Dynamic Modelling of a Metal Hydride Reactor During Discharge Through Artificial Neural Network Regression ^†

Abstract

With hydrogen as a clean but hazardous energy carrier, solid-state hydrogen storage in the form of a metal hydride has come forth as a safe and low-pressure storage solution with competitive volumetric energy density. This paper reports the modelling of a metal hydride reactor during its discharge state using neural network regression. This was done by generating a validated finite element model of the reactor, which was then used to generate dynamic operational data based on the desired pressure outlet and heating fluid temperature as independent variables. The best-performing neural network model validation using the experimentally observed data achieved a regression coefficient of 0.99 and a mean squared error of less than 10⁻⁴. This predictive model, with further refinement, can be implemented to allow for predictive control, which has always been a challenge through conventional means due to the batch nature of the system. Moreover, the hydrogen concentration as stored in a solid-state measurement would be too expensive for industrial applications.

1. Introduction

Solid-state hydrogen storage refers to the storage of hydrogen in a solid form, particularly through bulk absorption to form metal hydrides. These metal hydrides decompose during heating to release the stored hydrogen. This technology partially addresses the challenges associated with liquid and gas phase storage [1,2,3,4,5]. Furthermore, solid-state hydrogen storage has a very limited environmental impact, as reported by Costamagna et al. [6].

This can be seen in studies by Baldissin et al., Peng et al., and Chibani et al. [7,8,9]. The implementation of the finned heat sinks in a hydride reactor led to superior cooling compared to a reactor that did not have a heat transfer-aided design. Generally, the design of the metal hydride reactor is based on the heat transfer enhancement of the bed. As such, two common variants exist: first, with heat transfer from the outside, and second, with heat transfer from the inside [10,11,12].

A comprehensive literature review on metal hydride reactor design was published by Cui et al. [13], considering heat-sink and various internal cooling designs. Specifically, the internal configurations of radial tubes, helical tubes, and straight pipe heat exchangers were considered and compared. However, there is a clear lack of analysis on designs using both a cooling jacket and internal cooling. Similarly, there is a clear knowledge gap concerning the effects of scaling reactor diameter with internal cooling setups.

Additionally, the studies reviewed by Cui et al. [13] did not consider the trade-off of consuming some of the overall reactor volume with heat exchangers, lowering the effective bed volume and, ultimately, the capacity of the reactor. Lui et al. [14] did report the effects of the tube pitch of the internal straight pipe heat exchangers, albeit only for a singular pipe size, and there has not been a significant contribution to this area since.

Conventional control systems require a feedback approach that measures responses and adjusts accordingly. Alternatively, control systems require predictive control, which needs to be able to predict changes in the response parameters with changes in the input parameters. Solid-state hydrogen storage poses a unique challenge for control systems, being a complex batch system with changes being felt after a long delay. This is compounded by the cost of sensors that can measure hydrogen concentration in a solid state, which are too expensive to be economically feasible. Thus, the ability to design an effective control system around a metal hydride-based hydrogen storage system requires a predictive model to be referenced by the control system [15].

2. Method

The experimental equipment and experimental data necessary to validate this study were supplied by HySA-Systems under the umbrella of the University of the Western Cape [16]. Figure 1 presents a schematic diagram of the experimental setup filled with LaNi_4.9Sn_0.1 hydride-forming metal. For a larger version of Figure 1 with a full part number legend, as well as the COMSOL Multiphysics (Version 5.5) hydride bed geometry diagram and mesh diagram, see the Supplementary Materials supplied alongside this article. This unit is an industrial unit used to compress hydrogen, which is installed at a mine in the northwest. Thus, tests were performed under normal operational conditions so as not to disrupt the operation of the mine. For the discharging validation data, the desired gas pressure delivered by the unit was set at around 16 bar on the regulator. To achieve this, 135 °C to 145 °C steam was used.

For the finite element model, COMSOL Multiphysics was used with an ideal gas assumption. Considering the bed expansion, the assumption was made that 25% volumetric expansion is linearly proportional to concentration. To avoid large temperature difference computations in the first few minutes of the dynamic simulation, the initial temperature and pressure were set to the desired discharge pressure and heating fluid temperature. Finally, the gas in the bed transport was understood to follow Darcy’s law, and a porous heat transfer model was used. The flow of the hydrogen gas can be simplified via Darcy’s Law to account for the pressure drop over the porous bed, represented by Equations (1) and (2) [17].

$${\displaystyle \frac{\partial }{\partial t}}\left({\epsilon }_p\rho \right)+\nabla \left(\rho u\right)=Q_m$$

