the measured point locates in the Region 2, as shown in Fig. 6. According to the previous analysis, three thermal conductivities were measured at different S for normalization of kunsaturate . Actually, only kunsaturate at a certain S is needed because kdry and ksat
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Fig. 6. Effect of the measured points on the calculation results of the dimensionless TCC.
Fig. 7. TCC for typical clay based on the three regions.
0.38 to 0.63. He et al. [1] also investigated several models for kdry by the same experimental data and suggested that Eq. (7) performed best among them.
Table 3 R2 of the calculated dimensionless TCCs at various S. Soil S R2
0.05 0.87
0.25 0.60
0.50 0.99
0.75 0.87
0.95 0.64
can be calculated by several calculation models [1,44]. Therefore, the simplified method for calculation of the TCC consists of two equations and a measured point in Region 2. 4. Calculation for the thermal conductivity of soils
kdry ¼ 0:58n þ 0:50
ð7Þ
where n is the porosity. Two methods (Eqs. (8) and (9)) were used to calculate the thermal conductivity of solid particles (ks ). If mineral composition of a soil is known, Eq. (8) is used to calculate ks [44]; otherwise, Eq. (9) is used to calculate ks [31].
ks ¼
Y xj 9 kmj > > = j
X xj ¼ 1 > > ;
ð8Þ
j
4.1. Calculation for the thermal conductivity of soils at dried and saturated conditions Thermal conductivity of soils at dried condition was calculated by the model in Ref. [1], which is suitable for soils with a wide range of textures and quartz contents. Eq. (7) from Ref. [1] shows kdry will be a negative value when porosity (n) approaches 1. However, for 36 soils in this research, Eq. (7) is correct as n ranges from
where kmj is the thermal conductivity of mineral j; xj is the volumetric proportion of mineral j.
( ks ¼
2:01q 7:7q ; q > 0:2 3:01q 7:7q ; q 6 0:2
where q is the quartz content.
ð9Þ
J. Bi et al. / International Journal of Heat and Mass Transfer 123 (2018) 407–417
Thermal conductivity of saturated soils was calculated by the geometric mean method for the high accuracy and simplest form [15,44].
ksat ¼ k1n knw s
ð10Þ
where kw is the thermal conductivity of water.
413
put into Eq. (2). Subsequently, a was obtained by solving the equation. Then b, c could be calculated by Eqs. (5) and (6), respectively. (4) Thermal conductivity of soils at full range of S. TCC was obtained when kdry , ksat from step (1) and the parameters a, b, c from step (3) were put into Eq. (4).
4.2. Steps to calculate the TCC 4.3. Calculation example The TCC of soils can be simply calculated by two equations in the previous section and a measured point in Region 2 in four steps: (1) Thermal conductivities of soils at dried and saturated conditions. The thermal conductivities of dried soils, kdry , and saturated soils, ksat , can be calculated by Eqs. (7) and (10), respectively. If mineral composition of a soil is known, ks can be calculated by Eq. (8); otherwise, ks can be calculated by Eq. (9). (2) Ke at unsaturated condition. kunsaturate was measured in Region 2. Then Ke at unsaturated condition was obtained when kunsaturat was normalized by kdry , ksat from step (1). (3) Solve the equation. All the parameters, e.g. Ke at unsaturated condition from step (2), b from Eq. (5), c from Eq. (6), were
6 Chinese soils (soil 31 to soil 36) from Lu et al. [16] were selected to evaluate the applicability of the new calculation model, following the steps in ‘‘Steps to calculate the TCC” section. The calculation process of this new model can be illustrated by soil 31. The first step is to calculate kdry and ksat . Eq. (9) is used to determine ks because the mineralogical composition is unknown in the present study.
ks ¼ 2:010:47 7:70:47 ¼ 3:77 W=m K Thermal conductivity of the dried soil is:
kdry ¼ 0:58 0:51 þ 0:5 ¼ 0:20 W=m K Thermal conductivity of water equals 0.594 W=m K at 20 °C [16]. So the thermal conductivity of the saturated soil is:
Fig. 8. Calculation result of the new model for soil 31 [16].
Fig. 9. Calculation result of the new model for soil 32 [16].
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1:22 0:20 ¼ 0:80 1:47 0:20
ksat ¼ 0:5940:51 3:7710:51 ¼ 1:47 W=m K
Ke ¼
The second step is to calculate Ke at unsaturated condition. The thermal conductivity of the measured point is 1.22 W m1 K1 at the S of 0.53.
The third step is to put Eqs. (5) and (6) and Ke = 0.80 at the S of 0.53 into Eq. (2),
Fig. 10. Calculation result of the new model for soil 33 [16].
Fig. 11. Calculation result of the new model for soil 34.
Fig. 12. Calculation result of the new model for soil 35 [16].
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1 K e ¼ 0:80 ¼ a 1:83þ7:71a2:58 2:59þ3:06a0:40 ln e þ 0:53
Figs. 8–13 show the results of the calculated TCCs against the fitted TCCs (fitted by Eq. (4)) and the measured thermal conductivities for the 6 soils [16]. For most of the soils, the calculated TCCs match well with the fitted TCCs and the experimental results at full range of S. However, the new calculation model underestimates the thermal conductivities at low water contents for the soil 32 and soil 36. In general, the new model is able to provide accurate calculation result at full range of S. Besides, the 6 calculated TCCs all fall within the Wiener bounds and the HAS bounds [29,30], showing the good performance of the new model. Table 4 shows
The solution of the function is a = 0.28, then b = 2.50, c = 2.12. The fourth step is to obtain the equation of thermal conductivity for soil 31 at full range of S,
k¼
8 <
S¼0
0:20
: 0:20 þ 1:27 ln
2:50 2:12 eþð0:28 S Þ 1
S>0
Fig. 13. Calculation result of the new model for soil 36 [16].
