1. Introduction
Spent nuclear fuels emit strong radiative decay heat over long periods. Therefore, spent nuclear fuel must maintain its thermal integrity during longterm dry storage [1]. In the thermal analysis of dry storage systems, the fuel rod temperature has been evaluated conservatively by simplifying the spent fuel assembly to an effective thermal conductivity or a porous media model [2, 3]. Conservative evaluation of the fuel rod temperature has advantages in terms of thermal safety and licensing of dry storage systems. However, the conservation of spent fuel temperature leads to certain issues, such as reduced storage and disposal densities, longer storage periods, decreased transportation efficiency, and increased radial hydrides in the cladding [4]. Additionally, detailed temperature distributions of fuel cladding are required to evaluate the longterm integrity of spent fuel dry storage systems [5].
The Korea Atomic Energy Research Institute (KAERI) is developing a single fuel assembly test equipment with the same dimensions and shape as the PLUS7 fuel assembly to obtain detailed temperature distributions of the spent fuel in a dry storage system. The objective of this study is to obtain detailed temperature distributions of a fuel assembly through subchannel analysis using the COBRA SFS code in parallel with the thermal test of a single fuel assembly.
The Pacific Northwest National Laboratory developed the COBRASFS code [6] to predict fuel temperatures, fluid temperatures, and velocities under various flow conditions in spent fuel shipping and storage casks. COBRASFS was built based on the COBRAIIIC [7] developed for the subchannel analysis of fuel assemblies in the reactor core. It is an incompressible singlephase flow analysis code that uses semiimplicit methods to solve mass, momentum, and energy conservation equations. COBRASFS has been used for subchannel analysis of various spent fuel assemblies with different backfill gases in canisters. Furthermore, the code has been incorporated into the Used Nuclear FuelStorage, Transportation & Disposal Analysis Resource and Data System (UNFST&DARDS) tool [8] as a module devoted to the thermal analysis of spent fuel packages.
Several studies on the thermal analysis of dry storage casks using the COBRASFS were conducted more than 20 years ago to validate the first version of the code [911]. Recently, some institutions have been using the COBRASFS for validation of new code. T.E. Michener et al. [12] conducted verification and validation of COBRA SFS by comparing the COBRASFS temperature predictions with the experimental data at the Idaho National Laboratory for 4 types of storage casks. In this study, COBRASFS was shown to be capable of detailed and accurate temperature prediction of spent fuel assemblies in the storage casks. M. Sevecek et al. [13] conducted validation of CFD to COBRASFS using experimental data for the CASTOR 440/84 model. In this study, it was confirmed that the COBRASFS accurately predicts the PCT in dry storage cask.
The purpose of this study is to characterize COBRASFS modeling parameter for vertical storage cask. In this study, we aim to identify the most sensitive COBRASFS modeling parameters for PCT prediction. Identification of most sensitive parameters for PCT can be used inform the development of best estimate thermal model, both in more accurate estimation of uncertainties in PCT. The exact difference in PCT can only be applicable to this specific canister and fuel loading, but the sensitivity of each parameter will be applicable to any vertical dry storage cask.
2. Overview of Thermal Test Equipment
KAERI is developing thermal test equipment for a single fuel assembly to simulate the thermal flow in spent fuel assembly. As depicted in Fig. 1, the test equipment comprises a prototype pressurized water reactor (PWR) fuel assembly, which is electrically heated inside a representative fuel basket, and a cylindrical pressure vessel representing the canister. A simulated fuel assembly was manufactured with dimensions and shapes identical to those of the PLUS7 fuel assembly, which is primarily used in nuclear power plants in Korea. The PLUS7 assembly has a 16 × 16 rod configuration with 236 rods, 4 guide tubes, an instrument tube, 12 spacer grids, a top nozzle, and a bottom nozzle. A total of 64 thermocouples are installed on the heater rods to measure the surface temperature of the cladding. In the crosssection of the fuel assembly, 16 thermocouples were placed at four axial positions on the heater rods.
The canister will be filled with different backfills (air, helium, and vacuum) in both vertical and horizontal directions. The test equipment will be installed in the constant temperature chamber to maintain a constant ambient temperature, and the axial and radial temperatures will be measured. The thermal test equipment is currently being manufactured, and thermal tests using the test equipment will be performed from next year to obtain verification data for the COBRASFS analysis.
3. Subchannel Analysis of a Single Fuel Assembly
3.1 Subchannel Analysis Modeling
To predict the temperature distributions of the single fuel assembly canister, which is under development at KAERI, a threedimensional subchannel analysis model was established for the COBRASFS analysis.
Fig. 2 depicts a transverse crosssection of the COBRA SFS analysis model illustrating the subchannels and wall nodes. The model comprises two assemblies, namely the inner and outer assemblies. The inner assembly is a PLUS fuel assembly containing 256 fuel rods and 289 fluid subchannels. The outer assembly is a downcomer assembly with 16 subchannels for a large flow area between the basket and canister. The model contains 20 uniform axial nodes in the axial direction, and 236 heated and 20 unheated rods. Sixteen wall nodes (eight baskets and canisters) were used to describe the basket and canister structural members. Heat conduction between the solid nodes is modeled by the thermal conductance at the interfaces.
