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1. Introduction
Thermal dissipation of decay heat from a disposal canister containing high-level radioactive waste (HLW) is important for maintaining an efficient and safe repository; therefore, many studies have considered factors affecting the thermal conductivity of the bentonite buffer material [1-6]. The thermal conductivity of the bentonite buffer is one of the most important properties for determining the appropriate design temperature to ensure the safety of HLW disposal [5, 7, 8]. Hence, the thermal conductivity of the bentonite buffer material, which is primarily affected by various factors such as the dry density, water content, mineral composition, and temperature variation, must be evaluated [5, 8, 9, 10, 11]. In particular, the thermal conductivities of most minerals in bentonite such as quartz, cristobalite, and feldspar are relatively higher than that of montmorillonite, which constitutes the most significant proportion of bentonite; therefore, when the montmorillonite content is low, the thermal conductivity of the bentonite is typically high owing to the thermal conductivities of the other minerals. Conversely, when the montmorillonite content is high, the thermal conductivity of bentonite is typically low [12]. However, for bentonites with similar mineral compositions, their thermal conductivities may differ depending on the type of montmorillonite. It may be inferred that the exchangeable cations of bentonites affect the thermal conductivity.
In this study, the mechanism by which the thermal conductivity of bentonites varies was investigated based on the exchangeable cations of bentonites; subsequently, the findings were compared with those of previous relevant studies and the effect of bentonite type was comparatively analyzed and was verified through the effective medium model for thermal conductivity. For this study, GMZ07, and MX80 [12] which are Na-bentonites and KJ [6], and FEBEX [5] which are Ca-bentonites, which have been previously studied for thermal conductivity [1, 4, 5, 6, 8] were analyzed to compare the effect of bentonite type.
2. Effect of Montmorillonite
At the same temperature, dry density, and saturation, the thermal conductivities of four bentonites, GMZ07, MX-80, KJ, and FEBEX show a considerable difference one another (Fig. 1). The thermal conductivity of the FEBEX bentonite which has the highest content of montmorillonite with low thermal conductivity and the lowest content of quartz with high thermal conductivity is high, and the MX-80 bentonite with lower content of montmorillonite and higher content of quartz shows the lowest thermal conductivity (Table 1). Also, even though both the GMZ07 bentonite [12] and the KJ bentonite [6] have similar content of montmorillonite, and the GMZ07 bentonite contains more than twice as much of quartz and cristobalite than the KJ bentonite (Table 1), the thermal conductivity of the KJ bentonite is slightly higher than that of the GMZ07 bentonite.
Although the reasons for the different thermal conductivities are not clear, the possible explanation is a difference in the pore water chemistry of bentonite and the degree of contact among the bentonite constituting minerals. The heat conduction through the water saturated bentonite with a constant porosity (namely dry density) is composed of two contributions that are the heat conduction in pore water of the bentonite, and the heat conduction through the bentonite matrix. Therefore, the thermal conductivity of saturated bentonite consists of two components namely, the thermal conductivity of the pore water and the thermal conductivity of bentonite matrix.
The pore water of the bentonite can be considered as an electrolyte solution, and the thermal conductivity of an electrolyte solution depends primarily on the ionic strength of the electrolyte solution. It is reported that the ionic strength of Na-bentonite pore water is similar to that of Ca-bentonite pore water. For example, the ionic strength of the FEBEX bentonite pore water for dry density of 1.65 Mg‧m−3 is 0.3 M [13], and that of the MX-80 bentonite pore water for dry density between 1.2 to 1.6 Mg‧m−3 is ~ 0.3 M [14]. Therefore, the pore water thermal conductivities for Na- bentonite and Ca-bentonite seem to be similar each other. However, it has been reported that in addition to the ionic strength, the species of ion have also effects on the thermal conductivity of electrolyte solution. The wellknown expression for the individual ionic contributions to the thermal conductivity of an electrolyte solution was suggested by Riedel (1951);
where λe is the thermal conductivity of electrolyte solution (W‧mK−1), λo is the thermal conductivity of pure water (W‧mK−1), ci is the molar concentration of ion i and α i is the contribution of ion i. This equation can be used for dilute and moderately concentrated multicomponent electrolyte solution which is the case of bentonite pore water with good accuracy. Eq. (1) implies that the thermal conductivity of the electrolyte solutions with same ionic strength varies with ionic species. The coefficient α can be expressed as a function of temperature [16];
where T is the temperature in K, To = 273.15 K, and A is a universal constant equal to 0.023. The predominant cation in the pore water is Na+ for Na-bentonite, and Ca+2 for Ca-bentonite. The values of α i1 and α i2 for Na+ in water are 0.0 and 0.0, respectively, and the values of α i1 and α i2 for Ca2+in water are −0.052799 and 0.126519, respectively [16]. Therefore, the thermal conductivity of pore water for Ca-bentonite is higher than that for Na-bentonite even if their ionic strengths of the pore water are similar each other.