Table 4 Calculated parameters, fitted parameters for 6 soils. Soil No.
Calculated parameters
31 32 33 34 35 36
Fitted parameters
a
b
c
a
b
c
0.28 0.34 0.47 0.42 0.45 0.34
2.50 2.12 1.55 1.74 1.62 2.12
2.12 2.31 2.93 2.65 2.81 2.31
0.23 0.55 0.35 0.38 0.28 0.30
3.87 0.90 2.23 1.88 3.06 1.92
1.84 3.38 2.40 2.69 2.34 1.67
Table 5 Overall performance of the new model. Soil No.
RMSE
AD
MARD
R2
Soil No.
RMSE
AD
MARD
R2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0.07 0.10 0.14 0.09 0.10 0.07 0.09 0.05 0.05 0.05 0.10 0.09 0.09 0.11 0.08 0.06 0.09 0.24
0.05 0.07 0.10 0.04 0.03 0.04 0.01 0.01 0.02 0.03 0.04 0.04 0.05 0.06 0.04 0.02 0.03 0.05
0.06 0.06 0.10 0.07 0.07 0.05 0.07 0.08 0.11 0.12 0.21 0.06 0.05 0.11 0.07 0.04 0.08 0.10
0.98 0.97 0.95 0.98 0.95 0.99 0.98 0.99 0.99 0.98 0.95 0.97 0.97 0.90 0.98 0.98 0.97 0.87
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
0.06 0.14 0.07 0.04 0.07 0.06 0.05 0.03 0.07 0.10 0.09 0.09 0.06 0.08 0.07 0.05 0.06 0.14
0.01 0.10 0.04 0.01 0.04 0.00 0.03 0.01 0.03 0.08 0.07 0.06 0.02 0.04 0.04 0.04 0.03 0.09
0.16 0.08 0.06 0.07 0.05 0.13 0.07 0.06 0.07 0.17 0.17 0.08 0.07 0.10 0.10 0.07 0.09 0.13
0.97 0.95 0.99 0.99 0.99 0.98 0.98 0.99 0.97 0.91 0.93 0.96 0.98 0.97 0.97 0.98 0.98 0.93
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(2) Three regions were proposed to divide the TCC based on the study of Lu et al. [16]. The borders between Region 1 and Region 2 and between Region 2 and Region 3 were approximated by Sr and Sc, respectively. Sensitivity analysis showed that the measured point in the Region 2 gave the most reliable calculation result. (3) On the basis of two equations and a measured point at a certain S, a four-step calculating process was presented to determine the TCC. Compared with the experimental data, the new model could have good performance (R2 = 0.97) in determining the TCC of soils.
Acknowledgements
Fig. 14. Comparison of thermal conductivities between experiment and calculation from the new model.
the calculated parameters are slightly different from the fitted parameters, suggesting the difference between the calculated TCCs and the fitted TCCs. The root mean squared error (RMSE), average deviations (AD), mean of the absolute relative deviations (MARD), and R2 were used to evaluate the overall performance of the new model [1,19,45], as summarized in Table 5. The variations of R2 for the 6 soils are 0.93 to 0.98, showing the high performance of the calculation model. Fig. 14 also shows good performance (R2 = 0.97) of the new calculation model in calculating the thermal conductivity of soils. The results of 30 soils (soil 1 to soil 30) were also calculated by the new model. The low values of RMSE, MARD and high values of R2 indicate the good performance of the new model. The high accuracy of the new calculation model gives us confidence in the calculation of the TCC by the new model. Table 5 also shows that most values of AD are negative. It means that the new calculation model usually underestimates the thermal conductivity [1]. The suggested parameters of other calculation models are fixed for the same soil types. For example, Côté and Konrad model for Ke shows the suggested parameters for gravel and coarse sand, medium and fine sand, silty and clayey soils, silt and clay, and organic fibrous soils are 4.60, 3.55, 1.90, and 0.60, respectively [15]. Lu et al. model for Ke shows the suggested parameters for coarsegrained soils and fine-grained soils are 0.96 and 0.27, respectively [16]. However, the suggested parameter a for this new model isn’t fixed and is determined by two equations and a measured point. It means that the same soil type will lead to different parameter a. Besides, many other factors also influence the thermal conductivity of unsaturated soils, but only S was investigated in this new model. Other factors and microstructural features of soils will be considered in the further investigations. 5. Conclusions A new calculation model was developed to determine the TCC based on 30 Canadian soils. Then the applicability of the new calculation model was evaluated with 6 Chinese soils that had not been introduced during the establishment equation stage. The main conclusions can be summarized as follows. (1) A new three-parameter model was presented to model the thermal conductivity of fine-grained soils at full range of S. Three parameters of the new model were calculated by two equations and a measured point at a certain S.
This research was supported by the National Natural Science Foundation of China (Grant No. 41471063), the 100-Talent Program of the Chinese Academy of Sciences (Granted to Dr. Mingyi Zhang), the Program of the State Key Laboratory of Frozen Soil Engineering (Grant No. SKLFSE-ZT-23), the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences (QYZDY-SSWDQC015), and the STS Program of the Chinese Academy of Sciences (Grant No. HHS-TSS-STS-1502).
