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A homogenization-based model for the effective thermal conductivity of bentonite–sand-based buffer material
International Communications in Heat and Mass Transfer 68 (2015) 43–49
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A homogenization-based model for the effective thermal conductivity of bentonite–sand-based buffer material☆ Min Wang a, Yi-Feng Chen a,⁎, Song Zhou b, Ran Hu a, Chuang-Bing Zhou a,c a b c
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China Changjiang Institute of Survey, Planning, Design and Research, Changjiang Water Resources Commission, Wuhan 430010, China School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China
a r t i c l e
i n f o
Available online 4 September 2015 Keywords: Thermal conductivity Homogenization method Bentonite–sand mixture Buffer/backfill material
a b s t r a c t A homogenization-based effective thermal conductivity model was proposed for unsaturated compacted bentonite–sand-based buffer materials. The microstructure of the mixture was approximated with pores and sand particles of spheroidal shape and random orientation embedded in the homogeneous bentonite matrix. By virtue of the analytical solution to the inhomogeneous inclusion problem in heat conduction, the model was developed using homogenization techniques such as the Mori–Tanaka (MT) and interaction direct derivative (IDD) schemes for different consideration of the interactions between pores, sand particles, and the bentonite matrix. The proposed estimates are dependent on the thermal conductivities of the bentonite matrix, the liquid and gas phases and sand, porosity, the degree of saturation, sand content, and the aspect ratios of pores and sand particles. The proposed model was validated against four sets of laboratory measurement data on the Kunigel-V1, MX-80, Kyungju, and GMZM bentonite–sand mixtures. It is demonstrated that although with simplified considerations of the microstructural features and mechanisms, the model predicts the effective thermal conductivity of the bentonite–sand mixtures with a fairly good agreement. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Bentonite–sand mixtures have been widely considered as a potential engineered buffer that could be placed between high-level radioactive wastes and host rocks in the process of deep geological waste disposal. Compared to pure bentonite materials, bentonite–sand mixtures have better thermal conductivity, relatively higher strength and lower compressibility, and very low hydraulic conductivity. The effective thermal conductivity of bentonite–sand-based buffer materials is one of the physical properties that remarkably influence the transfer of decay heat and the resultant coupled thermo-hydro-mechanical-chemical (THMC) processes in the barrier system [1–3]. Experimental tests [4–10] were performed to examine the thermal conductivity of the compacted bentonite–sand mixtures, showing that the heat-conductive property is strongly dependent on the soil structure and phase composition, such as mineral constituents, dry density, porosity, and water content. Few efforts were made to develop theoretical or empirical models for generalizing the experimental results to any possible states. For instance, Ould-Lahoucine et al. [6] adopted the idea of random phase distribution to estimate the effective thermal
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y.-F. Chen).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.007 0735-1933/© 2015 Elsevier Ltd. All rights reserved.
conductivity of the bentonite–sand mixtures using different models such as Fricke's model [11], Maxwell's model [12], Bruggeman's model [13], and Johnson's model [14]. However, the microstructural features of the bentonite–sand mixtures were not properly considered in the existing models. It has been shown [15] that the bentonite–sand mixtures typically display a bimodal shape of pore size distribution (PSD) curve, as plotted in Fig. 1, with the dominant smaller pore size mode corresponding to the intra-particle pores and the larger one to the inter-particle and/or inter-aggregate pores [15–17]. Homogenization methods provide an alternative that possibly better relates the microstructural features to the effective thermal properties for materials with inhomogeneities. This approach has been frequently applied to saturated or partially saturated soil, rock, and concrete materials [18–22], but not to bentonite–sand mixtures. In this study, a homogenization-based effective thermal conductivity model was developed for unsaturated compacted bentonite–sand mixtures using homogenization techniques. The model was developed by using the analytical solution to the inhomogeneous inclusion problem in heat conduction [23] and by assuming the pores and sand particles to be of spheroidal shape and random orientation based on the microscopic observations [15]. The Mori– Tanaka (MT) scheme [24] and the interaction direct derivative (IDD) scheme [25] are adopted for different considerations of the interactions between the pores, sand particles, and the solid phase. The proposed model is validated against four sets of laboratory data on
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M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
2.2. A homogenization-based effective thermal conductivity model
the Kunigel-V1 [6], MX-80 [7], Kyungju [9], and GMZM [10] compacted bentonite–sand mixtures, showing good agreement between the model predictions and the laboratory measurements and an overall much better performance than Johnson's model [14].
This study considers the commonly applied homogenization schemes for estimation of the effective thermal properties, including the MT and IDD schemes. With the above conceptual model and by virtue of the analytical solution to the inhomogeneous inclusion problem in heat conduction [21–23,26–28], the effective thermal conductivity of the considered soil mixture is overall isotropic, and can be estimated using different homogenization schemes for consideration of the arrangements of pores and sand particles:
2. Model development 2.1. Conceptual model As conventional compacted bentonite materials [2,3], the bentonite– sand mixture is regarded as a three-phase medium, with the pores occupied by water and air at partially saturated states and the solid phase being composed of bentonite and sand particles, as shown in Fig. 2. Denoting the porosity and the degree of saturation of the mixture by ϕ and Sr, respectively, one obtains the following relationships: ϕg ¼ ϕð1−Sr Þ; ϕm ¼ 1−ϕ−ϕs
1 g ∮ 2 ½ðλw −λm ÞϕSr Aw α ðnÞ þ λg −λm ϕð1−Sr ÞAα ðnÞ 4π ‘ ð2Þ þðλs −λm Þϕs Asα ðnÞdS
where λαeff (α = MT or IDD) is the effective thermal conductivity estimated by the MT or IDD scheme, δ the second-order identity tensor, n the unit normal of the inclusions, ‘2 = {n||n| = 1} is the surface of a unit sphere, and Aβα (β = w, g, or s) the second-order concentration tensor associated with the pores occupied by water, the pores occupied by air and sand particles, respectively:
ð1Þ
where ϕm and ϕs are the volume fractions of bentonite matrix and sand particles, respectively, ϕw and ϕg the volume fractions of pores occupied by water and gas, respectively. In this study, both the pores and the sand particles of different size, shape, orientation and arrangement in the compacted bentonite–sand mixture are regarded as inclusions in the bentonite matrix, as shown in Fig. 2. Although the bentonite particles and the additive sand particles have been observed to be typically of ellipsoidal shape [15], the geometry of pores could be much more diverse. As a first approximation, this study assumes the inclusions to be of spheroidal shape for simplifying the model development, with ωp and ωs for overall representing the aspect ratios of the pores and the sand particles, respectively. The pores and the sand particles are assumed to be randomly oriented. According to the Young–Laplace equation, the pores with a smaller radius will be preferentially occupied by water and the rest by air. At the microscopic scale, the thermal conductivity of pores (λp) is assumed to be isotropic, and at a given degree of saturation Sr, it takes the thermal conductivity of water (λw) for the smaller pores of a volume fraction of ϕw = ϕSr and the thermal conductivity of gas (λg) for the rest of the pores ϕg = (1 − ϕ)Sr. The thermal conductivities of the sand material and the bentonite matrix are also assumed to be isotropic, and are denoted by λs and λm, respectively.
