Thermodynamic analysis for hydriding-dehydriding cycle of metal hydride system

Journal Pre-proof Thermodynamic analysis for hydriding-dehydriding cycle of metal hydride system Jinsheng Xiao, Liang Tong, Pierre Bénard, Richard Chahine PII:

S0360-5442(19)32230-3

DOI:

https://doi.org/10.1016/j.energy.2019.116535

Reference:

EGY 116535

To appear in:

Energy

Received Date: 24 July 2019 Revised Date:

2 November 2019

Accepted Date: 10 November 2019

Please cite this article as: Xiao J, Tong L, Bénard P, Chahine R, Thermodynamic analysis for hydriding-dehydriding cycle of metal hydride system, Energy (2019), doi: https://doi.org/10.1016/ j.energy.2019.116535. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

Thermodynamic analysis for hydriding-dehydriding cycle of metal hydride system Jinsheng Xiao a, b, *, Liang Tong a, b, Pierre Bénard b, Richard Chahine b a

Hubei Key Laboratory of Advanced Technology for Automotive Components and Hubei

Collaborative Innovation Center for Automotive Components Technology, School of Automotive Engineering, Wuhan University of Technology, Hubei 430070, China b

Hydrogen Research Institute, Université du Québec à Trois-Rivières, QC G9A 5H7, Canada

Abstract: Thermodynamic analyses for a hydriding-dehydriding cycle process of a hydrogen storage system are carried out. Assuming the mass flow rate and the mass source term are both constant, the analytical solutions for lumped temperature in the metal hydride hydrogen storage tank are obtained in the absorption, dormancy and desorption processes. The analytical solution is important to understand the hydriding-dehydriding process, which can be used as a benchmark to validate lumped/distributed parameter models for further study. A lumped parameter numerical model is developed on the Matlab/Simulink software platform. The numerical solutions of this model with constant source term coincide with the analytical solutions. Different analytical solutions are solved with various combinations of the hydrogen inflow/outflow temperature. The analytical solution in present work is compared with the reduced model in the reference, and shows more accurate than the reduced model. The variable source term is applied to the validated lumped parameter model. The results of parametric study show that the source term of metal hydride is roughly constant when the hydrogen storage capacity is not beyond about 90% of its limited capacity. Keywords: hydrogen storage; metal hydride; hydriding-dehydriding cycle; thermodynamics; lumped parameter; analytical solution *Corresponding author. Hydrogen Research Institute, Université du Québec à Trois-Rivières, QC G9A 5H7, Canada. Tel.: +1 819 376 5011x4478; fax: +1 819 376 5164. E-mails: [email protected] (J.Xiao), [email protected] (L.Tong), [email protected] (P.Bénard), [email protected] (R.Chahine).

1. Introduction Hydrogen can be considered as a fuel and an energy storage medium, which can play an important role in the future smart energy system [1]. Effective hydrogen storage is good for applying hydrogen energy widely, such as fuel cell vehicles. In general, there are three methods used for hydrogen storage, including compression, liquefaction and solid state hydrogen storages. The solid-state hydrogen storage includes chemical absorption and physical adsorption [2]. The atomic hydrogen can be absorbed into metal hydrides in the chemical absorption process. Metal hydride can be used in various fields, such as hydrogen storage [3, 4], thermal heat storage [5] and waste heat recovery [6]. It is an effective way to store hydrogen using metal hydrides due to low operation pressure and safety characteristics. Considering the low heat conductivity of metal hydride bed, a large amount of reaction heat in the hydriding and dehydriding processes has a significant influence on the performance of the hydrogen storage system. The structure of heat exchanger in the metal hydride system must be carefully designed. Many researchers have studied the mass and heat transfer phenomenon of metal hydride systems based on the distributed parameter models, including one-dimensional model [7-10], two-dimensional model [11-17] and three-dimensional model [18-21]. Various heat exchangers have been applied into the metal hydride system in order to improve the system performance, such as metal foam [7, 8], finned tube heat exchanger [16], metallic honeycomb structure heat exchanger [17] and phase change materials [21]. The lumped parameter model can be used to research the metal hydride system as a whole [22-28]. Besides the above numerical researches, the thermodynamic analysis is considered as a common method in various fields [29, 30]. In general, a mathematical model can be a lumped parameter model (0-dimensional in space, expressed as an ordinary differential equation on time) or a distributed parameter model (1-, 2- or 3-dimensional in space, expressed as a partial differential equation on time and space), and they can be solved numerically and sometimes analytically. Among them, the lumped parameter model has more possibility to be solved analytically. A numerical model is the mathematical model combining with numerical methods and algorithms, such as the Euler method or the Runge-Kutta method for ordinary differential equation (ODE) model, finite difference method or finite element method for partial differential equation (PDE) model. A numerical model, either a lumped parameter model or a distributed parameter model, can be validated by analytical solutions under the simplified boundary conditions and material properties. Considering an example of heat conduction of a metal hydride reactor, a distributed parameter model will reduce 2

