Design of metal hydride reactor with embedded helical coil using optimization methods

Hydrogen is a viable energy carrier option due to its high energy density and its clean combustion in environment offer. To store the hydrogen, solid-state hydrogen storage is considered among the best options due to the safety, cost and energy efficiency as compared to liquid or gaseous phases. In solid-state hydrogen storage, the hydrogen is absorbed and desorbed in metal hydride through exothermic and endothermic reactions. The difference in the supply pressure of hydrogen and equilibrium pressure of the metal hydride is the main driving force during absorption and desorption of hydrogen from metal hydride. The dependency of equilibrium pressure on temperature necessitates the removal and supply of heat quickly from metal hydride bed during absorption and desorption processes, which require efficient heat transfer system inside metal hydride bed to enhance the reaction kinetic [1].

In the last three decades, extensive numerical and experimental investigations have been conducted to explore the options to efficiently manage the heat transfer inside the metal hydride bed [2]. Gopal and Murthy [3] described the importance of higher thermal conductivity and small thickness of metal hydride bed to obtain faster kinetics. The improvement in the thermal conductivity of metal hydride was obtained using aluminum plates by Guo and Sung [4]. They also found that the thickness of metal hydride bed plays an important role to improve the heat transfer rate in the metal hydride bed. Further, Kim et al [5] introduces recompressed expanded graphite technique to improve thermal conductivity of metal hydride from 0.1 W m⁻¹-K⁻¹ to 6 W m⁻¹-K⁻¹. These expanded natural graphite show slightly higher reaction kinetics when compared with the system employed with aluminum foam by Sanchez et al [6]. Gopal and Murthy [7] further proposed to integrate water jacket around the metal hydride bed to improve the overall heat transfer coefficient. Kemal et al [8] in their numerical study found that the hydride formation was faster near the cooled boundary and slower at the core region and hence recommended the cooling fluid jacket around metal hydride bed. Muthukumar et al [9] also described the importance of overall heat transfer coefficient and found that the reaction kinetics increases with increase in overall heat transfer coefficient up to a certain limit (1250 W m⁻²-K⁻¹) and beyond that there is no effect of this parameter. Besides, water jackets some researchers use phase change material jacket around the metal hydride bed to store heat generated during absorption and utilizing it during desorption [10–12].

The heat transfer rate in metal hydride reactor was faster when fins are incorporated at the outer boundary of metal hydride bed as described by Mac Donald and Rowe [13]. Oi et al [14] uses internal fins to increase the heat transfer area inside the metal hydride bed and found reaction kinetics to be faster with this arrangement. Further, different arrangement of fins was studied by Garrison et al [15] and they found transverse fins to be more efficient than longitudinal fins. Chandra et al [16] numerically studied a metal hydride reactor and found that using conical fins improves heat transfer rate inside bed with funneling effect for better filling of material. Freni et al [17] introduce cooling tubes inside the metal hydride bed and reduces absorption time from 25 min to 15 min. Tiwari and Sharma [1] also studied effect of cooling tubes on reaction kinetics inside metal hydride bed and found that the reaction was faster with increase in number of tubes. Muthukumar et al [18] further observed that there exist an optimized number of tubes for a metal hydride bed diameter and beyond that increasing number of tubes did not affect the system performance. Mellouli et al [19] proposed spiral coil heat exchanger inside metal hydride bed and found that reaction rate increases when spiral coil heat exchanger was used. Futher, fins are attached on spiral coil by Dhou et al [20] and this arrangement was found better when compared with a system surrounded by water jacket. Later, Mellouli et al [21] further compared four design having single layer and double layer of spiral coil with and without fins and found out that double layer spiral coil with fins was best among all arrangements but this arrangement capture significant volume inside the bed. Souahlia et al [22] uses concentric tube with fins to reduce the occupied volume inside the bed. Raju et al [23] optimized the helical coil heat exchanger design inside the metal hydride reactor and found that helical coil heat exchanger is better compared to shell and tube arrangement heat exchanger. The smaller pitch of helical tube and higher heat transfer coefficient between heat transfer fluid and metal hydride are most suitable parameter for better rate of reaction as described by Wu et al [24]. Afzal and Sharma [25] also reduces the absorption time to 900 s by using multi-tubular shell and tube type storage system.