Thermal conductivity of bentonite matrix is related to the thermal conductivities of constituting minerals and the degree of contact among the minerals. If the bentonite matrix is structurally rigid and the constituting minerals are in close contact with one another, then the minerals are considered as layers perpendicular to the heat flow direction (series model) to determine the lower bound of the thermal conductivity of the bentonite matrix. The upper bound can be determined when the minerals are arranged parallel to the heat flow direction (parallel model). However, in a realistic geological medium, the minerals are randomly distributed throughout the matrix. Hence, Sass et al. (1971) suggested a geometric mean model that correlates the thermal conductivity of rock aggregates and the thermal conductivities of the individual minerals constituting a rock, as follows [17]:
where λ is the thermal conductivity of rock aggregates; λ1 ∙∙∙ λn are the thermal conductivities of the individual minerals constituting a rock; φ1 ∙∙∙ φn are the volume fractions occupied by the individual minerals. Brigaud and Vasseur (1989) applied the geometric mean model to estimate the thermal conductivity of argillaceous minerals using the measured thermal conductivity and mineralogy of a rock sample.
In bentonites, the matrix is not structurally rigid, and the constituting minerals are not in contact with one another. The thermal conductivity of the matrix can be calculated as shown in Eq. (3), which involves considerable uncertainties. Therefore, a modified geometric mean model has been proposed [19], i.e.,
where m1, m2, …, mn are the exponents that consider the unreality of the geometric mean model. If the bentonite matrix phase is structurally rigid and its arrangement is an ideal configuration, then the exponent mn is equal to 1. The exponents mn can be determined via regression based on experimental data. Because of the insufficient measured data, the quantitative values of the exponents mn cannot be obtained; however, their effects on thermal conductivity can be assessed qualitatively.
As montmorillonite comprises extremely soft phyllosilicate minerals, whereas feldspars comprise relatively less hard tectosilicate minerals, the particles of these minerals may be in close contact with one another, resulting in better heat conduction. By contrast, quartz particles, which are hard crystalline minerals composed of silicon dioxide, and cristobalite, which is also a hard mineral polymorph of silicon dioxide, may establish contact with one another more loosely, resulting in poor heat conduction. Therefore, the contribution of montmorillonite and feldspars to the overall thermal conductivity of bentonite is significantly higher than that of quartz and cristobalite.
The montmorillonite content of FEBEX bentonite [5] is approximately 29% higher than that of KJ bentonite (Table 1). The thermal conductivity of FEBEX bentonite is comparatively high for the same temperature, dry density, and saturation because of higher pore water thermal conductivity and better contact among the constituting minerals (Fig. 1). The thermal conductivity of pore water for Na-bentonite is lower than that for Ca-bentonite. Hence, the thermal conductivities of Na-bentonites, GMZ07 and MX80 [12] are lower than that of Ca-bentonite.
Comparing the thermal conductivities of different types of bentonites based on temperature for the same dry density, saturation conditions, and water content, the thermal conductivities of KJ bentonite [8], which is a Ca-bentonite and is used in the study of Yoon et al. 2018, were higher than those of GMZ07 and MX80, which are Na-bentonites regardless of temperature (Fig. 2). The thermal conductivities of GMZ07 and MX80 for water contents of 5.5–14.4% were similar to each other as temperature increased, in contrast to the thermal conductivity of KJ bentonite. This demonstrates the effect of bentonite type on whether the bentonite is Ca or Na type. The difference in thermal conductivity between the Na-bentonite and Ca-bentonite at similar water contents was 0.2–0.4 W‧mK−1, which indicates that Camontmorillonite is more efficient than Na-montmorillonite in terms of thermal dissipation.
Fig. 2
Thermal conductivity of different type bentonites with temperature (Legend name is expressed as bentonite type_water content).
