A generalized model for calculating the thermal conductivity of freezing soils based on soil components and frost heave

International Journal of Heat and Mass Transfer 150 (2020) 119166

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A generalized model for calculating the thermal conductivity of freezing soils based on soil components and frost heave Jun Bi a,b,c, Mingyi Zhang a,∗, Yuanming Lai a, Wansheng Pei a, Jianguo Lu a,d, Zhilang You a,d, Dongwei Li e a

State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou 730000, China Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education of China, Lanzhou University, Lanzhou 730000, China c School of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China d University of Chinese Academy of Sciences, Beijing 100049, China e School of Civil and Structural Engineering, East China University of Technology, Nanchang 330013, China b

a r t i c l e

i n f o

Article history: Received 1 August 2019 Revised 21 October 2019 Accepted 2 December 2019

Keywords: Thermal conductivity Freezing soils Generalized model Stages Connections

a b s t r a c t Thermal conductivity of freezing soils is an important parameter for the geotechnical engineering in cold regions. During a freezing process, unfrozen water freezes into ice. It changes soil components and induces frost heave, which will significantly increase the thermal conductivity of freezing soils. This study presents a generalized model for calculating the thermal conductivity of freezing soils with a consideration of soil components and frost heave. The generalized model for freezing soils was developed by different connections (e.g. series connection and parallel connection) between soil pores and solid grain and between unfrozen water and ice in the pores. This model was a function of unfrozen water content, frost heave, porosity, and initial water content. The proposed model was verified by measured data of eight silty clay samples with different dry densities and initial water contents. Results show that the calculated thermal conductivities agree well with measured data. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Thermal conductivity of soils is an important soil parameter because it is widely used in the numerical modeling of thermal stability of cold regions engineering [1–3]. The freezing process of a soil changes soil components and induces frost heave. The large difference of thermal conductivities between water (0.56 W m− 1 K−1 ) and ice (2.22 W m−1 K−1 ) indicates that soil components and frost heave have significant effects on the thermal conductivity of freezing soils. Not surprisingly, many experimental measurements and calculation models have been used to research the thermal conductivity of frozen soils. Kersten [4] measured the thermal conductivities of 19 soils in frozen and unfrozen states and divided the soils into two groups, namely, sands or sandy soils, and silt and clay soils. He proposed four equations to calculate the thermal conductivity of soils. Johansen [5] proposed a linear model to relate the thermal conductivity with degree of saturation (S) for frozen soils. In-

∗

Corresponding author. E-mail address: [email protected] (M. Zhang).

https://doi.org/10.1016/j.ijheatmasstransfer.2019.119166 0017-9310/© 2019 Elsevier Ltd. All rights reserved.

spired by the study of Johansen, Côté and Konrad [6] tested the thermal conductivities of base-course materials in frozen and unfrozen states and proposed a simple mathematical equation to establish the relationship between thermal conductivity and S for frozen base-course materials. Later, they proposed a generalized equation for soils of different types [7]. Pei et al. [8] tested the thermal conductivities of 11 soil-rock media and proposed a multiple linear regression equation to calculate the thermal conductivity of soil-rock media in cold regions. Lu et al. [9] measured the thermal conductivities of an aeolian sand with different dry densities and initial water contents and proposed a parallel-series mixed model to calculate the thermal conductivity of an aeolian sand during the freezing process. Zhang et al. [10] investigated the thermal conductivity of a silty clay during a freezing-thawing process. They evaluated the performances of three general models (e.g. weighted arithmetic mean model, weighted harmonic mean model, and weighted geometric mean model) and suggested that the weighted geometric mean model provides the best calculation results among the three models. Li et al. [11] measured the thermal conductivities of a clay at 5 different temperatures and evaluated the performances of 4 potential probability distribution models. It

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J. Bi, M. Zhang and Y. Lai et al. / International Journal of Heat and Mass Transfer 150 (2020) 119166

Fig. 1. A real structure and an equivalent structure of an unsaturated unfrozen soil sample [18].

