Numerical Analysis of Temperature Distribution During Charging Process of Vertically Installed Hydrogen Tanks

Abstract

Recent advancements in hydrogen tank technology have favored the use of composites over metals for weight reduction, although at the cost of a narrower operating temperature range. Therefore, this study aims to establish safety standards for charging tube skids—large tanks positioned vertically within hydrogen transport vehicles—by examining the variations in temperature distribution under different charging environments through numerical analysis. A two-dimensional axisymmetric model was developed based on ANSYS Fluent to incorporate the effect of buoyancy by applying gravitational conditions along the axial direction. The study analyzed the impact of various charging and environmental conditions (charging time (0~240 min), inlet temperature, ambient temperature, initial temperature (−20 °C~40 °C), and initial pressure (5~20 bar)) on temperature distribution, ultimately deriving a regression equation to predict the tank’s maximum temperature. The findings indicate that the initial temperature has the most significant correlation with the tank’s temperature, followed by charging time and ambient temperature. Inlet temperature and initial pressure demonstrated minimal influence within the study’s scope. The derivation of predictive formulas for the maximum temperatures of the tank’s three regions, all showing an R² value of 0.99 or higher, highlights the practicality of these results. This study’s insights are expected to contribute to further research aimed at optimizing charging conditions for tube skids and other hydrogen tanks.

1. Introduction

The environmental issues caused by carbon emissions have led to a concerted effort to mitigate greenhouse gas (GHG) emissions resulting from energy consumption. In particular, renewable energy technologies such as solar and wind power have demonstrated economic benefits through ongoing research and development [1]. Additionally, significant research has been directed toward carbon capture, utilization, and storage technologies to prevent the emission of GHGs produced by fossil fuel consumption into the atmosphere [2]. However, renewable energy sources face challenges in providing a stable electricity supply due to significant load fluctuations [3]. Moreover, integrating carbon capture equipment into transportation modes, such as automobiles, is impractical due to constraints on space and weight. Consequently, hydrogen fuel has emerged as a promising alternative to overcome the limitations of these carbon reduction technologies. Hydrogen produced through the electrolysis of water offers a medium to store energy generated from renewable sources and has the potential to replace conventional fossil fuels in the transportation sector, with only water as a byproduct after chemical reactions.

Currently, the prevalent hydrogen fuel technology involves compressing hydrogen to high pressures, approximately 350 to 700 bars, to address the issue of hydrogen gas’s low density [4]. This results in high pressure changes in hydrogen tanks during charging, compared to other gas fuels such as compressed natural gas [5], and leads to rapid temperature increases within the tank due to the enthalpy rise from gas compression during the filling process. Recent advancements in high-pressure hydrogen tanks, for instance, type IV tanks, have incorporated composites with a limited operational temperature range for weight reduction [4]. Thus, ensuring the safe usage of hydrogen tanks necessitates the establishment of specific charging and discharging conditions by analyzing the temperature distribution within hydrogen tanks under various charging scenarios.

Most previous studies on predicting temperature variations in hydrogen tanks have focused on fuel tanks for vehicles. For example, Yang [6] analytically estimated the temperature change during the hydrogen refueling process in hydrogen fuel-cell vehicles using ideal gas equations and applied the reduced Helmholtz free energy to correct the inaccuracies of these equations at high pressures.

Numerous studies have employed numerical methods to examine the temperature dynamics in hydrogen tanks designated for vehicle use, with a particular emphasis on rapid filling processes [7,8,9,10,11,12,13,14,15,16,17,18,19,20]. These investigations have explored the impact of various factors—such as state of charge, ambient temperature, charging duration, and tank geometry—on the temperature distribution within the tank throughout the charging procedure. Given that rapid filling typically occurs within a short period of a few minutes, some studies have modeled and analyzed the tank’s complete three-dimensional (3D) structure [7,8,9,10,11]. However, due to the extensive computational resources and time required to analyze compressible fluids, other research has limited the analysis to either half of the 3D geometry [12] or a simplified two-dimensional (2D) axisymmetric representation of the tank [13,14]. Notably, Melideo et al. [15] conducted a comparative analysis using both a 2D axisymmetric model and a full 3D model of a hydrogen tank. Their findings indicated that the results from the 2D and 3D models closely aligned during the charging and discharging phases, wherein hydrogen is injected into or released from the tank at high velocities, weakening the effects of buoyancy.