λ λm m n⊗n ðδ−n⊗nÞ þ Aβα ¼ 1−ς β λm þ ς β λβ 2ς β λm þ 1−2ς β λβ
g s λαeff ¼ λm þ ðλw −λm ÞϕSr Aw α þ λg −λm ϕð1−Sr ÞAα þ ðλs −λm Þϕs Aα ð4Þ with Aβdil ¼
2 λ 1 λm m þ 3 1−ς β λm þ ς β λβ 3 2ς β λm þ 1−2ςβ λβ
g s −1 AβMT ¼ Aβdil 1−ϕSr 1−Aw dil −ϕð1−Sr Þ 1−Adil −ϕs 1−Adil 0.40
0.14
PSD
Pore size density function
0.35
porosity
0.12
0.30 0.10 0.25 0.08
0.20
0.06
0.15
0.04
0.10
0.02
0.05
0.00 0.01
0.1
ð3Þ
where ςβ is a geometric parameter related to the aspect ratio ωβ of the inclusions [23]. The geometric parameters of pores occupied by water and gas are equal (ςw = ςg = ςp) for a uniform aspect ratio of pores ωp, as assumed in Section 2.1. 1 ∮ ‘2 n⊗ndS ¼ 13 δ, Eq. (2) is simplified as Using the identity 4π
1
10
100
Cumulative porosity
ϕw ¼ ϕSr ;
λαeff δ ¼ λm δ þ
0.00 1000
Entrance pore diameter ( μm) Fig. 1. Pore size distribution and cumulative porosity curve of the MX-80 bentonite–sand mixture (taken after Saba et al. [15]).
ð5aÞ
ð5bÞ
M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
ω=
c a λβ
n
Spheroidal inclusion
Bentonite matrix Sand particles
x2
a
45
λm
c Pores occupied x1 by gas
Pores occupied by water
Bentonite
Fig. 2. A conceptual model for the bentonite–sand mixtures.
AβIDD
8 2 3 e e < 1−2ς ðλw −λm Þ ς ð λ −λ Þ p 2 1 w m p β 4 5 þ ¼ Adil 1−ϕSr : 3 λm þ ς p ðλw −λm Þ 3 λm þ 1−2ςp ðλw −λm Þ 2 3 1−2ςep λg −λm ς ep λg −λm 2 1 þ 5 −ϕð1−Sr Þ4 3 λm þ ς p λg −λm 3 λm þ 1−2ςp λg −λm −ϕs
−1 1−2ς es ðλs −λm Þ 2 ς es ðλs −λm Þ 1 þ 3 λm þ ς s ðλs −λm Þ 3 λm þ ð1−2ς s Þðλs −λm Þ
ð5cÞ
where ςei (i = p or s) is the geometric parameter of an ellipsoidal effective environment cell in which a single pore or sand particle is embedded (see Fig. 2), as assumed in the IDD scheme [25]. ςei can be calculated from the corresponding aspect ratios ωei of the effective cell, and as suggested by Zhou et al. [29], ωei is determined from the aspect ratio ωi of pores or sand particles by equating the volume fraction ϕi to the volume fraction of a single pore with respect to the volume of its effective environment cell [21,22]:
1−ωei ωei
3 ¼ ϕi
ð1−ωi Þ3 ωi
ð6Þ
Eq. (4) is derived with the considerations of preferential occupation of wetting fluid in smaller pores and spatial arrangements of pores/sand particles by different homogenization schemes, which roughly account for the influences of microstructural features and mechanisms on the
effective thermal conductivity of the bentonite–sand mixture. Such considerations, although highly simplified, significantly reduce the number of parameters and lead to easier laboratory parameterization of the model. Furthermore, it is worth noting that although an overall isotropic thermal conductivity is predicted with the assumption of random orientations of inclusions, the anisotropy of the heat properties can be easily incorporated in the proposed model as long as the preferential orientations of pores and sand particles could be properly determined. 2.3. Model parameters The proposed thermal conductivity models contain the following parameters: the state variables (ϕ and Sr), the volumetric sand content (ϕs), the thermal conductivities of the four phases (λm, λw, λg and λs), and two variables that describe the geometries of pores and sand particles (ωp and ωs). The state variables (ϕ and Sr) are governed by the coupled THMC processes in the materials and could be obtained through laboratory measurements or numerical simulations [1–3]. The volumetric sand content ϕs is determined with consideration of the function and location of the mixtures in the nuclear waste repositories. A mixing ratio of 7:3 was suggested for buffer materials between the dry masses of bentonite and sand, and a ratio of 3:7 for backfill materials between the dry masses of bentonite and other aggregates, based on a number of laboratory tests [30]. The thermal conductivities of water and gas, λw and λg, depend on their state variables of respective phases, i.e. temperature and pressure
Table 1 Physical properties of the bentonite–sand mixtures. Bentonite–sand mixture
Kunigel-V1 [6]
MX-80 [7]
Kyungju [9]
GMZM [10]
Bentonite Admixture Quartz content in bentonite ϑ (wt. %) Thermal conductivity of admixture (W/m K) Specific gravity of bentonite Specific gravity of admixture Dry density ρd (g/cm3) Volume fraction of sand ϕs (%) Porosity ϕ Degree of saturation Sr (%) Testing method
Na+-bentonite (Japan) Silica sand 0.6 3.40 2.70 3.00 N/A 10.0–44.0 0.30–0.63 0.0–94.7 Thermistor probe method
Na+-bentonite (USA) Crushed granite 15 1.69 2.65 2.67 1.4–1.8 7.0–40.0 0.33–0.55 0.0–20.4 Line source method
Ca2+-bentonite (Korea) Quartz sand 1.0 4.30 2.40 2.94 1.6–1.8 5.0–20.0 0.31–0.42 57.3–94.4 Line source method