Fig. 3 illustrates the vertical crosssection of the analysis model. The three primary areas of the COBRASFS model include the upper plenum, channel zone, and lower plenum. The channel zone comprises fuel rods and subchannel representations of the fuel assembly inside wall nodes indicating the basket and canister. The PLUS7 fuel assembly has a 16 × 16 rod array. The total length of the fuel rods excluding the top and bottom nozzles is 4,094 mm, and the active length is 3,810 mm. Here, the upper and lower plenums were used to simulate the mixing volumes and heat transfer in the upper and lower cavities of the channel zone. In the plenums, onedimensional radial and axial heat transfer is evaluated without calculating the flow field. Typically, the plenum boundary temperature must be stipulated to construct the thermal model of the plenum regions. In this model, the environmental temperature was considered as 150°C in the channel zone, and the temperatures of the upper and lower plenums were assumed to be 170 and 130°C, respectively. The convective heat transfer coefficient at the canister surface was considered to be 5 W·m^{−2}·K^{−1}, and the emissivity of the stainless steel was 0.36.
COBRASFS provides a detailed modeling of the flow and heat transfer within the fuel assembly in the basket, simulating the heat transfer by conduction and convection. Furthermore, thermal radiation can be calculated using radiation exchange factors (rodtorod and rodtowall) for all fuel rods in the fuel assembly. The RADGEN [14] program calculates the radiation exchange factors for various spent fuel rod geometries. The emissivity values of 0.8 [15] and 0.36 [16] were selected for the oxidized zircaloy fuel cladding and stainless steel, respectively.
The Nusselt number (Nu) is a dimensionless number which quantifies convective heat transfer from a surface. The Nusselt number is defined as the ratio of heat transfer by convection to the heat transfer by conduction within a fluid. In general, the Nusselt number is considered as 3.66 for a fully developed laminar flow in a circular tube with a constant wall temperature [6]. In this analysis model, the Nusselt number was set to be 3.66 as the convective heat transfer coefficient.
In COBRASFS, the typical flow resistances affecting the momentum equation are the axial pressure losses caused by the wall friction factor, and local obstacles in the flow field. In general, the friction factor is a function of the Reynolds number (Re). The rod and wall frictions in the interior assembly were modeled with f = 100/Re [9] for a fully developed laminar flow between cylinders arranged in a square array. The friction factor inside the canister was modeled with f = 64/Re for a fully developed laminar flow in a circular tube. In COBRASFS, the pressure losses generated from the grid spacers of the fuel assembly can be considered for analysis. The pressure losses in the grid spacers are considered as K = 1 and 8 for the eggcrate and bartype spacers [17]. Eleven eggcrate grid spacers are installed in the PLUS7 fuel assembly. The pressure losses in the grid spacers were considered as K = 1 in the analysis model.
A subchannel analysis was performed with air as the backfill considering a vertical direction at a decay heat of 1.0 kW. Spent PWR fuel is typically cooled at least 5 years in the pool before transport to cask. The maximum decay heat of the fuel assembly with a cooling time of 5 years is about 1 kW. In this study, the decay heat of the fuel assembly was considered to be 1 kW.
3.2 Subchannel Analysis Results for a Single Fuel Assembly
As stated in Section 3.1, a subchannel analysis of a single assembly canister was performed with air backfill in the vertical direction at a decay heat of 1.0 kW. Fig. 4 shows the temperature contours of fuel rods. The fuel rod temperatures ranged from 170.7–262.9°C, with higher temperatures at the upper part. Fig. 5 depicts the maximum temperature distribution in the canister. The maximum temperatures of the canister and basket were determined to be 166 and 210°C, respectively. The maximum temperature of the fuel rods was between 223–263°C. The transverse temperature distribution in the crosssection of the canister was symmetrical on the left and right sides. Fig. 6 illustrates the axial temperature profiles of the main fuel rods. The lowest temperature at Rod #1 (corner rod) ranged from 171–223°C, whereas the highest temperature at Rod #104 (center rod) ranged from 186–263°C. The axial temperature distribution was higher in the upper part of the canister owing to the effect of natural convection.
Fig. 7 shows the velocity vectors in the fuel assembly. The fluid velocities ranged from 0.042–0.108 m·s^{−1}, with slightly higher values at the upper part. Fig. 8 depicts the maximum velocity distribution in the canister. The maximum velocity in the fuel assembly ranged from 0.048–0.108 m·s^{−1}. The lowest velocity was identified in subchannel 19 (close to the corner), from 0.042–0.048 m·s^{−1}, whereas, subchannel 9 (side position) exhibited the highest velocity, ranging from 0.094–0.108 m·s^{−1}. The flow velocities in the open downcomer region were negative. Therefore, a recirculation flow was established between the fuel assembly and the open downcomer region.
4. Sensitivity Analysis of Thermal Parameters Affecting the Peak Cladding Temperature
Emissivity, convective heat transfer coefficients, flow resistances, and thermal conductivities were considered for the sensitivity analysis. Table 1 summarizes all the thermal parameters used for the analysis.
Table 1
Thermal input parameters  Values used for the sensitivity analysis  