Conflict of interest The authors declare that there is no conflict of interest. References [1] H. He, Y. Zhao, M.F. Dyck, B. Si, H. Jin, J. Lv, J. Wang, A modified normalized model for predicting effective soil thermal conductivity, Acta Geotech. (2017) 1–20. [2] Y. Dong, J.S. McCartney, N. Lu, Critical review of thermal conductivity models for unsaturated soils, Geotech. Geol. Eng. 33 (2) (2015) 207–221. [3] V.R. Tarnawski, W.H. Leong, F. Gori, G.D. Buchan, J. Sundberg, Inter-particle contact heat transfer in soil systems at moderate temperatures, Int. J. Energy Res. 26 (15) (2002) 1345–1358. [4] V.R. Tarnawski, T. Momose, W.H. Leong, Assessing the impact of quartz content on the prediction of soil thermal conductivity, Géotechnique 59 (4) (2009) 331–338. [5] M. Gangadhara Rao, D.N. Singh, A generalized relationship to estimate thermal resistivity of soils, Canadian Geotech. J. 36 (4) (1999) 767–773. [6] D.N. Singh, K. Devid, Generalized relationships for estimating soil thermal resistivity, Exp. Therm. Fluid Sci. 22 (3) (2000) 133–143. [7] T.S. Yun, J.C. Santamarina, Fundamental study of thermal conduction in dry soils, Granular Matter 10 (3) (2007) 197. [8] F. Tong, L. Jing, R.W. Zimmerman, An effective thermal conductivity model of geological porous media for coupled thermo-hydro-mechanical systems with multiphase flow, Int. J. Rock Mech. Mining Sci. 46 (8) (2009) 1358–1369. [9] Y. Chen, S. Zhou, R. Hu, C. Zhou, A homogenization-based model for estimating effective thermal conductivity of unsaturated compacted bentonites, Int. J. Heat Mass Transfer 83 (Suppl. C) (2015) 731–740. [10] M. Wang, Y.F. Chen, S. Zhou, R. Hu, C.B. Zhou, A homogenization-based model for the effective thermal conductivity of bentonite-sand-based buffer material, Int. Commun. Heat Mass Transfer 68 (Suppl. C) (2015) 43–49. [11] Y. Chen, M. Wang, S. Zhou, R. Hu, C. Zhou, An effective thermal conductivity model for unsaturated compacted bentonites with consideration of bimodal shape of pore size distribution, Sci. China Technol. Sci. 58 (2) (2015) 369–380. [12] N. Zhang, Z. Wang, Review of soil thermal conductivity and predictive models, Int. J. Therm. Sci. 117 (2017) 172–183. [13] D. Barry-Macaulay, A. Bouazza, R.M. Singh, B. Wang, P.G. Ranjith, Thermal conductivity of soils and rocks from the Melbourne (Australia) region, Eng. Geol. 164 (2013) 131–138. [14] G. Cai, T. Zhang, A.J. Puppala, S. Liu, Thermal characterization and prediction model of typical soils in Nanjing area of China, Eng. Geol. 191 (2015) 23–30. [15] J. Côté, J.M. Konrad, A generalized thermal conductivity model for soils and construction materials, Canadian Geotech. J. 42 (2) (2005) 443–458. [16] S. Lu, T. Ren, Y. Gong, R. Horton, An improved model for predicting soil thermal conductivity from water content at room temperature, Soil Sci. Soc. Am. J. 71 (1) (2007) 8–14. [17] F. Gori, S. Corasaniti, New model to evaluate the effective thermal conductivity of three-phase soils, Int. Commun. Heat Mass Transfer 47 (2013) 1–6. [18] F. Gori, S. Corasaniti, Effective thermal conductivity of composites, Int. J. Heat Mass Transfer 77 (2014) 653–661.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A new model to determine the thermal conductivity of fine-grained soils Jun Bi a, Mingyi Zhang b,⇑, Wenwu Chen a, Jianguo Lu b, Yuanming Lai b a Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education of China and School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China b State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou 730000, China
a r t i c l e
i n f o
Article history: Received 5 September 2017 Received in revised form 8 February 2018 Accepted 10 February 2018
Keywords: Three-parameter model Thermal conductivity Fine-grained soils Regression analysis Sensitivity analysis
a b s t r a c t In the study, a three-parameter model was presented to calculate the thermal conductivity at full range of degree of saturation (S) for fine-grained soils based on the Fredlund and Xing model and the normalized thermal conductivity method. Three parameters (a, b, c) of the new model are determined by two equations and a measured point at a certain S. Two equations were obtained by the correlations between the parameters a, b, c and the basic properties of 30 Canadian soils by regression analysis. Moreover, the relationship between the thermal conductivity and S at full range of S was defined as the thermal conductivity curve (TCC) and was divided into 3 regions for the improvement of the calculation result. According to the sensitivity analysis, it is found that the calculated TCC is the most reliable when a point is measured in the Region 2. In addition, the new calculation model was also verified by 6 Chinese soils, suggesting the new model could present a good calculation result (R2 = 0.97) for the TCC. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Soil is a multi-phase system consisting of three phases, i.e. soil particle, gas, water [1,2]. Thermal conductivity of soils is one of the most important parameters because of its important role in environment, earth science, and engineering applications [1]. Results from Refs. [3–11] show thermal conductivity of soils is determined by many factors: mineralogical composition, particle size, gradation, packing geometry, dry density, porosity, water content, cementation, temperature, pore size, pore shape, pore orientation, and spatial arrangement of pores, etc. According to Ref. [12], these influencing factors can be classified into three types: (1) compositional factors, including mineralogical composition, particle size, shape, gradation, interparticle physical contact, etc. (2) environmental factors, including water content, density, temperature, etc. and (3) other factors, including properties of soil components, ions, salts, additives, and hysteresis effect, etc. Two methods, i.e. experimental measurement and model calculation, have been put forward to obtain the thermal conductivity of soils. The experimental measurement uses steady state method and transient state method to measure the thermal conductivity of soils [13,14]. However, it is usually time-consuming, expensive, limited, and only available in certain conditions [1,22]. Thus, a considerable number