Na+-bentonite (China) Quartz sand 11.7 4.30 2.66 2.65 1.7, 1.8, 1.9 6.0–36.0 0.30–0.45 0.0–100.0 Thermistor probe method
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M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
Table 2 Calibrated values of the aspect ratios of pores and sand particles for the bentonite–sand mixtures. Bentonite–sand mixtures MT method IDD method
ωp ωs ωp ωs
Kunigel-V1 [6]
MX-80 [7]
Kyungju [9]
GMZM [10]
0.0386 0.4593 0.0493 0.8151
0.0280 0.5995 0.0341 0.5954
0.0485 1.0000 0.0538 1.0000
0.0362 1.0965 0.0430 1.0002
imaging or μCT). However, the estimated values may vary from sample to sample given the inhomogeneous nature of soils. A more practical alternative is to calibrate the geometrical parameters by best-fitting the laboratory data, with the calibrated values overall representative of the shapes of pores and sand particles. In this study, the least square method is employed to back-calculate the geometrical parameters. 3. Model validation 3.1. Validation of the proposed model
[3]. The thermal conductivity of bentonite matrix, λm, depends on the mineral constituents and their spatial distribution. It could be estimated with geometric mean of the thermal conductivities of quartz λq and other minerals λo [31]: λm ¼ λϑq λ1−ϑ o
ð7Þ
where ϑ is the volume fraction of quartz. The most common values of λq and λo suggested for bentonites are, respectively, 7.7 and 2.0 W/m K [2,3]. The thermal conductivity of sand or other additives (e.g., graphite, chopped carbon fiber, and crushed granite), λs, can be easily measured in the laboratory for its relatively homogeneous mineral constituents. The aspect ratio of pores and sand particles, ωp and ωs, could be roughly estimated from microstructural observations (e.g., micro-CT
a
b 2.0 MT method IDD method Johnson's model
+10% +20% 10% 20%
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
Predicted thermal conductivity (W/m K)
Predicted thermal conductivity (W/m K)
2.5
2.0
The proposed models were verified against the laboratory experimental data published in the works of Ould-Lahoucine et al. [6] on Kunigel-V1, Tien et al. [7] on MX-80, Cho et al. [9] on Kyungju, and Ye et al. [10] on GMZM bentonite–sand mixtures. The physical properties of the bentonite–sand mixture samples, such as the quartz content ϑ, the thermal conductivity of sand λs, porosity ϕ, and the degree of saturation Sr, are listed in Table 1. The thermal conductivity values of water and gas were taken as λw = 0.611 W/m K and λg = 0.026 W/m K in the validation as suggested in [3]. The thermal conductivity of the bentonite matrix λm was estimated by Eq. (7). As listed in Table 2, the aspect ratios of pores and sand particles, ωp and ωs, were obtained by best curve-fitting of the laboratory data using Eq. (4) within a broad range [0, ∞]. It is interesting to observe from Table 2 that the back-calculated aspect ratio values of pores by the MT
2.5
MT method IDD method Johnson's model
d +10% +20% 10% 20%
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
Measured thermal conductivity (W/m K)
2.5
Predicted thermal conductivity (W/m K)
Predicted thermal conductivity (W/m K)
2.0
20%
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
Measured thermal conductivity (W/m K)
c MT method IDD method Johnson's model
+10% 10%
1.5
Measured thermal conductivity (W/m K)
2.5
+20%
2.5
2.0
MT method IDD method Johnson's model
+20%
+10% 10% 20%
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
Measured thermal conductivity (W/m K)
Fig. 3. Predicted thermal conductivity by the homogenization-based model versus measured thermal conductivity of (a) Kunigel-V1 (Ould-Lahoucine's data), (b) MX-80 (Tien's data), (c) Kyungju (Cho's data), and (d) GMZM (Ye's data) bentonite–sand mixtures.
M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
47
Table 3 Relative prediction errors of thermal conductivity of the bentonite–sand mixtures.a Bentonite–sand mixtures
Kunigel-V1 [6]
MX-80 [7]
Kyungju [9]
No. of data points
44
102
14
18
65.91% 18.18% 20.45% 79.55% 34.09% 45.45% 97.73% 93.18% 93.18%
10.78% 49.02% 51.96% 18.63% 82.35% 84.31% 45.10% 100.00% 100.00%
0.00% 42.86% 42.86% 7.14% 78.57% 78.57% 64.29% 100.00% 100.00%
5.56% 44.44% 50.00% 5.56% 83.33% 83.33% 33.33% 94.44% 94.44%
Percentage of data points with er ≤ 5%
Johnson's model MT method IDD method Johnson's model MT method IDD method Johnson's model MT method IDD method
Percentage of data points with er ≤ 10%
Percentage of data points with er ≤ 20%
a
GMZM [10]
er denotes the relative error at kth data point defined by |λexp,k − λe,k|/λexp,k × 100%.
