1. Introduction
According to data released by Korea Hydro & Nuclear Power in June 2022 as Fig. 1, the SNF (Spent Nuclear Fuel) storage capacity of the temporary storage facility at the nuclear power plant is expected to be saturated soon. Therefore, transportation from the current temporary storage facility to the intermediate storage facility is required.
Such transportation is largely divided into land transportation using vehicles and maritime transportation on the sea. According to the current status of domestic nuclear power plants provided by the Korea Atomic Industry Forum in October 2022 [2], all nuclear power plants are located on seaside. Considering this domestic environment, maritime transportation of radioactive waste for interim storage and disposal is an unavoidable management option. Evaluating the risk caused by accidents during the transportation of radioactive waste is a major item of radiation safety regulation technology [34].
In particular, risk assessment for transportation accidents of highlevel radioactive waste such as SNF is a key technology necessary to secure public safety from the radiological hazard of radioactive waste. One of the dangerous situations that can occur during maritime transportation is that the transportation cask loaded with radioactive waste is lost in the deep sea due to a ship accident, etc., and the contents of the transportation cask are released into the ocean. Evaluating how the released radioactive material will diffuse in the ocean and the extent to which the diffused nuclide will cause exposure to the public through various pathways including the food chain of ocean ecosystems are the key factors in risk assessment of the maritime transportation of radioactive waste. In foreign countries such as the United States, France, and Japan, studies to evaluate the risk of radioactive material transportation by sea and the sinking of the transportation cask have been conducted using codes such as MARINRAD, POSEIDON, and Barrier Effect Model [56].
There has been no reported case of selfdeveloped codes in Korea similar to the codes mentioned above, and related technology development is necessary. The abovementioned studies focus on the assessment of the ocean diffusion of radioactive materials and their effects. In MARINRAD and POSEIDON, the effect of barrier effect due to the engineered barrier existing in the transportation cask was ignored. In CRIEPI’s Barrier Effect Model, it was simplified by conservative assumptions and reflected in the calculation.
A typeB transportation cask is a very robust system and its containment function might not be completely lost even in the deep sea. Therefore, the risk assessment performed without properly considering the barrier effect is likely to give very conservative results. The goal of the second stage of this study is to reflect the barrier effect of the containment system of the transportation cask in the calculation of release rate of its radioactive contents with a scientific and engineering method using CFD. Through this, it would be possible to more realistically determine the release rate of radioactive material from the lost transportation cask in the deep sea.
As part of the model development, the goal of the first step is to develop detailed flow models inside and around the transportation cask to understand the detailed flow and heat transfer in and around the cask. The results will be coded and linked with the maritime diffusion assessment code and risk assessment code. Thus, it is intended to contribute to the development of an integrated maritime transportation risk assessment code of radioactive waste reflecting the domestic ocean environment.
2. CRIEPI’s Barrier Effect Model
2.1 CRIEPI’s Scenario of Radioactive Material Release
CRIEPI’s Barrier Effect Model is a model to calculate the release rate of radioactive material from a submerged transportation cask where the release rate (Ci·yr^{−1}) is kept smaller than the leaching rate of the contents due to the barrier effect of the transportation cask. The release scenario considered by CRIEPI is shown in Fig. 2.
It was assumed that all transportation casks could be salvaged in the near sea below 200 m depth and that there is no barrier effect by the transportation cask submerged in the sea deeper than 200 m. In addition, the barrier effect by the transportation cask was calculated assuming that the transportation cask was intact and only the sealing material was damaged in the sea of 200 m depth. In this case, the process of release rate calculation is as follows.

