Strength Analysis and Reinforcement of 120 m³ Liquid Helium Storage Tank for Transportation Case

Abstract

To ensure the safe transportation of a 120 m³ liquid hel ium storage tank, based on the analysis of its structure, material, and load characteristics, a finite element model for stress and strength analysis is established and solved by using the coupled thermo-mechanical analysis method for the accelerating and cornering cases experienced by the tank during transportation. The stress and deformation distributions in different transportation cases are analyzed in comparison to the finite element model, and the strength check is carried out for the tank. The study shows that under various working cases, the extreme value of stress of some parts is 267.5 MPa; this does not satisfy the strength requirement, and therefore requires structural reinforcement. The analysis of the reinforcing effect of the reinforcement structure shows that the stress extremes of the dangerous parts of the tanks after reinforcement decreased by 63.7% and 34.7%; the structural strength meets the standard requirements, and the new reinforcement structure only leads to an increase of 6% in the liquid nitrogen consumption of the cold screen, which is a better reinforcing effect.

1. Introduction

As a rare strategic resource, liquid helium plays an irreplaceable role in the fields of military defense, low temperature, low latent heat of vaporization, and high thermal conductivity due to its special physical properties, such as low temperature, low-temperature superconductivity, medical treatment, testing, and production of optoelectronic products. For example, it is used to pressurize the propellant tank of a spaceship, to be used as a refrigerant in nuclear fission reactions, and as a coolant in nuclear magnetic resonance. Due to its extremely low temperature, liquid helium is usually stored in low-temperature adiabatic tanks with complex structures. This paper deals with a 120 m³ fixed liquid helium storage tank. After the production of the liquid helium tank, it needs to be transported to the production unit for use, and a certain amount of liquid helium is kept in the tank during transportation to maintain the low temperature of the inner tank [1]. At this time, the components of the tank cannot be freely contracted at low temperatures under the joint action of thermal stresses and structural stresses caused by pressure, gravity, and other mechanical loads. There is a risk of failure under the coupled thermo-mechanical analysis of the tank [2], which is a serious threat to the safety of liquid helium tanks [3,4]. Therefore, it is of great significance to study the stress and deformation of liquid helium storage tanks during transportation to ensure safe transportation.

Up to now, many researchers have analyzed the strength of mobile LNG storage tanks during transportation and achieved certain results. For example, Xu et al. [5] studied the stress intensity distribution of a high vacuum multilayer adiabatic structure of LNG storage tanks during transportation and found that the stress intensity at the end of the tank support is the highest. Wang [6] addressed the strength evaluation problem of a cryogenic liquid transportation tanker, studied the influence of geometric parameters of the support structure on the structural strength, and found that the angle and distance of the support bar distribution had the greatest influence on the stress.

However, no research has been reported on the transportation of liquid helium tanks, resulting in the following two problems in the strength analysis of liquid helium tanks during transportation. First, the temperature of liquid helium is much lower than that of LNG [7], and, due to the significant contraction effect of the support system at low temperatures, compared to that of LNG, the thermal stresses generated are much higher, making it difficult to provide a reference for the strength analysis of liquid helium storage tanks during transportation around LNG storage tanks. Second, the existing research on the optimization of cryogenic transport tanks with structures is mainly focused on the stress intensity analysis caused by direct load (as opposed to thermal loads, which are indirect actions), ignoring the impact of optimized structure on the heat transfer of the tank, which may lead to structural optimization of the evaporation rate of the tank not meeting the requirements.

To solve the above problems, the heat transfer model of liquid helium storage tanks, a coupled thermo-mechanical analysis model, was created. Based on the finite element software (ANSYS 2022) platform for the model solution, the stress and deformation distribution of liquid helium tanks under the transportation cases was calculated.

2. Basic Modeling of Liquid Helium Tanks

2.1. Geometric Model of Liquid Helium Tanks

In this paper, a 120 m³ horizontal liquid helium storage tank is taken as the research object to carry out stress analysis based on coupled thermo-mechanical analysis. The liquid helium storage tank mainly consists of five parts: inner vessel, outer vessel, cold screen, insulating layer, and support system. The inner vessel is made of S316L stainless steel with a length of 18,784 mm, a diameter of 3000 mm, and a thickness of 6 mm, while the outer vessel is made of 16MnDR low alloy steel with a length of 21,085 mm, a diameter of 3500 mm, and a thickness of 16 mm. The inner and outer vessels are connected by a support system made of 304L stainless steel, and the support rods are made of S321 stainless steel. The support system mainly consists of inner support bars, which are cross-distributed, outer support bars, which are placed vertically and horizontally, axial support cylinders, which are connected to the outer tank, and inner tank support cylinders, which are connected to the inner tank. The support bars are connected to the support system by hinges. The model of the liquid helium storage tank and the local magnification model are shown in Figure 1 and Figure 2, respectively.