shows that the normal distribution model performs best among the 4 models. Besides, many researchers also developed some calculation models for frozen soils based on the de Vries model [12], such as Penner-de Vries model [13], Fuchs-de Vries model [14], Farouki-de Vries model [15], and Tian-de Vries model [16]. However, thermal conductivity models for freezing soils containing soil components and frost heave are quite scarce. Therefore, the objectives of the study are to: (1) propose a generalized model for calculating the thermal conductivity of freezing soils based on soil components and frost heave; (2) evaluate the generalized model by measured data from Zhang et al. [10]. 2. Method and model 2.1. Wiener bounds Wiener bounds are usually used to constrain the thermal conductivity models [17]. The series model represents the lower bound, while the parallel model represents the upper bound [17]. For the unfrozen unsaturated soils, it usually contains three components, e.g. solid grain, water, and air. Therefore, the series model and parallel model for the unfrozen unsaturated soils are given as,

 λsu = λpu =





ϕj λj

−1 =

ϕ

s0

λs

+

ϕa0 ϕw0 −1 + λa λw

(1)

ϕj λj = ϕs0 λs + ϕa0 λa + ϕw0 λw

(2)

where λsu and λu are the series and parallel models of the thermal conductivities at unfrozen unsaturated state; ϕ j is the volume  fraction of component j, ϕj =1; ϕ s0 , ϕ a0 , and ϕ w0 are the volume fractions of solid grain, air and water in the unfrozen state, respectively, ϕs0 +ϕa0 +ϕw0 = 1; λj is the thermal conductivity of component j; λs , λa , and λw are the thermal conductivities of solid grain, air, and water, respectively. The thermal conductivity of unfrozen unsaturated soils can be calculated by the mixture of the series model and parallel model with a weighting parameter, as shown in Eq. (3). p

λ = ηλpu + (1 − η )λsu

(3)

where η is a weighting parameter depending on pore structure of soils, degree of saturation (S) and temperature. Obviously, it should be in the range of 0 ≤ η ≤ 1. Substituting Eqs. (1) and (2) into Eq. (3), yielding Eq. (4).

λ=η (ϕs0 λs + ϕa0 λa + ϕw0 λw )+(1 − η )

ϕ

s0

λs

+

ϕa0 ϕw0 −1 + λa λw

(4)

Fig. 1 shows the real structure and the equivalent structure of a soil. Each volume fraction in Eq. (4) was shown in Fig. 1. Soil

Fig. 2. Variations of components in a soil sample during a freezing process.

pores were divided into two parts, and each part was filled with water and air. Fig. 1(b) shows that solid grain, water and air are in parallel connection, and it indicates η (ϕs0 λs + ϕa0 λa + ϕw0 λw ) in Eq. (4). Fig. 1(c) shows that solid grain, water and air are in series ϕ ϕ ϕ connection, and it indicates (1 − η )( λs0 + λa0 + λw0 )−1 in Eq. (4). s a w Tong et al. [18] also obtained the equivalent structure of unsaturated soils, and used it to calculate the thermal conductivity of unsaturated bentonites. However, in this study, we used the equivalent structure to propose a generalized thermal conductivity model for freezing soils. 2.2. A generalized thermal conductivity model for freezing soils In this model, a part of soil pores can be termed as “parallel pores” because the air and water in the soil pores are in parallel connection, as shown in Fig. 1(b). The other part of soil pores can be termed as “series pores” because the air and water in the soil pores are in series connection, as shown in Fig. 1(c). We assumed that “parallel pores” and “series pores” have the same pore size distribution, indicating that ice forms simultaneously in the “parallel pores” and “series pores”. Decreasing the subzero temperature leads to the formation of ice in the soil pores. The connections between ice and unfrozen water can be assumed to be parallel connection and series connection. Figs. 2 and 3 show the variations of the components and the corresponding volume fractions during the freezing process. The soil pores are originally filled with water and air in unfrozen state. It is a three-component mixture, e.g. solid grain, water and air (Figs. 2(a) and 3(a)). With the decrease of temperature, ice forms in the soil pores, and air is gradually expelled from the soil pores. The soil sample in Stage 1 is a four-component mixture, e.g. solid grain, water, ice and air. Further decrease of temperature leads to Stage 2. Stage 2 is a critical condition and lies between Stage 1 and Stage 3. In stage 2, soil pores are only filled with unfrozen water and ice, and frost heave does not occur. When the temperature continues to decrease, frost heave occurs, as shown in Figs.

