Design of a Metal Hydride Cartridge Heated by PEMFC Exhaust

Abstract

This study investigates the structure of a metal hydride (MH) cartridge as a hydrogen storage tank for small-scale fuel cells (FCs). This cartridge is designed to be stacked and used in layers, allowing flexible capacity adjustment according to demand. MH enables compact and safe hydrogen storage for small-scale fuel cell (FC) applications due to its high energy density and low-pressure operation. However, because hydrogen desorption from MH is an endothermic reaction, an external heat supply is required for stable performance. To enhance both the heat transfer efficiency and cartridge usability, we propose a heat supply method that utilizes waste heat from an air-cooled proton-exchange membrane fuel cell (PEMFC). The proposed cartridge incorporates four cylindrical MH tanks that require uniform heat transfer. Therefore, we proposed the tank arrangements within the cartridge to minimize the non-uniformity of heat transfer distribution on the surface. The flow of exhaust air from the PEMFC into the cartridge was analyzed using computational fluid dynamics (CFD) simulations. In addition, an empirical correlation for the Nusselt number was developed to estimate the heat transfer coefficient. As a result, it was concluded that the heat utilization rate of the exhaust heat flowing into the cartridge was 13.2%.

1. Introduction

The urgent challenges of climate change and environmental degradation have accelerated global efforts to develop sustainable energy solutions. Although wind and solar power show great potential, their intermittent nature limits their reliability as stable energy sources. However, hydrogen has emerged as a versatile and clean energy carrier because of its high energy density and environmentally friendly properties [1,2,3].

A significant advantage of using hydrogen is the variety of production methods [4,5], including steam-reforming fossil fuels, water electrolysis using renewable energy sources such as wind and solar, and biomass gasification. Among these, biomass gasification is particularly notable for its high efficiency and low environmental impact, making it a potential key renewable energy source [4,6].

Green hydrogen production from renewable sources is another promising approach. Waste biomass resources (e.g., sewage and food waste), which must be reduced owing to recycling initiatives, are especially plentiful and represent valuable resources expelled from local regions. Therefore, utilizing them as hydrogen sources can promote localized production and energy consumption. Although biomass-derived hydrogen generally exhibits higher energy and exergy efficiency than water electrolysis [7], its calorific value and conversion efficiency are approximately 50% of those of fossil fuels [8]. To address this issue, the integration of biomass-derived hydrogen with fuel cells (FCs), which are highly efficient energy conversion devices, has been proposed [9,10].

Proton-exchange membrane fuel cells (PEMFCs) are among the most widely used types of FCs. In addition, PEMFCs have high energy efficiency and can operate at relatively low temperatures [11,12], thus providing significant advantages for small-scale FCs intended for use within local regions. Owing to these advantages, PEMFCs are ideal for lightweight and small-scale urban mobility applications, such as bicycles and kick scooters [13,14]. These advantages have also led to research on FC-powered drones [15,16] and autonomous mobile robots used in factories and retail environments [17,18,19].

However, the primary hydrogen storage methods include high-pressure gas compression and liquefaction, which pose serious safety risks, including potential hydrogen leakage and tank rupture [20,21]. In Japan, the High-Pressure Gas Safety Act imposes strict regulations on gas containers operating above 1 MPa [22,23]. As such, dependence on high-pressure or liquefied hydrogen storage could limit the widespread adoption of hydrogen and FCs, especially in small-scale FC applications intended for public use.

In contrast, metal hydrides (MHs) represent a safer and more efficient alternative for hydrogen storage [24,25,26]. Certain MHs can absorb and release hydrogen at room temperature and low pressures (below 1 MPa) [27]. Their safety and exemption from the High-Pressure Gas Safety Act make MHs attractive hydrogen carriers for small-scale FC applications. However, hydrogen desorption from MHs is an endothermic reaction that requires a sufficient heat supply for stable performance. Therefore, numerous studies have focused on developing effective heat supply methods for MH systems [28,29,30,31]. Raju et al. [32] performed simulations to evaluate the performance of MH tanks equipped with axially arranged fins connected to heat transfer pipes. Although the internal design improved the thermal performance, the structural complexity remained. Segawa et al. [33] experimentally enhanced hydrogen desorption by incorporating 50 heat-exchange pipes into an MH tank; however, the large scale of the system (40 L) made it unsuitable for small-scale use. Hwang et al. [34] proposed a water-jacket heating system for light electric vehicles (LEVs). Although it could effectively maintain the hydrogen output, the added weight of the water jacket posed integration challenges. Similarly, Chung et al. [35,36] used internal heat pipes to improve the thermal conductivity in the MH tank and successfully enhanced both the hydrogen storage and release performance. However, the external fin protrusion still caused increased volume and usability concerns. Additionally, some research has proposed a thermal coupling system (TCS) for MHs and PEMFCs. As a fundamental study of TCS, Wilson et al. [37] experimentally demonstrated that two cylindrical metal hydride tanks with a hydrogen storage capacity of 280 NL can supply hydrogen to 1.0 kW PEMFCs. MacDonald et al. [38] also proposed three structural configurations of MH tanks for providing hydrogen to a 1.2 kW PEMFC. They demonstrated that the finned MH tank with a diameter of 103.5 mm and a length of 240 mm exhibited the highest hydrogen release performance. For small-scale applications, Davids et al. [39] demonstrated that a more compact MH tank could release 90% of the hydrogen from waste heat generated by a 130 W FC exhaust fan despite the tank design incorporating thermally expanded graphite (TEG) and protruding aluminum fins. Andreasen et al. [40] also integrated an MH tank with a diameter of 50 mm and a length of 200 mm into a PEMFC with a rated output of 300 W, demonstrating that the PEMFC could generate power for 140 min under a partial load operation of 50 W.