The LaNi₅ metal hydride is one of the most promising and easily available metal hydride and is utilized for various applications. Kang and Kuznetsov [26] carried out thermal modeling and analysis of a metal hydride chiller for air conditioning and used LaNi₅ as low plateau pressure material. The automobile air conditioner was also developed using the alloy of LaNi₅ by Qin et al [27]. Kim et al [28] developed metal hydride heat pump and used LaNi₅ metal hydride inside the reactor. Hopkins and Kin [29] developed dual stage metal hydride compression system using LaNi₅ and Ca_0.6Mm_0.4Ni₅ as working materials and achieve a compression ratio of 12. Further, a three stage metal hydride compressor was studied by Muthukumar et al [30] and achieve a pressure ratio of 28. Yang et al [31] numerically investigated metal hydride heat transformer using LaNi₅-LaNi_4.7Al_0.3 pair metal hydride and achieve a temperature boost of 6.8 K. Tao et al [32] proposed electrochemical compressor driven metal hydride heat pump using LaNi₅ as metal hydride. The LaNi₅ metal hydride is also used as low temperature metal hydride in thermal storage applications [33]. Malleswarao et al [33, 34] also studied metal hydride based thermal battery and thermal storage system using LaNi₅ as one of the metal hydride.

It can be observed from literature review that efforts have been made to improve the heat transfer rate inside metal hydride bed to further improve the reaction kinetics of the system. However, most of these study focuses on the system design based on the absorption process instead of desorption process. Further, most of the study focuses on the parametric studies to evaluate the effect of a single parameter on the system performance with no further guidelines to use them for system designs.

In this study an effort has been made to design a system embedded with helical tube for heat transfer improvement during desorption process. A stepwise procedure is described to design the MH reactor, which includes tank and helical tube design based on codal provisions, 3-D finite element mathematical modeling and optimization techniques. While the tank design is performed based on the ASME recommendations, the design of helical tube parameters i.e. diameter of helical tube (d_h), major diameter of helical tube (d_m), number of turns (N) and velocity of heat transfer fluid ( V _f ) is performed using Taguchi method and Grey relational optimization techniques. The system is designed to achieve a minimum desorption time (t_d) for 80% reaction completion and a minimum outlet temperature (T₀) of heat transfer fluid (HTF) coming out from helical tube.

The configuration of metal hydride based cooling system is shown in figure 1. In this system an embedded helical tube heat exchanger is utilized inside metal hydride bed as heat transfer arrangement. The HTF is passed through this helical tube during desorption of hydrogen from metal hydride bed. During desorption there is sudden drop in temperature of metal hydride bed. Hence, HTF when passed through the helical tube gets cooled by transferring its heat to metal hydride bed. This cooled HTF coming out from helical tube is then pass through the load where cooling is required. After desorption, the hydrogen is again filled inside the metal hydride bed using hydrogen cylinder. The HTF in this case is utilized to cool the metal hydride bed.

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This section explains the major components involved in the methodology and steps involved in the methodology for the design of MH reactor.

3.1. Tank design

The design of tank involves the evaluation of the shape, volume, thickness and material of tank. For the present study, metal hydride selection is based on the absorption capacity and working temperature range [21]. The volume of tank is calculated considering the porosity and expansion volume of metal hydride. The dimension of the tank is calculated by selecting a height to diameter ratio for efficient and fast heat transfer based on a detailed parametric study. The thickness of the tank for maximum working pressure and considering factor of safety (F.O.S.) is estimated on the basis of ASME code for pressure vessel. Finally, the material of tank is selected to uphold the working pressure and temperature based on the ASME code [25].

The helical tube heat exchanger arrangement was set up inside metal hydride bed to supply heat to it required during desorption process. To compensate the volume loss due to the presence of the helical coil type heat exchanger arrangement inside the tank, the length of the reactor is increased by maintaining the same diameter as [35]:

$$\begin{eqnarray}&&{V}_{ex}={V}_{h}\end{eqnarray} \tag{ 1 }$$

where, ${V}_{ex},\,{V}_{h}$ are the increased volume of reactor by extended length and volume of helical tube respectively.

The extended volume of reactor is given by:

$$\begin{eqnarray}&&{V}_{ex}=\displaystyle \frac{\pi }{4}\times {{d}_{r}}^{2}\times {l}_{ex}\end{eqnarray} \tag{ 2 }$$

where, ${d}_{r},{l}_{ex}$ are the diameter and extended length of reactor.

The volume of helical tube is calculated as

$$\begin{eqnarray}&&{V}_{h}=\displaystyle \frac{\pi }{4}\times {{d}_{h}}^{2}\times {L}_{a}\end{eqnarray} \tag{ 3 }$$

where, ${d}_{h},{L}_{a}$ are the diameter and actual length of helical tube.

