## 1. Introduction

Despite their environmental effect and depletion, fossil fuels are still the main energy source for many applications. In power plants and the transportation sector, only 20–30% of the chemical energy of fossil fuels is converted to electric power. The remaining (~60–70%) is wasted to the environment. Likewise, in industrial processes, part of the energy is wasted through flue gases or effluents [

1,

2]. Therefore, waste heat recovery methods allowing for energy efficiency improvement have gained prominence in recent years. These methods include organic Rankine cycle (ORC), thermoelectric generators, and thermal energy storage systems (TES). However, ORC and thermoelectric generators convert waste heat to electricity with very low energy efficiency in the range ca. 2–15% [

3,

4,

5,

6]. Besides, these methods need to be installed near the waste heat source, which can make the overall installation (e.g., heat source + heat recovery systems) very costly [

1]. For this reason, TES have become more appealing since they allow for a synchronization of heat source and demand. Moreover, they can be installed close to the waste heat source (on-site) or be transported to the users far away from the source (off-site). TES can be divided into three types: sensible heat storage, latent heat storage, and thermochemical storage. The latter two types are the most utilized in waste heat recovery applications since they can store a large energy density. Yu et al. [

7], Magro et al. [

8] investigated the coupling of phase change materials (PCM) with ORC to smooth the power generation given the high heat source fluctuation. The results indicated that coupling the PCM with ORC increased the thermal efficiency from 15.5 to 16.4%, while the energy density dramatically increased from 32 to 52%. Gopal et al. [

9] conducted the energy and exergy analyses of PCM-based TES coupled with a diesel engine. The results indicated a significant improvement in energy and exergy up to 34.14 and 27.4%, respectively, alongside more than 6.13% of fuel-saving. Dispatch-able or mobile TES have been proposed to circumvent the long distance between the heat source (industrial sites) and end-users (cities) [

10]. These TES predominantly used PCMs to store waste heat. Moreover, a techno-economic analysis of dispatch-able TES showed that its overall cost depends mostly on the price of the PCM and the transportation distance from the waste heat source to the end-users [

11]. However, due to its low maturity level, only a few studies have discussed the possibility of Mobile-TES using adsorption materials or thermochemical materials such as zeolite, salt hydrates, etc. [

12]. Thermochemical heat storage systems have an energy density within 2–10 folds higher than that of latent heat storage, which makes them even more appealing for solar energy and waste heat recovery applications. Many reviews on the current state-of-the-art of these systems are reported [

13,

14,

15]. Regardless of the intended application, one of the widely accepted TES configurations is the use of coupled beds of solid (sorbent)-gas (sorbate) reactions, where the sorbate is used as a working fluid among beds [

14,

16,

17]. One bed is filled with a sorbent that works at high temperatures, which is used for heat storage. The other bed contains a sorbent that reacts with the sorbate and rejects heat at low temperatures. There are different classes of thermochemical (sorbents) materials that fit the scope. In the literature, a great number of these thermochemical materials deals with ammoniated [

16,

17,

18,

19,

20], hydrated salts [

21,

22,

23,

24,

25] and metal hydrides [

26,

27,

28,

29,

30,

31]. Furthermore, each type of material has its own advantages and shortcomings.

An earlier study [

21] reported a comparison between salt ammoniates and hydrates for cooling and refrigeration applications. Due to the low equilibrium temperature of NH

_{3} (−77.7 °C) at atmospheric pressure, ammoniated salts are utilized for freezing (refrigeration) applications, while salt hydrates are utilized for cooling and heating applications. Furthermore, salt ammoniates possess low energy storage densities ranging from 0.166 to 0.51 kWh

_{th}/kg and 144–304 kWh

_{th}/m

^{3} [

16,

17,

18,

19] as compared to those of some promising salt hydrates such as SrBr

_{2}.H

_{2}O, with 0.65 kWh

_{th}/kg and 250 kWh

_{th}/m

^{3} [

23]. However, salt hydrates have practical low energy storage density and efficiency. This is due to their melting and particles agglomeration [

25] during the heat charging/discharging, which limit water permeation thereby reducing their cycling stability. Another critical issue in the use of salt hydrates is their corrosiveness towards metallic containers. Fernandez et al. [

26] tested the corrosiveness of two type of salt hydrates, namely MgSO

_{4}.6H

_{2}O and SrBr

_{2}.6H

_{2}O, on different containers made of carbon steel, aluminum, copper and stainless steel. The results showed that containers made of carbon steel corrode at a fast rate (0.038 mm/year) compared to that of (<0.008 mm/year) other kind of containers.

Interestingly, metal hydrides offer even higher energy density compared to other thermochemical materials on the same metal basis, e.g., Mg-based hydrides have energy density in the range 0.52–0.8 kWh

_{th}/kg and 305–877 kWh

_{th}/m

^{3} [

27,

28,

29]. As a result, there is a renewed interest in using Mg-based hydrides for high-temperature energy storage applications.