J. Bi, M. Zhang and Y. Lai et al. / International Journal of Heat and Mass Transfer 150 (2020) 119166

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Fig. 3. Variations of volume fractions of components in a soil sample during a freezing process. Fig. 5. An effective structure of a four-component soil sample during a freezing process when unfrozen water and ice are arranged in series connection

Fig. 4. An effective structure of a four-component soil sample during a freezing process when unfrozen water and ice are arranged in parallel connection.

2(d) and 3(d). In Stage 3, soil pores are only filled with unfrozen water and ice, and frost heave occurs.

Fig. 6. An effective structure of a three-component soil sample during a freezing process when unfrozen water and ice are arranged in parallel connection

2.2.1. Thermal conductivity model of the freezing soils in Stage 1 Case 1: Eq. (4) was used to model the thermal conductivity of unsaturated unfrozen soils. With the decrease of temperature, water freezes into ice in Stage 1. It contains four components, e.g. solid grain, unfrozen water, ice, and air. Unfrozen water, ice and air co-exist in the soil pores, as shown in Fig. 2(b). If unfrozen water and ice are assumed to be arranged in parallel connection (Fig. 4). Based on the Figs. 2(b), (c) and 4, the thermal conductivity of freezing soils in Stage 1 is given as,

λpf−Stage1 =ηϕs1 λs +ηϕw1 λw +ηϕi1 λi +ηϕa1 λa +(1 − η )  −1 ϕs1 (ϕw1 +ϕi1 )2 ϕa1 + + (5) λs ϕw1 λw +ϕi1 λi λa where λf−stage1 is the thermal conductivity of freezing soils in p

Stage 1 for the condition that ice and unfrozen water are arranged in parallel connection; λi is the thermal conductivity of ice; ϕ s1 is the volume fraction of solid grain in Stage 1; ϕ w1 is the volume fraction of unfrozen water in Stage 1; ϕ i1 is the volume fraction of ice in Stage 1; ϕ a1 is the volume fraction of air in Stage 1. The volume fractions of the four components in Fig. 3(b) are given as,

ϕs1 = ϕs0

(6)

ϕw1 = θu

(7)

ϕi1 = k(ϕw0 − θu )

(8)

ϕa1 = ϕa0 − (k − 1 )(ϕw0 − θu )

(9)

where θ u is the volumetric unfrozen water content; k is the expansion parameter of water-ice phase transition, defined as ρ w /ρ i ;

Fig. 7. An effective structure of a three-component soil sample during a freezing process when unfrozen water and ice are arranged in series connection

ρ w is the density of water, usually taken as 10 0 0 kg/m3 ; ρ i is the density of ice, usually taken as 917 kg/m3 . Substituting Eqs. (6)–(9) into Eq. (5), yielding Eq. (10),

λpf−Stage1 = ηϕs0 λs +ηθu λw + ηk(ϕw0 − θu )λi + η[ϕa0 − (k − 1 )(ϕw0 − θu )]

2 ϕ [θ +k(ϕw0 − θu )] λa +(1 − η ) s0 + u λs θu λw +k(ϕw0 − θu )λi −1 ϕa0 − (k − 1 )(ϕw0 − θu ) + λa

(10)

Case 2: If unfrozen water and ice are assumed to be arranged in series connection, as shown in Fig. 5. The thermal conductivity of freezing soils is given as,

ϕ ϕ −1 λsf−Stage1 =ηϕs1 λs +η (ϕw1 +ϕi1 )2 w1 + i1 λw λi ϕ ϕ ϕ ϕa1 −1 s1 w1 i1 +ηϕa1 λa +(1 − η ) + + + λs λw λi λa

(11)

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J. Bi, M. Zhang and Y. Lai et al. / International Journal of Heat and Mass Transfer 150 (2020) 119166

where λsf−stage1 is the thermal conductivity of freezing soils in Stage 1 for the condition that ice and unfrozen water are arranged in series connection. Substituting Eqs. (6)–(9) into Eq. (11), yielding Eq. (12),