This study focuses on a specific application of hydrogen tanks in “tube trailers”, which are specialized vehicles designed for the transportation of high-pressure hydrogen gas from production sites to refueling stations. These trailers house multiple tanks, known as “tube skids”, vertically installed and interconnected to store and transport significant quantities of hydrogen, as depicted in Figure 1. This configuration offers enhanced flexibility in vehicle integration compared to traditional horizontal tube skids, and it facilitates rapid hydrogen release upwards from the vehicle in case of an emergency.

Given the considerable duration required to fully charge a vertically structured tank, the numerical analysis had to accommodate a broader time range than that typically associated with rapid filling. Additionally, the process of hydrogen introduction at the tank’s bottom at comparatively slow velocities suggested a pronounced effect of buoyancy on the internal flow patterns within the tank. Accordingly, this study developed a numerical analysis model based on the commercial software ANSYS Fluent. The model is designed as a 2D axisymmetric framework to effectively simulate the entire charging process of a single tube skid by incorporating the influence of gravity and buoyancy effects.

Utilizing this numerical model, the study established criteria for charging tube skids, analyzing the impact of five charging and environmental conditions on the temperature distribution within the tube skids. The analysis yielded temperature data, from which the sensitivity of each variable to temperature increases in the tank was determined. Subsequently, temperature prediction formulas for different regions within the tank were derived based on the numerical analysis results.

2. Numerical Analysis Model

2.1. Geometry of Analysis Model

To efficiently simulate the charging process of a tube skid, we simplified the analysis model. Although a typical trailer houses around 40 hydrogen tanks, as depicted in Figure 1, analyzing each tank simultaneously would be impractical. Therefore, we focused our numerical analysis on a single tank, based on the assumption that the slow flow velocity of hydrogen—resulting from the tanks’ parallel configuration and their substantial combined storage capacity—ensures uniform charging across all tanks, rendering the analysis of a single tank representative of the entire system. Furthermore, we neglected the influence of the connecting conduit, considering it inconsequential to the overall charging dynamics.

The tank was modeled in 2D axisymmetric geometry for efficient analysis. Figure 2 illustrates the geometry and dimensions of an individual tank, where the inlet’s complex structure was simplified for numerical analysis. The materials constituting the inlet, shell, and liner were aluminum, carbon fiber-reinforced polymer (CFRP), and polyamide (PA), respectively. The tank measured approximately 2.6 m in length and 0.55 m in diameter, with a total volume of 400 L. With the cylinder positioned vertically within a tube trailer, hydrogen was introduced via an inlet at the bottom of the cylinder.

Given the tank’s vertical orientation, buoyancy aligned with the tank’s axis. Furthermore, the tank’s geometry and the primary forces acting on the hydrogen exhibited symmetry relative to the central axis, enabling a 2D axisymmetric analysis to accurately account for buoyancy effects. Figure 3 shows the numerical analysis domain and the generated mesh representing the central cross section of the 3D tank geometry for 2D axisymmetric analysis. Comprising approximately 40,000 quadrilateral cells—determined optimal through grid testing—the grid’s average quality was assessed at 0.71.

2.2. Governing Equations of Analysis Model

Analyzing the hydrogen charging process comprehensively required simulating both the turbulent flow within the tank, driven by the momentum and buoyancy of the incoming hydrogen, and the resultant temperature increase and heat transfer due to hydrogen compression. Accordingly, this study employed a set of governing equations, including a continuity equation for compressible flow simulation, a momentum equation incorporating gravity’s effect, and a shear stress transport (SST) turbulence model to simulate turbulent flow. The adoption of the turbulence model aligned with its prevalent use in numerous prior investigations into the charging dynamics of high-pressure hydrogen [8,11,14]. Additionally, an energy equation was applied to model heat transfer within the tank. The computational analysis utilized the coupled algorithm within a pressure-based solver framework. The governing equations used in this study are as follows.

Continuity equation:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \overrightarrow{u}\right)=0$$

(1)

Momentum equation:

$${\displaystyle \frac{\partial \left(\rho \overrightarrow{u}\right)}{\partial t}}+\nabla ·\left(\rho \overrightarrow{u}\overrightarrow{u}\right)=\rho \overrightarrow{g}-\nabla P+\nabla ·\left(\mu \nabla \overrightarrow{u}\right)$$

(2)

Energy equation:

$${\displaystyle \frac{\partial \left(\rho c_{p}T\right)}{\partial t}}+\nabla ·\left(\rho c_{p}T\overrightarrow{u}\right)=\nabla ·\left(k\nabla T\right)$$

(3)

Transport equation for k of the k–ω SST model:

$${\displaystyle \frac{\partial \left(\rho \kappa \right)}{\partial t}}+\nabla ·\left(\rho k\overrightarrow{u}\right)=\nabla ·\left({\Gamma }_k\nabla \overrightarrow{u}\right)+G_{\kappa }-Y_k$$