Minimum  Nominal  Maximum  


Cladding emissivity (ε_{c})  0.6  0.8  1.0 
Basket emissivity (ε_{b})  0.2  0.36  0.5 
Convective heat transfer coefficient (Nu)  1.0  3.66  5.0 
Wall friction factor (f)  64/Re  100/Re  150/Re 
Form loss coefficient (K)  0.0  1.0  10.0 
Thermal conductivity of air (k_{air})  0.8 k_{air}  1.0 k_{air}  1.2 k_{air} 
Conductivity of pellet and cladding (k_{fuel})  0.8 k_{fuel}  1.0 k_{fuel}  1.2 k_{fuel} 
Conductivity of basket and canister (k_{sts304})  0.8 k_{sts304}  1.0 k_{sts304}  1.2 _{sts304} 
4.1 Sensitivity Analysis of Surface Emissivity
As the spent fuel cladding temperature reaches approximately 300°C during dry storage, radiation is the dominant heat transfer mode. Thermal radiation from surface to surface is modeled using radiation exchange factors reported by Hottel and Saroffim [18]. The radiation heat transfer rate emitted from surface i to surface j, expressed as q_{ij}, is defined with the exchange factor F_{ij}.
Radiation exchange factors for enclosures in the subchannel model can be calculated from the RADGEN, an auxiliary program of COBRASFS. The linear equations are solved for each enclosure by the Hottel’s crossedstring method [19].
In the base model, the emissivity values of fuel cladding and stainless steel (basket) were set to 0.8 and 0.36, respectively. For the sensitivity analysis of the emissivity, the cladding emissivity was considered as 0.6, 0.8, and 1.0, whereas the basket emissivity was considered as 0.2, 0.36, and 0.5.
Table 2 lists the sensitivity of temperature to the emissivity of the fuel cladding and basket. As the emissivity values of the fuel cladding and basket increased, the temperatures of the fuel rods and basket decreased owing to the increased effect of radiative heat transfer.
Table 2
Emissivity  Maximum temperature (°C)  



