Geometric Model

The dimensions of the cask model and concrete shell were based on Holtec's MPC-32^[26]. We reduced the inner structure of the storage cask to a single basket, and the spent fuel assembly was simplified to helium through a porous media model.

According to the reference, the porosity of the solid component is 0.652 (solid volume/total volume=0.348) and approximately equal to the estimated value, whereas the correlation coefficient in the literature is obtained from experiments. The porosity in this study was 0.348, the viscosity coefficient was 687973 m^-2, and the inertia coefficient was 29.86 m^{-1 [26]}.

The overall model includes concrete cladding, an interior annular gap, a metal shell, and a basket (Fig.1). The inclination angle was tilted θ (15°, 45°, 75°, and 90°) based on the central axis of symmetry (Fig.1). Table 2 presents the specific structural dimensions of the models. Because the overall structure is center-symmetric, a 1/8 model was used to spare computing resources.

Fig. 1Full-screen View

(Color online) Cask 3D model.

Physical model

Gravity was considered in the z-axis direction. The acceleration of gravity is 9.8 m/s², the external air was 300 K and 101,325 Pa, and the internal helium was 294 K and 330,000 Pa. We made the following assumptions to simplify the calculation of the fluid working conditions:

(1) there is transient heat transfer of the 3D model;

(2) the fluid was steady, incompressible, and isotropic;

(3) all solid materials are isotropic; and

(4) the displacement at the liquid-solid coupling is the same.

According to the four aforementioned assumptions, the following governing equation can be obtained:

Continuous equation $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0.$ (1) Momentum conservation equation $u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial w}{\partial z}=-\frac{1}{\text{ρ}}\frac{\partial p}{\partial x}+v\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}w}{\partial z^2}\right),$ (2) $u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial w}{\partial z}=-\frac{1}{\text{ρ}}\frac{\partial p}{\partial y}+v\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}w}{\partial z^2}\right),$ (3) $\begin{array}{l}u\frac{\partial u}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial w}{\partial z}=-\frac{1}{\text{ρ}}\frac{\partial p}{\partial z}\\ +v\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}+\frac{{\partial }^{2}w}{\partial z^2}\right)-\text{gρ}\text{.}\end{array}$ (4)

Energy conservation equation ${\rho c}_{\text{p}}\frac{\partial T}{\partial \tau }=\lambda \left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\partial }^{2}T}{\partial z^2}\right)+q_v,$ (5) where $u,v$, and $w$ are the velocity components in the X, Y, and Z directions, respectively, $\text{m}/\text{s}$; $\text{g }$ is the acceleration of gravity, $\text{m}/{\text{s}}^2$; ${\text{c}}_{\text{p}}$ is the specific volume at constant pressure, J/ (kg ·K); $\text{ρ}$ is the density, $\text{kg}/{\text{m}}^3$; $T$ is the temperature function, K; $\tau$ is the time, $\text{s}$; and $q_v$ is the heat source, $\text{W}$.

Fluid-structure coupling equation: $d_{\text{f}}=d_{\text{s}},$ (6) $T_{\text{f}}=T_{\text{s}},$ (7) $q_{\text{f}}=q_{\text{s}},$ (8) $n_{\text{f}}·{\sigma }_{\text{f}}=n_{\text{s}}·{\sigma }_{\text{s}},$ (9) where f represents fluid; s represents solid; d is displacement; q is heat; and ${\sigma }_f \text{and} {\sigma }_s$ are the fluid and solid stresses, respectively.