In summary, water-cooled thermal systems are often too large and heavy, and the complexity of liquid piping limits their scalability. Even in conventional air-cooled thermal systems, external fins and heat pipes result in bulk, thereby hindering flexible capacity adjustments. In addition, previously proposed TCSs have limited general applicability due to the large capacity of the MH tanks. Even with smaller tanks, the presence of components such as fins makes it difficult to adjust the number of units used, resulting in insufficient flexibility for use in general fuel cell applications.

Therefore, compact and flexible MH tanks intended for various applications, such as bicycles, drones, and autonomous robots, have not yet been fully developed.

In this study, an MH cartridge with a simple heating structure was designed for use in a compact PEMFC, and the arrangement of MH tanks within the cartridge cover was investigated. The cartridge consists of four small MH tanks enclosed within a cylindrical cover. Hot exhaust air from the FC flows into the cartridge and passes around the tanks, thereby transferring heat via forced convection. Moreover, to efficiently release hydrogen from the MH, a uniform temperature distribution should be maintained within the MH material. Previous studies by Yamate et al. [41] and Shimogawa et al. [42] have shown that in cylindrical MH containers, a significant radial temperature gradient in the MH bed can reduce the amount of hydrogen released. Therefore, the cartridge proposed in this study features a non-uniform flow path for the hot exhaust air owing to its structural design—specifically, the varying gaps between the tanks and cover and between adjacent tanks. These variations in the flow path may locally reduce the heat transfer coefficient, leading to an uneven heat transfer around the tanks. Such non-uniform heat transfer may cause temperature gradients within the MH bed, thereby impeding efficient hydrogen release. To address this issue, the tank arrangement inside the cover must be designed so that the heat transfer rate around the tank becomes as uniform as possible. In this study, four different configurations of tank arrangements were set within a cover with an inner diameter of 100 mm. The heat flux distribution around the tank was analyzed using computational fluid dynamics (CFD) simulations. The inlet boundary conditions for the simulations were based on experimentally measured exhaust air velocity and temperature data from a 300 W rated PEMFC. Based on the CFD analysis, an optimal tank arrangement that enables uniform heat transfer to the tank was proposed. Furthermore, for the proposed cartridge design, the Nusselt number, representing the heat transfer performance in the fluid flow on the lateral surfaces of the MH tanks, was analyzed, and an empirical correlation was developed. This correlation enables the evaluation of heat transfer to the tanks based on the local velocity distribution of the hot exhaust air from the FC flowing around the tanks within the cartridge.

2. Materials and Methods

2.1. Structure of the Cartridge and Metal Hydride Tanks

This study focuses on developing an MH cartridge with a simple structure and scalable design that allows for flexible capacity adjustment. The optimal arrangement of tanks within the cartridge was investigated to ensure uniform heat transfer from the exhaust air from the FC to the tanks. The cartridge consists of four cylindrical tanks housed within a cylindrical cover with an inner diameter of 100 mm, as shown in Figure 1a. The design allows for capacity variations through the modular stacking of cartridge units. Each tank is filled with 95 g of LaNi₅, allowing the cartridge to store approximately 50 L of hydrogen. This concept enables flexible adjustment of the number of cartridges installed for specific applications, allowing for easy coordination of the driving range or output. Therefore, a simple structure is an essential element for meeting diverse requirements in smaller, general-purpose PEMFC applications. In this paper, the suitable conditions between hydrogen discharge and supplemental heat supply from the exhaust gas performance will be identified, contributing to the practical design.