and IDD estimates for different bentonite–sand mixtures fall in a rather narrow range between 0.02 and 0.05, which overall illustrates the representative geometry of pores in compacted bentonite–sand mixtures. The aspect ratio values of sand particles also vary in a rather narrow range between 0.46 and 1.10, which well represents the nearspherical geometry of sand particles observed in laboratory tests [15]. Fig. 3 shows a comparison between the predicted and measured thermal conductivity values of the Kunigel-V1, MX-80, Kyungju, and GMZM bentonite–sand mixtures, respectively. Also plotted in the figures are the results predicted by a model developed by Johnson [14], which was applied to estimate the effective thermal conductivity with a high-volume fraction of randomly oriented ellipsoids by OuldLahoucine et al. [6]. One observes from the plots that the predictions by the MT and IDD schemes are generally in good agreement with the experimental data on the Kunigel-V1, MX-80, Kyungju, and GMZM bentonite–sand mixtures (Fig. 3a–d). Especially for Tien's data on the MX-80 (Fig. 3b) and Cho's data on the Kyungju (Fig. 3c) bentonite– sand mixtures, the proposed models overcome the deficiency of Johnson's model, which overestimates the effect of the volume fraction of sand. Table 3 lists the prediction errors of the proposed estimates on the above four sets of experimental data, in which the relative error at kth data point is defined by |λexp,k − λe,k|/λexp,k × 100%, where λexp,k and λe,k are the kth thermal conductivity values of laboratory measurements and the corresponding model predictions, respectively. One observes from Table 3 that 18%–52% of the predictions have a discrepancy lower than 5% and most (93%–100%) of the predictions of both the homogenization schemes fall in between the 20% relative error lines. It can also be observed from Fig. 3 and Table 3 that the IDD scheme seems to have slightly better performance than the MT estimate, for its better consideration of the interaction between pores, sand particles,
a
and the bentonite matrix. But overall, the performance of the MT and IDD schemes is largely similar, and differs only in a rather narrow range. 3.2. Responses of the proposed model with respect to porosity, saturation, and sand content Without loss of generality, the MX-80 bentonite–sand mixture was taken for discussion of the responses of the proposed model to variations of porosity, saturation, and sand content. The material parameters are the same with those for the MX-80 bentonite–sand mixture given in Section 3.1. Fig. 4 plots the variations of the effective thermal conductivity predicted by the IDD scheme against porosity and saturation under a constant sand content (ϕs = 30%) and against saturation and volume fraction of sand under a constant porosity (ϕ = 40%). One observes from Fig. 4a that at any given degree of saturation, the predicted effective thermal conductivity reduces with increasing porosity at a decreasing rate; and at a given porosity, the effective thermal conductivity tends to increase at an increasing rate with increasing degree of saturation. For small values of porosity (e.g., b 5%), the effective thermal conductivity is mainly determined by the thermal conductivity of the bentonite matrix, and the influence of water content is rather weak; while for high values of porosity (e.g., N 90%), the effective thermal conductivity is mainly determined by the thermal conductivities of water and gas, with the lowest values bounded by the thermal conductivity of gas when the water content tends to zero. Fig. 4b shows that at any given volume fraction of sand, the predictions increase with increasing degree of saturation; and at a given saturation, the effective thermal conductivity tends to increase at an increasing rate with the increasing volume fraction of sand. Fig. 5 plots the variations of the effective thermal conductivity predicted by the IDD scheme against porosity and volume fraction of
b 1.8 Effective thermal conductivity (W/m K)
Effective thermal conductivity (W/m K)
2.5 2.0 1.5 1.0 0.5 0.0 1.0 0.8
φ s=30% 0 Sat .6 ura 0 tion .4 S r
0.2 0
1.0
0.8
0.6
P
0.4 yφ orosit
0.2
0
1.6 1.4 1.2 1.0 0.8 0.6 0.4 1.0 0.8 0 Sat .6 ura 0 tion .4 S r
φ =40%
0.2 0
0
0.2
0.4 me f
Volu
0.8 0.6 and φ s n of s ractio
1.0
Fig. 4. Evolution of the effective thermal conductivity predicted by the IDD scheme against (a) porosity and saturation at constant volume fraction of sand, ϕs = 30%, and (b) saturation and volume fraction of sand at constant porosity, ϕ = 40%.
48
M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
a
conductivity tends to vary less nonlinearly against porosity and the volume fraction of sand.
Effective thermal conductivity (W/m K)
2.5
4. Conclusions 2.0 1.5 1.0 0.5 0.0 0
Sr=10%
0.2 0.4 Por 0 osit .6 yφ
0.8
1.0
0
0.8 0.6 nd φ s a s 0.4 of ction 0.2 e fra m lu Vo
1.0
b
Effective thermal conductivity (W/m K)
2.5 2.0
Acknowledgments
1.5
Financial support from the National Natural Science Foundation of China (nos. 51222903 and 51179136) are gratefully acknowledged.
1.0 0.5 0.0 0
References Sr=50%
0.2 0.4 Por 0.6 osit yφ
0.8 1.0
0.2 0
0.8 0.6 and φ s s f o 0.4 ion fract lume
1.0
Vo
c 2.5
Effective thermal conductivity (W/m K)
The thermal conductivity of unsaturated compacted bentonite– sand-based buffer materials plays a vital role in optimization design and performance assessment of the engineered barrier systems. In this study, an isotropic homogenization-based effective thermal conductivity model was proposed for compacted bentonite–sand mixtures with simplified considerations of the microstructural features of mixtures such as the size, shape, orientation, and spatial arrangement of pores and admixtures. The proposed model was comprehensively verified by four sets of laboratory data on the Kunigel-V1, MX-80, Kyungju, and GMZM compacted bentonite–sand mixtures with different dry densities, water contents, sand contents, and mineralogical compositions, and good agreements were obtained between the laboratory measurements and the model predictions by the MT and IDD schemes. The verification studies showed that the MT and IDD schemes exhibit similar performance. Compared to Johnson's model, the proposed model not only displays overall better performance but also, as a great advantage, has much clearer physical mechanisms.
2.0 1.5 1.0 0.5 0.0 0
Sr=90%
0.2 0.4 Por osit y
φ
0.6 0.8 1.0
0
0.8 0.6 dφ s f san o 0.4 n ctio 0.2 e fra m lu Vo
1.0
Fig. 5. Evolution of the effective thermal conductivity predicted by the IDD scheme against the porosity and volume fraction of sand at various constant saturations: (a) Sr = 10%, (b) Sr = 50%, and (c) Sr = 90%.
sand at constant saturations (Sr = 10%, 50%, and 90%). It is interesting to observe that for small values of porosity (e.g., b10%), the effective thermal conductivity tends to reduce at a decreasing rate with an increasing volume fraction of sand because the thermal conductivity of the additive (crushed granite) in the MX-80 bentonite–sand mixture is lower than that of the pure bentonite, as listed in Table 1. It is also observed from Fig. 5a–c that with the increase of saturation, the effective thermal
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M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49 [18] O. Pierard, C. Friebel, I. Doghri, Mean-field homogenization of multi-phase thermoelastic composites: a general framework and its validation, Compos. Sci. Technol. 64 (10) (2004) 1587–1603. [19] A. Giraud, C. Gruescu, D.P. Do, F. Homand, D. Kondo, Effective thermal conductivity of transversely isotropic media with arbitrary oriented ellipsoidal inhomogeneities, Int. J. Solids Struct. 44 (9) (2007) 2627–2647. [20] W.Q. Shen, D. Kondo, L. Dormieux, J.F. Shao, A closed-form three scale model for ductile rocks with a plastically compressible porous matrix, Mech. Mater. 59 (2013) 73–86. [21] 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 Transf. 83 (2015) 731–740. [22] Y. Chen, M. Wang, S. Zhou, R. Hu, C. Zhou, An effective thermal conductivity model for unsaturated compacted bentonites with consideration of bimodal pore size distribution, Sci. China Technol. Sci. 58 (2) (2015) 369–380. [23] Y. Chen, D. Li, Q. Jiang, C. Zhou, Micromechanical analysis of anisotropic damage and its influence on effective thermal conductivity in brittle rocks, Int. J. Rock Mech. Min. Sci. 50 (2012) 102–116. [24] T. Mori, K. Tanaka, Averages stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall. 21 (5) (1973) 571–574.
