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1. Introduction
A deep geological disposal system has been widely recognized as the safest disposal method of high-level radioactive wastes (HLWs) including spent nuclear fuels (SNFs) globally. In general, an underground facility or a deep geological repository (DGR) of the system consists of an engineered barrier system (EBS) and the surrounding host rock. An EBS is composed of a disposal canister, the surrounding buffers, backfill materials and so on. One deposition hole contains one disposal canister loaded with four pressurized water reactor (PWR) type SNFs and the surrounding bentonite buffers [1]. A disposal canister is made of a cast iron insert component and a copper shell that encloses it. The radioactive decay heat from four SNFs in a disposal canister is emitted in all directions. The heat is transferred to a cast iron insert component and then to a copper shell. The heat emitted from a disposal canister is transferred to the bentonite buffers in its immediate vicinity. The heat emitted from a deposition hole is transferred directly to the host rock adjacent to it or to rock via the deposition tunnel. The temperature distribution of a DGR, especially a disposal canister, depends on the constant spacing between numerous deposition holes with the heat sources of SNFs and between many deposition tunnels.
The maximum temperature at the outermost surface of a disposal canister should not exceed 100 degrees Celsius (℃) to maintain long-term stability of the buffer. This is one of many technical design requirements of a DGR system. If the temperature of a bentonite buffer near a disposal canister is beyond 100℃, montmorillonite of the bentonite can transform into other minerals including illite and chlorite. This transformation leads to decreased swelling pressure and plasticity and increased hydraulic conductivity and diffusivity [2,3]. The maximum temperature of a bentonite buffer is consistent with that at the outermost surface of a disposal canister. Because the outermost surface of a disposal canister comes into contact with a bentonite buffer due to the swelling of bentonite. In POSIVA of Finland, the uncertainties in the thermal heat transfer calculation procedure were studied and the safety margin of 5℃ is being applied to the calculation procedure. The uncertainties include the inaccuracies in the decay heat calculation of nuclear fuel assemblies, the anisotropy of the thermal conductivity of the host rock, the geometric eccentricity of the argon-filled gap and so on. The peak temperature considering the uncertainties at the outermost surface of a disposal canister should not exceed 95℃ [4].
The DGR system considered in this paper is the Kärnbränslesäkerhet 3-Vertikal (KBS-3V) system in which disposal canisters are emplaced vertically. In addition, the “disposal cell” consists of one vertical deposition hole with one disposal canister and the surrounding buffers, one horizontal deposition tunnel with backfill materials, and the surrounding host rock. The type of SNF in the disposal canister is PLUS7 16×16 PWR fuel which has been irradiated in the most PWRs in South Korea. The initial enrichment and specific burnup of the SNFs are assumed to be 4.0wt% 235U and 45 GWD·MTU−1, respectively [1].
The main objectives of this paper are as follows:
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(1) to conduct the three-dimensional (3D) thermal analyses for the disposal cell in which numerous fuel rods of the SNF are heterogeneously modeled,
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(2) to evaluate whether the peak temperature at the outermost surface of the disposal canister is less than 95℃ over the disposal time, and
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(3) to assess the effect of the cooling time of the SNFs and the spacings between the deposition holes and tunnels on the temperature distribution of the disposal cell.
2. Preprocessing
COMSOL Multiphysics Version 6.1 was used in this paper. It is a very suitable computational code for performing 3D thermal analyses, based on the finite-element method. It provides an integrated environment that allows users to easily model a geometry, make a mesh, set boundary conditions and so on through a graphical user interface [5,6].
2.1 Geometry and Meshing of Disposal Cell
As the first step of the thermal analysis, the 3D geometry of the disposal cell for the KBS-3V system was modeled using COMSOL Multiphysics. A vertical deposition hole, one horizontal deposition tunnel, and the surrounding granitic rock were contained in the geometry to model the disposal cell. The EBS was assumed to be located at the depth of 500 m underground. The height of the overall geometry of the disposal cell ranged from 20 m to 980 m underground. The geometries of one disposal canister loaded with four PLUS7 SNFs and the surrounding bentonite buffers were drawn to model the deposition hole [7-12]. In particular, the geometries of the SNFs were heterogeneously drawn to improve the reality and accuracy of the thermal analysis [13]. One PLUS7 SNF was composed of 236 fuel rods and five guide tubes. The fuel rods were modeled with only UO2 fuel pellets as the radioactive decay heat sources. The zircaloy-4 claddings and guide tubes were modeled as the cylindrical shells because their thicknesses were very thin. The height of the disposal canister was set so that PLUS7 SNFs were able to be loaded in the canister. The whole geometry of the disposal cell that contained four PLUS7 SNFs was equally divided into four regions in the radial direction. The modeled geometry was thus drawn for a quarter of the whole geometry to minimize the computation time and calculate thermal modeling effectively.