⇑ Corresponding author. E-mail address: [email protected] (M. Zhang). https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.035 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
of researches have been focused on the development of calculation models for the thermal conductivity [15–22]. Dong et al. [2] suggested the calculation models for the thermal conductivity can be divided into the following three categories: (1) Mixing models (theoretical models), obtained by using the series model and parallel model to connect each component in the cubic cell or representative elementary volume (REV). (2) Mathematical models, derived from the predictive models of the other properties, such as dielectric permittivity, electrical conductivity, and hydraulic conductivity. (3) Empirical models, based on the relationship between the effective thermal conductivity and water content or degree of saturation (S). Mixing models are developed by the simplified derivation process with some assumptions, e.g. uniform particle shape, single particle size, series model and parallel model for heat transfer. However, the formula is very complex and it’s usually difficult to determine the parameters of the mixing models [12]. Mathematical models are developed by the predictive models of the other properties (e.g. dielectric permittivity, electrical conductivity, and hydraulic conductivity), by assuming the similarity between the thermal conductivity and the other soil properties without consideration of the influencing factors, such as particle size, particle shape, packing geometry, mineralogy composition, temperature, stress level and cementation [3,5,23–27]. Empirical models have been widely
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used to fit the experimental data by various mathematical formulas between the thermal conductivity and water content. They have simple formulas and high prediction accuracy [12,19]. Besides, there exists a class of models that considers the effect of microstructural features on the thermal conductivity of soils. Likos [28] obtained the thermal conductivity at full range of S from pore-scale thermal conductivity of a single unit pore by upscaling method. Chen et al. [9–11] proposed a homogenization-based model to calculate the thermal conductivity of unsaturated soils with considerations of the microstructural features of soils, e.g. size, shape, orientation, and spatial arrangement of pores. Compared with empirical models, more factors are considered in the models developed by the upscaling method or the homogenization-based method, leading to the complicate formulas. Therefore, the empirical model was chosen in this study to model the relationship between the thermal conductivity and S at full range of S (thermal conductivity curve, TCC). Moreover, the new model should be constrained by the Wiener bounds and Hashin-Shtrikman bounds (HAS bounds) [8,29]. The Wiener bounds, representing the lower and upper values of thermal conductivities, are usually used to constrain thermal conductivity models for any mixtures [12,29] while the HAS bounds proposed by Hashin and Shtrikman [30] are usually used to constrain thermal conductivity models for isotropic mixtures. Besides, the HAS bounds are narrower than the Wiener bounds [8,29]. The objective of the research is to present a new model for calculation of the thermal conductivity for fine-grained soils at full range of S. Three parameters of the new model were determined by two equations and a measured point at a certain S. Experimental data of 36 soils were collected from the Refs. [16,32,33], of which 30 Canadian soils were used to develop the equations while 6 Chinese soils were used to verify the new model.
between Ke and S is defined as the dimensionless thermal conductivity curve (dimensionless TCC).
h ¼ hs C w
1 w c
ln eþð a Þ
9 b > =
lnð1þw=C r Þ C w ¼ 1 lnð1þw 0 =C r Þ
ð1Þ
> ;
where C w is the correction factor; C r is the input value [37]; h is the volumetric water content; hs is the saturated volumetric water content; w is the matrix suction; w0 is the highest suction corresponding to zero water content, taken as 106 kPa [36]; e is the Euler’s number, taken as 2.71828 [38,39]; a, b and c are the curve-fitting parameters.
Ke ¼
8 <0
:
S¼0 1 c
ln eþðaSÞ
b
ð2Þ
S>0
where Ke is the dimensionless Kersten number [15]; S is the degree of saturation (S).
Ke ¼
k kdry ksat kdry
ð3Þ
where k is the thermal conductivity; kdry is the thermal conductivity at dried condition; ksat is the thermal conductivity at saturated condition.
8 < kdry k ¼ kdry þ ðksat kdry Þ :
S¼0 1
ln eþðaSÞ
c
b
S>0
ð4Þ
3. Parameters of the new model 3.1. Effects of the parameters on the dimensionless TCC
2. A new model for TCC Lu and Dong [34] proposed a TCC model based on the similarity between soil water characteristic curve (SWCC) and TCC, and then used the parameters of SWCC model to estimate TCC. He et al. [1] indicated the similar expressions of TCC and SWCC, and proposed a new model for calculation of TCC. Inspired by their work [1,34–36], the Fredlund and Xing model for SWCC (Eq. (1)) [36] was modified to model the relationship between Ke and S (Eq. (2)). Ke (Eq. (3)) is defined as the dimensionless thermal conductivity [31], combination of Eqs. (2) and (3) will lead to Eq. (4). Besides, the relationship
Effect of the parameter a on the dimensionless TCC can be seen in Fig. 1, where a increases from 0.1 to 0.5 when b = 1.5, c = 4. Clearly, the parameter a indicates the critical water content. Critical water content is defined to indicate the water content at which the thermal resistivity increases disproportionately with small reduction in water content [40,41]. Zhang et al. [27] also suggested the thermal resistivity will reach a stable region when the water content increases beyond the critical water content. As the thermal conductivity is in inverse proportion to the thermal resistivity [5], the thermal conductivity will also enter into a stable region when the water content increases beyond the critical water content.