3. Effects of Ca2+ and Na+ Cations in Montmorillonites
Bentonite swells when it comes into contact with water because of the relationship between the cation of montmorillonite and water. There are water and exchangeable cations between the particles of montmorillonite with a 2:1 structure of silica and gibbsite sheets. The place where water and exchangeable cations exist between these particles of montmorillonite is called interlayer. The cation in the interlayer draws water molecules, so that swelling by hydration occurs in the interlayer of the montmorillonites. At that time, it is more difficult for water to hydrate in the interlayer in the Ca-montmorillonite than in the Na-montmorillonite because the interaction between Ca2+ and the montmorillonite particle is greater than that between Na+ and the montmorillonite particle. Thus, the distance between Namontmorillonites when it is wet condition is longer than that between Ca-montmorillonites (Fig. 3). Also, the particles in Na-montmorillonites have higher number of stack contact with interacting diffuse double layer than those in Ca-montmorillonites [20], so that osmosis in diffuse double layers occurs more in Na-montmorillonite than in Ca-montmorillonites. Thus, the free water comes into the interlayers of the particles by osmosis makes Na-montmorillonites swell further comparing swelling in the Ca-montmorillonites. Therefore, the thermal conductivity of Ca-montmorillonite with shorter distance between montmorillonite particles might be higher than that of Na-montmorillonite.
Fig. 3
Hydration at interlayers of montmorillonites; (a) Na-montmorillonite, (b) Ca-montmorillonite.
4. Verification of Effect of Bentonite Type Using Effective Medium Model
The effective medium model [21], which accounts for the three phases of the soil structure, i.e., soil–air–water, and is suitable for silt and clay, was applied to the thermal conductivities of minerals shown in Table 1 to demonstrate the effect of the bentonite type as follows:
where λmineral is the thermal conductivity of the mineral, ƒmineral is a fraction of the mineral, λm is the thermal conductivity of montmorillonite, λq is the thermal conductivity of quartz, λc is the thermal conductivity of cristobalite, and λƒ is the thermal conductivity of feldspar. The thermal conductivity of the Ca-montmorillonite used was λm = 1.232 W‧mK−1 [22]. For the FEBEX bentonite, the fraction of cristobalite was assumed as 2%, considering that the thermal conductivity can be the highest for the mineral composition ratio. When the dry density is 1.6 g‧cm−3, the water content in the saturated state is approximately 24%, and when the specific gravity of bentonite is 2.71 [6], the porosity nmix is 39%. The saturation, Smix, is 1 when the bentonites are in a fully saturated state. The thermal conductivity of water, λwater, is approximately 0.61 [23] at room temperature. In the case of FEBEX, because the montmorillonite content is the highest, the thermal conductivity model value is the lowest because of the relatively insignificant effect of the composition of minerals other than that of montmorillonite (Fig. 4). Comparing the thermal conductivity measurement results in Fig. 1 with that of the effective medium model, the model values of Ca-bentonite as well as KJ and FEBEX bentonites indicated relative errors of 9% and 12% compared with the measured values, respectively. This shows that the effective medium model is reasonable considering the relative errors between the measured and predicted values of thermal conductivity from other geometric mean models [24], whereas Na-bentonites GMZ07 and MX80 demonstrated larger relative errors than KJ and FEBEX bentonites (Fig. 3). It was demonstrated that the thermal conductivity of the montmorillonite of Na-bentonite was lower than that of Ca-bentonite. To achieve a relative error of less than 15%, the thermal conductivity of the montmorillonite of Na-bentonite must be less than twice that of the montmorillonite of Ca-bentonite.
Fig. 4
Comparison of the thermal conductivities between the medium model results and the measured values.
5. Conclusions
In this study, the variation in the thermal conductivities of bentonites was evaluated based on the exchangeable cations. The exchangeable cations of bentonite might be regarded as an important factor in addition to the mineral composition of bentonite. Ca-bentonite indicated a higher thermal conductivity than Na-bentonite, even though its mineral composition was disadvantageous to the thermal conductivity. In the case of wet montmorillonite, the thermal conductivity of Ca-montmorillonite is higher than that of Na-montmorillonite, and it is thought that the distance between Na-montmorillonite particles is larger than the distance between Ca-montmorillonite particles due to hydration with predominant cations in the interlayer. Using the effective medium model, it was shown that the montmorillonite of Na-type bentonite exhibited a lower thermal conductivity than the montmorillonite of Ca-type bentonite. Hence, the development of high-functional bentonites using Ca-bentonite is expected to significantly improve the thermal conductivity of bentonites. Although it is necessary to proceed with further chemical experiments, the mechanism revealed through this study will contribute to the development of high-functional bentonite.