(4)

Transport equation for ω of the k–ω SST model:

$${\displaystyle \frac{\partial \left(\rho \omega \right)}{\partial t}}+\nabla ·\left(\rho \omega \overrightarrow{u}\right)=\nabla ·\left({\Gamma }_{\omega }\nabla \overrightarrow{u}\right)+G_{\omega }-Y_{\omega }+D_{\omega }$$

(5)

2.3. Properties and Boundary Conditions

Given the study’s focus on a tube skid designed to withstand a maximum charging pressure of 450 bars, the simulation of the hydrogen charging process had to incorporate hydrogen properties at pressures up to 450 bars. While the ideal gas law is commonly employed for its simplicity in calculating gas properties at low pressures, it fails to account for molecular attractions [21], rendering it inadequate for high-pressure applications. Consequently, the van der Waals equation, which considers both the size of molecules and intermolecular forces as shown in Equation (6), was developed. Moreover, various cubic equations of state have been modified from the van der Waals equation to enhance accuracy. Among these, the Redlich–Kwong model [22] has been utilized in prior studies to simulate the hydrogen charging process [5,7,8,9,10].

$$P={\displaystyle \frac{RT}{V-b}}-{\displaystyle \frac{a/\surd T}{V(V+b)}}$$

(6)

Park [23] evaluated the accuracy of three state equations—Redlich–Kwong, Soave–Redlich–Kwong, and Peng–Robinson—in predicting hydrogen’s properties, finding that the Peng–Robinson equation exhibited a relative error of up to 5% against the National Institute of Standards and Technology (NIST) database. Although the Redlich–Kwong and Soave–Redlich–Kwong equations demonstrated smaller errors than the Peng–Robinson model, they still showed a relative error of up to 2%. While such discrepancies might be negligible in specific contexts, this study opted to utilize NIST’s property data to enhance analysis accuracy by minimizing inherent errors in these equations. Therefore, the NIST real gas model [24], as provided by ANSYS Fluent v21.2, was employed to calculate hydrogen’s thermodynamic properties based on the NIST database. This model, previously applied in studies [13,14] simulating the hydrogen tank charging process, was chosen to represent the properties of high-pressure hydrogen accurately.

Regarding the materials of the tank shell, ANSYS Fluent v21.2’s default properties were used for aluminum, while property values for CFRP and PA were specified, as listed in

Table 1. Properties of tube skid materials.

Properties	Al6061	PA	CFRP
Density [kg/m3]	2719	1100	1540
Specific Heat [J/(kg K)]	896	1872	1040
Thermal Conductivity [W/(m K)]	167	0.334	1.2

.

To assess the impact of environmental and operational conditions on the temperature distribution during charging, initial and boundary conditions were established, as outlined in

Table 2. Numerical analysis list for operating conditions and environmental conditions.

Case	Charging Time	Inlet Temperature (°C)	Ambient Temperature (°C)	Initial Temperature (°C)	Initial Pressure (bar)
Ref	2 h (120 min)	20	20	20	15
1	1 h (60 min)	20	20	20	15
2	3 h (180 min)	20	20	20	15
3	4 h (240 min)	20	20	20	15
4	2 h (120 min)	−20	20	20	15
5	2 h (120 min)	0	20	20	15
6	2 h (120 min)	40	20	20	15
7	2 h (120 min)	20	−20	20	15
8	2 h (120 min)	20	0	20	15
9	2 h (120 min)	20	40	20	15
10	2 h (120 min)	20	20	−20	15
11	2 h (120 min)	20	20	0	15
12	2 h (120 min)	20	20	40	15
13	2 h (120 min)	20	20	20	5
14	2 h (120 min)	20	20	20	10
15	2 h (120 min)	20	20	20	20

. A total of five variables were considered to analyze their effect on the tank’s temperature rise. Seasonal variations, represented by summer (40 °C) and winter (−20 °C) conditions, informed the set values for inlet, ambient, and initial temperatures. In the initialization step of each case, both the fluid and solid regions were initialized to the initial temperature as shown in Table 2, and the pressure in the fluid region was specified as the initial pressure. The pressure of the inlet was regulated to increase linearly from the initial pressure to the maximum capacity of the tank (450 bar) during the charging time, and the temperature of the inlet stream was specified as the inlet temperature. It was assumed that convective heat transfer by the outside air continuously occurred on the outer wall of the tank, the temperature of the outside air was applied as the ambient temperature, and the convective heat transfer coefficient was set as 6 W/(m²·K) assuming minimum air flow. Moreover, walls in contact with different materials share the mesh geometry and mediate heat transfer under coupled conditions.