Tank material settings will have a greater impact on the tank stress analysis simulation. The material part of the parameters are shown in

Table 1. Material parameters of liquid helium storage tank.

Location	Materials	Densities /(g/cm3)	Poisson’s Ratio	Thermal Conductivity/(W/(m·K))	Yield Strength /MPa
Inner tank and reinforcement ring	S316LN	7.99	0.3	15.35	280
Outer tank and reinforcement ring	16MnDR	7.85	0.3	23.35	315
Cold screen and support system	304L	7.93	0.3	14.85	205
Support bar	S321	7.9	0.3	16.05	310

, where the thermal conductivity varies with temperature, for conservative considerations to take the maximum value within the working range of the tank [8]. The modulus of elasticity and coefficient of thermal expansion of the material will change with the change of temperature, as shown in Figure 3 and Figure 4, respectively.

2.2. Liquid Helium Tank Heat Transfer Modeling

Liquid helium storage tanks are transported with the heat transfer assumed to be steady state heat transfer. The heat transfer is analyzed using steady state thermal analysis, which explores how the temperature at various points in a heat transfer system varies only with differences in location and is not affected by the passage of time. In other words, the amount of heat passing through a heat transfer surface per unit time remains constant. This method of analysis is mainly used to study the thermal response of a structure when subjected to a steady thermal load. The problem of uneven temperature distribution of the medium loaded in the inner tank is neglected and the temperature of the inner tank is considered as a constant and unchanging temperature of the medium inside the tank. The governing equations for the solution of the temperature field of the overall structure of the liquid helium storage tank are set according to Equation (1), and the boundary cases for the heat transfer in the liquid helium storage tank are set according to Equation (2) for the boundary cases of the inner and outer tank walls.

$$\left\{\begin{array}{l}{\nabla }^{2}t=0\hfill \\ q_i=-{\displaystyle \sum _{k=1}^3}{\lambda }_{ij}{\displaystyle \frac{\partial t}{\partial j}}\hfill \end{array}\right.\left(i,j=x,y,z\right)$$

(1)

$$\left\{\begin{array}{l}{T}_{\mathrm{in}}=T_{\mathrm{c}}\hfill \\ -\lambda ({\displaystyle \frac{\partial T}{\partial n}}{)}_{\mathrm{out}}=h(T_{\mathrm{out}}-T_{\mathrm{f}})\hfill \end{array}\right.$$

(2)

where: T is the temperature, K; q_i is the heat flux component along each coordinate direction, J/(m²·s); λ_ij is the support thermal conductivity component, which indicates the size of the heat flux caused by the unit rate of change of temperature in the j direction in the i direction, W/(m·K); h is the convective heat transfer coefficient [9], W/(m²·K); T_in is the temperature of the inner surface of the inner tank, K; T_out is the temperature of the outer surface of the outer tank and K T_f is the ambient temperature, K; λ is the thermal conductivity of the outer tank, W/(m·K); n is the normal direction outside the boundary; T_c is the liquid helium temperature, K.

2.3. Coupled Thermal-Mechanical Modeling of Liquid Helium Storage Tanks

The 120 m³ liquid helium storage tanks are loaded with liquid helium at a temperature of only 4.2 K. Due to the limitations of the liquid helium storage tank structure, the tank structure is not free to contract at such low temperatures, which will generate large thermal stresses and affect the tank structure. The displacement and deformation of the tank structure under the action of the mechanical load and temperature load will affect the heat transfer of the tank and thus affect the distribution of the thermal stresses, and the heat transfer of the tank and the structural stress will affect each other. Mutual influence is also known as thermal-mechanical coupling. Considering that the tank in the force displacement deformation did not produce a new thermal bridge, even if the tank heat transfer has an impact, but the impact is very small, it can be considered that in the coupled thermo-mechanical analysis of the 120 m³ liquid helium storage tank for the unidirectional coupling, that is, the heat affects the structure, the structure does not affect the heat [10].

A coupled thermal-mechanical analysis model is established based on the heat transfer model [11,12]. According to the theory of elastic mechanics, the differential control equations of the structure are set according to Equation (3), and the boundary cases, which are to constrain all the degrees of freedom of the bottom surface of the right saddle, to release the degrees of freedom of the bottom surface of the left saddle along the axial direction of the tank, and to constrain the rest of the degrees of freedom, are set according to Equation (4).