to a lumped parameter model when a uniform boundary condition and an infinity thermal conductivity applied to the reactor. From the view of engineering, the numerical model should be further validated by experiments. The analytical solutions are solved from the thermodynamic model based on the principles of mass conservation and energy conservation under certain assumptions of conditions. Although these analytical solutions are applicable for ideal or theoretical problems only and not for real engineering problems, they are exactly correct under the assumed conditions and they can be used to validate the numerical methods and algorithms by comparing the numerical solutions of the numerical models for the same problems with same conditions. So the analytical solutions are not very necessary to be validated by experiments. Also, they are not so easy to be validated by experiments because the ideal conditions used for solving procedure are difficult to implement in the experiments. We have developed some validated lumped parameter models to improve the performances of hydrogen storage and purification systems using metal hydride [31]. In addition, some thermodynamics models and analytical solutions for the compressed and adsorptive hydrogen storage systems were studied in our previous works [32, 33]. The metal hydride hydrogen storage system is based on the hydriding-dehydriding chemical reaction, which is more complicated than the compressed and adsorptive hydrogen storage systems. The analytical solutions for the metal hydride hydrogen storage system are important to understand the hydriding-dehydriding process and the basic theory. The analytical solutions can be considered as a benchmark to validate the lumped parameter or distributed parameter numerical model, which is applied to further research the metal hydride hydrogen storage system from the perspective of time and space. In this work, based on the mass and energy balance equations, the thermodynamic analyses for the hydriding-dehydriding cycle of metal hydride hydrogen storage system are carried out for various processes, including hydriding, dormancy and dehydriding processes. The analytical solutions for the metal hydride hydrogen storage system are obtained from the thermodynamic model. The constant/variable inflow/outflow temperature of hydrogen is taken into consideration in the analytical solutions during hydriding and dehydriding. Then, the analytical solutions, in which the inflow temperature is assumed as constant while the outflow temperature is variable, are applied to validate the numerical model. A lumped parameter model is developed based on the mass and energy balance equations, the reaction kinetics equation, the equilibrium pressure equation and the equation of state. The analytical solutions are used as a reference to validate the lumped parameter numerical model with constant mass source term. Then, the validated numerical model with variable mass source term is studied, and the numerical results of the model with variable mass source term are compared with 3

those with constant mass source term. The effect of mass flow rate on the metal hydride hydrogen storage system is researched using the lumped parameter numerical model with variable mass source term. The limitedness of the analytical solution is also discussed.

2. Thermodynamic model of metal hydride system The thermodynamic model for the metal hydride hydrogen storage system includes the mass conservation equation, the reaction kinetics equation, the equilibrium pressure equation, the energy conservation equation, and the equation of state for the ideal gas, as shown below. The source term of metal hydride in the mass conservation equations are closely related to the reaction kinetics equation and the equilibrium pressure equation. 2.1. Mass balance The alloy LaNi is selected as the material for hydrogen storage in this work. The hydriding-dehydriding reaction equation can be written as LaNi + 3H ↔ LaNi H + Q. There are generally gaseous hydrogen and solid materials in the metal hydride hydrogen storage reactor. The solid materials include the alloy LaNi and the metal hydride LaNi H in this reactor, as presented in

Eq.(1). The mass of alloy  decreases and the mass of metal hydride  increases during

hydriding. In contrast, the alloy mass  increases and the metal hydride mass  decreases

during dehydriding. As a whole the solid mass  increases during hydriding and decreases during

dehydriding.

 =  + 

(1)

The saturated mass of solid materials is defined as  . When the whole alloy is completely reacted  with hydrogen, the metal hydride mass reaches its maximal value  .