Rod #1  Rod #8  Rod #56  Rod #104  Basket  


Fuel cladding  0.6  222.7  237  260.8  269.7  210.3 
0.8  222.6  235.7  255.4  262.9  210.2  
1.0  222.6  234.7  250.9  257.1  210.1  


Fuel basket  0.2  222.7  2370  260.8  269.7  210.3 
0.36  222.6  235.7  255.4  262.9  210.2  
0.5  222.6  234.7  250.9  257.1  210.1 
When the cladding emissivity increased from 0.6 to 1.0, the PCT (Rod #104) decreased by 12.6°C (4.9%), from 269.7 to 257.1°C. Additionally, when the basket emissivity was increased from 0.25 to 0.50, the PCT (Rod #104) decreased by 8.6°C (3.3%), from 267.6 to 259.0°C. The emissivity values of the fuel rod and basket significantly affected the PCT sensitivity. Therefore, the selection of the emissivity is crucial for accurately predicting the PCT.
4.2 Sensitivity Analysis of Convective Heat Transfer Coefficients
In COBRASFS, the convective heat transfer coefficient can be expressed as a dimensionless Nusselt number.
The Nusselt number is usually a function of the Reynolds number (Re) and the Prandtl number (Pr) as follows.
The energy equation for a fully developed laminar flow in a circular tube can be expressed as below, and the Nusselt number can be calculated using the energy equation.
The values of the Nusselt number are determined to be 4.36 and 3.66 for tubes with constant surface heat flux at the wall and constant wall temperature, respectively.
The Nusselt number was specified as 3.66 in the base model. In general, convective heat transfer can be ignored if the Nusselt number is 1.0. For the sensitivity analysis of convective heat transfer, the Nusselt numbers were set to 1.00, 3.66, and 5.00.
Table 3 lists the temperature sensitivity of rods and the fuel basket to the convective heat transfer coefficients. Depending on the Nusselt number, the PCT varied by more than 10°C. Typically, a convective heat transfer coefficient of Nu = 3.66 is used for a fully developed laminar flow in a circular tube, such as inside the spent fuel assembly. Therefore, using the Nu = 3.66 as the convective heat transfer coefficient, the axial convection effect can be sufficiently simulated.
Table 3
Nusselt number  Maximum temperature (°C)  



Rod #1  Rod #8  Rod #56  Rod #104  Basket  


1.00  227.2  242.6  263.8  271.9  210.8 
3.66  222.6  235.7  255.4  262.9  210.2 
5.00  220.9  233.1  251.9  259.1  209.9 
4.3 Sensitivity Analysis of Flow Resistance Coefficients
The variables affecting the momentum equation are the axial pressure losses caused by friction at the wall, and local obstacles such as grid spacers in the flow field. COBRA SFS considers the wall friction factors and form loss coefficients as input data. The wall friction factors can be expressed using the Darcy friction factor formulae, which can be defined as a pressure drop, such as a friction pressure gradient, in the axial direction [20].
In the cavity between the basket and canister, the wall friction factor was modeled using the typical friction factor for a fully developed laminar flow in a circular tube, with f = 64/Re. In the case of the interior fuel assembly, the wall friction factor was considered as f = 100/Re for laminar flow between cylinders arranged in a square array.
In the base model, the wall friction factor was set to f = 100/Re. Moreover, the wall friction factors considered during the sensitivity analysis were f = 64/Re, 100/Re, and 150/Re.
In the flow field with grid spacers, the pressure loss caused by the form drag on local obstacles can be calculated as:
The axial form loss coefficient can be expressed as:
The form loss coefficient in the grid spacers is considered as K = 1 for the eggcrate spacers. In the base model, the form loss coefficient was set to K = 1. During the sensitivity analysis, the form loss coefficients were considered as K = 0, 1, and 10.
Table 4 lists the temperature sensitivity to the flow resistance coefficients. The results of the sensitivity analysis indicated that the effect of the flow resistance coefficients such as friction factors and form loss coefficients on the PCT was almost negligible.
Table 4
Flow resistance coefficients  Maximum temperature (°C)  