θu k(ϕw0 − θu ) λ ηϕs0 λs +η[θu +k(ϕw0 − θu )] + λw λi +η[ϕa0 − (k − 1 )(ϕw0 − θu )]λa +(1 − η )  −1 ϕs0 θu k(ϕw0 − θu ) ϕa0 − (k − 1 )(ϕw0 − θu ) + + + λs λw λi λa

−1

2

s f−Stage1 =



(12)

(13)

where λf-stage1 is the thermal conductivity of freezing soils in Stage 1; κ is a weighting parameter, 0 ≤ κ ≤ 1. 2.2.2. Thermal conductivity model of the freezing soils in Stage 2 Figs. 2(c) and 3(c) show that soil pores are only filled with unfrozen water and ice, and air is completely expelled in Stage 2. Frost heave does not occur in this stage. Figs. 6 and 7 show two effective structures in Stage 2. Volume fraction of air is equal to 0, so the air item in Eqs. (10) and (12) is equal to 0.

ϕa0 − (k − 1 )(ϕw0 − θu ) = 0

(14)

Rearranging Eq. (14) leads to the critical volumetric unfrozen water content in Stage 2.

θu−Cri = ϕw0 −

ϕa0 (k − 1 )

(15)

where θ u-Cri is the critical volumetric unfrozen water content in Stage 2 when air is completely expelled, and frost heave does not occur. The thermal conductivity model in Stage 2 can be determined by the models in Stage 1, as shown in Eqs. (16) and (17).

λ

 −1 ϕ (ϕ +ϕ )2 ηϕs2 λs +ηϕw2 λw +ηϕi2 λi +(1 − η ) s2 + w2 i2 λs ϕw2 λw +ϕi2 λi

p = f−Stage2

(16)

ϕ ϕ −1 λsf−Stage2 =ηϕs2 λs +η (ϕw2 +ϕi2 )2 w2 + i2 λw λi ϕ

−1 ϕ ϕ s2 w2 i2 + (1 − η ) + + λs λw λi

λpf−Stage2 =ηϕs0 λs +ηθu−Cri λw +ηk(ϕw0 − θu−Cri )λi +

−1 2 +k(ϕw0 − θu−Cri )] ϕ [θ (1 − η ) s0 + u−Cri λs θu−Cri λw +k(ϕw0 − θu−Cri )λi λsf−Stage2 =ηϕs0 λs +η[θu−Cri +k(ϕw0 − θu−Cri )]2

Eqs. (10) and (12) were obtained using different connections between ice and unfrozen water. For freezing soils, each case may be available, so the thermal conductivity of freezing soils in Stage 1 can be expressed as the weighted arithmetic mean of the two models, as shown in Eq. (13).

λf−stage1 =κλpf−stage1 +(1 − κ )λsf−stage1

Substituting Eqs. (18)–(20) into Eqs. (16) and (17), yielding Eqs. (21) and (22), respectively.

(17)

where λf−Stage2 is the thermal conductivity of freezing soils in p

Stage 2 for the condition that ice and unfrozen water are arranged in parallel connection; λsf−Stage2 is the thermal conductivity of freezing soils in Stage 2 for the condition that ice and unfrozen water are arranged in series connection; ϕ s2 is the volume fraction of solid grain in Stage 2; ϕ w2 is the volume fraction of unfrozen water in Stage 2; ϕ i2 is the volume fraction of ice in Stage 2. The volume fractions of the three components in Fig. 3(c) are given as,

ϕs2 = ϕs0

(18)

ϕw2 = θu−Cri

(19)

ϕi2 = k(ϕw0 − θu−Cri )

(20)

 + (1 − η )

ϕs0 θu−Cri k(ϕw0 − θu−Cri ) + + λs λw λi

−1

θu−Cri k(ϕw0 − θu−Cri ) + λw λi

(21) −1

(22)

The thermal conductivity of freezing soils in Stage 2 can be expressed as the weighted arithmetic mean of the two models, as shown in Eq. (23)

λf−stage2 =κλpf−stage2 +(1 − κ )λsf−stage2

(23)

where λf-stage2 is the thermal conductivity of freezing soils in Stage 2. 2.2.3. Thermal conductivity model of the freezing soils in Stage 3 For the freezing soils in Stage 3, frost heave occurs, and the soil sample is a three-component mixture, e.g. solid grain, water, and ice. The thermal conductivity model in Stage 3 can be determined by the models in Stage 2, as shown in Eqs. (24) and (25).