3. Analysis of Temperature Distribution Inside Tube Skid According to Operating and Environmental Conditions

3.1. Verification of Analysis Model

The numerical model used in this study incorporates the properties of high-pressure hydrogen and various governing equations, utilizing a 2D geometric representation of the hydrogen tank. To verify the model’s reliability, we used a study by Zheng et al. [13], which similarly employed a 2D axisymmetric model alongside NIST property data. The tank geometry in Zheng et al.’s study was approximated using the digitizer function within Origin software. Given that Zheng et al.’s tank was oriented horizontally, our analysis model was adjusted by omitting the gravity component from the numerical model utilized in our research.

In Zheng et al.’s experiment, hydrogen pressure exhibited an approximately linear increase; accordingly, we set the inlet pressure’s rise from approximately 5 to 64 MPa over 180 s, based on the graph from the verification experiment presented in their paper. Zheng et al. did not specify values for the convective heat transfer coefficient of the tank’s outer surface; consequently, we assumed a convective heat transfer coefficient of 6 W/(m² K). The temperature rise of the tank predicted by our model was then compared with the temperature increase derived from the verification experiment graph in Zheng et al.’s paper, with results illustrated in Figure 4. Although an error margin of up to 6.6 °C was observed in the tank’s temperature rise—attributable to the reliance on estimated values from the graph—the overall trends between the two datasets were consistent, with an error of less than 1 °C at both the initial and final points. Because our study’s focus was on the tank’s maximum temperature at the end of charging, the error at the final charging point was considered a more critical evaluative criterion than the peak error observed throughout the charging process.

3.2. Analysis of Effect of Tube Skid Charging Time

The maximum charging rate for a tube skid is primarily governed by the compressor’s capacity to supply high-pressure hydrogen, whereas the minimum charging speed can be adjusted based on environmental conditions through a regulator. Given that tube skids are configured by interconnecting multiple tanks in parallel, hydrogen can be charged faster than the designated rate when only a few tanks are being filled. Therefore, examining the temperature distribution within the tube skid relative to charging duration is crucial for defining operational standards. This study conducted numerical analyses across four scenarios, ranging from 1 to 4 h, with a benchmark charging period of 2 h.

Figure 5 shows the temperature distribution across the central section of the tube skid following charging. Notably, the upper region exhibited higher temperatures compared to the lower region. This temperature discrepancy arose as the cooler hydrogen, introduced at the tube skid’s base, warmed up during compression and ascended due to buoyancy. The incoming hydrogen from the inlet ascended a certain distance propelled by its momentum before descending again due to density variations, resulting in unstable flow patterns near the skid’s bottom and a pronounced temperature gradient in the section where mixing occurred. Moreover, Zheng et al. [13] indicated that a temperature gradient could also be present along the axis of a horizontal tank if the ratio between its axial length and diameter was substantial.

Accordingly, the position within the tube skid significantly influences the disparity between the average temperature (mass-weighted average) and the maximum temperature (vertex maximum) of the hydrogen. Specifically, shorter charging intervals increase the temperature difference between the upper and lower sections due to a rapid pressure rise and diminished heat transfer duration. The difference between the peak and average temperatures was 20.7 °C for case 1 (the shortest charging duration) and 8.3 °C for case 4 (the longest charging duration).

Figure 6 illustrates the peak temperature within each region of the tube skid as a function of charging time, including the hydrogen’s maximum temperature inside the tube skid, the solid region’s (tank shell) peak temperature, and the outer surface’s maximum temperature. Overall, a prolonged charging period corresponded with a narrowing gap between the average internal hydrogen temperature and the outer surface’s temperature, decreasing from 19.9 to 8.5 °C, attributable to an extended heat transfer duration. The extended period also allowed for more effective heat dissipation to the external air, establishing an inverse relationship between the tube skid’s maximum temperature and the charging duration.

Table 3. Maximum temperature for each region of tube skid by charging time.

Case	Charging Time	Maximum Temperature (Hydrogen, °C)	Maximum Temperature (Tank Shell, °C)	Maximum Temperature (Outer Surface, °C)
1	1 h (3600 s)	96.1	91.7	79.0
Ref	2 h (7200 s)	79.1	76.4	67.7
2	3 h (10,800 s)	69.6	67.6	61.2
3	4 h (14,400 s)	62.8	61.3	56.3

presents numerical data detailing the maximum temperatures upon charging completion.