The actual length of helical tube is defined as the length of the straight tube drawn from the helical tube. The actual length of helical tube is defined as:

$$\begin{eqnarray}&&{L}_{a}=N\left[\,\sqrt[2]{{\left(\pi \times {d}_{m}\right)}^{2}+{p}^{2}}\right]\,\end{eqnarray} \tag{ 4 }$$

where, N is the number of turns of helical tube, ${d}_{m}$ is the major diameter of the helical tube and $p$ is the pitch of helical tube. The pitch of helical tube is:

$$\begin{eqnarray}&&p=\displaystyle \frac{L}{N}\end{eqnarray} \tag{ 5 }$$

where, $L$ is the length of the reactor which is given as:

$$\begin{eqnarray}&&L=\left({l}_{s}+{l}_{ex}\right)\end{eqnarray} \tag{ 6 }$$

where, ${l}_{s}$ is the selected length of reactor without helical tube.

3.2. Optimization methods

Optimization techniques are used in the present methodology to obtain the geometrical parameters of helical tube (N, ${d}_{m},$ ${d}_{h}$) and velocity of HTF ( V _f ) based on the desired optimization functions. Two optimization methods i.e. Taguchi and Grey-Relational Analysis are considered to develop the most accurate and computationally efficient methodology for helical tube design.

3.2.1. Taguchi method

Taguchi method is an approach, which allows obtaining the optimized system from minimum number of experiments within the permissible limit of factors and levels. In this method, an objective function is decided first. Next important step is to select the control factors (parameters) and to assign multiple values to these control factors, also known as levels. Then the orthogonal array is constructed based on the number of control factors and their levels. The orthogonal array is constructed in such a way that for any pair of column, all combinations of factor levels are present for an equal number of times [36]. The standard orthogonal array (L₉) for four factors with three levels is given in table 1.

Table 1. Orthogonal array.

Exp. No.	A	B	C	D
1	1	1	1	1
2	1	2	2	2
3	1	3	3	3
4	2	1	2	3
5	2	2	3	1
6	2	3	1	2
7	3	1	3	2
8	3	2	1	3
9	3	3	2	1

In the next step, the objective functions are calculated by experiments or simulation based on the combination of factors level arranged in orthogonal array. The calculated objective function is then used to calculate the signal to noise ratio (SNR) value for each experiment with 'Smaller the better' case defined as [37, 38].

$$\begin{eqnarray}&&SNR\left({t}_{d}\right)=-10{\mathrm{log}}_{10}\left({t}_{d}^{2}\right)\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&SNR\left({T}_{o}\right)=-10{\mathrm{log}}_{10}\left({T}_{o}^{2}\right)\end{eqnarray} \tag{ 8 }$$

The mean SNR value for each control factor level known as performance characteristic (PS) is then calculated using these SNR value. The level of each control factor is then selected using these PS. The higher the value of PS the better is the result.

3.2.2. Grey relational analysis

The grey relational analysis is a multi-objective functional analysis, which gives an optimized level of parameters by considering all the objective functions simultaneously [39]. In this method, the procedure from the selection of objective functions to the orthogonal array construction is similar to the Taguchi Method. Then the experiments/simulations are performed to estimate the values of objective functions for each experiment with their respective levels of factors. The original sequence is then normalized for each characteristic using condition 'Smaller the better' defined as [40, 41]:

$$\begin{eqnarray}&&{X}_{i}^{* }\left(k\right)=\displaystyle \frac{\max \,{X}_{i}\left(K\right)-{X}_{i}\left(K\right)}{\max \,{X}_{i}\left(K\right)-\,\min \,{X}_{i}\left(K\right)}\end{eqnarray} \tag{ 9 }$$

where $\min \,{X}_{i}\left(K\right),$ $\max \,{X}_{i}\left(K\right)$ are the minimum and maximum value of particular characteristic in all experiments; ${X}_{i}\left(K\right)\,$is the value of characteristic for a particular experiment; K = 1 denotes first objective function sequence and K = 2 denotes second objective function sequence and i (1 to 9) denotes the experiment number.

After this normalization, the deviation sequence $\left({{\rm{\Delta }}}_{oi}\left(K\right)\right)$ is calculated as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{oi}\left(K\right)={X}_{0}^{* }\left(k\right)-{X}_{i}^{* }\left(k\right)\end{eqnarray} \tag{ 10 }$$

where, ${X}_{0}^{* }\left(k\right)$ is the reference sequence; ${X}_{i}^{* }\left(k\right)$ is the normalized sequence.