ηϕ λ ηϕ λ ηϕ λ λpf−Stage3 = s3 s + w3 w + i3 i +(1 − η ) 1+ε 1+ε 1+ε  −1 ϕs3 (ϕw3 +ϕi3 )2 + (1 + ε )λs (1 + ε )(ϕw3 λw +ϕi3 λi ) ηϕ λ η (ϕw3 +ϕi3 )2 ϕw3 ϕi3 −1 λsf−Stage3 = s3 s + + 1+ε 1+ε λw λi  −1 ϕw3 ϕi3 ϕs3 + (1 − η ) + + (1 + ε )λs (1 + ε )λw (1 + ε )λi

(24)

(25)

where λf−stage3 is the thermal conductivity of freezing soils in p

Stage 3 for the condition that ice and unfrozen water are arranged in parallel connection; λsf−stage3 is the thermal conductivity of freezing soils in Stage 3 for the condition that ice and water are arranged in series connection; ϕ s3 is the volume fraction of solid grain in Stage 3; ϕ w3 is the volume fraction of unfrozen water in Stage 3; ϕ i3 is the volume fraction of ice in Stage 3. The volume fractions of the three components in Fig. 3(d) are given as,

ϕs3 =ϕs0

(26)

ϕw3 =(1 + ε )θu

(27)

ϕi3 =k[ϕw0 − (1+ε )θu ]

(28)

where ɛ is the frost heave Fig. 3(d) shows that the sum of the volume fractions of the three components in Stage 3 is equal to 1 + ε , as shown in Eq. (29). Substituting Eqs. (26)–(28) into Eq. (29), yielding Eq. (30).

ϕs3 +ϕw3 +ϕi3 = 1 + ε

(29)

ϕs0 + (1+ε )θu + k[ϕw0 − (1+ε )θu ] = 1+ε

(30)

Rearranging Eq. (30), we obtained ɛ,

ε=

k(ϕw0 − θu ) + ϕs0 + θu − 1 ( k − 1 )θu + 1

(31)

J. Bi, M. Zhang and Y. Lai et al. / International Journal of Heat and Mass Transfer 150 (2020) 119166

Substituting Eqs. (26)–(28) into Eqs. (24) and (25), yielding Eqs.(32) and (33), respectively.

ϕ  η λpf−Stage3 = ϕs0 λs +ηθu λw +ηk w0 − θu λi +(1 − η ) 1+ε 1+ε    ϕw0 2 −1 θu +k 1+ε − θu ϕs0   + (1+ε )λs θu λw +k 1ϕ+w0ε − θu λi

(32)

 ϕ  2 η w0 λsf−Stage3 = ϕs0 λs + η θu + k − θu 1+ε 1+ε    ϕw0  −1  ϕw0  −1 k − θ k 1+ u θu ϕs0 θu 1+ε ε − θu + + (1 − η ) + + (33) λw λi λi (1 + ε )λs λw k (ϕ

−θ )+ϕ +θ −1

u u s0 where ε = w0(k−1 . )θu +1 The thermal conductivity of freezing soils in Stage 3 can be expressed as the weighted arithmetic mean of the two models, as shown in Eq. (34).

λf−stage3 =κλpf−stage3 +(1 − κ )λsf−stage3

(34)

where λf- stage3 is the thermal conductivity of freezing soils in Stage 3. From the above analysis, a generalized model is suggested based on the relationship between volumetric unfrozen water content and critical volumetric unfrozen water content in Stage 2:

⎧ ⎨λf−stage1 λ = λf−stage2 ⎩λ f−stage3

a0 θu >θu−Cri = ϕw0 − (kϕ−1 ) a0 θu = θu−Cri = ϕw0 − (kϕ−1 ) a0 θu < θu−Cri = ϕw0 − (kϕ−1 )

(35)

3. Determination of the parameters 3.1. Parameter λa Kannuluik and Carman [19] experimentally investigated λa using the hot-wire method and proposed a quadratic equation to model λa from −183 to 218 °C, which is expressed as,

  λa =0.0241 1 + 0.00317T − 0.0000021T 2

(36)

where T is temperature in Celsius, ranging from −183 to 218 °C. 3.2. Parameter λw Many models were proposed to relate λw with some soil properties, but most models were used to calculate λw at unfrozen state. In this study, Eq. (37) was used to calculate λw during the freezing process [20].