3.3. Analysis of Effect of Inlet Temperature

The inlet temperature of hydrogen introduced into the tube skid is primarily influenced by external environmental conditions. However, this temperature may increase as the hydrogen is compressed and fed into the tube skid. In scenarios involving rapid filling, precooling measures are often implemented prior to hydrogen injection to mitigate excessive temperature rise within the hydrogen tank [15]. Thus, an investigation was conducted to examine the effect of the inlet temperature on the temperature distribution within the tube skid.

Figure 7 illustrates the temperature distribution across the central section of the tube skid following the charging process according to the inlet temperature. Contrary to the observations in Figure 6, where temperature varied throughout the tank depending on the case, alterations in the inlet temperature predominantly affected the temperature distribution in the tank’s lower section, with no significant impact on the higher-temperature region at the top of the tank. This phenomenon is attributed to the increased influence of buoyancy, a consequence of the low flow velocity and diminished momentum of hydrogen over extended charging periods. As the inlet temperature decreases, the density difference between the incoming hydrogen and that within the tank’s upper portion increases, suggesting that changes in inlet temperature primarily influence the temperature difference between the tank’s upper and lower sections without markedly affecting the upper region’s temperature. The regional maximum temperature analysis, as depicted in Figure 8 and detailed in

Table 4. Maximum temperature for each region of tube skid by inlet temperature.

Case	Inlet Temperature	Maximum Temperature (Hydrogen, °C)	Maximum Temperature (Tank Shell, °C)	Maximum Temperature (Outer Surface, °C)
4	−20 °C	79.1	76.3	67.7
5	0 °C	79.2	76.5	67.9
Ref	20 °C	79.1	76.4	67.7
6	40 °C	79.6	76.8	68.1

, further corroborated that the tank’s maximum temperature varied by only 0.5 °C in response to different inlet temperatures.

Figure 8 also reveals a distinctive pattern in case 6 during the initial phase of charging on the tank’s outer surface, diverging from other cases. This anomaly arose because the peak surface temperature initially appeared in the aluminum section of the inlet, which was in direct contact with hydrogen at 40 °C. However, as charging advanced, the location of the highest surface temperature shifted to the upper part of the tube skid, aligning with the trends observed in other cases.

3.4. Analysis of Effect of Ambient Temperature

The thermal conductivity of the tube skid was notably low, attributable to its composite-based construction. In cases other than rapid filling scenarios, the tube skid is exposed to air for extended durations during the charging process, leading to an increase in heat dissipation to the surrounding environment. To investigate the influence of ambient temperature, an analysis was conducted while holding the convective heat transfer coefficient constant at 6 W/(m² K) and varying the ambient temperature. The results of this analysis are presented in Figure 9 and Figure 10.

Figure 9 illustrates the effect of ambient temperature on the overall temperature distribution within the tube skid, subsequently affecting the maximum temperature by region, as depicted in Figure 10. The data from Figure 10 reveal that ambient temperature exerted minimal impact at the onset of charging when the hydrogen temperature experienced a rapid increase. However, the discrepancy in maximum temperature across different scenarios progressively widened after approximately 10 min from the start of charging, attributable to differences in heat transfer to the external air. These findings highlight the importance of considering seasonal variations when establishing charging rates for tube skids. The internal hydrogen’s maximum temperature exhibited fluctuations of up to 25 °C depending on the ambient temperature, with comprehensive details provided in

Table 5. Maximum temperature for each region of tube skid by ambient temperature.

Case	Ambient Temperature	Maximum Temperature (Hydrogen, °C)	Maximum Temperature (Tank Shell, °C)	Maximum Temperature (Outer Surface, °C)
7	−20 °C	62.6	59.8	50.3
8	0 °C	70.6	67.8	58.3
Ref	20 °C	79.1	76.4	67.7
9	40 °C	87.6	85.0	77.6

.

3.5. Analysis of Effect of Initial Temperature

The initial temperature of the tube skid acted as an environmental variable because both the tube skid and the internal hydrogen, after prolonged exposure to outdoor air, tended to reach thermal equilibrium with the ambient temperature. However, the tube skid temperature changed when hydrogen was either injected into or released from it. Consequently, operations involving frequent charging or discharging could lead to notable variations in the initial temperature at subsequent recharging instances. The impact of the initial temperature was examined, with results presented in Figure 11 and Figure 12 and

Table 6. Maximum temperature for each region of tube skid by initial temperature.