In the metal hydrides-based TES presented so far, the low-temperature metal hydrides (LTMH) are limited to two types: AB

_{5} and AB

_{2} hydrides, for which their heat of reaction is within the range 20–30 kJ/mol-H

_{2}. If used in two-tank TES with Mg-based hydride, they generate or need a tremendous amount of energy, say up to 40% of the heat stored in the HTMH (Mg: 75 kJ/mol-H

_{2}). Therefore, the need for efficient thermal management is crucial to make such systems viable. To evade this problem of heat management, two solutions were proposed. The first is to replace the LTMH bed with a compressed H

_{2} bottle [

30]. However, energy is required for compression which also adds to the energy efficiency of the system. The other solution is to use passive heat management based on PCM [

31,

32,

33]. In this scenario, the PCM liquefies or solidifies during the heat charging and discharging process, respectively. PCMs have been proposed and successfully applied to the thermal management of single hydrogen storage tanks [

32,

33], and more recently to two-tank TES for waste heat recovery applications [

31]. It was shown that the thermo-physical properties of PCMs play a pivotal role in the performance of such systems. Two properties have stood out as the most crucial: the melting point and the thermal conductivity. Several authors suggested that the melting point

T_{m} should be located in the midpoint of H

_{2} absorption/desorption operating temperatures of the LTMH [

33]. On the other hand, PCMs have low thermal conductivities, e.g., 0.24 W/mK for organic-based PCM [

34] which is nearly five times lower than that of metal hydrides (e.g., 1.32 W/mK for LaNi

_{5}). As a result, this low thermal conductivity drastically affects the heat transfer between the LTMH and the PCM jacket. To increase the effective thermal conductivity of PCM, two methods are popularly proposed: the addition of nano-particles of carbon-based materials, or the insertion of metal foams and fins. Several results showed that the insertion of expanded graphite by 2.8 to 11.4 vol.% can improve the thermal conductivity by 170–190% [

35]. Even more interestingly, the addition of metallic foams drastically enhances the effective thermal conductivity of PCM by 393.4 up to 12,300% [

36]. Overall, the addition of these inert materials (graphite and metal foams) not solely improves the thermal conductivity, but also decreases the energy storage performance. For example, Ling et al. [

37] showed that adding 25 and 30 wt.% of graphite in a paraffin leads to a diminishing of its latent heat from 226 kJ/kg to 168.1 and 152.5 kJ/kg, respectively. Others issues deriving from the addition of graphite or metal foams may be the alteration of the melting point due to the increase in the heat capacity of the PCM [

35].

The analysis of previous works indicated that some aspects have not been taken into consideration and need to be addressed. First, every PCM shows a hysteresis behavior, meaning the melting point is different from the solidification/freezing point by 1 to 5 °C; see, for example,

Figure 1 in [

38]. This difference in temperature might affect the performance of the TES and will be explored in this study. Second, to what extent the addition of metal foams or carbon-based materials balances the improvement of PCM thermal conductivity and the decrease in energy storage capacity. Third, according to some studies, the melting point should be at the mid-range of the absorption/desorption temperatures [

32,

33]; however, no studies have pinpointed the PCM melting point that could correspond to the optimal performance behavior of the heat storage systems. Therefore, the objective of this work is to address the three issues mentioned above. To accomplish this objective, a 2D mathematical model studying the heat and mass transfer of the thermochemical heat storage system is formulated and analyzed in detail. Furthermore, a multi-objective optimization is performed to find the thermo-physical properties and size of the PCM that maximize the performance indicators of two-bed thermochemical heat storage systems using a PCM system to recover low temperature heat.

## 2. Mathematical Model

In this study, the thermochemical energy storage is comprised of a HTMH (Mg

_{2}Ni)/LTMH (LaNi

_{5}) pair for which the computational domain is presented in

Figure 1. To understand the working principles of this proposed heat storage system, the reader should refer to our previous work [

31]. Each metal hydride bed has the same thickness of 15 mm and the same length of 500 mm, with stainless steel (SS316L) walls of 2 mm thickness.

Assuming a porosity of 0.5, the weight of HTMH filled in the reactor is 0.791 kg, whereas that of LTMH is 2.078 kg. As a result, a maximum of 28.5 g of hydrogen can be exchanged between the beds. To scale up the TES, similar subsystems can be assembled to reach a desired energy storage density. The LTMH bed is enveloped with a PCM jacket to store/restore its heat of reaction. Furthermore, the jacket is fully insulated to limit the heat loss from the outer wall. The commercial-grade paraffin-based PCM has been chosen since it has several interesting attributes, such as low melting point (near room temperature, suitable for this application), low volumetric/thermal expansion (less than 10%) [

33], and high range of melting point with constant thermophysical properties, among others [

34,

35]. As paraffin-based PCMs possess low thermal conductivity (0.24 W/mK), in this study we use graphite powder in different proportions to augment its thermal conductivity. Details of the system design and material properties are listed in

Table 1.

#### 2.1. Governing Equations

The following simplifications are made for metal hydride reactors [

28,

29,

31] and PCM:

The thermo-physical properties of hydride are independent of temperature and concentration.

The thermal equilibrium between the gas and solid is established.

The radiative heat transfer is neglected.

The hysteresis in the equilibrium pressure is negligible for any material under study.