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[25] Q.S. Zheng, D.X. Du, An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution, J. Mech. Phys. Solids 49 (11) (2001) 2765–2788. [26] J.D. Eshelby, Elastic inclusions and heterogeneities, Progress in Solid 2. , North-Holland, Amsterdam, 1961. [27] B. Shafiro, M. Kachanov, Anisotropic effective conductivity of materials with nonrandomly oriented inclusions of diverse ellipsoidal shapes, J. Appl. Phys. 87 (12) (2000) 8561–8569. [28] C. Gruescu, A. Giraud, F. Homand, D. Kondo, D.P. Do, Effective thermal conductivity of partially saturated porous rocks, Int. J. Solids Struct. 44 (3) (2007) 811–833. [29] C. Zhou, K. Li, X. Pang, Effect of crack density and connectivity on the permeability of microcracked solids, Mech. Mater. 43 (12) (2011) 969–978. [30] D. Sun, H. Cui, W. Sun, Swelling of compacted sand–bentonite mixtures, Appl. Clay Sci. 43 (3) (2009) 485–492. [31] J.K. Carson, J.P. Sekhon, Simple determination of the thermal conductivity of the solid phase of particulate materials, Int. J. Heat Mass Transf. 37 (9) (2010) 1226–1229.
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
A homogenization-based model for the effective thermal conductivity of bentonite–sand-based buffer material☆ Min Wang a, Yi-Feng Chen a,⁎, Song Zhou b, Ran Hu a, Chuang-Bing Zhou a,c a b c
State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China Changjiang Institute of Survey, Planning, Design and Research, Changjiang Water Resources Commission, Wuhan 430010, China School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, China
a r t i c l e
i n f o
Available online 4 September 2015 Keywords: Thermal conductivity Homogenization method Bentonite–sand mixture Buffer/backfill material
a b s t r a c t A homogenization-based effective thermal conductivity model was proposed for unsaturated compacted bentonite–sand-based buffer materials. The microstructure of the mixture was approximated with pores and sand particles of spheroidal shape and random orientation embedded in the homogeneous bentonite matrix. By virtue of the analytical solution to the inhomogeneous inclusion problem in heat conduction, the model was developed using homogenization techniques such as the Mori–Tanaka (MT) and interaction direct derivative (IDD) schemes for different consideration of the interactions between pores, sand particles, and the bentonite matrix. The proposed estimates are dependent on the thermal conductivities of the bentonite matrix, the liquid and gas phases and sand, porosity, the degree of saturation, sand content, and the aspect ratios of pores and sand particles. The proposed model was validated against four sets of laboratory measurement data on the Kunigel-V1, MX-80, Kyungju, and GMZM bentonite–sand mixtures. It is demonstrated that although with simplified considerations of the microstructural features and mechanisms, the model predicts the effective thermal conductivity of the bentonite–sand mixtures with a fairly good agreement. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction Bentonite–sand mixtures have been widely considered as a potential engineered buffer that could be placed between high-level radioactive wastes and host rocks in the process of deep geological waste disposal. Compared to pure bentonite materials, bentonite–sand mixtures have better thermal conductivity, relatively higher strength and lower compressibility, and very low hydraulic conductivity. The effective thermal conductivity of bentonite–sand-based buffer materials is one of the physical properties that remarkably influence the transfer of decay heat and the resultant coupled thermo-hydro-mechanical-chemical (THMC) processes in the barrier system [1–3]. Experimental tests [4–10] were performed to examine the thermal conductivity of the compacted bentonite–sand mixtures, showing that the heat-conductive property is strongly dependent on the soil structure and phase composition, such as mineral constituents, dry density, porosity, and water content. Few efforts were made to develop theoretical or empirical models for generalizing the experimental results to any possible states. For instance, Ould-Lahoucine et al. [6] adopted the idea of random phase distribution to estimate the effective thermal
☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (Y.-F. Chen).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.08.007 0735-1933/© 2015 Elsevier Ltd. All rights reserved.
conductivity of the bentonite–sand mixtures using different models such as Fricke's model [11], Maxwell's model [12], Bruggeman's model [13], and Johnson's model [14]. However, the microstructural features of the bentonite–sand mixtures were not properly considered in the existing models. It has been shown [15] that the bentonite–sand mixtures typically display a bimodal shape of pore size distribution (PSD) curve, as plotted in Fig. 1, with the dominant smaller pore size mode corresponding to the intra-particle pores and the larger one to the inter-particle and/or inter-aggregate pores [15–17]. Homogenization methods provide an alternative that possibly better relates the microstructural features to the effective thermal properties for materials with inhomogeneities. This approach has been frequently applied to saturated or partially saturated soil, rock, and concrete materials [18–22], but not to bentonite–sand mixtures. In this study, a homogenization-based effective thermal conductivity model was developed for unsaturated compacted bentonite–sand mixtures using homogenization techniques. The model was developed by using the analytical solution to the inhomogeneous inclusion problem in heat conduction [23] and by assuming the pores and sand particles to be of spheroidal shape and random orientation based on the microscopic observations [15]. The Mori– Tanaka (MT) scheme [24] and the interaction direct derivative (IDD) scheme [25] are adopted for different considerations of the interactions between the pores, sand particles, and the solid phase. The proposed model is validated against four sets of laboratory data on
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M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
2.2. A homogenization-based effective thermal conductivity model
the Kunigel-V1 [6], MX-80 [7], Kyungju [9], and GMZM [10] compacted bentonite–sand mixtures, showing good agreement between the model predictions and the laboratory measurements and an overall much better performance than Johnson's model [14].