Case	Initial Temperature	Maximum Temperature (Hydrogen, °C)	Maximum Temperature (Tank Shell, °C)	Maximum Temperature (Outer Surface, °C)
10	−20 °C	55.6	53.4	47.4
11	0 °C	66.9	64.4	57.1
Ref	20 °C	79.1	76.4	67.7
12	40 °C	91.8	88.9	79.1

.

Given an initial charging pressure of 15 bars, as specified in Table 2, the hydrogen mass constituted less than 5% of the total capacity. Owing to the substantial heat retention by the solid shell region, fluctuations in the tank’s initial temperature significantly altered the tube skid’s overall temperature distribution, as depicted in Figure 11. Notably, a comparison between Figure 10 and Figure 12 reveals that, compared to the changes in ambient temperature, variations in the initial temperature exerted a more pronounced effect on the maximum temperature. While the ambient temperature change resulted in a maximum hydrogen temperature change of approximately 25 °C, this variation increased to approximately 36 °C with the change in initial temperature.

In Figure 12, the discontinuity observed in the maximum hydrogen temperature for case 10 is attributed to the rapid temperature change when hydrogen at 20 °C was introduced into the region initially set at −20 °C. Subsequently, the maximum temperature remained stable for approximately 15 min due to the heat capacity before resuming its increase as the enthalpy across the entire hydrogen region sufficiently rose.

3.6. Analysis of Effect of Initial Pressure

The residual hydrogen quantity and pressure within a tube skid following unloading may differ based on when the release is halted, implying that the initial pressure might vary for each subsequent charging cycle. Alterations in the initial pressure can influence the enthalpy and temperature of the internal hydrogen at the end of charging, as they affect the compression ratio needed to reach the maximum pressure. Therefore, an analysis was conducted to determine the impact of initial pressure variations, with the assumption that the tank was nearly depleted to set a benchmark for maximum discharge. The investigation focused on observing any significant temperature disparities resulting from initial pressure adjustments within the 5 to 20 bars range.

According to Figure 13, which illustrates the central section’s temperature distribution upon charging completion, variations in the initial pressure from 5 to 20 bars did not significantly affect the final temperature distribution, with discrepancies under 4 °C. This phenomenon likely occurred because a lower initial pressure necessitated more compression as the tank filled, yet the relevance of compressing the initially present hydrogen diminished as its initial mass decreased. As shown in the case 13 graph in Figure 14a, the temperature initially surged when starting at a lower pressure, but then the rate of increase moderated, aligning with other cases as more hydrogen was introduced. Consequently, Figure 14 confirms a marginal uptrend in maximum temperature with decreasing initial pressure, although with relatively low sensitivity to the pressure changes. Detailed results are provided in

Table 7. Maximum temperature for each region of tube skid by initial pressure.

Case	Initial Pressure	Maximum Temperature (Hydrogen, °C)	Maximum Temperature (Tank Shell, °C)	Maximum Temperature (Outer Surface, °C)
13	5 bar	81.5	78.6	69.5
14	10 bar	80.1	77.2	68.4
Ref	15 bar	79.1	76.4	67.7
15	20 bar	78.1	75.3	66.7

.

4. Sensitivity Analysis for Factors and Temperature Relationship Derivation

Following the analysis of the five variables critical to the charging characteristics of vertically installed hydrogen tanks, a design of experiments approach was employed to derive equations predicting the maximum temperature within the internal hydrogen, shell, and outer surface of the tank. Initially, key factors influencing the tank’s maximum temperature were identified through sensitivity analysis using a factorial design. Subsequently, equations for predicting these maximum temperatures were formulated via response surface methodology.

Given the varied materials and permissible temperature ranges across different sections of the tube skid, the tank was segmented into hydrogen, shell, and outer surface regions for analysis. The validity of the charging variables for each region’s maximum temperature was assessed through factorial design. The charging time and temperature ranges were maintained at 60 to 240 min and −20 to 40 °C, respectively, consistent with the previous sections. The experimental combinations derived from factorial design are listed in

Table 8. Range of parameters for factorial analysis.

Parameter	Low Limit Value	Upper Limit Value
Charging Time	60	240
Inlet Temperature	−20	40
Ambient Temperature	−20	40
Initial Temperature	−20	40
Initial Pressure	5	20

and

Table 9. Factorial analysis for hydrogen tank filling conditions.