Numerical calculations require an appropriate turbulence model to reflect the actual situation. In this study, the air calculation domain was turbulent, and the helium gas inside was a natural convection. Hence, the k-omega turbulence model was selected from the turbulence model provided by Fluent. The SST-k-omega model considers the turbulent shear stress based on the general k-omega model; thus, it is more accurate than the general model ^[16]. Therefore, the SST-k-omega turbulence model was adopted in the simulation in this study, and the equation for this model is as follows: $\frac{\partial \left(\text{ρ}k\right)}{\partial t}+\frac{\partial \left(\text{ρ}k{u}_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left({\Gamma }_k\frac{\partial k}{\partial x_j}\right)+G_k-Y_k+S_k,$ (10) $\frac{\partial \left(\text{ρ}\omega \right)}{\partial t}+\frac{\partial \left(\text{ρ}\omega u_i\right)}{\partial x_i}=\frac{\partial }{\partial x_j}\left({\Gamma }_{\omega }\frac{\partial \omega }{\partial x_j}\right)+G_{\omega }-Y_{\omega }+S_{\omega }.$ (11)

In contrast with standard k-omega, turbulent viscosity ${\mu }_{\text{t}}$ was calculated as follows: ${\mu }_t=\frac{\text{ρ}k}{\omega }\frac{1}{\text{max}\left[\frac{1}{{\alpha }^{*}},\frac{S{F}_2}{{\alpha }_1\omega }\right]},$ (12) $F_2=\tanh {\varnothing }_2^2,$ (13) ${\varnothing }_2=\text{max}\left[2\frac{\surd k}{0.09\omega y},\frac{500\mu }{\text{ρ}{y}^2\omega }\right],$ (14) where $G_k$ represents the turbulent kinetic energy generated by the average velocity gradient, and ${\Gamma }_k$ and ${\Gamma }_{\omega }$ represent the effective diffusion coefficients of k and ω, respectively. $Y_k$ and $Y_{\omega }$ represent the dissipations of k and ω under the action of turbulence, respectively. $S_k$ and $S_{\omega }$ are user-defined source items (zero in this model). ${\sigma }_k$ and ${\sigma }_{\omega }$ are the turbulent Prandtl numbers of k and ω, respectively. $y$ is the distance to the adjacent surface, ${\alpha }_1$= 0.31.

For spent fuel casks, the temperature distribution is principally affected by convection and radiant heat, with radiation being the main heat transfer mode for spent fuel assemblies. Commonly used radiation models include the Surface to Surface (S2S) model, Discrete Ordinates (DO) model, and ray tracing model. Among these radiation heat transfer models, the calculation accuracy of the S2S model is not as good as that of the DO or ray tracing models. To save computing resources, we did not adopt the ray tracing model. Additionally, the DO model is more appropriate than others when the geometric model used is comparatively complex.

The transient calculation time step calculated by the formula (∆t=$1/3·L/V$, L: Characteristic size, V: Volume) was approximately 0.025 s, the convergence speed of the heat transfer calculation was slow, and the accuracy was high, but the calculation results were divergent when the same time step was used in the structure set. Finally, when the step size was 1 s, the heat transfer process converged faster than before. The resulting error was within an acceptable range, and the calculation convergence of the structural module was guaranteed.

Boundary conditions and meshing

The SIMPLE algorithm was adopted based on the pressure-velocity coupling, and the energy equation was discretized using second-order upwind. The outlet was set as the pressure outlet, according to the calculation of the natural convection Reynolds number on the external metal shell, which was approximately 6000, and the inlet speed, which was approximately 0.46 m/s. The emissivity of the metal wall was 0.85, and the model calculation grid is shown in Fig.2. Because of limited computing resources, 6.7 million grids were used to save computing time; the temperature is similar to 8.76 million grids and 10 million grids. The minimum orthogonal mass and skewness were used to test the element quality. The skewness of the grid element was ≤0.75, which is acceptable in the calculation.

Fig. 2Full-screen View

(Color online) Grid diagram and independence validation.

In this study, thermal stress was simulated by varying the power and tilt angle of the model. The total power ($Q$) remained unchanged. The spent fuel grid model is divided into inner and outer zones (Fig.3). The dark area is the inner zone, the remainder is the outer zone, the inner zone power is $q_{\text{in}}$, and the outer zone power is $q_{\text{out}}$. The $\frac{q_{\text{in}}}{q_{\text{out}}}$ ratio was changed (Table 3) to estimate thermal stress.

Fig. 3Full-screen View

(Color online) Diagram of internal and external zones.