Then coefficient i.e. Grey Relational Coefficient expressing the relationship between the ideal and actual normalized experimental result is calculated as

$$\begin{eqnarray}&&{G}_{i}(K)=\displaystyle \frac{\min \,{{\rm{\Delta }}}_{oi}\left(K\right)+\zeta \,\max \,{{\rm{\Delta }}}_{oi}\left(K\right)}{{{\rm{\Delta }}}_{oi}\left(K\right)+\zeta \,\max \,{{\rm{\Delta }}}_{oi}\left(K\right)}\end{eqnarray} \tag{ 11 }$$

where, ζ is distinguishing and identification coefficient. Its value is 0.5 for equal preference of all the parameters [40]. The grey relational coefficient for each experiment in L₉ orthogonal array is calculated using equation (11).

Finally, the overall evaluation of the multi objective characteristics based on grey relational grade is calculated as

$$\begin{eqnarray}&&{\Upsilon }_{i}=\,\displaystyle \frac{1}{n}\displaystyle \sum _{1}^{n}{G}_{i}\left(K\right)\end{eqnarray} \tag{ 12 }$$

where, ${\Upsilon }_{i}$ is grey relational grade for i^th experiment and n is the number of performance characteristics. The higher value of grey relational grade represents that the corresponding experimental result is closer to the ideally normalized value.

3.3. Mathematical modelling

The objective functions for each combination of control factor and their level described in orthogonal array is determined using mathematical modeling. In this study all the simulations were carried out for desorption cycle with the assumptions mentioned in literature [16]. Mathematical modelling is carried out in Comsol 5.3 a with the governing differential equations given below.

• a)  
  Mass equation for the bed:

  $$\begin{eqnarray}&&\left(1-\varepsilon \right)\displaystyle \frac{\partial {\rho }_{s}}{\partial t}=\dot{{\rm{m}}}\end{eqnarray} \tag{ 13 }$$
• b)  
  Mass equation for hydrogen gas

  $$\begin{eqnarray}&&\varepsilon \displaystyle \frac{\partial {\rho }_{g}}{\partial t}+{\rm{\nabla }}.\left({\rho }_{g}{{\boldsymbol{V}}}_{{\boldsymbol{g}}}\right)=-\dot{{\rm{m}}}\end{eqnarray} \tag{ 14 }$$

  The ideal gas eqaution is valid for hydrogen gas and its density is defined as:

  $$\begin{eqnarray}&&{\rho }_{g=\displaystyle \frac{{P}_{g}{M}_{g}}{RT}}\end{eqnarray} \tag{ 15 }$$
• c)  
  Darcy's lawThe velocity of hydrogen gas is determined by Darcy's law:

  $$\begin{eqnarray}&&{\boldsymbol{Vg}}=-\displaystyle \frac{K}{{\mu }_{{\boldsymbol{g}}}}{\rm{\nabla }}{P}_{{\boldsymbol{g}}}\end{eqnarray} \tag{ 16 }$$
• d)  
  Energy equation for the bed

  $$\begin{eqnarray}&&{\left(\rho {C}_{p}\right)}_{e}\displaystyle \frac{\partial {T}_{s}}{\partial t}+{\rho }_{g}{C}_{pg}{{\boldsymbol{V}}}_{{\boldsymbol{g}}}.{\rm{\nabla }}{T}_{s}={\rm{\nabla }}.\left({K}_{eff}{\rm{\nabla }}{T}_{s}\right)+\displaystyle \frac{\dot{{\rm{m}}}{\rm{\Delta }}H}{{M}_{g}}\end{eqnarray} \tag{ 17 }$$

  The first term on the left hand side denotes the change of heat energy with time inside the reactor. The second term on the left hand side is the measure of advection by hydrogen gas through metal hydride bed. The velocity of hydrogen gas is negligible inside the bed and hence making second term negligible. However, to make the study comprehensive, this term was included in the study. The first term in the right hand side denotes the heat conduction inside the bed and the second term denotes the heat generation [16]. Effective thermal conductivity is defined as:

  $$\begin{eqnarray}&&{K}_{eff}=\varepsilon {K}_{g}+\left(1-\varepsilon \right){K}_{m}\end{eqnarray} \tag{ 18 }$$

  Effective volumetric heat capacity is expressed as follows:

  $$\begin{eqnarray}&&{\left(\rho {C}_{p}\right)}_{e}=\varepsilon {\left(\rho {C}_{p}\right)}_{g}+\left(1-\varepsilon \right){\left(\rho {C}_{p}\right)}_{m}\end{eqnarray} \tag{ 19 }$$
• e)  
  Reaction kinetics

  $$\begin{eqnarray}&&\dot{{\rm{m}}}={C}_{d}\exp \left(-\displaystyle \frac{{E}_{a}}{R{T}_{s}}\right)\left(\displaystyle \frac{{P}_{g}-{P}_{eq}}{{P}_{eq}}\right)\left({\rho }_{s}-{\rho }_{emp}\right)\end{eqnarray} \tag{ 20 }$$