The cartridge cover has an inner diameter of 100 mm and a length of 76 mm and is made of polyvinyl chloride. Based on the findings of Askri et al. [43], who reported that a length-to-radius (L/R) ratio of 2 in the MH bed enhances the hydrogen desorption performance, Hara et al. [44] adopted compact tanks with a length of 30 mm and radius of 15 mm (Figure 1b) to achieve a more uniform radial temperature distribution during hydrogen release. Thus, the tanks implemented here have a wall thickness of 2.5 mm along the cylindrical section, an outer diameter of 35 mm, and a length of 41.5 mm. It should be noted that the bottom and top parts of the tank are thicker, with wall thicknesses of 6 mm and 5.5 mm, respectively. Each tank has a built-in metal filter to prevent internal leakage. The four tanks were arranged adjacent to the cartridge covers.

The cartridge is placed downstream of the FC cooling fan to utilize its hot exhaust air as a heat source (Figure 1c). The cartridge cover directs the exhaust airflow from the FC and provides thermal insulation.

Hara et al. [44] proposed a design policy for the tank size. However, they did not address the heat supply to a cartridge composed of four tanks and a cover. Since cartridges are installed in a stacked configuration, the exhaust heat temperature may vary between layers. In our previous study, Shimogawa et al. [42] measured the temperature of the exhaust air flowing through a dummy model consisting of 10 stacked cartridges. They revealed that the rate of temperature increases in the air surrounding the cartridges located farther from the cooling fan was slower than that in cartridges situated closer to it.

Furthermore, a temperature difference was observed between the air flowing through the center of the cartridge and that near the inner wall, and it became more significant as the cartridge’s distance from the fan increased. These findings suggest that uneven heat distribution may occur within and between cartridges, potentially resulting in insufficient heat transfer to certain tanks. However, the cartridge they used was a dummy and did not involve validation using actual MH. Therefore, Miao et al. [45] experimentally evaluated the effectiveness of supplying exhaust heat using a prototype cartridge filled with MH. They also tested two different duct configurations connecting the PEMFC to the cartridge and investigated a more effective exhaust strategy. However, their study did not address the arrangement of tanks inside the cartridge cover. Investigating the tank arrangement is essential for establishing an effective cartridge design, as non-uniform heat transfer to the tanks must be avoided. From these terms, we focused on designing the hydrogen cartridge to meet the practical conditions for mobile use in this paper.

Specifically, in areas where the flow path of hot air becomes narrow, such as between the inner cartridge wall and the tanks or adjacent tanks, the airflow may be obstructed, leading to localized reductions in the heat transfer rate. Variations in the heat transfer rate may result in a non-uniform temperature distribution within the MH filling layer inside the tanks that may affect the hydrogen desorption performance. Previous studies have shown that a significant drop in the central temperature in a cylindrical MH tank decreased the hydrogen desorption performance [41,42]. Therefore, achieving a uniform temperature distribution within the filling layer is critical, and heat should be uniformly supplied from around the tank periphery. This study focused on lateral heat transfer to the tanks and investigated the optimal tank arrangement within the cartridge to minimize non-uniformities in heat transfer.

The tanks inside the cartridge cover were arranged symmetrically around the central axis of the cover, with each offset by 90°, and placed adjacent to each other. Narrow flow pass in the cartridge may hinder the hot exhaust air, potentially affecting the local heat flux. For example, the cartridge diameter is restricted to 100 mm because of structural limitations in the fuel cell-assisted bicycle for which the cartridge is intended. In this size, four tanks are set in the cartridge. On this scale, considering the four tanks’ set-up position, that is, the hydrogen discharge and supplemental heat utilization, the following gaps of 1, 3, 5, or 7 mm or more would be possible. Note that when contacting the inner wall of the cover with a tank, the heat transfer from exhaust heat at that contact area becomes zero despite the approximately 11.0 mm distance between the tanks. Likewise, when tanks are in contact with each other, the heat transfer from exhaust heat at the contact area also becomes zero, even though the distance between the tanks and the cover is approximately 7.8 mm. Therefore, four tank configuration patterns, Types A–D, were proposed. In these configurations, the cover diameter was fixed at 100 mm while the minimum distance between the tanks and cover was set to 1, 3, 5, and 7 mm for Types A, B, C, and D, respectively (Figure 2a–d). These variations in clearance also resulted in different distances between adjacent tanks.

2.2. Analysis of Heat Flux Using CFD Simulation

To analyze the flow of FC exhaust air around the tanks inside the cartridge and the distribution of heat flux on the tank surfaces, a CFD simulation was conducted. The CFD simulations in this study were conducted using ANSYS Fluent 2025 R1, based on the finite volume method. The default numerical schemes in Fluent were applied. Specifically, the gradients were calculated using the least-squares cell-based method, and the momentum and energy equations were discretized using the second-order upwind scheme. In addition, the coupled scheme was employed for pressure–velocity coupling. Hot exhaust air from the FC entered the cartridge through its inlet opening. The conservation of momentum equation in the governing equations accounted for gravitational acceleration. At the inlet boundary, the air velocity was set to 1.0 m/s and the inlet temperature was set to 40 °C based on experimental measurements of the FC exhaust air at an output of approximately 250 W.