Fig. 1. Relationships between Ke and S when varying the parameter a, b = 1.5, and c = 4.
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Fig. 2. Relationships between Ke and S when varying the parameter b, a = 0.1, and c = 2.
Fig. 3. Relationships between Ke and S when varying the parameter c, a = 0.2, and b = 3.
Effect of the parameter b on the dimensionless TCC is shown in Fig. 2, where b increases from 1.1 to 10 when a = 0.1, c = 2. The parameter b influences the finer content of the soil sample. A small value of b indicates the dimensionless TCC belongs to the coarse-grained soils with sharp increase of Ke at low S (S < 0.1) while a large value of b suggests the dimensionless TCC belongs to the fine-grained soils with small changes in the Ke at low S (S < 0.1) [16,34]. Effect of the parameter c on the dimensionless TCC is shown in Fig. 3, where c increases from 2 to 5 when a = 0.2, b = 3. It is obvious that the parameter c influences the slope of the dimensionless TCC. The slope of the dimensionless TCC increases with the increase of c. 3.2. Methods to determine the parameters 30 fine-grained soils (soil 1 to soil 30) were selected initially from the Refs. [32,33] with P200 (the percentage of soil passing standard sieve No. 200) > 30% [42,43]. The thermal conductivity of each soil sample was measured at full range of S (S = 0.00, 0.10, 0.25, 0.50, 0.70, and 1.00). The 30 soils were used to establish the equations. In addition, 6 Chinese soils (soil 31 to soil 36) from Ref. [16] were used to evaluate the new calculation model. Tarnawski and Leong [32] also collected experimental data from Ref. [16], and used to evaluate the thermal conductivity model for unsaturated soils. The physical properties of these soils are shown in Table 1.
Briefly, the solving process for the model parameters can be summarized as follows: (1) Thermal conductivity at full range of S and the basic soil properties from Refs. [32,33] were initially imported to Microsoft Excel 2003 with P200 > 30% [42,43]. (2) From the above selection, the thermal conductivities of 30 soils were curve-fitted with the new model (Eq. (4)) by MATLAB in order to obtain 30 groups of fitted parameters (a, b, c). (3) After step (2), regression analysis was used to establish the correlations between the fitted parameters a, b, c and the basic soil properties, such as clay content, silt content, sand content, thermal conductivity of soil particles, thermal conductivities of soils at dried and saturated conditions, quartz content, particle density, and porosity. (4) Once the correlations were obtained, the parameters a, b, c can be expressed as functions of the basic soil properties in terms of the maximum correlation coefficients. 3.3. Analysis of parameters Table 2 shows the parameters b and c are poorly correlated with the physical properties, and all the correlation coefficients between the parameters and the soil properties range from 0.71 to 0.77.
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Table 1 Physical properties of soils used for establishing and evaluating the new model. Soil No.
Soil name
Particle size distribution (PSD) Clay
Silt
Sand
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
a
0.10 0.05 0.05 0.06 0.12 0.08 0.09 0.15 0.17 0.10 0.10 0.33 0.08 0.18 0.07 0.14 0.14 0.24 0.21 0.26 0.06 0.15 0.05 0.10 0.42 0.42 0.30 0.41 0.33 0.10 0.22 0.19 0.27 0.32 0.30 0.09
0.57 0.34 0.37 0.38 0.67 0.42 0.39 0.82 0.83 0.66 0.64 0.67 0.56 0.75 0.37 0.54 0.69 0.55 0.76 0.74 0.27 0.83 0.28 0.52 0.58 0.58 0.70 0.59 0.67 0.58 0.51 0.70 0.54 0.60 0.38 0.41
0.32 0.61 0.57 0.56 0.22 0.50 0.51 0.03 0.00 0.24 0.26 0.00 0.37 0.07 0.56 0.32 0.17 0.22 0.03 0.00 0.67 0.02 0.68 0.38 0.00 0.00 0.00 0.00 0.00 0.32 0.27 0.11 0.19 0.08 0.32 0.50
NS-1 a NS-2 a NS-3 a NS-6 a NS-7 a PE-1 a PE-2 a NB-1 a NB-2 a NB-3 a NB-4 a NB-5 a ON-1 a ON-2 a ON-5 a ON-7 a MN-1 a MN-2 a MN-3 a SK-1 a SK-2 a SK-3 a SK-5 a AB-1 a BC-1 a BC-2 a BC-3 a BC-4 a BC-5 a BC-6 b #5 b #6 b #7 b #8 b #9 b #11
ks (W m1 K1)
kdry (W m1 K1)
ksat (W m1 K1)
q quartz content
qs particle density (kg m3)
n porosity
4.23 4.99 5.08 5.28 4.24 4.75 4.50 4.05 3.75 3.20 3.22 4.09 3.29 2.26 3.30 2.97 4.01 5.31 2.63 4.45 4.07 2.91 4.57 3.80 2.43 2.40 2.88 2.16 2.32 3.62 – – – – – –
0.18 0.25 0.29 0.21 0.18 0.26 0.29 0.18 0.16 0.13 0.15 0.20 0.26 0.20 0.25 0.24 0.19 0.26 0.15 0.28 0.23 0.19 0.23 0.17 0.21 0.20 0.20 0.19 0.19 0.19 – – – – – –
1.46 1.93 2.17 1.76 1.40 1.92 1.94 1.46 1.35 1.14 1.31 1.46 1.60 1.16 1.74 1.46 1.42 2.19 1.05 1.97 1.73 1.27 1.84 1.39 1.20 1.21 1.30 1.12 1.14 1.43 – – – – – –
0.51 0.61 0.63 0.65 0.34 0.66 0.58 0.57 0.56 0.55 0.60 0.39 0.28 0.17 0.36 0.25 0.38 0.20 0.21 0.48 0.61 0.37 0.63 0.55 0.21 0.19 0.27 0.17 0.17 0.37 0.47 0.62 0.34 0.36 0.41 0.57
2708 2711 2680 2684 2781 2636 2663 2590 2540 2569 2588 2707 2704 2758 2754 2760 2685 2788 2739 2693 2703 2702 2677 2640 2740 2718 2713 2782 2775 2757 – – – – – –
0.55 0.45 0.40 0.51 0.57 0.44 0.42 0.54 0.56 0.62 0.54 0.54 0.43 0.51 0.38 0.45 0.55 0.41 0.63 0.41 0.45 0.53 0.45 0.55 0.51 0.50 0.51 0.52 0.53 0.52 0.51 0.52 0.52 0.52 0.52 0.49
Note: a Data from Tarnawski et al. [32,33] were used to establish the new model. b Data from Lu et al. [16] and Tarnawski and Leong [32] were used to evaluate the new model.