Figs. 1(a) and (b) show the whole and modeled 3D geometries of the disposal cell, respectively. Fig. 1(c) indicates the radial cross-sectional geometry of the disposal cell with 236 fuel rods of the SNF. In this figure, the red regions represent the cylindrical fuel rods of the SNF. The black, cyan, yellow, blue, violet, and light green regions express the fuel-loaded channel wall, cylindrical cast iron insert, copper shell, bentonite buffers, deposition tunnel, and granitic rock, respectively. The light gray region in the fuelloaded channel wall is argon gas, the light blue gap between the copper shell and bentonite buffers is air, and the white gap between the bentonite buffers and granitic rock is water.
Fig. 1 focused on the EBS in detail rather than the overall geometry of the disposal cell. The mesh elements of the 3D geometry of the disposal cell were generated with the total mesh number of 1,422,594 and an average quality of about 0.8036. This value was able to be regarded as statistically reasonable [14,15], which made the convergence better and faster for the thermal analysis. The inside of the fuel-loaded channel with many fuel rods of the SNF was made of numerous extremely fine mesh elements. Because the inside of the fuel-loaded channel was a very important region to directly release decay heat from all fuel rods to channel interior. The other regions with simpler geometries were composed of coarser mesh elements.
Figs. 2(a) and (b) show the 3D and radial cross-sectional mesh elements of the disposal cell, respectively. Fig. 2 (c) indicates the mesh element quality distribution of the mesh generation. In Fig. 2(c), the green and yellow areas represent that the quality of the mesh element are 1(best) and 0.5, respectively.
2.2 Thermal Properties of Materials
As the second step of the thermal analysis, the thermal properties of the materials that were composed of the disposal cell system were applied to the corresponding regions of the 3D geometry of the system. The materials of the PLUS7 SNF were the UO2 fuel pellet and zircaloy-4 cladding [1]. The materials of the fuel-loaded channel wall, insert component and shell of the disposal canister were structural steel, cast iron and copper, respectively. The materials of the buffers in the vertical deposition hole and of the horizontal deposition tunnel were bentonite and Friedland clay [16-17]. In particular, the granitic rock in South Korea was applied as the material of the host rock [18-19]. The dominant heat transfer mechanism of these materials was heat conduction because they were in a solid state. The thermal conductivities (ksolid), mass densities (ρsolid), and specific heat capacities (Cp,solid) of these solids were thus applied according to Eq. (1) [20].
In Eq. (1), T stands for the temperature, t the disposal time, and Q the decay heat source. The thermal conductivity of the UO2 fuel pellet especially depends on its temperature and specific burnup. The higher the specific burnup of the UO2 fuel pellet was, the lower its thermal conductivity was [21-23]. The most sensitive thermal property of the thermal analysis is generally the thermal conductivity of a material. Figs. 3(a) and (b) show the temperature-dependent thermal conductivities of the UO2 fuel pellet with a specific burnup of 45 GWD·MTU−1 and zircaloy-4 cladding, respectively [21].
Fig. 3
Thermal conductivities of (a) UO2 fuel pellet with 45 GWD·MTU−1 and (b) zircaloy-4 cladding [21].
The temperature-dependent thermal conductivities in Fig. 3 were set using the equations provided in Ref. 21. Table 1 shows the thermal conductivities, mass densities, and specific heat capacities of the solids [16-20].
Table 1
| Solid materials | Thermal conductivity (W·(m−1·K−1)) | Mass density (kg·m−3) | Specific heat capacity (J·(kg−1·K−1)) |
|---|---|---|---|
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| UO2 fuel pellet [21] | Fig. 3(a) | 10,950 | 250 |
| Zircaloy-4 cladding [21] | Fig. 3(b) | 6,500 | 281 |
| Structural steel [17] | 52 | 7,850 | 500 |
| Cast iron [17] | 36 | 7,200 | 461 |
| Copper [17] | 391 | 8,940 | 394 |
| Bentonite [16] | 0.906 | 1,830 | 1,287 |
| Friedland clay [16] | 0.544 | 1,720 | 1,208 |
| Granitic rock [18,19] | 3.398 | 2,580 | 831 |
The materials inside the fuel-loaded channel and between the cast iron insert component and the copper shell in the disposal canister were both argon gas. The material of the inner gap between the copper shell and the buffers in the deposition hole was dry air. The heat transfer mechanisms of these materials was thermal radiation as well as heat conduction because they were in a gas state. The emissivities (ε) emitted by the solid surfaces in contact with these gases as well as their thermal conductivities (kgas), mass densities (ρgas), and specific heat capacities (Cp,gas) were thus applied according to Eq. (2) [16].