The materials composing the cartridge cover and tanks were modeled using thermal property data from the ANSYS materials database to reflect realistic thermal behavior. The MH inside the tanks was defined using independently specified effective thermal conductivity, specific heat capacity, and density, as listed in

Table 1. Thermophysical properties of MH [36,46].

Parameter	Value
Effective heat conductivity [W/m K]	1.32
Specific heat [J/kg K]	419
Density [kg/m3]	8175

[36,46].

The MH temperature was set to a constant value of 14.4 °C. This temperature corresponds to the condition under which the equilibrium hydrogen desorption pressure $p_{\mathrm{e}\mathrm{q},\mathrm{d}}$ [Pa] exceeds 0.045 MPa (minimum hydrogen supply pressure required for power generation in the FC used in this study) when the hydrogen storage amount in a single cartridge decreases to 10 L. The temperature $T$ [K] was calculated using the equilibrium pressure Equations (1)–(3) for MH desorption [46].

$$\begin{array}{c}\ln \left(p_{\mathrm{e}\mathrm{q},\mathrm{d}}/p_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)=\left(A_{\mathrm{d}}-B/T+a·{\left.\mathrm{H}/\mathrm{M}\right|}_{0.3 \mathrm{w}\mathrm{t}\%}\right)\times \left[1-{\displaystyle \frac{{\left(\mathrm{H}/\mathrm{M}-{\left.\mathrm{H}/\mathrm{M}\right|}_{0.3 \mathrm{w}\mathrm{t}\%}\right)}^2}{{\left(\mathrm{H}/{\left.\mathrm{M}\right|}_{\mathrm{e}\mathrm{m}\mathrm{p}}-{\left.\mathrm{H}/\mathrm{M}\right|}_{0.3 \mathrm{w}\mathrm{t}\%}\right)}^2}}\right], \mathrm{H}/\mathrm{M}<{\left.\mathrm{H}/\mathrm{M}\right|}_{0.3 \mathrm{w}\mathrm{t}\%} \end{array}$$

(1)

$$\ln \left(p_{\mathrm{e}\mathrm{q},\mathrm{d}}/p_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)=\left(A_{\mathrm{d}}-B/T+a·{\left.\mathrm{H}/\mathrm{M}\right|}_{1.1\mathrm{w}\mathrm{t}\%}\right),\hspace{1em}{\left.\mathrm{H}/\mathrm{M}\right|}_{0.3\mathrm{w}\mathrm{t}\%}\le \mathrm{H}/\mathrm{M}\le {\left.\mathrm{H}/\mathrm{M}\right|}_{1.1 \mathrm{w}\mathrm{t}\%}$$

(2)

$$\ln \left(p_{\mathrm{e}\mathrm{q},\mathrm{d}}/p_{\mathrm{r}\mathrm{e}\mathrm{f}}\right)=\left(A_{\mathrm{d}}-B/T+a·{\left.\mathrm{H}/\mathrm{M}\right|}_{1.1\mathrm{w}\mathrm{t}\%}\right)+\left(D+{\displaystyle \frac{b}{T}}\right){\left(\mathrm{H}/\mathrm{M}-{\left.\mathrm{H}/\mathrm{M}\right|}_{1.1\mathrm{w}\mathrm{t}\%}\right)}^2,\hspace{1em}{\left.\mathrm{H}/\mathrm{M}\right|}_{1.1\mathrm{w}\mathrm{t}\%}\le \mathrm{H}/\mathrm{M}$$

(3)

where $\mathrm{H}/\mathrm{M}$ is the hydrogen-to-metal atomic ratio and ${\left.\mathrm{H}/\mathrm{M}\right|}_{\mathrm{e}\mathrm{m}\mathrm{p}}$ represents the total amount of hydrogen that can be reversibly desorbed from the metal and was assumed to be zero. The piecewise structure in Equations (1)–(3) describes the desorption equilibrium pressure in the PCT diagram of LaNi₅. Equation (1), $\mathrm{H}/\mathrm{M}<{\left.\mathrm{H}/\mathrm{M}\right|}_{0.3 \mathrm{w}\mathrm{t}\%}$, represents the desorption equilibrium pressure when the hydrogen stored in the MH is 0.3 wt% or less. Equation (2), ${\left.\mathrm{H}/\mathrm{M}\right|}_{0.3 \mathrm{w}\mathrm{t}\%}\le \mathrm{H}/\mathrm{M}\le {\left.\mathrm{H}/\mathrm{M}\right|}_{1.1 \mathrm{w}\mathrm{t}\%}$, corresponds to the desorption equilibrium pressure when the stored hydrogen is between 0.3 wt% and 1.1 wt%, which represents the plateau region in the PCT diagram. Equation (3), ${\left.\mathrm{H}/\mathrm{M}\right|}_{1.1 \mathrm{w}\mathrm{t}\%}\le \mathrm{H}/\mathrm{M}$, describes the desorption equilibrium pressure when the hydrogen content exceeds 1.1 wt%.