In the mechanical calculation, we calculated the shell and basket, suppressed the other models, and set fixed supports on the upper and lower parts of the basket. In this study, the calculated thermal loads of the shell and basket were introduced into the model in a unidirectional coupling manner, and the calculation step was consistent with the heat transfer process.

Model verification

The size of the model was analogous to that of Herranz et al. ^[26], and the accuracy of the computation was verified by comparison with its temperature field. We compared the temperature distribution from the center to the outside of the model at a height of approximately 4.8 m with that in the literature and discovered that the temperature difference in the basket zone was 0.5%-6.2%, and the maximum temperature difference was 0.8%, at the height of 4.6-4.8 m. Although the materials used in the literature are undefined, the error in the metal container is within an acceptable range. In Fig.4, in the annular gap outside the container, the temperature in the literature is lower than the simulated temperature because only the base of the concrete layer is set as the adiabatic condition in the simulation process (consistent with the literature); however, the computation results illustrate that the temperature of the outer wall of the container is close to the temperature of the concrete, which is approximately 550 K. This paper principally discusses the temperature and stress changes in the metal cask; therefore, the influence of the external concrete layer temperature on the subsequent calculation in the metal container can be disregarded.

Fig. 4Full-screen View

(Color online) Model validation: (a) model radial temperature distribution, (b) basket zone temperature deviation.

Figure 5 shows that the temperature in the metal container after steady-state computation is concentrated at the middle part, and the temperature in the inner zone of the component is higher than that in the outer zone, with the highest temperature of 621.81 K on assembly 14. The maximum temperature of the external concrete layer is approximately 550 K. We compared Fig.5 and Fig.6 and found that the base has a slight change in the external heat dissipation process of the container. All concrete layers should be set as adiabatic conditions to be consistent with the reference. In addition, Fig.6 reveals that in the case of partial convergence, the internal high temperature is mainly distributed in the middle of the component. With the natural convection of helium in the container and heat conduction of the assemblies, Fig.5 displays that the high-temperature part is transferred upward. The model is accurate based on the verification.

Fig. 5Full-screen View

(Color online)Temperature distribution of cask and assembly (steady state): (a) cask temperature distribution, (b) basket and assembly temperature distribution.

Fig. 6Full-screen View

(Color online) Temperature distribution of cask and assembly (transient state): (a) cask temperature distribution, (b) basket and assembly temperature distribution.

Thermal stress

This study focuses on the stress changes caused by the temperature field, and the stress problem on the basket can be regarded as an axial plane stress problem, which is assumed to fulfill the four basic assumptions of elasticity ^[27]. This is: $\{\begin{array}{l}{\sigma }_y={\tau }_{xy}={\tau }_{zy}=0\hfill \\ {\sigma }_x={\sigma }_x\left(x,z\right)\hfill \\ {\sigma }_z={\sigma }_z\left(x,z\right)\hfill \\ {\tau }_{xz}={\tau }_{xz}\left(x,z\right)\hfill \end{array}$ (15) where ${\sigma }_x,{\sigma }_y,{\sigma }_z,{\tau }_{xy},{\tau }_{zy},\text{and }{\tau }_{xz}$ are stress components. Consequently, the general physical equation of thermal stress can be simplified as $\{\begin{array}{l}{\epsilon }_x=\frac{1}{\text{E}}\left({\sigma }_x-\text{ν}{\sigma }_z\right)+\alpha T\hfill \\ {\epsilon }_z=\frac{1}{\text{E}}\left({\sigma }_z-\text{ν}{\sigma }_x\right)+\alpha T\hfill \\ {\gamma }_{xz}=\frac{2\left(1+\text{ν}\right)}{\text{E}}{\tau }_{xz}\hfill \end{array}$ (16) where ${\epsilon }_x,{\epsilon }_z,\text{and }{\gamma }_{xz}$ are the strain components, $\text{ν}$ is Poisson's coefficient, $\alpha$ is the thermal expansion coefficient, $\text{E}$ is the elastic modulus, and $T$ is the temperature variable. Substituting the plane equation into equation (16) yields $\{\begin{array}{l}{\sigma }_x=\frac{\text{E}}{1-{\text{ν}}^2}\left(\frac{\partial i}{\partial x}+\text{ν}\frac{\partial k}{\partial z}\right)-\frac{\text{E}\alpha }{1-\text{ν}}T\hfill \\ {\sigma }_z=\frac{\text{E}}{1-{\text{ν}}^2}\left(\frac{\partial k}{\partial z}+\text{ν}\frac{\partial i}{\partial x}\right)-\frac{\text{E}\alpha }{1-\text{ν}}T\hfill \\ {\tau }_{xz}=\frac{E}{2\left(1+\text{ν}\right)}\left(\frac{\partial i}{\partial z}+\frac{\partial k}{\partial x}\right)\hfill \end{array}$ (17) where $i$ and $k$ represent the displacements in the X-and Y directions. Substituting equation (17) into the equilibrium equation yields $\{\begin{array}{c}\frac{{\partial }^{2}i}{\partial x^2}+\frac{1-\text{ν}}{2}\frac{{\partial }^{2}i}{\partial z^2}+\frac{1+\text{ν}}{2}\frac{{\partial }^{2}k}{\partial x\partial z}-\left(1+\nu \right)\alpha \frac{\partial T}{\partial x}=0\\ \frac{{\partial }^{2}k}{\partial z^2}+\frac{1-\text{ν}}{2}\frac{{\partial }^{2}k}{\partial x^2}+\frac{1+\text{ν}}{2}\frac{{\partial }^{2}i}{\partial x\partial z}-\left(1+\nu \right)\alpha \frac{\partial T}{\partial x}=0\end{array}$ (18)