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial {\sigma }_i}{\partial i}}+{\displaystyle \frac{\partial {\tau }_{ji}}{\partial j}}+{\displaystyle \frac{\partial {\tau }_{ki}}{\partial k}}+f_i=0\hfill \\ {\displaystyle \frac{\partial u_i}{\partial i}}+{\displaystyle \frac{\partial u_j}{\partial j}}-{\displaystyle \frac{{\tau }_{ij}}{G_i}}=0\hfill \\ {\displaystyle \frac{\partial u}{\partial x}}-{\displaystyle \frac{1}{E_i}}\left[{\sigma }_i-\mu ({\sigma }_j+{\sigma }_k)\right]-{\alpha }_{i}T=0\hfill \end{array}\right.\left(i,j,k=x,y,z\right)$$

(3)

$$\left\{\begin{array}{l}{(u_i)}_{\mathrm{L}}=0(i=x,y,x)\hfill \\ {(u_i)}_{\mathrm{R}}=0(i=y,z)\hfill \end{array}\right.$$

(4)

where: σ_i is the positive stress, MPa; τ_i is the tangential stress, MPa; u_i is the displacement along the three coordinate directions, mm; f_i is the mass force, m/s²; α is the coefficient of thermal expansion, 1/K; μ is the Poisson’s ratio; E_i is the modulus of elasticity, GPa; G_i is the shear modulus, $G_i={\displaystyle \frac{E_i}{2(1+\mu )}}$, GPa; and the subscripts L and R stand for the bottom surface of the left hand side and the bottom surface of the right hand side, respectively.

2.4. Finite Element Modeling

2.4.1. Meshing

In this paper, a finite element model of a liquid helium storage tank is established, as shown in Figure 5. Due to the great difference in size between the thickness and length of the inner and outer tank and the cold screen, the use of solid elements for meshing will lead to the problem of too much grid or too low grid quality. Therefore, the shell element is used to mesh the inner and outer tank and the cold screen, and the other structures are meshed by solid elements [13,14]. The results are shown in Figure 6.

To determine the number of grids that do not affect the calculation accuracy and take into account the calculation cost, the authors verified the grid independence, finding five different numbers of grids: 563,600, 748,840, 860,563, 103,166, and 128,479. The stress distribution at the connection between the head of the outer tank and the cylinder is taken as the benchmark, and the validation of the irrelevance of different numbers of grids is carried out. The irrelevance validation of different numbers of meshes was carried out.

Figure 7 shows the results of stress calculation for different number of meshes at the head and cylinder of the outer tank. It can be determined that when the number of grids increases from 103,166 to 128,479, the maximum stress value at this place does not change much; the maximum deviation is 5.65% and the average deviation is 2.3%. Therefore, the number of meshes of 128,479 is selected and the number of mesh nodes is 263,215.

2.4.2. Load and Calculation Setup

When the temperature field is not considered, the stress distribution of the tank can be obtained by hydrostatic analysis. The tank thermal structure coupling is realized by the indirect coupling method of Workbench platform, i.e., the temperature field of the tank is solved first, and then the temperature field is applied to the hydrostatic analysis as a load for the coupling solution [15,16]. The boundary cases and loads of the temperature field are as follows: the wall temperature of the inner container is 4.2 K, and the ambient temperature of the outer tank is 298.15 K. The boundary cases and loads of the two working cases before and after the consideration of thermal stresses are as follows in

Table 2. Liquid helium storage tank boundary cases and loads.

Case	Boundary Condition	Load Specifics
1	Fixed restraints on the bottom surface of the saddle	Internal pressure 0.168 MPa, external pressure 0.1 MPa, self-weight, temperature load	Inertial load: 2 g in transportation direction
2			Inertia load: 1 g horizontally in the vertical transportation direction
3			Inertia load: 2 g vertically downward
4			Inertia load: 2 g vertically upwards

.

Liquid weight and inertia force inside the tank are applied by the equivalent density method; the equivalent density method is used to adjust the density of the inner tank material, the mass of the medium filled in the inner tank as an additional mass to be added to the inner tank, and then, according to the direction of the specified acceleration for the overall structure of the whole, to impose an overall acceleration. If the mass of the inner tank is m₁, the density of the inner tank is ρ₀, and if the mass of the medium is m₂, then the equivalent density ρ of the inner tank is calculated as follows:

$$\rho ={\displaystyle \frac{{\rho }_0\left(m_1+m_2\right)}{m_1}}$$

(5)

Ansys is used for the sequential solution of the thermal-mechanical coupled stress model of liquid helium storage tanks; the results of the temperature field solution of liquid helium storage tanks are used as the temperature loads for the stress analysis, and the stresses and deformations of the storage tanks are calculated.