In Eq. (2), εinner stands for the emissivity of the inner surface, εouter the emissivity of the outer surface, σ the Stefan- Boltzman constant (5.6697×10−8 W·m−2·K−4), Δl the gap thickness. The surface emissivities of the fuel cladding, channel wall, insert component and buffer were set to 0.7, 0.6, 0.6 and 0.8, respectively. In addition, the emissivities of the inner and outer surface of the disposal canister were set to 0.1 and 0.3, respectively [24]. The material of the outer gap between the buffers of the deposition hole and the granitic rock was water.
2.3 Initial and Boundary Conditions
As the third step of the thermal analysis, the initial and boundary conditions of the disposal cell system were applied to the corresponding regions of the 3D geometry of the system. For the initial conditions, the temperature of the top surface of the granitic rock was set to 14.1℃ [25]. In addition, the temperature of the bottom surface of the granitic rock was set to 40.38℃ because the geothermal gradient of the granitic rock in South Korea was 27.37℃·km−1 and the temperature rose as the depth under the ground increased [18]. For the boundary conditions, four lateral sides of the geometry were set to be symmetrical. In particular, the decay heat rate emitted from the PLUS7 SNF was important in the thermal analysis because it was the only heat source in the disposal cell system. In Ref. 23, the SNF was considered to be discharged at the Hanbit Unit 3 nuclear reactor in South Korea, and its average specific burnup and axial burnup profile were estimated. The initial enrichment and average specific burnup of the SNF were assumed to be 4.0wt% 235U and 45 GWD·MTU−1, respectively. In addition, the cooling time of the SNF contained in the disposal canister emplaced in the disposal cell was assumed to be three cooling time cases of 40, 50 and 60 years [1]. The initial enrichment, average specific burnup and cooling time of the SNF were used to calculate its average decay heat rate with time by the SCALE 6.1/ORIGEN-S code. Fig. 4 shows the average decay heat rate of the PLUS7 SNF with its cooling time. The horizontal axis represents the cooling time of the SNF and the vertical axis is its average decay heat rate.
The average decay heat rate in Fig. 4 were set using the equations fitted with three exponential functions. The average decay heat rate of the SNF decreased exponentially as its cooling time proceeded. The axial decay heat profile of the SNF was assumed to be the same as its axial burnup profile because its decay heat was in proportion to its burnup. The axial decay heat profile of the SNF over the cooling time was identical because its axial burnup profile varied only in a PWR operation but did not change during its cooling time. Fig. 5 shows the height-dependent decay heat profile of the PLUS7 SNF applied in this paper [26]. As the vertical axis denotes the percentage (%) of the SNF’s height, “0%” represents the bottom of the SNF and “100%” the top. The horizontal axis stands for the ratio of the height-dependent decay heat rate to the overall average decay heat rate.
The height-dependent decay heat profile in Fig. 5 were set using the table provided in Ref. 26. The product of the average decay heat rate and height-dependent decay heat profile of the PLUS7 SNF with time was set to the heat source condition of the thermal analysis.
3. Results and Analyses
All thermal analyses were based on the time-dependent simulation from the emplacement time of the disposal canister with the SNF in the disposal cell to 100,000 years later. The emplacement time was assumed to be the same as the cooling time of the SNF and three different cooling times of 40, 50 and 60 years were considered. In particular, the spacing between the deposition holes and between the deposition tunnels had a significant effect on the thermal analysis results. Five different spacings between the deposition holes and five different spacings between the deposition tunnels were thus considered. In other words, the thermal analysis cases in this paper were performed for three cooling time cases, five different spacing cases between the deposition holes and five different spacing cases between the deposition tunnels. Thus, the thermal analyses for a total of 27 cases were conducted to assess the temperature distributions of the disposal cell.
3.1 Reference Design Case
As the reference design case, the cooling time of the SNF was set to be 40 years, the spacing between the deposition holes was 8 m and the spacing between the deposition tunnels was 30 m. Fig. 6 indicates the reference design of the disposal cell. The dimensions in this figure represent the half of the diameter of the disposal canister and the bentonite buffers and the half of the spacing between the deposition holes and tunnels because the whole geometry of the disposal cell was modeled in a quarter.