The geometric, physical, and equilibrium equations refer to the theory of elasticity ^[27]. The corresponding stress component can be calculated by substituting the temperature boundary conditions into Eqs. (17) and (18). In this study, SS304 steel was used as the wall surface of the metal cask in the simulation calculation, and zirconium was used as the lattice material. Table 4 presents the specific parameters.

Influence of internal and external power

The transient calculation results are as follows. Assuming that the initial temperature of the metal container is 550 K after the fuel is stored for some time, we used the results of the approximate convergence (temperature fluctuation ± 2 K) for comparison, and the transient temperature field was used as the subsequent stress calculation load.

Fig.7 illustrates the temperature distribution when the power ratio in the inner and outer zones was 0.5 and 2. At a power ratio of 0.5, the maximum temperature was 592.59 K, located on assembly 14. Nevertheless, the power of assembly 14 in the inner region was lower than that in the outer region. Moreover, the outward radiation power of component 3 was greater than that of the adjacent components (approximately two times), owing to the strong convective heat transfer on the front wall of component 3, located on the same side of the inlet and outlet. Fig.7(b) shows that when the power ratio is 2, the maximum temperature is 600.54 K, located in assembly 14. In contrast with the literature, the internal temperature of the component is higher than that of the external component before reaching the steady state, owing to the external component being more susceptible to the natural convection of helium gas and air in the external annular channel than the internal component. After changing the power, the temperature became lower than the power ratio of 1 (609 K). The temperature distribution in Fig.7(a) is closer to the container wall than that in Fig.7(b), and the temperature in Fig.7(a) is significantly higher than Fig.7(b) near the top of the container in the inner region, which also reflects that when the internal power is smaller than the external power, the outer component not only dissipates heat but also conducts heat inward.

Fig. 7Full-screen View

(Color online) Transient temperature distribution of different power. (a) Power ratio=0.5 transient temperature distribution; (b) Ratio=2 transient temperature distribution.

Figure 8 illustrates the temperature distribution of the frame and the component with the highest temperature in the model under the analyzed power. Figure 9 shows the axial temperature distribution of the internal and external assemblies (14, 3) under different powers. Fig.8(a) transverse temperature change at the lower side near the bottom of the container is due to the natural convection influence of helium in the container. The exterior temperature is lower than the interior temperature, and helium flows inward from the outer wall. In Fig.8(b)(c), power distribution is more uneven than that in Fig.8(a), which is affected by helium convection, and the high temperature shifts upward.