3. Results and Discussion

3.1. Stress Analysis

To facilitate the subsequent analysis, seven paths were set up as the structural hazards. Path P₁ was set up at the connection between the outer tank head and axial support cylinder; path P₂ was set up at the connection between the inner tank support cylinder; paths P₃ and P₄ were set up at the connection between the support system and the vertical (horizontal) support bar, respectively, and paths P₅ and P₆ were set up at the support bar 1 and support bar 2, respectively; path P₇ was set up at the axial support cylinder. The schematic diagram of path setting (the red arrows in the diagram indicate the direction of the path from the start point to the end point) is shown in Figure 8.

3.1.1. Distribution of Stress and Displacement Deformation of the Outer Tank

Figure 9 shows the stress distribution of the outer tank for each transportation case, and Figure 9a–d show the stress distribution of the outer tank from Condition 1 to Condition 4, in turn. From the analysis, it can be seen that the middle of the head of the outer tank is the place with the highest stress under each case. Comparing the stress distribution here under case 2, case 3, and case 4 with non-axial inertia load, it can be found that the stress distribution here is not uniform, and the stress in the direction of inertia load is larger than the stress in other directions. Among the four cases, the stress in the outer tank is larger in cases 1 and 3, and the stress extremes are 247.2 MPa and 146.9 MPa, respectively.

Figure 10 shows the displacement and deformation distribution of the outer tank under each transportation case, and Figure 10a–d show the displacement and deformation distribution of the outer tank from Condition 1 to Condition 4, in turn. It can be seen that in the vertical direction, the displacement and deformation of the outer tank decrease gradually from the top to the bottom [17]; except for the case 1, the displacement and deformation distribution of the remaining cases in the horizontal direction are symmetrically distributed along the center of the tank, and the displacement and deformation distribution of the case 1 show a gradual increase in the direction of the inertial load due to the application of the inertial load in the axial direction. Among the four transportation cases, the displacement and deformation value of the outer tank, in case 2, was 4.0 mm.

Figure 11 shows the stress distribution curve along the middle of the outer tank head at the path P₁; the analysis shows that in path P₁, the stress of case 1 is much higher than the rest of the cases. This is because the tank is subjected to the inertia load of axial acceleration of two times gravity in case 1, and here the outer tank head is connected with the axial support tube as the main stressing part in this direction. In case 1, the stress extreme value is 247.2 MPa, and the average stress is 222.0 MPa.

3.1.2. Distribution of Stress and Displacement Deformation in the Inner Tank

Figure 12 shows the stress distribution of the inner tank for each transportation case. Figure 12a–d show the stress distribution of the inner tank from case 1 to case 4, in turn. From the analysis, it can be seen that the middle of the head of the inner tank is the place with the highest stress under each case. Due to the presence of reinforcement, the stress of the inner container head at the reinforcement is small [18], the stress distribution is affected by inertial load, and the stress in the direction of inertial load is larger than the other side. Among them, the stresses in case 1 and case 3 are larger, and the stress extremes are 154.8 MPa and 195.7 MPa, respectively.

Figure 13 shows the displacement and deformation distribution of the inner tank for each transportation case, and Figure 13a–d show the displacement and deformation distribution of the inner tank from Condition 1 to Condition 4, in turn. It can be analyzed that the displacement and deformation on the side without the axial support cylinder is larger than that on the other side for all four cases. This is because of the asymmetric structure of the tanks, and there is no constraint of axial support cylinder on the left side. Comparing the displacement and deformation of the inner tank in case 4 and the rest of the cases, it is found that the displacement and deformation of the inner tank in case 4 increase gradually from the top to the bottom in the vertical direction and decrease gradually in the rest of the cases, which is due to the fact that the inertial load of 2 g in the vertical upward direction of case 4 is partially canceled by gravity and the direction of inertial load is upward in the vertical direction [19]. Comparing the displacement and deformation distributions of the inner tanks under the four transportation cases, the displacement and deformation of the outer tank of the liquid helium storage tank in case 1 is the largest with a value of 12.2 mm.

Figure 14 shows the stress distribution curve of the inner tank along path P₂, and it can be analyzed that at path P₂, the stress extremes of each case appear in different locations; this is because the direction of the inertia load is different. The stress extremes of case 1, case 2, and case 3 are 154.8 MPa, 106.3 MPa, and 120.0 MPa, respectively.

3.1.3. Stress and Displacement-Deformation Distribution of the Support System

The stress distribution cloud diagrams of the support system for each transportation case are shown in Figure 15a–d, respectively. It can be analyzed that under the four transport cases, the connection between the support system and the vertical (horizontal) support bar, the support bar and the axial support cylinder are the main stress parts. Second, under different inertia loads, the maximum stresses of the support system are different; for example, when the tank is subjected to an inertial load of twice the acceleration of gravity in the horizontal direction in case 2, the stresses at the connection between the support system and the support bar 2 are greater than those at other parts.