  The equilibrium pressure of metal hydride is given by

  $$\begin{eqnarray}&&{P}_{eq}=\exp \left[\left(\displaystyle \frac{{\rm{\Delta }}S}{R}\right)-\left(\displaystyle \frac{{\rm{\Delta }}H}{R\,\square {T}_{s}}\right)\right]\end{eqnarray} \tag{ 21 }$$
• f)  
  Energy equation for embedded tube walls

  $$\begin{eqnarray}&&{\rho }_{w}{C}_{w}\displaystyle \frac{\partial {T}_{w}}{\partial t}={\rm{\nabla }}.\left({K}_{w}{\rm{\nabla }}{T}_{w}\right)\end{eqnarray} \tag{ 22 }$$
• g)  
  Momentum equation for the heat transfer fluidThe Continuity equation for heat transfer fluid is defned as:

  $$\begin{eqnarray}&&{\rm{\nabla }}.{V}_{f}=0\end{eqnarray} \tag{ 23 }$$

  The flow of fluid inside the helical tube is defined by Navier–Stokes equation as:

  $$\begin{eqnarray}&&{\rho }_{f}\left(\displaystyle \frac{\partial {V}_{f}}{\partial t}+{V}_{f}.{\rm{\nabla }}{V}_{f}\right)=-{\rm{\nabla }}p+{\rm{\nabla }}.\left(\mu \left({\rm{\nabla }}{V}_{f}+{\left({\rm{\nabla }}.{V}_{f}\right)}^{T}\right)-\displaystyle \frac{2}{3}\mu \left({\rm{\nabla }}.{V}_{f}\right)I\right)+F\end{eqnarray} \tag{ 24 }$$

  In this equation the first term on the left hand side is the inertial force term. The first term in the right hand side denotes the pressure force, the second term denotes the viscous force and the third term denotes the applied force.
• h)  
  Energy equation for the heat transfer fluid

  $$\begin{eqnarray}&&{\rho }_{f}{C}_{f}\displaystyle \frac{\partial {T}_{f}}{\partial t}+{\rho }_{f}{C}_{f}{{\boldsymbol{V}}}_{{\boldsymbol{f}}}.{\rm{\nabla }}{T}_{f}={\rm{\nabla }}.\left({K}_{f}{\rm{\nabla }}{T}_{f}\right)\end{eqnarray} \tag{ 25 }$$
• i)  
  Initial and boundary conditions

Initially (at t = 0), hydride density concentration are assumed to be constant as:

$$\begin{eqnarray}&&{\rho }_{s}\left(x,y,z,t=0\right)={\rho }_{sat}\end{eqnarray} \tag{ 26 }$$

The temperature of metal hydride bed, the HTF inside the tube and the supply temperature of HTF entering inside helical tube is assumed to be constant as:

$$\begin{eqnarray}&&T\left(x,y,z,t=0\right)={T}_{0i}={T}_{s}=303\,K\end{eqnarray} \tag{ 27 }$$

Pressure inside the metal hydride bed reaches the desorption pressure almost instantaneously as the process is started. Hence, it is considered that the desorption reaction started throughout the reactor at desorption pressure (P_d). All the simulations in this study are done at a desorption pressure of 1 mbar.

$$\begin{eqnarray}&&P\left(x,y,z,t=0\right)={P}_{0}={P}_{d}=1\,mbar\end{eqnarray} \tag{ 28 }$$

The velocity of heat transfer fluid at the inlet of helical tube is defined as:

$$\begin{eqnarray}&&u=v=w={U}_{in}\end{eqnarray} \tag{ 29 }$$

The no slip condition is assumed at the inner wall of helical tube:

$$\begin{eqnarray}&&u=v=w=0\end{eqnarray} \tag{ 30 }$$

The outlet boundary of heat transfer fluid is defined as:

$$\begin{eqnarray}&&{P}_{f}=0\end{eqnarray} \tag{ 31 }$$

The temperature condition at the outlet of heat transfer fluid is defined as:

$$\begin{eqnarray}&&{\rm{\nabla }}.\left({K}_{f}{\rm{\nabla }}{T}_{f}\right)=0\end{eqnarray} \tag{ 32 }$$