λw =0.11455+1.6318 × 10−3 (273.15+T )

(37)

3.3. Parameter λi

λi was influenced by salinity and temperature [21]. In the study, the effect of salinity was ignored because of a lack of measured data. The method for calculating λi was given as [20],

λi =0.4685+

488.19 273.15+T

(38)

3.4. Parameter θ u Many equations were proposed to relate θ u with subzero temperature, such as Xu et al. model [22], Liu and Yu model [23], Bai et al. model [24]. A modified Fredlund and Xing-Clapeyron model was used in this study, as shown in Eq. (39).

θu = C ( T ) 



θ0

wT ln 2.718 + − 273Lρ.15 aFX

nFX mFX

(39a)

C (T ) = 1 −

 

ln 1 − ln 1 +

Lρw T 273.15Cr 10 0 0 0 0 0 Cr

 

5

(39b)

where aFX , mFX , and nFX are the fitted parameters of modified Fredlund and Xing-Clapeyron model [25]; L is the latent heat of fusion of water, usually taken as 334 KJ/kg; Cr is the suction corresponding to the residual water content, taken as 1500 kPa. 3.5. Weighting parameter κ It is difficult to determine the weighting parameter κ . Chen et al. [26] used the model to fit the measured data, and then they obtained the 4 weighting parameters in the model. However, the objective of this study is to obtain a calculation model for the thermal conductivity of freezing soils. So the fitting process is not adopted. Base on the previous analysis, the two cases can be available. Therefore, it’s reasonable to postulate that κ is equal to 0.5. 3.6. Parameter η Chen et al. [26] suggested that η for dry soils depends on the pore structure of soils and proposed an exponential equation to relate the parameter with porosity. However, Tong et al. [18] indicated that η depends on the pore structure of solid-gas mixture, and it depends on the pore structure, degree of saturation and temperature for a three-component mixture, and proposed an equation to calculate the parameter without a consideration of temperature. However, temperature has a significant effect on the soil properties during the freezing process in this study compared to the other factors. Therefore, we assumed that only temperature has a significant effect on the parameter η in freezing soils because of a lack of measured data. An empirical relationship between η and temperature is given as,

η=

−aT + b −T + 1

(40)

where T is the temperature; a and b are empirical parameters determined by the measured data. 4. Model verification The measured data from reference were used to verify the proposed generalized model. Zhang et al. [10] used 8 silty clay columns with a diameter of 63 mm and a height of 72 mm to research the thermal conductivity of freezing soils. Step freezing was applied to cool the soil samples at 8 different subzero temperatures, namely, −0.5 ◦ C, −1 ◦ C, −1.5 ◦ C, −2 ◦ C, −5 ◦ C, −8 ◦ C, −13 ◦ C and −19 ◦ C. In order to reach thermal equilibrium, the soil specimens were kept for 3–5 h under each subzero temperature. After thermal equilibrium, thermal conductivity of soils was measured using a QL-30 Thermophysical Instrument. Moreover, they also measured the volumetric unfrozen water content of a soil sample with an initial volumetric water content of 34.84% (19.91% by gravity) and a dry density of 1750 kg/m3 . Thus, in this study, we used 5TM to measure the dielectric permittivities of the other 7 soil samples. Volumetric water content with an accuracy of ±3% was obtained when dielectric permittivity was substituting into the Topp equation. The unfrozen water contents of the eight soil samples were shown in Fig. 8. Table 1 lists all the parameters in this study. Fig. 9 shows that the calculated thermal conductivities match well with the measured thermal conductivities for the eight soil samples. With the decrease of temperature, the calculated thermal conductivity increases fast in the range of 0 and −2 ◦ C. It can be explained by the fact this temperature range is the dramatic phase transformation zone, and unfrozen water content is significant reduced in this

6

J. Bi, M. Zhang and Y. Lai et al. / International Journal of Heat and Mass Transfer 150 (2020) 119166 Table 1 Soil properties and parameters in this study. Soil No.