transportation cask submergence → Oring breakage → seawater inflow → exposure of radioactive contents to seawater → leaching of contents → increase of radioactive material concentration → seawater outflow from transportation cask
In this case, radioactive material can be released through diffusion, but the effect of convection is dominant. The driving force of the seawater flow into and out of the transportation cask is the buoyant force generated by the temperature difference. When the flow rate is not large, the concentration of radioactive material in the transportation cask reaches the solubility limit and the release rate is kept below the leaching rate of the radioactive material.
2.2 Release Rate Calculation
CRIEPI’s barrier effect model is a computational model established based on the conservation of mass for individual nuclides assuming that flow occurs due to natural convection. The phenomenon in which the outflow rate of nuclides due to convection is smaller than the leaching rate of nuclides is called the barrier effect. Depending on the damage to the transportation cask, it is determined whether the barrier effect is applicable or not. As shown in Fig. 5, the transportation cask must maintain the integrity of the containment boundary against designbased accidents such as 9 m drop, fire, puncture, and water immersion [34]. When the containment boundary of the transportation cask is damaged and gap occurs, the size is expected to be very small and it is reasonable to consider the barrier effect. However, the CRIEPI’s model used simple formulas for the concentration of nuclides and the flow rate of seawater inside the transportation cask, rather than using precise computational analysis. Therefore, the following supplements to the CRIEPI’s barrier effect model are suggested.

 Reflection of changes in seawater temperature due to changes in the ocean environment

 Reflecting the temperature change in the transportation cask considering the residual amount of nuclides

 Addition of a calculation module for input factors related to flow specifications according to the condition of the transportation cask

 Reflection of fuel damage ratio (FDR) in the calculation

 Calculation and reflection of the critical depth of seawater that can generate damages to the transportation cask and fuel cladding
2.3 Modified Scenarios of Nuclide Release From a Submerged Cask
In this section, a modified scenario of radionuclide release into seawater is proposed in Fig. 6. It is a regulatory requirement that a transportation cask should maintain its structural integrity under seawater of depth 200 m. Therefore, in the transportation cask whose containment boundary is not damaged, there is no release of contents until time passes the corrosion resistance life in the ocean environment at the depth of 200 m or less [4]. If the transportation cask is lost in the seawater of depth exceeding 200 m, it is necessary to obtain the critical depth at which the integrity of the containment boundary is compromised through the evaluation of the pressure limit of the transportation cask. If the transportation cask experienced a beyonddesignbasis accident which causes a permanent damage to the containment function, the seawater can flow into the transportation cask and initiate the leaching of the nuclides which are exposed to the seawater. Then the concentration of nuclides in the seawater in the transportation cask cavity increases. When this seawater leaks to the outside, the nuclide is released into the ocean. If the release rate of nuclides through the containment boundary is less than the rate at which the nuclides are leached into the cavity seawater, the seawater in the cavity is saturated and the leaching rate is controlled by the flow rate of the seawater through the containment boundary.
3. Analysis of Internal Flow of SNF Transportation Cask
3.1 SNF Model Selection
The SNF, which is used as the basis for the development of the internal flow analysis model of the transportation cask, was selected. According to the paper [10] published by Cho et al., the PLUS7 occupies the biggest portion of 20.7% of the total SNF storage and it is reasonable to consider PLUS7 as the reference for the development.
3.2 Development of a Simplified Model of Spent Fuel Assembly
If the PLUS7 fuel assembly is modeled in detail and used for analysis, it is inefficient in terms of calculation time and cost. Therefore, a simplified model based on the porous media proposed by the U.S. DOE (Department of Energy) was used [11].
The porous model is a model configured to conduct heat equally based on thermal conductivity as shown in Fig. 8. To develop a porous model, anisotropic thermal conductivity to each direction is required.

 The formula for effective thermal conductivity in the x, y axis direction is as follows [11].
: It is the heat diffusion equation for the steady temperature of a square media with heat generation. Assuming isotropic thermal conductivity it is as follows.
: The maximum temperature occurring at the center of the crosssectional area (x = y = 0) is as follows.
: Due to the strong divergence of cosh, the sum converges quickly and is as follows.
: Assuming the wall temperature is not zero, and substituting q''' = Q/4a^{2}L_{a} gives as follows.
: The thermal conductivity of PLUS7 is as follows.

 An areaweighted average method was used for the formula for effective thermal conductivity in the z axis direction.