Figure 16 shows the stress distribution of the support system along the paths, where Figure 16a shows the stress distribution curve along path P₃ and Figure 16b shows the stress distribution curve along path P₄. It can be analyzed that the stress extremes of case 1 and case 3 at P₃ are much larger than those of the rest of the cases at P₃, and the stress extremes of case 2 at P₄ are much larger than those of the rest of the cases at P₄, and there is a greater risk of structural failure in the above areas.

3.1.4. Distribution of Stress and Displacement Deformation of Support Bars

Figure 17 shows the stress distribution of the support bar under each transportation case, and Figure 17a–d show the stress distribution of the support bar from Condition 1 to Condition 4, in turn. It can be analyzed that the stress on support bar 1 is the largest under each transport case, and support bar 1 is the main load-bearing part of the support bar. Among them, support bar 1 is subjected to the greatest stress in working case 3, and the stress extreme value is 212.6 MPa.

Figure 18 shows the displacement and deformation distribution of the support bar for each transportation case, Figure 18a–d are the displacement and deformation distribution of the support bar from case 1 to case 4, and Figure 19 shows the maximum displacement and deformation distribution of the support bar for each transportation case. It can be analyzed that the maximum displacement and deformation of each support bar under working case 1 is larger than that of other working cases, and only the maximum displacement and deformation of support bar 2 is slightly smaller than that of working case 2, which indicates that the direction of inertial load that has the greatest influence on the displacement and deformation distribution of the support bar is the axial direction, followed by the horizontal direction perpendicular to the transportation direction. The maximum displacement deformation of support bar 4 in case 1 is 7.39 mm, which is the largest displacement deformation of the support bar in the transportation case.

Figure 20 shows the distribution of stresses along the paths of the four types of support bars. Figure 20a shows the stress distribution of support bar 1 along path P₅ and Figure 20b shows the stress distribution of support bar 2 along path P₆. It can be analyzed that the stress of support bar 1 is greater than for the rest of the support bars in each working case, which indicates that support bar 1 is the most important load-bearing part of the support bars during the transportation of liquid helium tanks. In addition, in case 3, the stresses on the support bars, especially support bar 1 and support bar 2, are much higher than for the rest of the cases, which indicates that the inertia load in the vertical direction has a greater influence on the stresses on the support bar.

3.1.5. Axial Support Cylinder Stress Distribution

Figure 21 shows the stress distribution of the axial support cylinder for each transportation case, and Figure 21a–d shows the stress distribution of the axial support cylinder for case 1 to case 4, respectively. It can be analyzed that the stress distribution of the axial support cylinder is similar under different transportation cases, and the stress at the connection with the head of the outer tank is larger than for other parts. The extreme value of the stress at this place is 525.6 MPa in case 1, which is very likely to have the risk of structural failure during transportation.

Figure 22 shows the stress distribution of the axial support cylinder along path P₇. From the analysis, it can be seen that due to the inertial load of two times the gravitational acceleration in the axial direction in case 1, the stress on the axial support cylinder is much larger than that in the other cases [20]. Considering that case 3 is also subjected to an inertial load of two times the gravitational acceleration, but the stress in the axial support cylinder is still much smaller than that in case 1, combined with the previous stress analysis, it is assumed that the structural design of the tank in the axial direction is not reasonable enough, which leads to the lack of uniformity of the stress.

3.2. Strength Checks

The 120 m³ liquid helium storage tank strength checking method adopts the stress classification method based on the fourth strength theory stipulated in the standard JB4732-1995 “Steel Pressure Vessel-Analysis and Design Standard” [21].

According to the principle of determining the stress classification line, stipulated in the standard, combined with the results of stress calculations for the structural characteristics of 120 m³ liquid helium storage tanks, we can determine the stress classification line. Figure 23, Figure 24 and Figure 25 present the stress classification line for the tank components.

Based on the results of the above liquid storage case stress calculations, the stress classification method is used to carry out structural strength calibration of liquid helium storage tanks. According to the strength calibration basis in the previous section, the permissible values of stress intensity of each material of liquid helium storage tanks under the design temperature of the tanks are shown in

Table 3. Allowable stress strength of each material at design temperature.

Materials	S316LN	16MnDR	304L	S321
KSm/MPa	137	188	118	168
1.5KSm/MPa	205.5	282	177	252
3Sm/MPa	411	564	354	504

. Permissible values of stress intensity of each material under the design temperature (K = 1.0, S_m is the design stress intensity [21]).