The temperature condition at the outer wall of helical tube:

$$\begin{eqnarray}&&{K}_{w}\displaystyle \frac{\partial {T}_{w}}{\partial {\boldsymbol{n}}}={h}_{ex}\left({T}_{w}-{T}_{s}\right)\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&{K}_{eff}\displaystyle \frac{\partial {T}_{s}}{\partial {\boldsymbol{n}}}=\,{h}_{ex}({T}_{s}-{T}_{w})\end{eqnarray} \tag{ 34 }$$

The boundary walls of the hydride bed vessel are assumed to be thermally insulated and hence the energy equations for these walls are:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {T}_{M{H}_{w}}}{\partial {\boldsymbol{n}}}=0\end{eqnarray} \tag{ 35 }$$

where, n is the unit normal vector out of the walls.

The amount of hydrogen desorbed is given by a dimensionless number and is described as:

$$\begin{eqnarray}&&C\left(1\right)=\displaystyle \frac{{\rho }_{sat}-{\rho }_{s}}{{\rho }_{sat}-{\rho }_{emp}}\left(for\,desorption\right)\end{eqnarray} \tag{ 36 }$$

The outlet temperature of HTF is described as:

$$\begin{eqnarray}&&{T}_{0}=\displaystyle \frac{1}{n}\displaystyle \sum _{t=0}^{10800}{T}_{f}\end{eqnarray} \tag{ 37 }$$

where, n is the total number of values of ${T}_{f}$ measured after every 10 s for a time period of 3 h at the outlet surface face of helical tube.

The Reynold's number inside the helical tube is defined as:

$$\begin{eqnarray}&&{R}_{e}=\,\displaystyle \frac{{\rho }_{f}\,\times \,{V}_{f}\times {d}_{h}}{{\mu }_{f}}\end{eqnarray} \tag{ 38 }$$

3.4. Steps involved for designing metal hydride reactor

This section explains the steps involved in the proposed methodology. Proposed methodology involves two major parts–Part 1 involves the design of metal hydride tank based on ASME code; Part 2 involves the design of helical tube heat exchanger embedded inside tank for efficient heat transfer. The optimization techniques (Taguchi and Grey relational) are used to design helical tube heat exchanger. Steps involved in the proposed methodology are shown in the flowchart in figure 2.

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Step 1. Tank design

The reactor is to be designed to store 2.5 kg of metal hydride having porosity of 0.5. The metal hydride used in this study is LaNi₅ due to its excellent hydrogen absorption capacity of 1.5 wt % and fast kinetics [21]. The volume of metal hydride reactor was estimated to be 0.00076 m³ considering porosity and expansion volume of metal hydride. Shape of the tank is chosen to be cylindrical due to its sufficient heat transfer capacity and ease of handling. The height to diameter ratio of tank is taken as 3:1 evaluated through a detailed parametric study for efficient and fast heat transfer. Based on this ratio, the height and diameter of the tank are estimated as 205.0034 mm and 69.342 mm respectively. The thickness of the reactor is calculated as 7.112 mm (≈0.3 inch) for a maximum working pressure of 60 bar considering F.O.S of 3 [25]. Based on the calculated parameters, a SS-316 L cylinder with Nominal Pipe Size (NPS) of 76.2 mm and height of 177.8 mm is selected with the schedule of 80 s.

Step 2. Design of heat exchanger

Helical tube is used as the heat exchanger in the present study due to its efficient heat transfer properties. Table 2 shows the considered control factors for helical tube determined using a detailed parametric study and corresponding levels of these control factors.

The inner diameter and the outer diameter of helical tube with nominal pipe size of 3.175 mm, 6.35 mm and 9.525 mm with schedule of 80 s used in the calculations and simulations are given in table 3.

These levels (table 2) are assigned to a standard orthogonal array L₉ (3⁴) by maintaining the orthogonality as shown in table 4. Further the extended and modified lengths of the reactor, pitch of the helical tube and Reynold number (Re) are also calculated for these designed experiments provided in orthogonal array as shown in table 4.