This study considers the commonly applied homogenization schemes for estimation of the effective thermal properties, including the MT and IDD schemes. With the above conceptual model and by virtue of the analytical solution to the inhomogeneous inclusion problem in heat conduction [21–23,26–28], the effective thermal conductivity of the considered soil mixture is overall isotropic, and can be estimated using different homogenization schemes for consideration of the arrangements of pores and sand particles:
2. Model development 2.1. Conceptual model As conventional compacted bentonite materials [2,3], the bentonite– sand mixture is regarded as a three-phase medium, with the pores occupied by water and air at partially saturated states and the solid phase being composed of bentonite and sand particles, as shown in Fig. 2. Denoting the porosity and the degree of saturation of the mixture by ϕ and Sr, respectively, one obtains the following relationships: ϕg ¼ ϕð1−Sr Þ; ϕm ¼ 1−ϕ−ϕs
1 g ∮ 2 ½ðλw −λm ÞϕSr Aw α ðnÞ þ λg −λm ϕð1−Sr ÞAα ðnÞ 4π ‘ ð2Þ þðλs −λm Þϕs Asα ðnÞdS
where λαeff (α = MT or IDD) is the effective thermal conductivity estimated by the MT or IDD scheme, δ the second-order identity tensor, n the unit normal of the inclusions, ‘2 = {n||n| = 1} is the surface of a unit sphere, and Aβα (β = w, g, or s) the second-order concentration tensor associated with the pores occupied by water, the pores occupied by air and sand particles, respectively:
ð1Þ
where ϕm and ϕs are the volume fractions of bentonite matrix and sand particles, respectively, ϕw and ϕg the volume fractions of pores occupied by water and gas, respectively. In this study, both the pores and the sand particles of different size, shape, orientation and arrangement in the compacted bentonite–sand mixture are regarded as inclusions in the bentonite matrix, as shown in Fig. 2. Although the bentonite particles and the additive sand particles have been observed to be typically of ellipsoidal shape [15], the geometry of pores could be much more diverse. As a first approximation, this study assumes the inclusions to be of spheroidal shape for simplifying the model development, with ωp and ωs for overall representing the aspect ratios of the pores and the sand particles, respectively. The pores and the sand particles are assumed to be randomly oriented. According to the Young–Laplace equation, the pores with a smaller radius will be preferentially occupied by water and the rest by air. At the microscopic scale, the thermal conductivity of pores (λp) is assumed to be isotropic, and at a given degree of saturation Sr, it takes the thermal conductivity of water (λw) for the smaller pores of a volume fraction of ϕw = ϕSr and the thermal conductivity of gas (λg) for the rest of the pores ϕg = (1 − ϕ)Sr. The thermal conductivities of the sand material and the bentonite matrix are also assumed to be isotropic, and are denoted by λs and λm, respectively.
λ λm m n⊗n ðδ−n⊗nÞ þ Aβα ¼ 1−ς β λm þ ς β λβ 2ς β λm þ 1−2ς β λβ
g s λαeff ¼ λm þ ðλw −λm ÞϕSr Aw α þ λg −λm ϕð1−Sr ÞAα þ ðλs −λm Þϕs Aα ð4Þ with Aβdil ¼
2 λ 1 λm m þ 3 1−ς β λm þ ς β λβ 3 2ς β λm þ 1−2ςβ λβ
g s −1 AβMT ¼ Aβdil 1−ϕSr 1−Aw dil −ϕð1−Sr Þ 1−Adil −ϕs 1−Adil 0.40
0.14
PSD
Pore size density function
0.35
porosity
0.12
0.30 0.10 0.25 0.08
0.20
0.06
0.15
0.04
0.10
0.02
0.05
0.00 0.01
0.1
ð3Þ
where ςβ is a geometric parameter related to the aspect ratio ωβ of the inclusions [23]. The geometric parameters of pores occupied by water and gas are equal (ςw = ςg = ςp) for a uniform aspect ratio of pores ωp, as assumed in Section 2.1. 1 ∮ ‘2 n⊗ndS ¼ 13 δ, Eq. (2) is simplified as Using the identity 4π
1
10
100
Cumulative porosity
ϕw ¼ ϕSr ;
λαeff δ ¼ λm δ þ
0.00 1000
Entrance pore diameter ( μm) Fig. 1. Pore size distribution and cumulative porosity curve of the MX-80 bentonite–sand mixture (taken after Saba et al. [15]).
ð5aÞ
ð5bÞ
M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
ω=
c a λβ
n
Spheroidal inclusion
Bentonite matrix Sand particles
x2
a
45
λm
c Pores occupied x1 by gas
Pores occupied by water
Bentonite
Fig. 2. A conceptual model for the bentonite–sand mixtures.
AβIDD
8 2 3 e e < 1−2ς ðλw −λm Þ ς ð λ −λ Þ p 2 1 w m p β 4 5 þ ¼ Adil 1−ϕSr : 3 λm þ ς p ðλw −λm Þ 3 λm þ 1−2ςp ðλw −λm Þ 2 3 1−2ςep λg −λm ς ep λg −λm 2 1 þ 5 −ϕð1−Sr Þ4 3 λm þ ς p λg −λm 3 λm þ 1−2ςp λg −λm −ϕs
−1 1−2ς es ðλs −λm Þ 2 ς es ðλs −λm Þ 1 þ 3 λm þ ς s ðλs −λm Þ 3 λm þ ð1−2ς s Þðλs −λm Þ
ð5cÞ
where ςei (i = p or s) is the geometric parameter of an ellipsoidal effective environment cell in which a single pore or sand particle is embedded (see Fig. 2), as assumed in the IDD scheme [25]. ςei can be calculated from the corresponding aspect ratios ωei of the effective cell, and as suggested by Zhou et al. [29], ωei is determined from the aspect ratio ωi of pores or sand particles by equating the volume fraction ϕi to the volume fraction of a single pore with respect to the volume of its effective environment cell [21,22]:
1−ωei ωei
3 ¼ ϕi
ð1−ωi Þ3 ωi
ð6Þ
Eq. (4) is derived with the considerations of preferential occupation of wetting fluid in smaller pores and spatial arrangements of pores/sand particles by different homogenization schemes, which roughly account for the influences of microstructural features and mechanisms on the