Table 2 Coefficients of correlation R between the fitted parameters of the new model and the basic soil properties.
a b c Clay content Silt content Sand content ks kdry ksat q
qs n
a
b
c
Clay content
Silt content
Sand content
ks
kdry
ksat
q
qs
n
1.00 0.87 0.84 0.77 0.49 0.71 0.47 0.27 0.46 0.60 0.20 0.32
1.00 0.65 0.56 0.34 0.51 0.28 0.08 0.22 0.42 0.13 0.10
1.00 0.75 0.42 0.65 0.58 0.42 0.60 0.51 0.10 0.47
1.00 0.44 0.79 0.57 0.18 0.47 0.70 0.36 0.20
1.00 0.90 0.48 0.58 0.63 0.43 0.12 0.62
1.00 0.61 0.48 0.66 0.63 0.09 0.52
1.00 0.48 0.86 0.70 0.23 0.40
1.00 0.83 0.14 0.27 0.94
1.00 0.52 0.03 0.80
1.00 0.71 0.14
1.00 0.23
1.00
Nevertheless, both of them are strongly correlated with the parameter a, and the correlation coefficients reach 0.87 and 0.84, respectively. The exponential functions are attempted to model the relationship between b and a and between c and a (Eqs. (5), (6)). Figs. 4 and 5 show that the parameter b decreases with the increase of the parameter a while the parameter c increases with the increase of the parameter a. Both coefficients of determination (R2) are larger than 0.72, indicating good performance of the exponential functions.
b ¼ 2:59 þ 3:06a0:40
ð5Þ
c ¼ 1:83 þ 7:71a2:58
ð6Þ
All the three parameters can be expressed as functions of the parameter a. If a was obtained, the other two parameters can be calculated by Eqs. (5), (6), respectively. In order to obtain a, three thermal conductivities (kdry ; kunsaturate ; ksat ) were measured at dried, unsaturated, and saturated conditions, respectively. Then Ke at
J. Bi et al. / International Journal of Heat and Mass Transfer 123 (2018) 407–417
411
Fig. 4. Relationship between the parameter a and the parameter b.
Fig. 5. Relationship between the parameter a and the parameter c.
unsaturated condition was obtained when kunsaturate was normalized by kdry and ksat . After normalization, all the parameters were put into Eq. (2), and the dimensionless TCC was forced to pass through or come close to the Ke at unsaturated condition, which yielded the parameter a. Obviously, kunsaturate has significant influences on the calculation results. Fig. 6 shows the dimensionless TCCs are completely different based on the five kunsaturate (at S = 0.05, 0.25, 0.50, 0.75, 0.95). Therefore, it’s necessary to determine which region of the TCC (or dimensionless TCC) provides the most reliable calculation result. Lu et al. [16] indicated the TCC of fine-grained soils can be characterized by three regions in terms of slope of each region, suggesting S = 0.13 and 0.30 are the dividing points. Actually, the dividing points are not constant and they will change with soil texture [2]. Therefore, a new three-region was proposed to divide the TCC based on the study of Lu et al. [16], as shown in Fig. 7. Region 1 shows the thermal conductivity seldom changes with the increase of water content. When the water content increases in Region 2, it leads to the significant increase of the thermal conductivity until it reaches the border between Region 2 and Region 3. After that, the water content has hardly effect on the thermal conductivity in Region 3. The border between Region 1 and Region 2 can be
approximated by the residual water content (Sr) [34] while the border between Region 2 and Region 3 can be approximated by the critical water content (Sc), showing the thermal conductivity (thermal resistivity) will reach a stable region when the water content increases beyond a certain water content [27,40,41]. In order to determine which region gave the most reliable calculation result, sensitivity analysis was performed to research the effects of the measured points on the calculation results of the dimensionless TCCs. The fitted dimensionless TCC was used because of the variability of the measured thermal conductivity. Five values of S (S = 0.05, 0.25, 0.50, 0.75, 0.95) were selected to calculate the dimensionless TCCs by the previous calculation process. Among the five points, 0.05 is in Region 1, 0.50 is in Region 2, and 0.95 is in Region 3. Moreover, R2 was recommended to judge the calculation results of the proposed method for the dimensionless TCCs. Table 3 shows values of coefficient of determination (R2) for the five points, which indicates the measured point at the S of 0.50 gives the best calculation result for the dimensionless TCC. It also reveals that the measured point locates in the Region 2, as shown in Fig. 6. According to the previous analysis, three thermal conductivities were measured at different S for normalization of kunsaturate . Actually, only kunsaturate at a certain S is needed because kdry and ksat
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Fig. 6. Effect of the measured points on the calculation results of the dimensionless TCC.