 The finally obtained anisotropic effective thermal conductivity is summarized in Table 2.
Based on the obtained anisotropic thermal conductivity, SNF PLUS7 was simplified and used for CFD.
3.3 CFD Model of Flow Inside the Spent Fuel Cask
FLUENT 2021 R1, a generalpurpose thermal fluid analysis code, was used for the analysis. Because the flow path created at the containment boundary due to a beyond design basis accident is expected to be very small and the flow rate through this gap is limited, the thermal fluidic analysis inside the transportation cask was performed assuming intact containment boundary. In this analysis model, the standard kε model commonly used for turbulence analysis is utilized. In addition to the NavierStokes equations of motion and continuity equations, related expression is [12]:

 Dissipation rate
$$\begin{array}{l}\frac{\partial}{\partial {x}_{i}}\left(\rho \in {u}_{i}\right)=\frac{\partial}{\partial {x}_{j}}\left[\left(\mu +\frac{{\mu}_{t}}{{\sigma}_{k}}\right)\frac{\partial \in}{\partial {x}_{j}}\right]+{C}_{1\in}\frac{\in}{k}\left({G}_{k}+{C}_{3}{G}_{b}\right)\\ {C}_{2}\rho \frac{{\in}^{2}}{k}+{S}_{\in}\end{array}$$(17)
In this study, as a preparatory stage for the development of the CFD model simulating the barrier effect, a threedimensional 1/2 full model with symmetry was created by referring to the transportation cask capable of transporting 21 PWR (Pressurized Water Reactor) SNF as shown in Fig. 9.
The body of the referenced PWR transportation cask is made of carbon steel (SA350 LF3), and the upper part of the transportation cask is sealed with a double lid. A neutron shielding material (NS4FR) is installed on the outer shell of the transportation cask. The material of the shock absorber installed on the upper and lower parts of the transportation cask was assumed to be Balsa wood. For the reference fuel considered in this transportation cask analysis, the data of decay heat according to the burnup and cooling period of PLUS7 presented in were referred as in Fig. 10 [13]. In this study, a burnup rate of 45 GWd/MTU and a cooling period of 10 years were assumed. The decay heat of 1 bundle of nuclear fuel assembly was 800 W, and since 21 assemblies were loaded, the total decay heat was set to 16.8 kW. The vertically upright posture of the transportation cask is assumed where the boundary layer is expected to develop the most. The natural convection situation was assumed by applying gravitational acceleration toward the bottom of the transportation cask. In addition, it was assumed that the inside of the transportation cask was filled with seawater due to the accident condition. The external condition is the temperature of the seawater outside the transportation cask (T_{∞} = 15℃). 300 [W·m^{−2}− K^{−1}] was entered for the convective heat transfer coefficient to determine the temperature of the surface of the transportation cask [15]. The pressure inside and outside the container was assumed the same due to the breached containment boundary.
The material properties for the flow analysis inside the transportation cask are shown in Table 3~Table 8 [1620].
4. Analysis of External Flow Around Transportation Cask
The posture of the transportation cask submerged in the deep sea is random. Considering the shape of SNF transportation cask and the posture during the transportation, the horizontal position where the long side of cask is horizontal is most likely. In this study, the seawater flow and heat transfer around the submerged transportation cask are evaluated with CFD code, Fluent. Because the shape of the seabed is diverse, various postures can be made. However, as in the drop test of a transportation cask, vertical and horizontal postures are two limiting cases in terms of release rate. Therefore, the analyses are performed for the two postures, the horizontal and the vertical.
Purpose of these analyses are as follows:

 Determine the analysis area for simulating the detailed seawater flow around the transportation cask submerged in the deep sea.

 Evaluate how the convective heat transfer coefficient between the transportation cask and the surrounding seawater changes depending on environmental factors (seawater temperature, seawater velocity, and cask posture).

 Creating a boundary layer in the vicinity of lid gap and identifying the detailed flow in this region.