effective thermal conductivity of the bentonite–sand mixture. Such considerations, although highly simplified, significantly reduce the number of parameters and lead to easier laboratory parameterization of the model. Furthermore, it is worth noting that although an overall isotropic thermal conductivity is predicted with the assumption of random orientations of inclusions, the anisotropy of the heat properties can be easily incorporated in the proposed model as long as the preferential orientations of pores and sand particles could be properly determined. 2.3. Model parameters The proposed thermal conductivity models contain the following parameters: the state variables (ϕ and Sr), the volumetric sand content (ϕs), the thermal conductivities of the four phases (λm, λw, λg and λs), and two variables that describe the geometries of pores and sand particles (ωp and ωs). The state variables (ϕ and Sr) are governed by the coupled THMC processes in the materials and could be obtained through laboratory measurements or numerical simulations [1–3]. The volumetric sand content ϕs is determined with consideration of the function and location of the mixtures in the nuclear waste repositories. A mixing ratio of 7:3 was suggested for buffer materials between the dry masses of bentonite and sand, and a ratio of 3:7 for backfill materials between the dry masses of bentonite and other aggregates, based on a number of laboratory tests [30]. The thermal conductivities of water and gas, λw and λg, depend on their state variables of respective phases, i.e. temperature and pressure
Table 1 Physical properties of the bentonite–sand mixtures. Bentonite–sand mixture
Kunigel-V1 [6]
MX-80 [7]
Kyungju [9]
GMZM [10]
Bentonite Admixture Quartz content in bentonite ϑ (wt. %) Thermal conductivity of admixture (W/m K) Specific gravity of bentonite Specific gravity of admixture Dry density ρd (g/cm3) Volume fraction of sand ϕs (%) Porosity ϕ Degree of saturation Sr (%) Testing method
Na+-bentonite (Japan) Silica sand 0.6 3.40 2.70 3.00 N/A 10.0–44.0 0.30–0.63 0.0–94.7 Thermistor probe method
Na+-bentonite (USA) Crushed granite 15 1.69 2.65 2.67 1.4–1.8 7.0–40.0 0.33–0.55 0.0–20.4 Line source method
Ca2+-bentonite (Korea) Quartz sand 1.0 4.30 2.40 2.94 1.6–1.8 5.0–20.0 0.31–0.42 57.3–94.4 Line source method
Na+-bentonite (China) Quartz sand 11.7 4.30 2.66 2.65 1.7, 1.8, 1.9 6.0–36.0 0.30–0.45 0.0–100.0 Thermistor probe method
46
M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
Table 2 Calibrated values of the aspect ratios of pores and sand particles for the bentonite–sand mixtures. Bentonite–sand mixtures MT method IDD method
ωp ωs ωp ωs
Kunigel-V1 [6]
MX-80 [7]
Kyungju [9]
GMZM [10]
0.0386 0.4593 0.0493 0.8151
0.0280 0.5995 0.0341 0.5954
0.0485 1.0000 0.0538 1.0000
0.0362 1.0965 0.0430 1.0002
imaging or μCT). However, the estimated values may vary from sample to sample given the inhomogeneous nature of soils. A more practical alternative is to calibrate the geometrical parameters by best-fitting the laboratory data, with the calibrated values overall representative of the shapes of pores and sand particles. In this study, the least square method is employed to back-calculate the geometrical parameters. 3. Model validation 3.1. Validation of the proposed model
[3]. The thermal conductivity of bentonite matrix, λm, depends on the mineral constituents and their spatial distribution. It could be estimated with geometric mean of the thermal conductivities of quartz λq and other minerals λo [31]: λm ¼ λϑq λ1−ϑ o
ð7Þ
where ϑ is the volume fraction of quartz. The most common values of λq and λo suggested for bentonites are, respectively, 7.7 and 2.0 W/m K [2,3]. The thermal conductivity of sand or other additives (e.g., graphite, chopped carbon fiber, and crushed granite), λs, can be easily measured in the laboratory for its relatively homogeneous mineral constituents. The aspect ratio of pores and sand particles, ωp and ωs, could be roughly estimated from microstructural observations (e.g., micro-CT
a
b 2.0 MT method IDD method Johnson's model
+10% +20% 10% 20%
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
Predicted thermal conductivity (W/m K)
Predicted thermal conductivity (W/m K)
2.5
2.0
The proposed models were verified against the laboratory experimental data published in the works of Ould-Lahoucine et al. [6] on Kunigel-V1, Tien et al. [7] on MX-80, Cho et al. [9] on Kyungju, and Ye et al. [10] on GMZM bentonite–sand mixtures. The physical properties of the bentonite–sand mixture samples, such as the quartz content ϑ, the thermal conductivity of sand λs, porosity ϕ, and the degree of saturation Sr, are listed in Table 1. The thermal conductivity values of water and gas were taken as λw = 0.611 W/m K and λg = 0.026 W/m K in the validation as suggested in [3]. The thermal conductivity of the bentonite matrix λm was estimated by Eq. (7). As listed in Table 2, the aspect ratios of pores and sand particles, ωp and ωs, were obtained by best curve-fitting of the laboratory data using Eq. (4) within a broad range [0, ∞]. It is interesting to observe from Table 2 that the back-calculated aspect ratio values of pores by the MT
2.5
MT method IDD method Johnson's model
d +10% +20% 10% 20%
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
Measured thermal conductivity (W/m K)
2.5
Predicted thermal conductivity (W/m K)
Predicted thermal conductivity (W/m K)
2.0
20%
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
Measured thermal conductivity (W/m K)
c MT method IDD method Johnson's model
+10% 10%
1.5
Measured thermal conductivity (W/m K)
2.5
+20%
2.5
2.0
MT method IDD method Johnson's model
+20%
+10% 10% 20%
1.5
1.0
0.5
0.0 0.0
0.5
1.0
1.5
2.0
2.5
Measured thermal conductivity (W/m K)
Fig. 3. Predicted thermal conductivity by the homogenization-based model versus measured thermal conductivity of (a) Kunigel-V1 (Ould-Lahoucine's data), (b) MX-80 (Tien's data), (c) Kyungju (Cho's data), and (d) GMZM (Ye's data) bentonite–sand mixtures.
M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
47
Table 3 Relative prediction errors of thermal conductivity of the bentonite–sand mixtures.a Bentonite–sand mixtures
Kunigel-V1 [6]
MX-80 [7]
Kyungju [9]
No. of data points
44
102
14
18
65.91% 18.18% 20.45% 79.55% 34.09% 45.45% 97.73% 93.18% 93.18%
10.78% 49.02% 51.96% 18.63% 82.35% 84.31% 45.10% 100.00% 100.00%
0.00% 42.86% 42.86% 7.14% 78.57% 78.57% 64.29% 100.00% 100.00%
5.56% 44.44% 50.00% 5.56% 83.33% 83.33% 33.33% 94.44% 94.44%
Percentage of data points with er ≤ 5%
Johnson's model MT method IDD method Johnson's model MT method IDD method Johnson's model MT method IDD method
Percentage of data points with er ≤ 10%
Percentage of data points with er ≤ 20%
a
GMZM [10]
er denotes the relative error at kth data point defined by |λexp,k − λe,k|/λexp,k × 100%.