Through the finite element calculations and linearization of the results of numerical calculations of equivalent stresses, the values of stresses in each part of the tank under transportation cases, when the thermal stresses are considered, are obtained with the calibration results as shown in

Table 4. Strength check of tank for case 1.

Locations	Stress Type	Stress/MPa	Sm/MPa	Results
A-A (Middle of inner tank head)	SI	92.0	137	pass
	SIII	154.8	205.5	pass
B-B (Middle of the outer tank head)	SI	173.2	188	pass
	SIII	292.3	282	fail
C-C (Where the bracing system connects to the support bar 1)	SIII	122.5	177	pass
	SIV	174.8	354	pass
D-D (Where the support system is connected to the support bar 2)	SIII	67.9	177	pass
	SIV	68.5	354	pass
E-E (Support bar 1)	SIII	105.5	252	pass
	SIV	105.5	504	pass
F-F (Support bar 2)	SIII	40.2	252	pass
	SIV	40.2	504	pass
G-G (Axial support cylinder)	SIII	525.6	177	fail
	SIV	525.6	354	fail

,

Table 5. Strength check of tank for case 2.

Locations	Stress Type	Stress/MPa	Sm/MPa	Results
A-A (Middle of inner tank head)	SI	48.0	137	pass
	SIII	71.2	205.5	pass
B-B (Middle of the outer tank head)	SI	14.8	188	pass
	SIII	27.2	282	pass
C-C (Where the bracing system connects to the support bar 1)	SIII	106.4	177	pass
	SIV	106.4	354	pass
D-D (Where the support system is connected to the support bar 2)	SIII	434.0	177	fail
	SIV	434.0	354	fail
E-E (Support bar 1)	SIII	98.2	252	pass
	SIV	98.2	504	pass
F-F (Support bar 2)	SIII	33.0	252	pass
	SIV	33.0	504	pass
G-G (Axial support cylinder)	SIII	130.5	177	pass
	SIV	130.5	354	pass

,

Table 6. Strength check of tank for case 3.

Locations	Stress Type	Stress/MPa	Sm/MPa	Results
A-A (Middle of inner tank head)	SI	57.7	137	pass
	SIII	134.6	205.5	pass
B-B (Middle of the outer tank head)	SI	35.3	188	pass
	SIII	72.6	282	pass
C-C (Where the bracing system connects to the support bar 1)	SIII	169.2	177	pass
	SIV	213.1	354	pass
D-D (Where the support system is connected to the support bar 2)	SIII	102.4	177	pass
	SIV	102.4	354	pass
E-E (Support bar 1)	SIII	211.14	252	pass
	SIV	211.14	504	pass
F-F (Support bar 2)	SIII	87.8	252	pass
	SIV	87.8	504	pass
G-G (Axial support cylinder)	SIII	155.9	177	pass
	SIV	155.9	354	pass

and

Table 7. Strength check of tank for case 4.

Locations	Stress Type	Stress/MPa	Sm/MPa	Results
A-A (Middle of inner tank head)	SI	40.4	137	pass
	SIII	44.3	205.5	pass
B-B (Middle of the outer tank head)	SI	44.1	188	pass
	SIII	52.1	282	pass
C-C (Where the bracing system connects to the support bar 1)	SIII	57.9	177	pass
	SIV	57.9	354	pass
D-D (Where the support system is connected to the support bar 2)	SIII	93.5	177	pass
	SIV	93.5	354	pass
E-E (Support bar 1)	SIII	39.8	252	pass
	SIV	39.8	504	pass
F-F (Support bar 2)	SIII	40.2	252	pass
	SIV	40.2	504	pass
G-G (Axial support cylinder)	SIII	88.6	177	pass
	SIV	88.6	354	pass

.

Through the strength calibration of each part of the tank, it was found that there were several places that did not meet the strength requirements and required structural reinforcement. Combined with the structural characteristics of the tanks, it is assumed that it may be more effective to place the reinforcement structure at the support system.

3.3. Structural Reinforcement

Through the stress analysis of the four working cases under the transportation case, it is found that the force of the support system in the vertical direction is much larger than those in the other directions, which is presumed to be due to the uneven transfer of the force generated by the loads on the tanks, especially gravity load and inertial load, resulting in the support bar 2, as the main load-bearing component in the horizontal direction, not being directly stressed. That leads to the stresses on the remaining components being too large. To solve this problem, it is necessary to increase the reinforcement structure to make the force of the support system more uniform. It is also necessary to study the stress change of the tank before and after the reinforcement.

3.3.1. Reinforcement Program

To improve the stress on the support system and make it more uniform in both the horizontal and vertical directions, reinforcement was added between the support system and the cold screen connection [22], consisting of a trapezoidal plate and a rectangular plate, as shown in Figure 26. Two reinforcement bars are added to each side of the support system, for a total of four reinforcement bars.