Fig. 8Full-screen View

(Color online) Temperature distribution of basket and assembly 14. (a) Power ratio=1 basket and assembly temperature distribution; (b) Power ratio=2 basket and assembly temperature distribution; (c) Power ratio=0.5 basket and assembly temperature distribution.

Fig. 9Full-screen View

(Color online) Component axial temperature distribution under different power ratios.

Figure 9 displays that in the case of even power distribution, component gradient varieties in the inner and outer zones are primarily located at the upper and lower sides of the component. When the power distribution is different, the variation in the temperature gradient in the inner and outer zones of the upper side is significantly smaller than that in the case of uniform distribution. On the lower side, because of the influence of heat transfer in the outer zone under different power conditions, the internal temperature is always higher than that in the outer zone. The change in the temperature gradient in the outer zone is greater than that in the inner zone. The axial temperature gradient is caused by the filling gas in the cask and the cooling air in the concrete cladding.

Figure 10 illustrates the partial temperature distribution outside the metal cask and through the shell at the height of approximately 4.8 m. The radial temperature gradient through the exterior concrete shell is minimal, which is different from the steady-state situation, owing to the minimal thermal conductivity of concrete and affected by air convection; thus, the temperature change is not obvious. The radial temperature changes along the cask wall and the annular gap are evident. The gradient between the basket and the cask wall is caused by the inactivity of helium gas and its heat conduction to the cask wall. The difference between the steady and the transient states is attributed to a distinct temperature increase at the annular gap due to the steady-state mainstream temperature being higher than the temperature of the outer boundary flow.

Fig. 10Full-screen View

(Color online) Radial temperature distribution.

In this study, the stress distribution inside the model was estimated by the equivalent force, the physical meaning of which is that when the equivalent force at a point of the stressed object exceeds the yield strength of the material, the material begins to undergo plastic deformation, which is defined by the following equation: ${\sigma }_{\text{e}}=\sqrt{\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]}$ (19) where ${\sigma }_{\text{e}}$ is the equivalent force and ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ are the first, second, and third principal stresses, respectively.

For this model, the residual stress is disregarded, and only the thermal stress is considered. After calculations, we found that the thermal stresses on the shell surface and the basket changed little (varying by approximately 0.01 Pa) when the power ratios in the inner and outer zones differed (Fig.11 and Table 5).

Fig. 11Full-screen View

(Color online) Shell stress along the path with different power ratios.

As shown in Fig.12, where the maximum equivalent stress on the shell was 5269.5 Pa located at the intersection of the shell and concrete, the maximum circumferential stress was -6325.9 Pa (counterclockwise is positive) located on the lower side of the container wall with a total deformation of 2.59 × 10^-8m. The maximum equivalent stress (31.62 Pa) of the basket was located at the bottom. Fig.12 shows that the stress was concentrated on the lower side of the shell.

Fig. 12Full-screen View

(Color online) Distribution diagram of shell stress and temperature.

The temperature stratification on the lower side was primarily affected by the convection of the inlet air. Under the influence of the annular gap, the interior flow velocity of the air below is less than the inlet flow velocity, which cannot drive the inner air, and forming a vortex on the inner side is easy. However, a fluid with high speed at the inlet moves upward when heated. Regarding the thermal stress and physical parameters of the material (e.g., modulus of elasticity, coefficient of thermal expansion) and temperature change function, the choice of material physical parameters was relatively stable; thus, the total power remained unchanged, and the temperature gradient in the container directly affected thermal stress. The basket was affected by the temperature of the assembly, and the high temperature was concentrated above the basket, but the temperature difference between the upper and lower was minor (maximum of approximately 10 K), and the equivalent stress was comparatively small (approximately 6 Pa).