Rod #1  Rod #8  Rod #56  Rod #104  Basket  


Friction factor (f)  64/Re  222.4  235.6  255.3  262.8  209.9 
100/Re  222.6  235.7  255.4  262.9  210.2  
150/Re  222.8  235.7  255.4  262.9  210.4  


Form loss coefficient (K)  0  222.6  235.7  255.4  262.9  210.2 
1  222.6  235.7  255.4  262.9  210.2  
10  222.6  235.7  255.4  262.9  210.1 
4.4 Sensitivity Analysis of Thermal Conductivities
Typically, heat conduction in a dry storage canister occurs within the internal filling gas, fuel rods, and wall nodes indicating the fuel basket and canister. In COBRASFS, fluid conduction is only considered in the lateral direction; axial heat conduction of the gas is neglected. Conduction between adjacent solid nodes that represent continuous material is calculated by the heat conduction equation. The contact conductance at the interface must be properly considered for adjacent nodes representing different materials or structures.
Air was considered as the fluid inside the single fuel assembly canister, and stainless steel was used as the fuel basket and canister. The fuel rods comprise UO_{2} pellets and Zirlo cladding. In this study, the effect of variations (±20%) in the thermal conductivity of air, fuel rod (pellet and cladding), and stainless steel (basket and canister) on the PCT were evaluated.
Table 5 summarizes the temperature sensitivity of the rods and fuel basket to the thermal conductivities of the materials. When the thermal conductivity of air changed by ±20%, the PCT exhibited a difference of approximately 5°C. Therefore, the accurate thermal conductivity of the fluid is required to predict the actual PCT. The effect of the thermal conductivities of the fuel rod and stainless steel on the PCT was almost negligible.
Table 5
Thermal conductivity (k)  Maximum temperature (°C)  



Rod #1  Rod #8  Rod #56  Rod #104  Basket  


Air  0.8 k_{air}  223.4  236.9  256.9  264.5  210.3 
1.0 k_{air}  222.6  235.7  255.4  262.9  210.2  
1.2 k_{air}  220.9  233.2  252.1  259.3  209.7  


Fuel rod (pellet and cladding)  0.8 k_{fuel}  222.6  235.7  255.4  262.9  210.2 
1.0 k_{fuel}  222.6  235.7  255.4  262.9  210.2  
1.2 k_{fuel}  222.6  235.7  255.4  262.9  210.2  


Stainless steel (basket and canister)  0.8 k_{sts304}  222.7  235.8  255.4  262.9  210.3 
1.0 k_{sts304}  222.6  235.7  255.4  262.9  210.2  
1.2 k_{sts304}  222.6  235.7  255.3  262.8  210.1 
5. Conclusions
In this study, a subchannel analysis model was developed for the single fuel assembly canister using the COBRA SFS. Furthermore, thermal parameters affecting the PCT were selected and sensitivity analyses were performed to quantify the effects of these parameters on the PCT. The conclusions of the study can be summarized as follows.
The transverse temperature distribution was symmetrical on both the left and right sides of the canister. The axial temperature distributions were higher in the upper part of the canister because of natural convection. The air velocities in the open downcomer region were negative, and a recirculation flow was observed between the fuel assembly and downcomer region.
Sensitivity analysis according to the thermal parameters showed that the PCT was affected by the emissivity of the fuel cladding and basket, convective heat transfer coefficient, and thermal conductivity of the fluid. However, the effects of the wall friction factor of the canister, form loss coefficient of the grid spacers, and thermal conductivities of the solid materials, on the PCT were predominantly ignored.
The sensitive parameters for PCT obtained from this study can be used inform the development of best estimate thermal model, both in more accurate estimation of uncertainties in PCT. The exact difference in PCT can only be applicable to this specific canister and fuel loading, but the sensitivity of each parameter will be applicable to any vertical dry storage cask. The results of this study will be used as basic data for the thermal performance testing and analysis validation of the single fuel assembly canister to be performed in the future.