3.3.2. Stress Analysis of Reinforcing Structures

(1) Analysis of reinforcing effect of working case 1.

Figure 27 shows the stress distribution along the path P₁, i.e., the middle of the outer head, before and after reinforcement. It can be analyzed that the stress at the head of the outer tank decreases significantly compared to that before the reinforcement, and the extreme value of the stress at the middle of the head of the outer tank after reinforcement is 202.9 MPa, which has decreased by 44.3 MPa, compared to that before the reinforcement. Figure 28 shows the stress distribution along path P₇, the connection between the axial support cylinder and the head of the outer tank, before and after reinforcement. It can be analyzed that the stress here decreased significantly after the reinforcement compared to that before the reinforcement, and the stress extreme value at the middle of the head of the outer tank after the reinforcement is 96.5 MPa, which has decreased by 169.3 MPa compared to that before the reinforcement.

Figure 29 shows the stress distribution at the connection between the support system and the support bar. Figure 29a shows the comparison of the stress distribution along path P₃, i.e., the connection between the support system and support bar 1, before and after the reinforcement, and Figure 29b shows the comparison of the stress distribution along path P₄, i.e., the connection between the support system and support bar 2, before and after the reinforcement. It can be analyzed that the stress at Figure 29a has decreased significantly and the stress at Figure 29b increased slightly, which indicates that due to the reinforcement structure, the support system is more uniformly stressed in both vertical and horizontal directions, which leads to a decrease in the stress at the connection between the support system and support bar 1 and an increase in the stress at the connection between the support system and support bar 2. The stress extreme value at the connection with support bar 1 after the reinforcement is 174.8 MPa, which is 92.7 MPa lower than that before the reinforcement; the stress extreme value at the connection with support bar 2 after the reinforcement is 142.5 MPa, which is 73.9 MPa higher than that before the reinforcement.

Figure 30 shows the stress comparison of the four support bars before and after the reinforcement, and Figure 30a,b show the stress distribution along the paths P₅ and P₆, respectively. The analysis shows that the stress of support bar 1 decreased and the stress of support bar 2 increased significantly. The stress extreme value of support bar 2 is 114.1 MPa, which is 94.4 MPa higher than that before the reinforcement, which indicates that due to the existence of the reinforcement structure, the force of the support system is improved, and the force of the support system is relatively more homogeneous, which leads to a decrease in the stress of support bar 1 and an increase in the stress of support bar 2.

(2) Analysis of the reinforcing effect of working case 2.

Figure 31 shows the stress distribution along path P₁, i.e., the middle of the outer head, before and after the reinforcement. It can be analyzed that the stress at the head of the outer tank decreased significantly compared to that before the reinforcement, and the extreme value of the stress at the middle of the head of the outer tank after reinforcement is 44.9 MPa, which has decreased by 25.0 MPa, compared to that before the reinforcement. Figure 32 shows the stress distribution along path P₇, i.e., the connection between axial support cylinder and head of the outer tank, before and after the reinforcement. It can be analyzed that the stress here decreased significantly after the reinforcement compared to that before the reinforcement, and the stress extreme value at the middle of the head of the outer tank is 32.5 MPa after the reinforcement, which has decreased by 31.8 MPa, compared to that before the reinforcement.

Figure 33 shows the stress distribution at the connection between the support system and the support bar. Figure 33a shows the comparison of the stress distribution along path P₃, i.e., the connection between the support system and support bar 1, before and after the reinforcement, and Figure 33b shows the comparison of the stress distribution along path P₄, i.e., the connection between the support system and support bar 1, before and after the reinforcement. The analysis shows that the stress at path P₃ decreased significantly, and the stress at path P₄ increased slightly, which indicates that due to the reinforcement structure, the support system is more uniformly stressed in the vertical and horizontal directions, which leads to a decrease in the stress at the connection between the support system and support bar 1 and an increase in the stress at the connection between the support system and support bar 2. The stress extreme value at the connection with support bar 1 after the reinforcement is 128 MPa, which is increased by 65.9 MPa compared to that before the reinforcement; the stress extreme value at the connection with support bar 2 after the reinforcement is 71.1 MPa, which is decreased by 181.6 MPa compared to that before the reinforcement.

Figure 34 shows the stress comparison of the two kinds of support bars before and after the reinforcement, and Figure 34a,b show the stress distribution along the paths P₅ and P₆. It can be analyzed that the stress distribution of support bar 1 is more uniform after the reinforcement, and the stress extreme value is 64.8 MPa, which is 48.9% lower than that before the reinforcement, and the stress extreme value of support bar 2 is 83.9 MPa, which is 46.9% lower than that before the reinforcement.