Step 3. Defining objective functions

It is important to mathematically define objective functions to optimize the system. In this study two objective functions are set:

• a)  
  To minimize the desorption time for 80% of hydrogen desorption i.e.

  $$\begin{eqnarray}&&Min.\,{t}_{d}\end{eqnarray} \tag{ 39 }$$
• b)  
  To minimize the average HTF temperature at the outlet of helical tube i.e.

$$\begin{eqnarray}&&Min.\,{T}_{0}\end{eqnarray} \tag{ 40 }$$

Step 4. Mathematical modeling

The mathematical model used in this study is validated by comparing the absorption and desorption rate with the experimental result of Laurencelle and Goyette [43]. In validation the metal hydride reactor having radius of 3.175 mm and height of 25.40 mm is considered. The porosity of metal hydride inside the reactor is assumed to be 0.5. The absorption is done at a supply pressure of 5.956 atm and desorption at an outlet pressure of 0.068 atm. The comparisons are shown in figure 3 and it can be seen that the maximum percentage difference in the results of numerical simulation and experiments is 2.4%, and hence this simulation model was used further for the detailed analysis.

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The next step is to done the grid independence test for the current mathematical model. The four different meshes having 3295 (extremely course), 16177 (coarser), 75662 (normal) elements. The reacted fraction (C₀) of the metal hydride bed was simulated using these meshes and it was observed that the reaction fraction curve using coarser and normal mesh overlap each other as shown in figure 4. Hence, Coarser mesh was used in further studies.

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After grid independency, the validated mathematical model is used to simulate each experiment of orthogonal array and the time corresponds to 80% of desorption (t_d) and the outlet temperature of HTF (T₀) were calculated for each experiment. The thermophysical parameters used in these simulations are given in table A1 of appendix A.1 . The Reynold number (Re) of all the orthogonal array experiments are included in table 4. The value of Re for all these experiments is found to be less than 2400, hence the laminar flow model was used to simulate the fluid flow inside the helical tube. The simulations were done for a time period of 10800 s using a time step of 10 s with coarser meshing on the domains. All the calculated parameters are given in table 5.

Step 5. Estimation of optimized helical tube parameters using Taguchi method

SNR values are estimated for the designed experiments using the estimated values of objective functions in table 5. Table 6 shows the estimated SNR values.

Then the performance static (PS) value of different levels of each parameter is estimated by evaluating the arithmetic average of SNR corresponding to each level. For example, the PS for A1 is calculated by averaging SNR values for Case 1 through Case 3, in which factor A is at level 1. Likewise, the PS for A2 is calculated by averaging SNR values for Case 4 through 6 and the PS for A3 is calculated by averaging SNR values for Case 7 through 9. The SNR response for desorption time (t_d) is given in table 6. The highest value of performance characteristic for a level corresponding to particular parameter is the optimum level for that parameter. It can be observed from table 7 that to achieve minimum time (t_d) for 80% of reaction completion, the Level 3 of all the parameters need to be selected i.e. A3B3C3D3.

Similarly, the performance characteristic for the outlet temperature is also calculated as shown in table 8. It can be observed that the minimum temperature (T₀) of HTF at the outlet of helical tube is obtained due to decrease in the volume flow rate of HTF when the level 1 of diameter of helical tube and velocity of HTF were selected. It was also observed that when the major diameter is 25.4 mm, in that case the outer part of metal hydride bed is away from helical tube and due to lower thermal conductivity of metal hydride the heat transfer is slow inside the bed and hence the cooling of HTF inside the helical tube is slow. Similarly, the inner portion is not heated properly when the major diameter of helical tube is selected as 50.8 mm. Hence, the minimum temperature of HTF is obtained when major diameter is 38.1 mm providing heating effect effectively at the outer and the inner part of metal hydride bed. Hence, the optimum level of parameters to achieve minimum outlet temperature of HTF is A1B2C1D1.

Step 6. Estimation of optimized helical tube parameters using Grey relational analysis.

Helical tube parameters simultaneously fulfilling the objectives of lower desorption time and outlet temperature of heat transfer fluid are estimated in this analysis. Initial steps up to the construction of orthogonal array are same as that of Taguchi method. Then the sequence series of objective functions were normalized and results are shown in table 9. Deviation sequence is then calculated and results are provided in table 9. After processing the deviation sequence, the grey relational coefficient (G_i(K)) were calculated and results are shown in table 9.

The two objective functions were then combined into a single objective function by calculating grey relational grade (${\Upsilon }_{i}$). The calculated grade for each experiment is given in table 10.

Table 10. Grey relational grade for each experiment.