As shown in Fig.12, the stress change was apparent at the temperature gradient that altered the most. Therefore, we selected the specified path in Figs. 13, 14, 15 to calculate the equivalent stress, normal stress (positive for tensile stress and negative for compressive stress), and circumferential stress. Figure 13(b) implies that when the power ratio of the inner and outer zones differs, the temperature variation trend is identical, and the temperature mutation is caused by the temperature accumulation on the upper side of the cask, which corresponds to Fig.13(a) Stress mutation is observed, but the trend is slight, and then all stresses increase evenly. Figure 13(a) the end of the path is also affected by the bottom stress, which makes the stress near the edge abrupt.

Fig. 13Full-screen View

(Color online) Shell temperature and stress along the path: (a) axial stress distribution, (b) axial temperature distribution. The path location is in the middle of the shell's outside surface with maximum gradient change.

Fig. 14Full-screen View

(Color online) Shell stress along the path. The path location is on the bottom edge of the shell.

Fig. 15Full-screen View

(Color online) Basket stress along the path. The path location is in the middle of the assembly.

As shown in Fig.14, the path was at the bottom edge of the shell, and the stress was influenced by temperature fluctuations. The stress trend on the path first grew and then declined along the path direction, corresponding to its temperature distribution, and the temperature difference between the two sides was lower than the middle difference.

Fig.15 illustrates that the temperature above the assembly was high, and the variety was minor. As the temperature decreased, stress increased. The temperature gradient changes in the inner assembly were small; thus, stress change was relatively stable.

Therefore, we concluded that the temperature gradient is the primary factor of thermal stress on the shell, and the variation in the power ratio in the inner and outer zones with a constant total power only affects the temperature distribution of the assemblies, which has little effect on stress. The maximum equivalent stress estimated in the shell is less than the yield strength of the material (205 MPa); therefore, the container fulfills the safety requirements under this hypothetical condition.

Effect of inclination angle

In the process of spent fuel shipping and storage, the inclination is bound to occur. Taking the internal and external power ratio of 1.0 as the basis case (θ=0°calculated above), we calculated the temperature field and the corresponding stress at inclination θ=15°, 45°, 75°, and 90° as follows:

Figure 16 shows that shell temperature presents a wavy distribution affected by the tilt angle, with the peak and valley shifting from right to left and the peak appearing at the left edge. The bottom temperature is lower than the top temperature, and the top temperature slowly decreases when the tilt angle becomes larger. The leftmost velocity vector diagram displays that the inlet produces obvious vortices on the left side below θ=90°, near the left side of the velocity is higher than the right side. When this is close to the top of the container, the velocity of the air on the left side decreases under the influence of the annular gap and gradually increases toward the right outlet.

Fig. 16Full-screen View

(Color online) Temperature graduations of shell and velocity vector diagram at θ= 90°: (a) θ= 15° temperature graduations of shell, (b) θ= 45° temperature graduations of shell, (c) θ= 75° temperature graduations of shell, (d) θ= 90° temperature graduations of shell, (e) velocity vector diagram at θ= 90°.

Combined with Fig.12, the container tilt causes variations in the internal air distribution, and air in the annular gap due to the thermal pressure influences the upward flow. Part of the hot air impinged on the wall, decreasing speed, which could not drive the inlet’s cold air to form a vortex below. With a gradual increase in the tilt angle, the flow velocity on the left side (the top in the vertical direction) gradually increased, and velocity stratification appeared, which enhanced the heat transfer on the shell, exhibiting the aforementioned varieties.

Figure 17 demonstrates that in the tilting process, the stress was predominantly concentrated on the top and bottom edges of the shell. When the container was placed horizontally (θ=90°), the deformation was uniformly distributed from the middle to the left, and the basket deformation with the increase in tilt angle gradually shifted from the bottom to upward. The basket maximum deformation emerged in the outermost surface. The outermost side of the basket was affected by the natural convection of the inner and outer sides, and the temperature was distributed on the left side (top of the vertical direction).

Fig. 17Full-screen View

(Color online) Deformation of the shell and basket with different tilt angles. (a) θ= 15° deformation of the shell and basket; (b) θ= 45° deformation of the shell and basket; (c) θ= 75° deformation of the shell and basket; (d) θ= 90° deformation of the shell and basket.