(1)

$$u=-{\displaystyle \frac{K}{\mu }}\nabla p$$

(2)

Likewise, the flow of heat in the porous bed can be simplified via a porous heat conduction law, which combines convective and conductive heat transfer, represented by Equations (3) and (4) [17].

$${\left(\rho C_p\right)}_{eff}{\displaystyle \frac{\partial }{\partial t}}+\rho C_{p}u·\nabla T+\nabla q=Q+Q_{vd}$$

(3)

$$q=-k_{eff}\nabla T$$

(4)

The rate of hydrogen sorption in the metal hydride (adapted from [8,18,19,20,21]) is represented by Equation (5).

$$\dot{m}=C_{D}exp\left(-{\displaystyle \frac{E_D}{RT}}\right){\displaystyle \frac{P_g-P_{eq}}{P_{eq}}}({\rho }_s-{\rho }_0)$$

(5)

To solve the equilibrium pressure, the computational methodology laid out by Faurie & Premlall [22] was implemented, along with the fitting parameters reported by Faurie & Premlall [22] using the script [23] reported by Faurie et al. [24]. The parameter tables of the simulation can be found in the Supplementary Materials to this article, as reported by Faurie et al. [25]. The boundary assumptions are that the ends are thermally isolated, and the reactor is heated through the inner and outer heat exchanger surfaces. Hydrogen is only fed through one end of the tank, and the other end is set as a dead end for hydrogen. The finite element simulated temperature contour plots can be seen in the Supplementary Materials to this article.

Figure 2 shows the performance of the model. This model was then used to generate training data by varying the operational input parameters and measuring the dynamic output. This data was then used to train an artificial neural network using the desired gas pressure, heating fluid temperature, and time as inputs and concentration as the variable the neural network would predict. These neural networks would vary in layer count and hidden neuron count using a ReLU activation and the Levenberg–Marquardt training algorithm. For this, both MathWorks MATLAB (R2025a) and TensorFlow (Version 2.19) were used with a 70–15–15 training–testing–validation split.

The trained models would then be re-validated against the experimental data, and performance would be analysed in terms of R-squared and mean-squared-error, which can be considered measures of the precision and accuracy of the model. Using simulated data to train the neural networks bypasses the need for extensive and expensive experimental trials.

3. Results and Discussion

Figure 3 shows the final mean squared error of the nine different neural network architectures when evaluated on the experimental data regarding the discharge state. This is a fully connected neural network with one to three hidden layers and 5, 10, or 20 neurons on each hidden layer. While the mean squared error of all nine neural network architectures lies in the same range, the lowest observed mean squared error architecture with the least level of complexity is the two-layer architecture, with ten hidden neurons on each hidden layer. It should, however, be noted that the single-layer architecture is a contender as well, only being beaten by the two-layer architecture at five neurons, then outperforming the two-layer architecture at ten and twenty neurons on each hidden layer.

Figure 4 represents the R-squared performance of the nine neural network architectures when tested on the experimental data regarding the discharge state. This R-squared performance is observed during the linear regression of model-predicted values and the experimentally observed values. During this application, this may be referred to as the adjusted R-squared statistic as it does not measure the fit of the model on the dynamic data but only considers the predicted and observed data. This R-squared statistic measures how closely the model-predicted and experimentally observed data reflect each other; thus, a value closer to 1 is desired. It can be observed that the three-layer architecture with twenty neurons on each hidden layer has overfitted the data, performing much worse than the other architectures in this analysis. The other architectures performed equally well in this analysis.

Figure 5 represents the linear regression of the model-predicted values against a sample of the experimentally observed data. Specifically, for the two hidden layers, ten neurons on each hidden-layer neural network architecture was determined to represent the best-performing neural network architecture for desorption. The R-squared in this instance was determined to be 0.99039. The two distinct curves that formed seem to denote the two different experimental trials, having differing degrees of accuracy for the whole of the dataset. Within the bounds of accuracy, this indicates that the model is not perfect.

4. Conclusions

The best-performing artificial neural network model achieved a regression coefficient of 0.9999 and a mean squared error of less than 10⁻⁵ during training. Likewise, the best-performing neural network model validation using the experimentally observed data achieved a regression coefficient of 0.99 and a mean squared error of less than 10⁻⁴. This proves that neural networks can model the complexity of metal hydride reactors during discharge, specifically the HySA-systems metal hydride reactor prototype.

This work serves as proof of concept that a predictive digital twin model can be generated for the discharge state of metal hydride reactors with further refinement. This refinement will come in the form of further computational fluid dynamics model validation, such as that of partial discharge capacity, and increasing the number of situation permutations that would be used to train the artificial neural network.