Other descriptions and values of the constants used in Equations (1)–(3) are shown in

Table 2. Coefficients of the desorption rates [36].

Parameter	Value
Reference pressure, $p_{\mathrm{r}\mathrm{e}\mathrm{f}}$	1000 Pa
Desorption plateau pressure coefficient, $A_{\mathrm{d}}$	18.3
Plateau pressure coefficient, $B$	3704.6 K
a phase coefficient, $a$	0.0819
b phase coefficient, $b$	330 K
b phase coefficient in desorption, $D$	0.7522

[36].

The inlet air velocity used in the CFD simulation was 1.0 m/s, and the Reynolds number for the airflow inside the cylindrical duct with an inner diameter of 100 mm was approximately 6700, indicating that the flow was turbulent. Considering the balance between computational efficiency and the level of accuracy required for the analysis, the standard k-ε turbulence model, widely used in turbulent flow analysis, was adopted in this study. This model is known for its robustness and computational efficiency across a wide range of turbulent flow scenarios, employed in several studies aimed to provide engineering insights into heat exchangers [47,48,49]. The primary objective of this study is to compare the relative differences in heat transfer performance among different tank arrangements rather than to analyze detailed heat transfer phenomena near the walls with high precision. Therefore, the standard k-ε model was deemed suitable and adequate for appropriately capturing the necessary fluid characteristics.

CFD simulations were conducted for all tank configurations (Types A–D), with the total number of mesh cells ranging from approximately 5.6 million to 5.9 million (Type A: 5,597,774; Type B: 5,892,256; Type C: 5,828,724; and Type D: 5,625,403). A mesh independence study was conducted, and it confirmed that the results were not sensitive to the mesh resolution, thereby validating the reliability of the CFD analysis. The other simulation conditions are summarized in

Table 3. CFD analysis conditions for heat flux around the tank.

Items	Condition
Calculation software	ANSYS Fluent 2025R1
Calculation method	Steady state
Governing equations	Conservation of energy Continuity equation Conservation of momentum equation
Turbulence model	$\mathrm{k}-\epsilon$
Convergence conditions	Velocity, continuity, k and $\epsilon$ criteria for residual: 1 × 10−3 Energy: 1 × 10−6 Iteration: 1000 times
Fluid gas	Air
Boundary conditions	Inlet: 1.0 m/s, 40 °C Outlet: total pressure 0 PaG Wall boundary condition: non-slip

.

This analysis did not aim to capture transient thermal phenomena within the tanks. Instead, it focused on a steady-state evaluation of airflow and heat flux under the assumed operating conditions of the cartridge. CFD analysis was conducted on the surfaces of the four tanks inside the cartridge to examine the heat flux distribution.

The analysis was focused on the tank surface located 55 mm downstream from the cartridge inlet along the direction of the hot exhaust airflow. The heat flux in each tank was evaluated at eight circumferential positions on the side surface spaced at 45° intervals in the clockwise direction. Heat flux was calculated at each measurement point as a surface-averaged value over a circular area with a diameter of 10 mm, as shown in Figure 3.

2.3. Empirical Equation for the Nusselt Number

An empirical correlation for the Nusselt number was derived based on the simulation results for the tank configuration that exhibited the least variation in heat flux around the tank surface. The heat transfer coefficient $h$ [W/m²·K] at each measurement point #1–8 was calculated from the heat flux $q$ [W/m²] obtained from the CFD simulation.

According to Newton’s law of cooling, the heat flux is expressed by Equation (4):

$$q=h(T_{\mathrm{\infty }}-T_s)$$

(4)

where $T_s$ [°C] is the area-averaged surface temperature of the tank at each point, obtained from the CFD simulation, and $T_{\infty }$ [°C] is the fluid temperature, set at 40 °C.

The Nusselt number $Nu$ is defined by Equation (5):

$$Nu={\displaystyle \frac{hL}{k}}$$

(5)

where $k$ [W/m·K] is the thermal conductivity of air (evaluated at 40 °C) and $L$ [m] is the characteristic length, defined as the hydraulic diameter of the cartridge cross-section. The hydraulic diameter is given by Equation (6):

$$L={\displaystyle \frac{4A}{P}}$$

(6)

where $A$ [m²] is the cross-sectional area of the flow passage 55 mm downstream from the cartridge inlet and $P$ [m] is the wetted perimeter, defined as the sum of the inner perimeter of the cartridge cover and the outer perimeter of the tanks.