S1 S2 S3 S4 S5 S6 S7 S8

Dry density

Volumetric water content

Fredlund and Xing-Clapeyron model

η

kg/m3

cm/cm3

aFX

mFX

nFX

κ

a

b

1500 1500 1500 1500 1750 1750 1750 1750

0.298 0.272 0.241 0.195 0.348 0.316 0.281 0.226

5694.81 85647.61 476.00 961.82 336.01 449.48 448.38 387.71

2.49 5.58 0.75 0.43 0.21 0.14 0.13 0.16

0.41 0.38 0.86 1.92 8.14 29.12 27.31 4.14

0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

0.90 0.81 0.74 0.66 0.03 0.80 0.73 0.72

0.44 0.40 0.39 0.32 0.38 0.47 0.44 0.40

Fig. 8. Relationship between volumetric unfrozen water content and temperature for 8 silty clay soil samples during a freezing process.

range [10,27]. It shows that the proposed model usually underestimates the thermal conductivity in this temperature range. Further decrease of temperature leads to the slowly increase of the thermal conductivity. When the temperature is lower than −2 ◦ C, the calculated thermal conductivities match well with the measured data. Generally, the proposed generalized model gives good agreement with measured data. Fig. 10 compares the calculated thermal conductivities against the measured thermal conductivities for the eight soil samples. The root mean squared error (RMSE), coefficient of determination (R2 ), and average deviation (AD) were used to evaluate the overall performance of the generalized model [17]. Table 2 indicates that the proposed model exhibits good performance. The AD value is

Fig. 9. Comparison of calculated and measured thermal conductivity.

0.01, suggesting that the proposed model overestimates the thermal conductivity. In order to evaluate the proposed model in different temperature ranges, the RMSE, R2 and AD were calculated in this study. The temperature criteria proposed by Zhang et al. [10] was used to divide the temperature into two different ranges (e.g. T < −2 ◦ C, and −2 ≤ T ≤ 0 ◦ C). RMSE values and R2 values in Table 2 revealed that the warmer temperature range (−2 ≤ T ≤ 0 ◦ C) provided poor calculated thermal conductivity compared to the colder temperature range (T < −2 ◦ C). AD values revealed that the proposed model usually overestimated the thermal conductivity in the warmer temperature range (−2 ≤ T ≤ 0 ◦ C), while the proposed model usually underestimated the thermal conductivity in the colder temperature range

J. Bi, M. Zhang and Y. Lai et al. / International Journal of Heat and Mass Transfer 150 (2020) 119166 Table 2 Values of RMSE, R2 , and AD at different temperature ranges. Temperature

RMSE

R2

AD

T < −2 ◦ C −2 ≤ T ≤ 0 ◦ C Full range

0.06 0.12 0.10

0.94 0.85 0.91

−0.01 0.02 0.01

Fig. 10. Validation of measured versus calculated thermal conductivity at various temperatures.

(T < −2 ◦ C). In summary, the proposed generalized model provided good fit to the measured data. 5. Conclusions This study presents a generalized model for the thermal conductivity of freezing soils based on soil components and frost heave. Soil components and frost heave were introduced to split the freezing process into three stages: (1) Stage 1, a soil sample does not have frost heave and contains four components, e.g. unfrozen water, ice, solid grain, and air; (2) Stage 2, a soil sample does not have frost heave and contains three components, e.g. unfrozen water, ice, and solid grain; (3) Stage 3, a soil sample has frost heave and contains three components, e.g. unfrozen water, ice, and solid grain. In each stage, the generalized model was developed by different connections (e.g. series connection and parallel connection) between soil pores and solid particle and between unfrozen water and ice in the pores, and was a function of unfrozen water content, frost heave, porosity, and initial water content. The model was verified by measured data of eight soil samples from literature. Results show good agreements (R2 = 0.91) between calculated and measured thermal conductivity. Declaration of Competing Interest We declare that there is not any interest conflict about the paper. Acknowledgments This research was supported by the National Science Fund for Distinguished Young Scholars (Grant No. 41825015), the Program of the State Key Laboratory of Frozen Soil Engineering (Grant No. SKLFSE-ZT-23), and the Key Research Program of the Frontier

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