Case	Charging Time	Inlet Temperature	Ambient Temperature	Initial Temperature	Initial Pressure
1	60	−20	−20	−20	20
2	240	−20	−20	−20	5
3	60	40	−20	−20	5
4	240	40	−20	−20	20
5	60	−20	40	−20	5
6	240	−20	40	−20	20
7	60	40	40	−20	20
8	240	40	40	−20	5
9	60	−20	−20	40	5
10	240	−20	−20	40	20
11	60	40	−20	40	20
12	240	40	−20	40	5
13	60	−20	40	40	20
14	240	−20	40	40	5
15	60	40	40	40	5
16	240	40	40	40	20

, with analysis results presented in

Table 10. Factorial analysis for maximum temperature of inner gas.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Model	5	10,368	2074	11.96	0.001
Linear	5	10,368	2074	11.96	0.001
Charging Time	1	3978	3978	22.94	0.001
Inlet Temperature	1	113	113	0.65	0.438
Ambient Temperature	1	2382	2382	13.74	0.004
Initial Temperature	1	3888	3888	22.43	0.001
Initial Pressure	1	7	7	0.04	0.846
Error	10	1734	173
Total	15	12,102

,

Table 11. Factorial analysis for maximum temperature of tank shell.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Model	5	10,486	2097	14.32	0.000
Linear	5	10,486	2097	14.32	0.000
Charging Time	1	3621	3621	24.73	0.001
Inlet Temperature	1	63	63	0.43	0.526
Ambient Temperature	1	2736	2736	18.68	0.002
Initial Temperature	1	4042	4042	27.61	0.000
Initial Pressure	1	24	24	0.16	0.695
Error	10	1464	146
Total	15	11,951

and

Table 12. Factorial analysis for maximum temperature of outer surface.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Model	5	9222	1844	15.61	0.000
Linear	5	9222	1844	15.61	0.000
Charging Time	1	2077	2077	17.58	0.002
Inlet Temperature	1	118	118	1.000	0.341
Ambient Temperature	1	3539	3539	29.96	0.000
Initial Temperature	1	3488	3488	29.52	0.000
Initial Pressure	1	0	0	0.000	0.985
Error	10	1181	118
Total	15	10,403

. Adj SS represents the temperature variance analyzed statistically, while Adj MS is derived by dividing Adj SS by the degrees of freedom (DF). The F-value correlates with the ratio of each variable’s Adj MS to the error’s Adj MS, and the p-value inversely relates to the F-value. A p-value < 0.05 generally indicates a factor’s significance. Applying this criterion, charging time, ambient temperature, and initial temperature were identified as significant factors, whereas inlet temperature and initial pressure were considered nonsignificant, as evidenced in the tables. The analysis results are detailed in Figure 15.

Further, a response surface methodology analysis was conducted to establish a regression equation for predicting the tank’s maximum temperature using the three significant factors identified. A face-centered design was utilized, with data from an additional 15 cases analyzed, as shown in

Table 13. Response surface analysis for hydrogen tank filling conditions.

Case	Charging Time (min)	Ambient Temperature (°C)	Initial Temperature (°C)
1	60	−20	−20
2	240	−20	−20
3	60	40	−20
4	240	40	−20
5	60	−20	40
6	240	−20	40
7	60	40	40
8	240	40	40
9	60	10	10
10	240	10	10
11	150	−20	10
12	150	40	10
13	150	10	−20
14	150	10	40
15	150	10	10

.

This design facilitated the derivation of the regression equation (Equation (7)) from the analysis results. In the regression equation, the validity of the terms can be identified by applying the reference value of the p-value (0.05) to each term, and the terms with no influence can be excluded from the equation. Here, X_i represents the design factors, Z denotes the result, and β_i and β_ij are the coefficients for each term.

$$Z={\beta }_0+{\mathsf{\Sigma }}_{i=1}^k{\beta }_i{X}_i+{\mathsf{\Sigma }}_{i=1,j\le i}^k{\beta }_{ij}{X}_i{X}_j$$

(7)

The response surface methodology’s analysis results for the tube skid are summarized in

Table 14. Response surface analysis for maximum temperature of inner gas.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Model	6	8205	1368	1370	0.000
Linear	3	7246	2415	2421	0.000
Charging Time (tchr)	1	2649	2649	2655	0.000
Ambient Temperature (Tamb)	1	1808	1808	1812	0.000
Initial Temperature (Tini)	1	2789	2789	2795	0.000
Square	1	171	171	171	0.000
tchr × tchr	1	171	171	171	0.000
Interaction	2	788	394	395	0.000
tchr × Tamb	1	365	365	366	0.000
tchr × Tini	1	423	423	424	0.000
Error	13	13	1
Total	19	8218

,

Table 15. Response surface analysis for maximum temperature of tank shell.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Model	6	7650	1275	1549	0.000
Linear	3	6785	2262	2748	0.000
Charging Time (tchr)	1	2233	2233	2713	0.000
Ambient Temperature (Tamb)	1	1866	1866	2267	0.000
Initial Temperature (Tini)	1	2686	2686	3263	0.000
Square	1	137	137	167	0.000
tchr × tchr	1	137	137	167	0.000
Interaction	2	727	364	442	0.000
tchr × Tamb	1	340	340	412	0.000
tchr × Tini	1	388	388	471	0.000
Error	13	11	1
Total	19	7661

and

Table 16. Response surface analysis for maximum temperature of outer surface.