Fig. 7. TCC for typical clay based on the three regions.
0.38 to 0.63. He et al. [1] also investigated several models for kdry by the same experimental data and suggested that Eq. (7) performed best among them.
Table 3 R2 of the calculated dimensionless TCCs at various S. Soil S R2
0.05 0.87
0.25 0.60
0.50 0.99
0.75 0.87
0.95 0.64
can be calculated by several calculation models [1,44]. Therefore, the simplified method for calculation of the TCC consists of two equations and a measured point in Region 2. 4. Calculation for the thermal conductivity of soils
kdry ¼ 0:58n þ 0:50
ð7Þ
where n is the porosity. Two methods (Eqs. (8) and (9)) were used to calculate the thermal conductivity of solid particles (ks ). If mineral composition of a soil is known, Eq. (8) is used to calculate ks [44]; otherwise, Eq. (9) is used to calculate ks [31].
ks ¼
Y xj 9 kmj > > = j
X xj ¼ 1 > > ;
ð8Þ
j
4.1. Calculation for the thermal conductivity of soils at dried and saturated conditions Thermal conductivity of soils at dried condition was calculated by the model in Ref. [1], which is suitable for soils with a wide range of textures and quartz contents. Eq. (7) from Ref. [1] shows kdry will be a negative value when porosity (n) approaches 1. However, for 36 soils in this research, Eq. (7) is correct as n ranges from
where kmj is the thermal conductivity of mineral j; xj is the volumetric proportion of mineral j.
( ks ¼
2:01q 7:7q ; q > 0:2 3:01q 7:7q ; q 6 0:2
where q is the quartz content.
ð9Þ
J. Bi et al. / International Journal of Heat and Mass Transfer 123 (2018) 407–417
Thermal conductivity of saturated soils was calculated by the geometric mean method for the high accuracy and simplest form [15,44].
ksat ¼ k1n knw s
ð10Þ
where kw is the thermal conductivity of water.
413
put into Eq. (2). Subsequently, a was obtained by solving the equation. Then b, c could be calculated by Eqs. (5) and (6), respectively. (4) Thermal conductivity of soils at full range of S. TCC was obtained when kdry , ksat from step (1) and the parameters a, b, c from step (3) were put into Eq. (4).
4.2. Steps to calculate the TCC 4.3. Calculation example The TCC of soils can be simply calculated by two equations in the previous section and a measured point in Region 2 in four steps: (1) Thermal conductivities of soils at dried and saturated conditions. The thermal conductivities of dried soils, kdry , and saturated soils, ksat , can be calculated by Eqs. (7) and (10), respectively. If mineral composition of a soil is known, ks can be calculated by Eq. (8); otherwise, ks can be calculated by Eq. (9). (2) Ke at unsaturated condition. kunsaturate was measured in Region 2. Then Ke at unsaturated condition was obtained when kunsaturat was normalized by kdry , ksat from step (1). (3) Solve the equation. All the parameters, e.g. Ke at unsaturated condition from step (2), b from Eq. (5), c from Eq. (6), were
6 Chinese soils (soil 31 to soil 36) from Lu et al. [16] were selected to evaluate the applicability of the new calculation model, following the steps in ‘‘Steps to calculate the TCC” section. The calculation process of this new model can be illustrated by soil 31. The first step is to calculate kdry and ksat . Eq. (9) is used to determine ks because the mineralogical composition is unknown in the present study.
ks ¼ 2:010:47 7:70:47 ¼ 3:77 W=m K Thermal conductivity of the dried soil is:
kdry ¼ 0:58 0:51 þ 0:5 ¼ 0:20 W=m K Thermal conductivity of water equals 0.594 W=m K at 20 °C [16]. So the thermal conductivity of the saturated soil is:
Fig. 8. Calculation result of the new model for soil 31 [16].
Fig. 9. Calculation result of the new model for soil 32 [16].
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1:22 0:20 ¼ 0:80 1:47 0:20
ksat ¼ 0:5940:51 3:7710:51 ¼ 1:47 W=m K
Ke ¼
The second step is to calculate Ke at unsaturated condition. The thermal conductivity of the measured point is 1.22 W m1 K1 at the S of 0.53.
The third step is to put Eqs. (5) and (6) and Ke = 0.80 at the S of 0.53 into Eq. (2),
Fig. 10. Calculation result of the new model for soil 33 [16].
Fig. 11. Calculation result of the new model for soil 34.
Fig. 12. Calculation result of the new model for soil 35 [16].
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1 K e ¼ 0:80 ¼ a 1:83þ7:71a2:58 2:59þ3:06a0:40 ln e þ 0:53
Figs. 8–13 show the results of the calculated TCCs against the fitted TCCs (fitted by Eq. (4)) and the measured thermal conductivities for the 6 soils [16]. For most of the soils, the calculated TCCs match well with the fitted TCCs and the experimental results at full range of S. However, the new calculation model underestimates the thermal conductivities at low water contents for the soil 32 and soil 36. In general, the new model is able to provide accurate calculation result at full range of S. Besides, the 6 calculated TCCs all fall within the Wiener bounds and the HAS bounds [29,30], showing the good performance of the new model. Table 4 shows
The solution of the function is a = 0.28, then b = 2.50, c = 2.12. The fourth step is to obtain the equation of thermal conductivity for soil 31 at full range of S,
k¼
8 <
S¼0
0:20
: 0:20 þ 1:27 ln
2:50 2:12 eþð0:28 S Þ 1
S>0
Fig. 13. Calculation result of the new model for soil 36 [16].