The thermal analysis for the reference design case of the disposal cell was conducted after the geometry modeling of Fig. 6. Fig. 7 shows the 3D temperature distributions with the disposal time of the disposal cell for the reference case. Red displays the hottest temperature and blue indicates the coldest temperature in the rainbow-colored legend. The value of the regular triangle above the legend is the maximum temperature of the disposal cell and that of the inverted triangle below the legend is the minimum temperature. The unit of the temperature in Fig. 7 is degree Celsius (℃).
In Fig. 7, all peak temperatures at the outermost surface of the disposal canister for 100,000 years were located at the axial center of the SNF. Hence, Fig. 8 indicates the radial cross-sectional temperature distributions with the disposal time at the axial center of the SNF for the reference case.
As seen in Figs. 7 and 8, the decay heat of the SNF in the disposal canister was emitted in all directions. The heat was transferred to the cast iron insert component and then to the copper shell. The heat emitted from the disposal canister was transferred to the bentonite buffers. The heat emitted from the deposition hole was transferred to the granitic rock. In Fig. 8, the peak temperatures at the outermost surface of the disposal canister changed over 100,000 years. Hence, Fig. 9 shows the peak temperature with the disposal time at the outermost surface of the disposal canister for the reference case. The horizontal axis represents the disposal time, while the vertical axis indicates the maximum temperatures at the outermost surface of the disposal canister. The yellow dash line denotes the maximum temperature of 100℃ regulated for long-term stability of the buffer and the green dash line is the peak temperature of 95℃ considering the safety margin of 5℃ due to the uncertainties.
In Fig. 9, the peak temperature at the outermost surface of the disposal canister increased by the disposal time of 10 years, reached its highest value at 10 years, and then decreased thereafter. Because the emitted radioactive decay heat from the SNF in the disposal canister was sufficient to raise the overall temperature of the disposal cell by 10 years. However, the emitted decay heat rate of the SNF decreased exponentially with the disposal time and then its decay heat was not sufficient after 10 years.
However, the peak temperature at the outermost surface of the disposal canister at 10 years was calculated as 95.979℃. When the safety margin of 5℃ was added to this temperature, the peak temperature considering the uncertainties was 100.979℃. Therefore, the design of the disposal cell did not meet the thermal design requirement because this temperature was greater than the regulated temperature of 100℃. Thus, the design of the disposal cell had to be modified with respect to the cooling time of the SNF or the spacing between the deposition holes and tunnels.
3.2 Modified Design Cases
As the modified design cases, three SNF cooling time cases of 40, 50 and 60 years, five spacing cases of 6 m, 7 m, 8 m, 9 m and 10 m between the deposition holes and five spacing cases of 20 m, 25 m, 30 m, 35 m and 40 m between the deposition tunnels were applied in combination one by one. For the first modified case in which the spacing between the deposition tunnels was set to be 30 m, three cooling times of 40, 50 and 60 years and five spacings of 6 m, 7 m, 8 m, 9 m and 10 m between the deposition holes were considered to conduct the sensitivity analyses of peak temperature at the outermost surface of the disposal canister. Fig. 10 and Table 2 show the peak temperature at the canister surface that was the highest over the disposal time depending on the spacing between the deposition holes for the first modified case. The horizontal axis represents the spacing between the deposition holes and the vertical axis the peak temperatures at the canister surface. The red solid line represents the results for the cooling time of 40 years, the black those for 50 years and the blue those for 60 years.
Fig. 10
Peak temperature at canister surface with spacing between deposition holes for first modified case.
Table 2
Peak temperature at canister surface for first modified case
| Cooling time | Spacing between deposition holes | ||||
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| 6 m | 7 m | 8 m | 9 m | 10 m | |
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| 40 years | 102.130℃ | 99.583℃ | 95.979℃ | 93.359℃ | 92.201℃ |
| 50 years | 94.146℃ | 89.633℃ | 86.884℃ | 84.093℃ | 83.772℃ |
| 60 years | 86.257℃ | 81.454℃ | 79.160℃ | 76.665℃ | 75.946℃ |
In Fig. 10, the design conditions of the disposal cell below the green dash line met the thermal design requirement, implying that the disposal cell’s designs with the spacings of 9 m and 10 m between the deposition holes for the cooling time of 40 years had a peak temperature of less than 95℃ considering the safety margin of 5℃. The designs with all five spacings for 50 and 60 years showed a value of less than 95℃. As a result, the peak temperature at the canister surface decreased as the spacing between the deposition holes increased.