Source	DF	Adj SS	Adj MS	F-Value	p-Value
Model	7	6242	892	2367	0.000
Linear	3	5668	1889	5016	0.000
Charging Time (tchr)	1	1213	1213	3220	0.000
Ambient Temperature (Tamb)	1	2244	2244	5957	0.000
Initial Temperature (Tini)	1	2212	2212	5872	0.000
Square	2	78	39	104	0.000
tchr × tchr	1	32	32	85	0.000
Tini × Tini	1	4	4	11	0.000
Interaction	2	495	248	657	0.000
tchr × Tamb	1	229	229	607	0.000
tchr × Tini	1	167	267	708	0.000
Error	12	5	0
Total	19	6246

, with Figure 16 illustrating the correlations between each variable and the tank’s maximum temperature. The charging time is represented by t_chr, the ambient temperature by T_amb, and the initial temperature by T_ini. By isolating only the impactful terms, the regression equation for the internal hydrogen’s maximum temperature is expressed as Equation (8), with a coefficient of determination (R²) of approximately 0.9977—surpassing the often-referenced benchmark of 0.8 [25]. This high R² value suggests that the regression equation reliably predicted the maximum temperature, facilitating the establishment of charging standards for tube skids without necessitating further analysis.

Similarly, regression equations for the shell and outer surface temperatures were derived, resulting in Equations (9) and (10). The R² values for these equations exceeded 0.99, confirming their accuracy in capturing the analytical trends.

$$\begin{gathered} T_{gas}=96.69-0.3956·t_{chr}+0.073·T_{amb}+0.9605·T_{ini}+0.000722·t_{chr}^2 \\ +0.002501·t_{chr}·T_{amb}-0.002692·t_{chr}·T_{ini} \end{gathered}$$

(8)

$$\begin{gathered} T_{shell}=90.63-0.3586·t_{chr}+0.0934·T_{amb}+0.9330·T_{ini}+0.000647·t_{chr}^2 \\ +0.002413·t_{chr}·T_{amb}-0.002578·t_{chr}·T_{ini} \end{gathered}$$

(9)

$$\begin{array}{cc}{T}_{surface}=71.047-0.2381·& t_{chr}+0.2023·T_{amb}+0.791·T_{ini}+0.000391·t_{chr}^2\\ & +0.001269·T_{ini}^2+0.00198·t_{chr}·T_{amb}-0.002138·t_{chr}·T_{ini}\end{array}$$

(10)

5. Conclusions

To evaluate the effects of charging conditions on the temperature distribution within a 400L-class type IV hydrogen tank, referred to as tube skid, vertically positioned in a hydrogen transport vehicle, a 2D axisymmetric model incorporating the effect of buoyancy was developed. Comprehensive numerical analyses were performed, utilizing property data from NIST. The study examined the influence of five charging parameters (charging time, inlet temperature, ambient temperature, initial temperature, and initial pressure) on the tube skid’s maximum temperature, yielding the following insights:

• The inlet temperature predominantly affected the temperature at the lower section of the tank due to buoyancy, as hydrogen was introduced from the bottom. This condition increased the temperature disparity between the tank’s upper and lower sections without significantly altering the maximum temperature at the top of the tank.
• The initial pressure exhibited minimal impact on the maximum temperature, as the initial mass of hydrogen was relatively small within the examined pressure range of 5 to 20 bars.
• Sensitivity analysis through factorial design further confirmed the exclusion of inlet temperature and initial pressure as significant factors. Consequently, three out of the five analyzed factors were identified as closely associated with the tank’s maximum temperature. The initial temperature exhibited the strongest correlation, followed by charging time and ambient temperature, highlighting the importance of these three parameters in the hydrogen charging process.
• Utilizing the three identified critical factors, a regression equation for predicting the tank’s maximum temperature upon charging completion was formulated via response surface methodology. With the derived equation exhibiting an R² value of 0.99 or higher, it facilitated highly accurate predictions of temperature across different regions of the tank using these parameters.

Thus, the findings from this study are poised to contribute to the development of charging standards for vertically installed tube skids and may inform subsequent research aimed at optimizing charging conditions.