 Identify severer posture in terms of release rate of contents.
Finally, this analysis result is linked with the detailed heat flow evaluation model of the transportation cask in Section 3 and the assumptions made for the analyses are verified. Seawater flow into and out of the transportation cask will be evaluated in our future study and the results obtained in current work will be used as basic data for flow evaluation in microchannels created at the containment boundary.
4.1 Simplification of Spent Nuclear Fuel Transportation Cask
In this analysis, the transportation cask was modeled as a cylinder to which the surface temperature was given. The cuboidal seawater area around the transportation cask was selected as the area of analysis. In addition, the material properties for analysis are the same as those of the internal model [1620].
4.2 Setting of Analysis Area for Flow Evaluation Around Transportation Cask
In general, when performing external flow analysis on a model with a height of H and a width of W, the height of the external analysis area should be at least 5 times the height of the model and 10 times the depth of the model [21]. After setting the size according to the above criteria, the value of the convective heat transfer coefficient between the transportation cask and seawater was calculated, and the size was reduced to the extent that this value does not change significantly.
4.3 External Flow Model of the SNF Transportation Cask
FLUENT 2021 R1, a generalpurpose thermal fluid analysis code, was used for the analysis. For the flow model outside the transportation cask, the flow at the transportation cask surface is important. kω SST (shear stress transport) model with relatively accurate results in the wall boundary region was used. In addition to the NavierStokes equations of motion and continuity equations, related expression is [2223]:

 Dissipation Rate
$$\begin{array}{l}\frac{\partial \omega}{\partial t}+{U}_{j}\frac{\partial \omega}{\partial {x}_{j}}=a{S}^{2}\beta {\omega}^{2}+\frac{\partial}{\partial {x}_{j}}\left[\left(v+{\sigma}_{\omega}{v}_{T}\right)\frac{\partial \omega}{\partial {x}_{j}}\right]\\ +2\left(1{F}_{1}\right){\sigma}_{\omega}^{2}\frac{1}{\omega}\frac{\partial k}{\partial {x}_{i}}\frac{\partial \omega}{\partial {x}_{i}}\end{array}$$(21)
This model is a mixture of the two models to apply the kω model for flows near the wall and the kε model for flows away from the wall. This model can be used without problems for multiple target problems. In general, it is known to provide good results near the wall boundary [24].
In addition, a wall function [25] was used to determine whether all the flow on the wall is expressed in the boundary mesh layer to accurately simulate the flow on the wall as shown in Fig. 11. The wall function describes the empirical behavior on the wall as a dimensionless velocity and vertical distance from the wall and is expressed as follows.
The dimensionless normal distance on the surface is as follows [25].
Among them, the condition of y^{+} ≤ 5, which corresponds to a viscous sublayer that reflects the effect on viscosity at the boundary layer wall well, was used as shown in Fig. 12. Mesh was created using an inflation layer on the boundary around the cylindrical surface. When checking the dimensionless distance from the boundary layer wall, it was confirmed that y^{+} = 0.648 and the condition of the viscous sub layer was satisfied.
In this model, a steady state in vertical and horizontal orientations was considered. For the condition of the outside of the transportation cask, the temperature of the external seawater T_{∞} = 15℃ was assigned to the same as the internal model in Section 3. As shown in Fig. 13, the external seawater was assumed to be in a forced convection situation with a flow velocity of v = 0.1 m·s^{‒1}.
5. Results
5.1 Analysis Result of Internal Flow Model of SNF Transportation Cask
The temperature distributions inside and on the surface of the transportation cask are depicted in Fig. 15. The maximum temperature of the internal seawater rises to 102℃ as shown in Fig. 15 and the temperature difference between the inside and outside of the transportation cask reaches 87℃. The average temperature of the transportation cask surface is evaluated as 20℃.
The flow in the transportation cask is shown in Fig. 16. The flow is generated by natural convection due to the decay heat of fuel assemblies. From the simulation results, it was confirmed that turbulence occurred in the middle of the transportation cask and the flow circulates to the top and bottom of the transportation cask. As a result, a flow velocity of up to 1.72 m·s^{‒1} occurred at the bottom of the transportation cask.
5.2 Appropriate Size of Analysis Area for the External Flow Simulation
The transportation cask has a large diameter and the flow around it has a big Reynolds number. This implies the development of turbulent flow paths. Therefore, the analysis area for the downstream was set large to account for the cylindrical wake well. If the analysis is performed in a flow field smaller than the above conditions, an error such as a backflow occurs at the outlet, and proper convergence may not be achieved. Therefore, for the flow field simulation, it is reasonable to set the height of analysis area to twice the height of the transportation cask, the depth to be eight times the height of the transportation cask, the upstream to twice the height of the transportation cask, and the downstream to twelve times. Therefore, the size of the flow field used in the analysis is Width: 28,000 mm, Depth: 16,000 mm, and Height: 10,000 mm.
5.3 Analysis Result of External Flow Model of SNF Transportation Cask
The results of the external flow simulation are shown in Figs. 17‒20. The surface heat transfer coefficient is calculated as h = 362.8 [W·m^{−2}− K^{−1}] in the vertical orientation, and h = 313.2 [W·m^{−2}− K^{−1}] in the horizontal orientation. The heat transfer coefficient is smaller when the transportation cask is in horizontal orientation and a bigger temperature difference between the inside and outside the transportation cask is expected. This will result in bigger buoyant force inside the transportation cask and eventually a large flow rate through the breached containment boundary of the transportation cask.
In the internal flow model of the SNF transportation cask, the heat transfer coefficient was input as 300 [W·m^{−2}− K^{−1}] on the surface. When the decay heat of nuclear fuel is transferred to the surface of the transportation cask and thermal equilibrium is achieved with the temperature of the seawater outside the transportation cask with T_{∞} = 15℃, the surface temperature is T_{s} = 20℃. The heat transfer coefficient calculated in this external flow model does not show significant difference with the input value 300 [W·m^{−2}− K^{−1}] for the internal flow model. Therefore, it is shown that the input parameters of the internal and external flow models are physically reasonable and can be used in the development of the detailed flow model around the gap created at the containment boundary of the transportation cask.
It is expected that when the transportation cask is in horizontal position, bigger flow rate into and out of cask is expected than when in vertical position. In horizontal position, the bottom and top of the lid gap become the inlet and outlet of buoyant flow and the whole lid gap is used as flow passage. In vertical position, the inlet and outlet of flow share the lid gap and the effective flow area becomes smaller as shown in Fig. 21.
6. Discussion and Conclusions
This study was carried out as part of the development of a CFD model simulating the barrier effect. In this study, a modified scenario of release of radionuclides from a submerged SNF transportation cask was presented. A detailed CFD model was built to simulate the fluid flow inside the transportation cask and simplified models were developed to simulate the fluid flow and heat transfer around the submerged transportation cask in the ocean.
 A modified radioactive material release scenario was presented in consideration of the condition of the lost transportation cask and the depth of the sea where the cask is located. If the extent of damage is limited to the loss of Oring function of the transportation cask, the barrier effect is significant, and a much smaller release rate than the leaching rate of nuclides is predicted.

 A detailed analysis model of the transportation cask was constructed as the preparatory step to perform the fluid analysis of the transportation cask with damage to the containment boundary.

 The external flow model of transportation cask was developed. The proper analysis area around the transportation cask was identified based on the flow characteristics and resulting heat transfer from the transportation cask.

 It is confirmed that the temperature difference between the inside and outside of the transportation cask is bigger in the case of the horizontal position because the heat transfer coefficient is lower. It will create a bigger buoyancy inside the transportation cask and a larger flow velocity through the damaged containment boundary. Therefore the horizontal posture is expected to be a conservative posture in terms of the release rate and this will be confirmed in our future study.
The fluid flow through the gap in the containment boundary of the transportation cask should be evaluated by combining the fluid flow inside and outside the transportation cask. In our future research, the two models developed in this work will be combined to accurately simulate the fluid flow in the gap of microsize and eventually calculate accurate release rate of nuclides from the transportation cask to the ocean.