The Nusselt number $Nu$ is commonly correlated with the Reynolds number $Re$ and Prandtl number $Pr,$ as shown in Equation (7):

$$Nu=C·{Re}^m·{Pr}^n$$

(7)

where $C$, $m$, and $n$ are the empirical coefficients and exponents obtained from experiments or numerical simulations. However, because the working fluid was air and no significant temperature gradients developed within the cartridge under the simulated conditions, the influence of the Prandtl number on the Nusselt number was considered negligible. Therefore, the Prandtl number term in Equation (7) was omitted in this study.

The Reynolds number at each tank measurement point #1–8 was calculated using the flow velocity $u_x$ [m/s], which was measured at a point 1 mm perpendicular to the tank surface in the CFD simulation, as described in Equation (8):

$${Re}_x={\displaystyle \frac{{\rho u}_{x}L}{\mu }}$$

(8)

where $\rho$ [kg/m³] and $\mu$ [Pa·s] are the density and dynamic viscosity of dry air at 40 °C, respectively. The thermophysical properties are summarized in

Table 4. Physical properties of dry air at 40 °C.

Parameter	Value
Thermal conductivity, $k$ [W/m·K]	0.027076
Density, $\rho$ [kg/m3]	1.1275
Specific heat, $c_p$ [J/kg·K]	1006.9
Dynamic viscosity, $\mu$ [kg/m·s]	1.9150 × 10−5

.

2.4. Estimation of the Cartridge Thermal Utilization Rate

The heat transfer rate to the MH tanks was estimated using the heat transfer coefficients obtained from the empirical Nusselt number correlation derived via regression analysis. Based on this, the thermal utilization efficiency of the cartridge, which is defined as the ratio of the heat transferred from the hot exhaust air to the tanks to the total heat carried by the hot exhaust air from the FC stack, was evaluated.

Firstly, the heat input rate $Q_{inlet}$ [J/s] carried into the cartridge by the hot exhaust air from the FC stack can be expressed as follows (Equations (9) and (10)):

$$Q_{inlet}=\int m_{air}{C}_{air}\left(T_{out}-T_{in}\right)dt$$

(9)

$$m_{air}={\rho }_{air}{A}_{inlet}v$$

(10)

where $m_{air}$ [kg/s] denotes the mass flow rate of air, $C_{air}$ [J/kg·K] is the specific heat of air, ${\rho }_{air}$ [kg/m³] is the air density, $A_{inlet}$ [m²] is the inlet cross-sectional area of the cartridge, and $v$ [m/s] is the velocity of the exhaust air from the FC. The exhaust temperature $T_{out}$ [°C] and the ambient temperature $T_{in}$ [°C] were taken as 40 °C and 25 °C, respectively, based on experimental measurements.

The heat transfer rate to the tanks $Q_{in}$ [J/s] is expressed by Equation (11) based on the heat flux $q$ [W/m²] on the tank surface and lateral surface area $A$ [m²] of the tank:

$$Q_{in}={qA}_{tank}$$

(11)

The heat flux on the tank surface was calculated using the local heat transfer coefficients $h_x$ [W/m²·K] at measurement points #1 to #8, which were estimated based on the Nusselt number derived and the corresponding local tank surface temperatures $T_x$ [°C], as shown in Equation (12):

$$q={\displaystyle \frac{1}{8}}\sum _{x=1}^1{h}_x(T_{\infty }-T_x)$$

(12)

Finally, based on Equations (9)–(12), the thermal utilization efficiency $\eta$ of the energy supplied from the FC stack and transferred to the tanks via forced convection can be calculated:

$$\eta ={\displaystyle \frac{Q_{in}}{Q_{inlet}}}$$

(13)

3. Results and Discussion

3.1. Distribution of Heat Flux Around the Tanks According to the CFD Simulation

The heat flux at measurement points #1 to #8 was analyzed by CFD simulations for tank configuration Types A to D. The results are shown in Figure 4.

Among the tank configuration Types A to D, Type C exhibited the smallest difference between the maximum and minimum heat fluxes around the tank, with a value of 200 W/m². The next smallest difference was observed for Type B at 276 W/m². For Type C, the heat flux was lower at measurement points #1 and #3 compared with the other points. For Type B, a decrease in the heat flux was observed at points #4 and #8. These variations in the locations of reduced heat flux were dependent on the tank arrangement and resulted from changes in the flow of hot exhaust air from the FC, which caused differences in the velocity distribution around the tanks.

In Type A, a significant reduction in heat flux was observed at point #2, while in Type D, reductions were observed at points #5 and #7. The difference between the maximum and minimum heat flux was 857 W/m² for Type A and 820 W/m² for Type D. Measurement point #2 corresponded to the narrowest gap between the tank and cartridge cover, which is 1 mm in the case of Type A. Points #5 and #7 were located between adjacent tanks, where the spacing was approximately 1.1 mm in Type D. In such regions where the flow passage became extremely narrow, the flow of hot exhaust air from the FC was hindered, resulting in a significant reduction in heat transfer. The maximum and minimum heat flux values for Types A to D are included in

Table 5. The heat flux values (max. and min.) for Types A to D.