For the second modified case in which the spacing between the deposition holes was set to be 8 m, three cooling time cases and five spacings of 20 m, 25 m, 30 m, 35 m and 40 m between the deposition tunnels were applied to perform the sensitivity analyses of peak temperature at the canister surface. Fig. 11 and Table 3 show the peak temperature at the canister surface depending on the spacing between the deposition tunnels for the second modified case. The horizontal axis represents the spacing between the deposition tunnels.
Fig. 11
Peak temperature at canister surface with spacing between deposition tunnels for second modified case.
Table 3
Peak temperature at canister surface for second modified case
| Cooling time | Spacing between deposition tunnels | ||||
|---|---|---|---|---|---|
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| 20 m | 25 m | 30 m | 35 m | 40 m | |
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| 40 years | 105.250°C | 98.787°C | 95.979°C | 94.860°C | 94.589°C |
| 50 years | 95.715°C | 91.172°C | 86.884°C | 85.179°C | 84.968°C |
| 60 years | 87.822°C | 82.062°C | 79.160°C | 78.292°C | 77.990°C |
In Fig. 11, the conditions below the green dash line met the thermal design requirement, implying that the designs with the spacings of 35 m and 40 m between the deposition tunnels for the cooling time of 40 years had a peak temperature of less than 95℃. The designs with the spacings of 25 m, 30 m, 35 m and 40 m for 50 years showed a value of less than 95℃. The designs with all five spacings for 60 years resulted in a value of less than 95℃. As a result, the peak temperature at the canister surface decreased as the spacing between the deposition tunnels increased. In addition, the peak temperature at the canister surface decreased as the cooling time increased for the first and second modified case.
4. Conclusion
The 3D thermal analyses for the disposal cell in which SNF was heterogeneously modeled were performed. The analyses were based on the time-dependent simulation from the emplacement time of the disposal canister to 100,000 years later. As the reference case, the cooling time of the SNF was set to be 40 years, the spacing between the deposition holes was 8 m and the spacing between the deposition tunnels was 30 m. The 3D temperature distributions with the disposal time of the disposal cell were then calculated, and they were used to evaluate whether the peak temperature at the outermost surface of the disposal canister is less than 95℃ over the disposal time. As the modified cases, three SNF cooling time cases of 40, 50 and 60 years, five spacing cases of 6 m, 7 m, 8 m, 9 m and 10 m between the deposition holes and five spacing cases of 20 m, 25 m, 30 m, 35 m and 40 m between the deposition tunnels were applied in combination. The effect of the cooling time of the SNFs and spacings between the deposition holes and tunnels on the temperature distribution of the DGR system was then assessed. From the above results and analyses, the following conclusions were made:
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For the reference case, the peak temperature at the outermost surface of the disposal canister increased by the disposal time of 10 years, reached its highest value at 10 years, and then decreased thereafter. Because the emitted radioactive decay heat from the SNF in the disposal canister was sufficient to raise the overall temperature of the disposal cell by 10 years. However, the emitted decay heat rate of the SNF decreased exponentially with the disposal time and then its decay heat was not sufficient after 10 years.
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The peak temperature at the outermost surface of the disposal canister at 10 years was calculated as 95.979℃. When the safety margin of 5℃ was added to this temperature, the peak temperature considering the uncertainties was 100.979℃. Therefore, the design of the disposal cell did not meet the thermal design requirement and had to be modified.
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For the first modified case, the disposal cell’s designs with the spacings of 9 m and 10 m between the deposition holes for the cooling time of 40 years yielded a peak temperature of less than 95℃ considering the safety margin of 5℃. The designs with all five spacings for 50 and 60 years provided a peak temperature of less than 95℃. As a result, the peak temperature at the canister surface decreased as the spacing between the deposition holes increased.
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For the second modified case, the designs with the spacings of 35 m and 40 m between the deposition tunnels for the cooling time of 40 years resulted in a peak temperature of less than 95℃. The designs with the spacings of 25 m, 30 m, 35 m and 40 m for 50 years yielded a value of less than 95℃. The designs with all five spacings for 60 years provided a value of less than 95℃. As a result, the peak temperature at the canister surface decreased as the spacing between the deposition tunnels increased. In addition, the peak temperature at the canister surface decreased as the cooling time increased for the first and second modified case.
This study contributes to the improvement of the modeling technique for the thermal analysis of a disposal cell. In addition, it contributes to the optimization of the spacings between the deposition holes and tunnels of the disposal cell using the improved modeling technique.
In the future, it is necessary to study the thermal safety margin due to the uncertainties about the environment of South Korea. The uncertainties include the inaccuracies in the decay heat calculation of Korean nuclear fuel assemblies, the anisotropy of the thermal conductivity of the Korean host rock and so on.