and IDD estimates for different bentonite–sand mixtures fall in a rather narrow range between 0.02 and 0.05, which overall illustrates the representative geometry of pores in compacted bentonite–sand mixtures. The aspect ratio values of sand particles also vary in a rather narrow range between 0.46 and 1.10, which well represents the nearspherical geometry of sand particles observed in laboratory tests [15]. Fig. 3 shows a comparison between the predicted and measured thermal conductivity values of the Kunigel-V1, MX-80, Kyungju, and GMZM bentonite–sand mixtures, respectively. Also plotted in the figures are the results predicted by a model developed by Johnson [14], which was applied to estimate the effective thermal conductivity with a high-volume fraction of randomly oriented ellipsoids by OuldLahoucine et al. [6]. One observes from the plots that the predictions by the MT and IDD schemes are generally in good agreement with the experimental data on the Kunigel-V1, MX-80, Kyungju, and GMZM bentonite–sand mixtures (Fig. 3a–d). Especially for Tien's data on the MX-80 (Fig. 3b) and Cho's data on the Kyungju (Fig. 3c) bentonite– sand mixtures, the proposed models overcome the deficiency of Johnson's model, which overestimates the effect of the volume fraction of sand. Table 3 lists the prediction errors of the proposed estimates on the above four sets of experimental data, in which the relative error at kth data point is defined by |λexp,k − λe,k|/λexp,k × 100%, where λexp,k and λe,k are the kth thermal conductivity values of laboratory measurements and the corresponding model predictions, respectively. One observes from Table 3 that 18%–52% of the predictions have a discrepancy lower than 5% and most (93%–100%) of the predictions of both the homogenization schemes fall in between the 20% relative error lines. It can also be observed from Fig. 3 and Table 3 that the IDD scheme seems to have slightly better performance than the MT estimate, for its better consideration of the interaction between pores, sand particles,
a
and the bentonite matrix. But overall, the performance of the MT and IDD schemes is largely similar, and differs only in a rather narrow range. 3.2. Responses of the proposed model with respect to porosity, saturation, and sand content Without loss of generality, the MX-80 bentonite–sand mixture was taken for discussion of the responses of the proposed model to variations of porosity, saturation, and sand content. The material parameters are the same with those for the MX-80 bentonite–sand mixture given in Section 3.1. Fig. 4 plots the variations of the effective thermal conductivity predicted by the IDD scheme against porosity and saturation under a constant sand content (ϕs = 30%) and against saturation and volume fraction of sand under a constant porosity (ϕ = 40%). One observes from Fig. 4a that at any given degree of saturation, the predicted effective thermal conductivity reduces with increasing porosity at a decreasing rate; and at a given porosity, the effective thermal conductivity tends to increase at an increasing rate with increasing degree of saturation. For small values of porosity (e.g., b 5%), the effective thermal conductivity is mainly determined by the thermal conductivity of the bentonite matrix, and the influence of water content is rather weak; while for high values of porosity (e.g., N 90%), the effective thermal conductivity is mainly determined by the thermal conductivities of water and gas, with the lowest values bounded by the thermal conductivity of gas when the water content tends to zero. Fig. 4b shows that at any given volume fraction of sand, the predictions increase with increasing degree of saturation; and at a given saturation, the effective thermal conductivity tends to increase at an increasing rate with the increasing volume fraction of sand. Fig. 5 plots the variations of the effective thermal conductivity predicted by the IDD scheme against porosity and volume fraction of
b 1.8 Effective thermal conductivity (W/m K)
Effective thermal conductivity (W/m K)
2.5 2.0 1.5 1.0 0.5 0.0 1.0 0.8
φ s=30% 0 Sat .6 ura 0 tion .4 S r
0.2 0
1.0
0.8
0.6
P
0.4 yφ orosit
0.2
0
1.6 1.4 1.2 1.0 0.8 0.6 0.4 1.0 0.8 0 Sat .6 ura 0 tion .4 S r
φ =40%
0.2 0
0
0.2
0.4 me f
Volu
0.8 0.6 and φ s n of s ractio
1.0
Fig. 4. Evolution of the effective thermal conductivity predicted by the IDD scheme against (a) porosity and saturation at constant volume fraction of sand, ϕs = 30%, and (b) saturation and volume fraction of sand at constant porosity, ϕ = 40%.
48
M. Wang et al. / International Communications in Heat and Mass Transfer 68 (2015) 43–49
a
conductivity tends to vary less nonlinearly against porosity and the volume fraction of sand.
Effective thermal conductivity (W/m K)
2.5
4. Conclusions 2.0 1.5 1.0 0.5 0.0 0
Sr=10%
0.2 0.4 Por 0 osit .6 yφ
0.8
1.0
0
0.8 0.6 nd φ s a s 0.4 of ction 0.2 e fra m lu Vo
1.0
b
Effective thermal conductivity (W/m K)
2.5 2.0
Acknowledgments
1.5
Financial support from the National Natural Science Foundation of China (nos. 51222903 and 51179136) are gratefully acknowledged.
1.0 0.5 0.0 0
References Sr=50%
0.2 0.4 Por 0.6 osit yφ
0.8 1.0
0.2 0
0.8 0.6 and φ s s f o 0.4 ion fract lume
1.0
Vo
c 2.5
Effective thermal conductivity (W/m K)
The thermal conductivity of unsaturated compacted bentonite– sand-based buffer materials plays a vital role in optimization design and performance assessment of the engineered barrier systems. In this study, an isotropic homogenization-based effective thermal conductivity model was proposed for compacted bentonite–sand mixtures with simplified considerations of the microstructural features of mixtures such as the size, shape, orientation, and spatial arrangement of pores and admixtures. The proposed model was comprehensively verified by four sets of laboratory data on the Kunigel-V1, MX-80, Kyungju, and GMZM compacted bentonite–sand mixtures with different dry densities, water contents, sand contents, and mineralogical compositions, and good agreements were obtained between the laboratory measurements and the model predictions by the MT and IDD schemes. The verification studies showed that the MT and IDD schemes exhibit similar performance. Compared to Johnson's model, the proposed model not only displays overall better performance but also, as a great advantage, has much clearer physical mechanisms.
2.0 1.5 1.0 0.5 0.0 0
Sr=90%
0.2 0.4 Por osit y
φ
0.6 0.8 1.0
0
0.8 0.6 dφ s f san o 0.4 n ctio 0.2 e fra m lu Vo
1.0
Fig. 5. Evolution of the effective thermal conductivity predicted by the IDD scheme against the porosity and volume fraction of sand at various constant saturations: (a) Sr = 10%, (b) Sr = 50%, and (c) Sr = 90%.
sand at constant saturations (Sr = 10%, 50%, and 90%). It is interesting to observe that for small values of porosity (e.g., b10%), the effective thermal conductivity tends to reduce at a decreasing rate with an increasing volume fraction of sand because the thermal conductivity of the additive (crushed granite) in the MX-80 bentonite–sand mixture is lower than that of the pure bentonite, as listed in Table 1. It is also observed from Fig. 5a–c that with the increase of saturation, the effective thermal
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