Type	Max Heat Flux [W/m2]	Min Heat Flux [W/m2]
A	1067 (#7)	210 (#2)
B	1128 (#7)	852 (#8)
C	1067 (#7)	867 (#3)
D	1055 (#6)	235 (#5)

.

Figure 5 shows the velocity distribution of the exhaust flow cross-section at the heat flux measurement points. As shown, the air velocity decreased in regions where the heat flux around the tanks decreased.

3.2. Analysis of the Nusselt Number Around the Tanks

In Section 3.1, the Type C tank configuration was shown to exhibit the smallest difference between the maximum and minimum heat fluxes. Therefore, for the Type C configuration, the heat transfer coefficients at each measurement point were calculated from the heat flux obtained from CFD simulation using Equation (4), and the Nusselt numbers were then determined. These are the CFD-derived Nusselt numbers. Additionally, the exhaust air velocity in the flow direction at a position 1 mm perpendicular to the tank surface from each measurement point was obtained through CFD simulation, and the Reynolds number was then calculated. Furthermore, based on Equation (7), shape factor $C$ and Reynolds number exponent $m$, which constitute the empirical correlation for the Nusselt number, were determined by performing a linear regression analysis. The values used in the analyses are listed in

Table 6. CFD-derived analytical values of the Nusselt number ($Nu$), exhaust air velocity in the flow direction around the tank ($u_x$), exhaust air temperature around the tank ($t_x$), and Reynolds number ($Re$) for Type C.

Analysis Point	$\mathit{N}\mathit{u}$	${\mathit{u}}_{\mathit{x}}$ [m/s]	${\mathit{t}}_{\mathit{x}}$ [°C]	$\mathit{R}\mathit{e}$
#1	27.4	1.23	29.1	1530
#2	32.7	1.48	30.3	1860
#3	27.3	1.24	28.9	1560
#4	31.1	1.30	30.5	1630
#5	33.6	1.58	31.8	1980
#6	32.2	1.46	30.5	1820
#7	33.6	1.53	31.5	1920
#8	30.9	1.33	29.6	1660

.

Based on the above results, the coefficients and exponents of the correlation-based Nusselt number were determined via linear regression analysis. During the analysis, it was found that the temperature difference in the exhaust heat around the tanks was approximately 2.9 °C. The corresponding difference in the Prandtl number was within 5.1 × 10⁻⁴, which has a negligible effect on the Nusselt number correlation.

The resulting values were $C=0.085$ and $m=0.79$, leading to the following empirical correlation expressed in terms of the shape factor and Reynolds number:

$$Nu=0.085·{Re}^{0.79}$$

(14)

The relative error between the Nusselt numbers estimated by Equation (14) and those obtained from the CFD results is shown in

Table 7. Comparison of CFD and estimated Nusselt numbers with relative errors.

Analysis Point	Nu (CFD)	Nu (Estimated)	Relative Error [%]
#1	27.4	28.1	2.4
#2	32.7	32.7	0.0
#3	27.3	28.4	4.2
#4	31.1	29.5	−5.2
#5	33.6	34.4	2.2
#6	32.2	32.2	0.1
#7	33.6	33.6	−0.2
#8	30.9	29.9	−3.2

.

The maximum relative error was −5.2%, indicating that the Nusselt number could be estimated with high accuracy from the Reynolds number around the tanks.

The heat transfer coefficients calculated using the correlation-based Nusselt number obtained from Equation (14) were compared with the analytically calculated heat transfer coefficients obtained from the CFD simulation, as shown in Figure 6.

The maximum difference between the analytically calculated heat transfer coefficient obtained from the CFD simulation and numerically calculated heat transfer coefficient derived from the empirical correlation was 1.46 W/m² K. The correlation coefficient was 0.93, thus demonstrating the validity of the proposed empirical correlation.

In addition, Figure 7 shows the variations in the heat transfer coefficient and the Nusselt number based on the empirical correlation when the inlet air velocity fluctuates by ±25% relative to the baseline value of 1.0 m/s, with the variation range determined from experimental measurements of the FC exhaust velocity.

The bars represent the range of variation from the estimated value at the reference inlet velocity of 1.0 m/s. When the velocity decreased by 25% to 0.75 m/s, both the heat transfer coefficient and Nusselt number decreased at all measurement points, with change rates ranging from 19% to 27%. Conversely, when the velocity increased by 25% to 1.25 m/s, both values increased at all measurement points, with change rates ranging from 9% to 23%.

3.3. Utilization Rate of Fuel Cell Exhaust Heat Transferred to the Tanks

The amount of heat transferred per unit time from the FC stack exhaust air to the cartridge was calculated using Equations (9) and (10), and the value was 137.1 J/s.