The thermo-physical properties (density, solid–liquid specific heat, thermal conductivity.) of the phase change materials are assumed constant.

The latent heat of phase change is temperature-independent.

The natural convection is disregarded since the system is laid horizontally. Therefore, gravity effect on the PCM is neglected.

The PCMs experience negligible or small (<5%) volume expansion. As a result, the density of PCM is constant in liquid and solid phase [

33].

Energy balance

The average temperature of the metal hydride bed is computed by as follows:

where the effective heat capacity and the thermal conductivity are calculated by assuming the volumetric phase mixing rule:

The energy balance of the reactor wall is expressed as follows:

The energy balance of the PCM reads:

The apparent heat capacity of the mixture is a linear function of the melting fraction of the phase change material, defined as follows:

To improve the thermal conductivity, PCMs are composited with inert materials such as fins, metal foams and graphite. In that case, the effective heat capacity and thermal conductivity of the composite read as follows:

where

${\phi}_{EG}$ is the volume fraction of expanded graphite/metal foam added to the pristine PCM.

Mass balance

where the gas speed in the porous bed is described by Darcy’s law:

However, in the connecting pipe, the hydrogen speed is governed by the transient Navier–Stokes momentum equation as follows:

The reaction kinetics of metal hydrides considered here, LTMH (LaNi

_{5}) and HTMH (Mg

_{2}Ni) adopt the first order kinetic model [

29,

39] as follows:

The equilibrium pressure is expressed as follows:

The liquid fraction of PCM can be modeled by a smoothed Heaviside function as [

40,

41]:

where

T_{on}, Δ

T_{tr}, and

T_{m} are the onset melting temperature (beginning of the mushy zone), the mushy zone interval and the melting temperature (which is the peak temperature of the melting profile of a PCM). The relation between these three parameters is as follows:

T_{m} =

T_{on} + Δ

T_{tr}/2 [

40].

Initial and boundary conditions

At t = 0, T_{0} = 20 °C (293 K), P_{0} = 1.96 bar, α_{HTMH} = 1, α_{LTMH} = 0, f = 0.

Two types of heat transfer continuity are considered:

Conduction/conduction heat continuity between two domains 1 and 2

Conduction/convection heat continuity at the reactor wall

The hydrogen flow continuity across the interface MH/connecting pipe

The adiabatic (axis-symmetry) boundary conditions are:

#### 2.2. Performance Indexes of the Heat Storage System

In this study, we will be interested in heat recovery performance. To assess this, three performance indexes are formulated as follows [

31]:

The volumetric energy storage density during the heat discharging:

With

${A}_{MH}=2\pi \left({r}_{MH}+\delta \right){L}_{MH}$, where

V_{T} is the total volume of the heat storage components accounting for the MH beds and the PCM jacket,

t_{d} is the heat discharging time,

The specific power during the heat-discharging step:

where

m_{HTMH} is the weight of the high temperature metal hydride.

The energy storage efficiency, which is the ratio between the useful heat output to the heat input:

#### 2.3. Optimization Procedure

From the parametric analysis [

31], it was shown that all the thermo-physical properties of the PCM affect the performance of the TES system. Therefore, it is judicious to apply an optimization exercise to find the ideal PCM for this specific application that maximize the performance indexes, simultaneously. Since in this study we have defined three of these indexes, maximizing each of these, leads to a multi-objective optimization. Therefore the problem formulation is given as follows:

Here, the design parameters are the thermo-physical parameters of the PCM, the PCM jacket size, the amount of expanded graphite,

${\phi}_{EG}$ to increase the thermal conductivity and the duration of heat charging and discharging,

t_{c/d}. Because of the slow melting/freezing kinetics compared to that of LaNi

_{5},

t_{c/d} has been included as optimization variable in order to improve the amount of H

_{2} exchanged between beds, during the heat charging/discharging process. The range of some of these optimization variables are listed in

Table 2 along with the main simulation parameters. In addition, the graphite volume fraction

${\phi}_{EG}$ is limited to 30% due to its restricted effect on the power output beyond a certain value, as will be shown in the following. On the other hand, the heat charging discharging is allowed to vary in 2 to 6 h range.

There are two main methods for solving multi-objective optimization: the weighted sum which combines the multiple objectives in one single objective or the evolutionary algorithm-based multi-objective optimization. The latter is generally popular to generate a set of optimal solutions (Pareto front). The problem defined above is solved by an evolutionary-based multi-objective optimization NSGA-II (Non-dominated Sorting Genetic Algorithm) algorithm developed by Deb et al. [

42] as implemented in MATLAB utilizing the optimization toolbox “gamultiobj”. Since the optimization tool receives entries from a finite element analysis software (COMSOL) by solving the governing equations, Equations (1)–(19), the population and generations sizes should be limited due to time constraints. In this study, the basic parameters for the optimization algorithm are as follows: a population size of 10 individuals and maximum generation size of 60 have be chosen; the crossover and mutation rates are 0.80 and 0.05, respectively. Finally, it was observed that the optimization exercise took about 189 h using on our work station: DELL XPS Intel i7–8700 hexacore (12 threads) CPU @3.20 GHz (University of the Western Cape, Cape Town, South Africa).