Exp No.	A	B	C	D	${\Upsilon }_{i}$	Rank
1	3.175	25.4	8	0.05	0.684	4
2	3.175	38.1	9	0.1	0.620	7
3	3.175	50.8	10	0.15	0.703	3
4	6.35	25.4	9	0.15	0.417	9
5	6.35	38.1	10	0.05	0.723	2
6	6.35	50.8	8	0.1	0.646	6
7	9.525	25.4	10	0.1	0.445	8
8	9.525	38.1	8	0.15	0.667	5
9	9.525	50.8	9	0.05	0.742	1

The higher the value of grey relational grade, the closer the experiment to the best possible case [41]. In this study exp. no. 9 have highest grey relational grade and hence closest to the best optimized case. Now, to find out the optimal level of each factor, the mean of grade for particular factor at particular level was calculated. For example, the mean grade for factor A for level 1, 2 and 3 was calculated by averaging the grade value for experiment 1–3, 4–6, 7–9 respectively. Similarly, calculation is made for factors B, C and D and results are shown in table 11. The maximum value of mean grade for a parameter corresponds to the optimum level for that parameter. In this study, the optimum level of parameters corresponding to minimum desorption time and minimum outlet temperature of HTF is A1B3C1D1.

Step 7. Confirmation test

The efficacy of the optimized system (A3B3C3D3) was evaluated using mathematical modeling corresponding to the desorption time (t_d) with respect to the average performance of the nine experiments mentioned in orthogonal array. The average desorption time for 80% of hydrogen for nine experiments given in orthogonal array was 1923.33 s. As compared to this, the optimized system takes only 680 s i.e. 64.65% lesser than the average value. Similar test was conducted corresponding to the other objective function T₀. The average outlet temperature of HTF for the designed experiments coming out from helical tube was 298.76 K. The outlet temperature of HTF (T₀) for optimized system i.e. A1B2C1D1 was calculated to be 282.72 K i.e. 5.22% lesser than the average value. This confirmation test proves that the optimized systems perform well for the optimization function considered in the design.

The desorption time and outlet temperature for the system optimized using GRA i.e. A1B3C1D1 were 2000s and 287.77 K. Desorption time was 3.98% higher while the outlet temperature was 3.52% lesser than the average of the nine experiments. The results of all the experiments of orthogonal array along with the three optimized systems is shown in figure 5.

Step 8. Comparison of optimized systems

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A comparison of the performance of the three optimized systems for both the desorption time and outlet temperature is summarized in table 12. Among the considered optimization methodologies, it was observed that the optimized system ensures the superior performance in terms of the objective function considered while designing the systems, however it cannot necessarily ensure the superior performance corresponding to any other objective function. For example, it can be observed from table 12, for Taguchi based t_d optimization, performance of the designed system in terms of outlet temperature is inferior to the designed systems using other optimization methods and also to the mean of the experimental array. This limits the applicability of the single objective function-based optimization methods like Taguchi method to design a system requiring performance in terms of multiple objective functions. In these types of the problems, multi-objective optimization methods like grey relational method are significantly superior to the single-objective optimization methods like Taguchi method. It can be observed from table 12, designed system performance is satisfactory in terms of both t_d and T₀.

In this study, a metal hydride reactor having helical tube as heat exchanger arrangement was designed using optimization techniques. The optimization of helical tube parameters was done to obtain minimum desorption time (t_d) for 80% desorption and minimum outlet temperature (T₀) of HTF coming out from helical tube. Two optimization techniques i.e. Taguchi analysis and Grey relational analysis were used for the analysis. The following major conclusions are drawn from this study:

• A desorption time of as low as 680 s and temperature as low as 282.72 K was obtained when system was optimized individually for these objective functions using Taguchi method while the desorption time and the outlet temperature of HTF were found to be 2000 s and 287.77 K for the system designed using Grey-Relational Analysis considering both the objective functions simultaneously.
• While the described methodology is computationally-efficient and accurate, the methodology should be used with caution. It is found out that the optimized system ensures the superior performance in terms of the objective function considered, however it cannot necessarily ensure the superior performance corresponding for any other objective function as observed for this study.
• The systems optimized by these techniques are utilized for different applications such as—(1) T₀ optimized system can be used to provide continuous cooling, (2) t_d optimized system can be used for vehicular as well as stationary applications where faster desorption rate is required, (3) GRA based optimized system can be utilized for all applications described above simultaneously and can also be utilized as low temperature metal hydride tank in the two tank metal hydride reactor systems such as heat pump, heat transformer, thermal storage systems.

The authors are thankful to Prof. Suneet Singh, Department of Energy Science and Engineering, IIT Bombay for providing us the computational facility. Authors are thankful to the Department of Science and Technology, Government of India, for financially supporting the research vide project number DST/TMD/MECSP/2K17/14 i.e. DST-IIT Bombay Energy Storage Platform on Hydrogen.

All data that support the findings of this study are included within the article (and any supplementary files).

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Appendix. A.1 Thermophysical properties for simulations