Additionally, the tank surface temperatures of Type C obtained from the CFD simulations are listed in

Table 8. Tank surface temperature in the case of Type C.

Analysis Point	Surface Temp. of Tank [℃]
#1	15.03
#2	15.05
#3	15.06
#4	15.08
#5	15.10
#6	15.11
#7	15.10
#8	15.07

.

Based on the temperatures and calculations in Equations (11) and (12), the total heat transfer rate to the four tanks inside the cartridge was 18.03 J/s. Accordingly, Equation (13) was applied, and it revealed that the tanks utilized 13.2% of the exhaust heat energy from the FC. In our previous study, the exhaust heat generated during 65 W power generation was supplied to the cartridge. The hydrogen flow rate released from the cartridge was 325 sccm, and the heat utilization efficiency was calculated to be 16.2%. Therefore, the analysis in this study captures the heat utilization efficiency, and the calculation results are considered reasonable. In addition, when the inlet velocity varied by ±25% from the reference value of 1.0 m/s, the resulting thermal utilization efficiency was 13.7% for an inlet velocity of 0.75 m/s and 12.3% for 1.25 m/s.

4. Conclusions

In this study, a cartridge was proposed that consists of four compact, adjacent MH tanks and cylindrical cover using the exhaust heat expelled from the cooling fan of a small-scale PEMFC for heat supply. To establish the design direction of the cartridge, four different tank arrangements, Types A to D, were proposed, with each featuring different minimum distances between the tank surfaces and the inner wall of the cover. Using CFD simulations, the heat flux at eight locations around the tank was analyzed to evaluate the uniformity of heat transfer. Among the configurations, Type C, with a minimum gap of 5 mm between the tanks and cover, exhibited the smallest difference between the maximum and minimum heat flux values.

Furthermore, for the Type C configuration, the Nusselt numbers at eight measurement points around the tanks were evaluated, and an empirical correlation was developed to accurately estimate the heat transfer coefficient based on the local exhaust air velocity distribution. This correlation enabled the estimation of the total heat transferred to the tanks from the surface temperature and local exhaust air velocity distribution, thereby enabling an assessment of the utilization rate of waste heat energy from the hot exhaust air of the FC flowing into the cartridge. The results showed that 13.2% of the heat carried by the exhaust air from the FC was effectively transferred to the interior of the tanks. Additionally, the sensitivity analysis of the exhaust air velocity flowing into the cartridge revealed that a decrease in velocity led to a reduction in the heat transfer coefficient, while an increase in velocity resulted in an enhancement of the heat transfer coefficient. On the other hand, the thermal utilization efficiency showed the opposite trend, with higher velocities leading to a decrease in utilization efficiency. These findings provide important insights into the optimal thermal design of cartridges and highlight the significance of effectively directing the FC’s exhaust air, with a particular emphasis on the importance of its velocity distribution.

On the other hand, a limitation of this study is that the inlet air velocity was assumed to be uniform, and the velocity distribution was not considered. It should be noted that when introducing exhaust heat from the PEMFC in practice, the velocity distribution is not necessarily uniform. Additionally, in the simulation, the cartridge cover was assumed to be perfectly insulated, and heat loss from the surface of the cover’s outside was not considered. Heat loss occurs through radiation and conduction from the outer surface of the cover, and therefore, the actual heat utilization rate per cartridge may be less than 13.2%. However, the cartridge is designed to allow flexible adjustment of the number of units installed by stacking them according to the specific demands of the application, such as the required travel range or output. In other words, the heat not utilized by the first cartridge can be effectively used by subsequent cartridges. Therefore, the value of thermal utilization efficiency should not necessarily be considered low. When multiple cartridges are used in a modular system, the overall thermal utilization efficiency is expected to improve. Moreover, although an increase in inlet air velocity tends to decrease the thermal utilization efficiency, it also suggests the potential for accommodating more stacked cartridges. Regarding the heat transfer phenomena, we will investigate the performance using the cartridge apparatus. Furthermore, the cartridge consists of multiple cylindrical tanks enclosed within a cover, resulting in a straightforward structure. This simplicity allows for more flexible integration compared to conventional MH tanks with complex heat transfer mechanisms, such as those proposed in references [35,37].

Finally, the simulation conducted in this study was based on a steady-state analysis, as the objective was to analyze the flow field of the exhaust heat. In addition, the operating condition of the FC remains constant in general. This means that load following is worse. In actual operation, however, the temperature distribution inside the tank changes transiently during the release of hydrogen. Therefore, future research will include an analysis of the transient heat transfer phenomena within the tank, considering the temporal variation in the internal temperature distribution. Furthermore, the effect of stacking cartridges on heat transfer in each layer will be investigated to support the comprehensive design of a multi-layer cartridge module.