Table 4 Calculated parameters, fitted parameters for 6 soils. Soil No.
Calculated parameters
31 32 33 34 35 36
Fitted parameters
a
b
c
a
b
c
0.28 0.34 0.47 0.42 0.45 0.34
2.50 2.12 1.55 1.74 1.62 2.12
2.12 2.31 2.93 2.65 2.81 2.31
0.23 0.55 0.35 0.38 0.28 0.30
3.87 0.90 2.23 1.88 3.06 1.92
1.84 3.38 2.40 2.69 2.34 1.67
Table 5 Overall performance of the new model. Soil No.
RMSE
AD
MARD
R2
Soil No.
RMSE
AD
MARD
R2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
0.07 0.10 0.14 0.09 0.10 0.07 0.09 0.05 0.05 0.05 0.10 0.09 0.09 0.11 0.08 0.06 0.09 0.24
0.05 0.07 0.10 0.04 0.03 0.04 0.01 0.01 0.02 0.03 0.04 0.04 0.05 0.06 0.04 0.02 0.03 0.05
0.06 0.06 0.10 0.07 0.07 0.05 0.07 0.08 0.11 0.12 0.21 0.06 0.05 0.11 0.07 0.04 0.08 0.10
0.98 0.97 0.95 0.98 0.95 0.99 0.98 0.99 0.99 0.98 0.95 0.97 0.97 0.90 0.98 0.98 0.97 0.87
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
0.06 0.14 0.07 0.04 0.07 0.06 0.05 0.03 0.07 0.10 0.09 0.09 0.06 0.08 0.07 0.05 0.06 0.14
0.01 0.10 0.04 0.01 0.04 0.00 0.03 0.01 0.03 0.08 0.07 0.06 0.02 0.04 0.04 0.04 0.03 0.09
0.16 0.08 0.06 0.07 0.05 0.13 0.07 0.06 0.07 0.17 0.17 0.08 0.07 0.10 0.10 0.07 0.09 0.13
0.97 0.95 0.99 0.99 0.99 0.98 0.98 0.99 0.97 0.91 0.93 0.96 0.98 0.97 0.97 0.98 0.98 0.93
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(2) Three regions were proposed to divide the TCC based on the study of Lu et al. [16]. The borders between Region 1 and Region 2 and between Region 2 and Region 3 were approximated by Sr and Sc, respectively. Sensitivity analysis showed that the measured point in the Region 2 gave the most reliable calculation result. (3) On the basis of two equations and a measured point at a certain S, a four-step calculating process was presented to determine the TCC. Compared with the experimental data, the new model could have good performance (R2 = 0.97) in determining the TCC of soils.
Acknowledgements
Fig. 14. Comparison of thermal conductivities between experiment and calculation from the new model.
the calculated parameters are slightly different from the fitted parameters, suggesting the difference between the calculated TCCs and the fitted TCCs. The root mean squared error (RMSE), average deviations (AD), mean of the absolute relative deviations (MARD), and R2 were used to evaluate the overall performance of the new model [1,19,45], as summarized in Table 5. The variations of R2 for the 6 soils are 0.93 to 0.98, showing the high performance of the calculation model. Fig. 14 also shows good performance (R2 = 0.97) of the new calculation model in calculating the thermal conductivity of soils. The results of 30 soils (soil 1 to soil 30) were also calculated by the new model. The low values of RMSE, MARD and high values of R2 indicate the good performance of the new model. The high accuracy of the new calculation model gives us confidence in the calculation of the TCC by the new model. Table 5 also shows that most values of AD are negative. It means that the new calculation model usually underestimates the thermal conductivity [1]. The suggested parameters of other calculation models are fixed for the same soil types. For example, Côté and Konrad model for Ke shows the suggested parameters for gravel and coarse sand, medium and fine sand, silty and clayey soils, silt and clay, and organic fibrous soils are 4.60, 3.55, 1.90, and 0.60, respectively [15]. Lu et al. model for Ke shows the suggested parameters for coarsegrained soils and fine-grained soils are 0.96 and 0.27, respectively [16]. However, the suggested parameter a for this new model isn’t fixed and is determined by two equations and a measured point. It means that the same soil type will lead to different parameter a. Besides, many other factors also influence the thermal conductivity of unsaturated soils, but only S was investigated in this new model. Other factors and microstructural features of soils will be considered in the further investigations. 5. Conclusions A new calculation model was developed to determine the TCC based on 30 Canadian soils. Then the applicability of the new calculation model was evaluated with 6 Chinese soils that had not been introduced during the establishment equation stage. The main conclusions can be summarized as follows. (1) A new three-parameter model was presented to model the thermal conductivity of fine-grained soils at full range of S. Three parameters of the new model were calculated by two equations and a measured point at a certain S.
This research was supported by the National Natural Science Foundation of China (Grant No. 41471063), the 100-Talent Program of the Chinese Academy of Sciences (Granted to Dr. Mingyi Zhang), the Program of the State Key Laboratory of Frozen Soil Engineering (Grant No. SKLFSE-ZT-23), the Key Research Program of Frontier Sciences of the Chinese Academy of Sciences (QYZDY-SSWDQC015), and the STS Program of the Chinese Academy of Sciences (Grant No. HHS-TSS-STS-1502).
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