3.3.3. Heat Transfer Analysis of Reinforcing Structures

The reinforcement structure is located between the cold screen and the outer tank, and the newly added thermal bridge causes the ambient temperature to be transferred inward, thus leading to an increase in liquid nitrogen consumption in the liquid nitrogen cold screen. Since the heat passing through the reinforcement structure is absorbed by the cold screen, the reinforcement structure will not lead to an increase in the evaporation rate of liquid helium from the inner tank, but it will inevitably lead to an increase in the liquid nitrogen consumption of the liquid nitrogen cold screen. To quantitatively analyze the influence of the reinforcement structure on liquid nitrogen consumption, the heat leakage of the reinforcement structure can be calculated, and the corresponding increase in the liquid nitrogen consumption can be calculated based on the latent heat of vaporization of liquid nitrogen.

The amount of heat leakage through the reinforced structure can be calculated by the following equation:

$$Q=4Q_1=4A_1{q}_1$$

(6)

where Q is the heat leakage through all reinforcing structures, W; Q₁ is the heat leakage through a single reinforcing structure, W; A₁ is the surface area of a single reinforcing structure, mm²; and q₁ is the average heat flow density of a single reinforcing structure, W/mm².

The additional liquid nitrogen consumption can be calculated by:

$$m={\displaystyle \frac{Q}{H_{\mathrm{v}}}}$$

(7)

where m is the mass of liquid nitrogen consumed per second, kg; H_v is the latent heat of vaporization of liquid nitrogen, kJ/kg.

Figure 35 shows the heat flux distribution of the reinforcement in general and along the x, y, and z directions, and it can be analyzed that the heat flux of the reinforcement is smaller, that is, the heat transferred from the environment to the cold screen through the reinforcement is small. According to Equation (6), it can be calculated that the heat leakage of a single reinforcing bar is 1.7 W. Four reinforcing bars are added to the whole tank, and the total heat leakage of the reinforcing bars is 6.8 W. According to Equation (7), it can be calculated that due to the existence of the reinforcing bars, the liquid nitrogen cooling screen needs to consume an additional 3.4 × 10⁻⁵ kg of liquid nitrogen per second, which accounts for 6% of the liquid nitrogen design dosage of the liquid nitrogen cooling screen.

4. Conclusions

In this paper, the stresses of 120 m³ liquid helium storage tanks are calculated under four different transportation cases, and the stress distribution results of the four cases are obtained by applying the inertial loads to the tanks using the equivalent density method. According to the results of the calculation, the tanks were calibrated for strength, the weak points of the structure were reinforced, and the tanks after reinforcement were analyzed by the same method of stress calculation. The analysis results are as follows:

(1)

Among the four transportation cases, the stress extremes at the connection between the axial support cylinder and the head of the outer tank and at the connection between the support system and support bar 1 are 265.8 MPa and 267.5 MPa, respectively, for case 1, i.e., when subjected to inertia load of two times the gravitational acceleration along the direction of transportation; and the stress extremes at the connection between the support system and support bar 2 are 252.7 MPa, when subjected to the inertia load of one time the gravitational acceleration in the horizontal direction perpendicular to the direction of transportation. In case 2, i.e., when subjected to an inertia load of one time the horizontal acceleration of gravity perpendicular to the transportation direction, the stress extremes at the connection between the support system and the support bar 2 are 252.7 MPa. These locations are the structural weaknesses of the tanks during transportation, which do not satisfy the strength requirements under the corresponding cases, and therefore need to be structurally reinforced.

(2)

Aiming at the structural characteristics of the 120 m³ liquid helium storage tank, combined with the previous stress analysis, a structural reinforcement proposal is put forward. By comparing the stress distribution of each part of the tank before and after the reinforcement, it is found that the reinforcement structure makes the stress of the support system more uniform, and the stress extremes at the connection between the axial support cylinder and the head of the outer tank, the connection between the support system and the support bar 1, and the connection between the support system and the support bar 2 have decreased by 63.7%, 34.7%, and 71.9%, respectively. After calibration, the structural strength of the reinforced tank meets the requirements during transportation.

(3)

The reinforcement bars used for the structural reinforcement of the tank will lead to an increase in the consumption of liquid nitrogen in the cold screen. Four reinforcement bars are added to the whole tank, and the total heat leakage of the reinforcement bars is 6.8 W. Due to the existence of the reinforcement bars, the liquid nitrogen cold screen needs to consume an additional 3.4 × 10⁻⁵ kg of liquid nitrogen per second, which accounts for 6% of the liquid nitrogen design of the liquid nitrogen cold screen.