Tables 6 and 7 show the maximum deformation, equivalent stresses, normal stresses (positive along the shell upward and negative downward), circumferential stresses (positive counterclockwise), and tensile stresses (negative compressive stresses) for the shell and basket.

We determined that the temperature gradient varieties directly involved stress. Figure 18 illustrates the circumferential and equivalent stresses along the path on the outer surface of the shell. Figure 18(a) shows the temperature variation along the path. When θ=45° and 75°, the temperature change on the path was distinct, and the temperature gradient on the lower side grew abruptly near the container. The equivalent stress initially decreased, then tended to be a steady value of approximately 30 kPa (θ=90°, approximately 60 Pa), and quickly increased at the end. The trends of the different inclination angles were identical. The circumferential stress shifted slowly from top to bottom, and the circumferential stress increased with an increase in the temperature gradient at the end of the path.

Fig. 18Full-screen View

(Color online) Shell stress along the path. Path location on the outside surface of the shell with maximum gradient change: (a) the temperature variation along the path, (b) the equivalent stress along the path, (c) the circumferential stress along the path.

The change in wall temperature was mainly caused by the distinction of the air distribution, which leads to a distinction of the gradient below, increasing toroidal stress. The equivalent stress at the end of the path differs from the circumferential stress because the former is affected by the stress components in other directions. The end of the path is also influenced by the stress on other surfaces when it is located at the intersection of two surfaces.

Figure 19(b) displays the temperature variations on the path of the assembly, and Fig.19(a) illustrates the path stress change on the corresponding basket. During the tilt process (except θ=0°, 90°), the temperature variation trend was identical, and the upper and lower temperature distinction was tiny (approximately 1-2 ℃). Due to the influence of the inclination angle, the external air distribution changes, and heat transfer is enhanced; thus, the basis case (θ=0°) temperature was higher than in other conditions.

Fig. 19Full-screen View

(Color online) Basket stress and temperature along the path: (a) axial stress distribution, (b) axial temperature distribution. The path location is in the middle of the assembly.

When θ=90° (the container was placed horizontally), the fluid in the component could only flow to both sides, and the convective heat transfer ability was limited. Therefore, the overall temperature of the component higher than θ=15°, 45°, 75°, and stress was the least. With increasing tilt angle, the normal stress of inner assembly 14 (positive along the basket) decreased progressively and increased gradually along the radial direction of the path. The stress in the basket increased approximately with an increase in the temperature gradient.

Figure 17 shows that the stress was concentrated at the top and bottom of the shell. Combined with the bottom deformation of θ=75° in Fig.20, we found that the stress is concentrated near the edge. Therefore, the path in Fig.20 was selected as the top and bottom edges. In Fig.20(a), the equivalent stress of path1 (bottom edge) exhibited a slight fluctuation and was smaller than that of path2. The equivalent stresses of Path1 and Path2 first declined and then grew with fluctuation variety. Fig.20(b) shows that the circumferential stress changed in a trough shape, and the circumferential stress of path2 was less than that of path1 at 0.3-0.4 m of the path. Path1 and path2 were at the intersection of surfaces, and the stress was not only affected by temperature variation but also by the stress of adjacent surfaces, leading to stress fluctuations.

Fig. 20Full-screen View

(Color online) Shell stress along the path. The path location is on the bottom (top) edge of the shell.

The aforementioned analysis of the tilt condition demonstrates that the horizontal state (θ=90°) differed from the other states, and the stress magnitude was near to the basic case (θ=0°). During the tilt process, the deformation steadily increased from below and was eventually uniformly distributed on the shell. When θ=45°, the maximum equivalent stress was 202 MPa (less than the yield stress of 205 MPa), and the container fulfills the safety requirements in the tilt process.

The tilt predominantly alters the air distribution in the annular gap, which results in variations in the shell temperature distribution and various stresses. Therefore, it is concluded that altering the air distribution is a major cause of temperature distribution and stress variation. Notably, air distribution is also affected by the location, size of the inlet and outlet, fluid velocity, and shape of the air passage, which requires follow-up research.