  = 0 + 

(2)

The solid mass also can be written as the sum of the initial mass  +   and the hydrogen

mass absorbed into the metal hydride  , as described in Eq.(3).  and  mean the initial alloy

mass and the initial adsorbed hydrogen mass, respectively. When there is only alloy in the solid

materials at the initial state, the mass of adsorbed hydrogen  can be set as 0. Of course, the initial

mass  +   can be considered to be constant, and the mass of absorbed hydrogen  is variable, which increases during hydriding and decreases during dehydriding.  =  +  + 

(3)

4

The saturated mass of solid materials also can be presented as the sum of the initial alloy mass   and the maximal value of absorbed hydrogen mass  , shown as following:

  =  + 

(4)

The relationship between the mass of absorbed hydrogen  and the mass of metal hydride  is presented as:

 =  ψ ⁄

(5)

where the alphabet ψ is defined as the stoichiometric coefficient of hydrogen per metal hydride.  and  are the molecular weight of metal hydride and hydrogen, respectively.

A general form of mass balance equation for a lumped parameter model can be expressed as: 

!

= " + #"

(6)

where " is the net mass flow rate and #" is the net mass generation rate or the mass source term. The mass balance equation can be used for metal hydride, gaseous hydrogen and solid materials in the metal hydride hydrogen storage system.  $% !  % !  & !

" = #

(7)

" = " + # 

(8)

" = −# 

(9)

" is a positive quantity during absorption and negative The mass source term of metal hydride #

" is a quantity during desorption. As for the gaseous hydrogen in the reactor, its mass source term #  negative quantity during absorption and is a positive quantity during desorption. The hydrogen mass flow rate " is equal to the difference between the mass inflow rate "() and the mass outflow

rate "*+ , and the subscript in/out refers to the inflow/outflow stage. We assume that "*+ = 0 during hydriding and " () = 0 during dehydriding.

" and the gaseous hydrogen The relational expression between the metal hydride mass source term #

" can be written as # " = − # "  .,ψ -. The mass source term of the mass source term #   

metal hydride is related to the reaction kinetics equation of metal hydride and the equilibrium pressure equation. 2.2. Reaction kinetics and equilibrium pressure During absorption and desorption, the mass source terms of metal hydride can be written as: " = /0 e234 ⁄56 ln,9 ⁄9:;0 -  −  , for 9 > 9:;0 #   5

(10a)

" = /= e23> ⁄56 ?,9 − 9:;= -.9:;= @ , for 9 < 9:;= #  

(10b)

 . E means the activated where  is the saturated mass of solid materials and it is equal to 

energy, and the subscript a/d refers to the absorption/desorption stage. The hydrogen pressure 9 is above the equilibrium pressure 9:;0 during hydriding and below the equilibrium pressure 9:;=

" is set as 0 when 9 is between 9:;0 during dehydriding. The mass source term of metal hydride #

" is a positive and 9:;= . Based on Eqs. (10a) and (10b), the mass source term of metal hydride #

" is a negative quantity and the metal hydride mass  increases during hydriding, whereas #

quantity and  decreases during dehydriding. The equilibrium pressure 9:;0/= is presented as: DEF4/>

ln C

DG

H=−

∆J4⁄> 56

+

∆K4⁄> 5

+ LD C

$% &MN &

O

− H

(11)



where the reference pressure 9 is set as 0.1 MPa. The reaction enthalpy ∆P and the reaction entropy ∆# are set as positive quantities in this work. The symbol LD is the plateau slope coefficient.

2.3. Energy balance The heat exchangers can be used to improve the heat transfer efficiency of the metal hydride reactor. For simplification, the metal hydride reactor without a heat exchanger is studied in this section. The wall of the metal hydride reactor is ignored. The energy balance equation of the metal hydride hydrogen storage system is shown as: , % QR 6S & Q& 6!

= " TD UV + WX Y ,UX − U- −

K"% ∆J4/> Z %

(12)

where  means the mass of gaseous hydrogen in metal hydride reactor, T is the specific heats of

solid material; UV is the inflow or outflow temperature; WX is the heat transfer coefficient; Y is the

surface area of the reactor; the ambient temperature UX is set as the cooling/heating water temperature. The left term of Eq. (12) means the rate of internal energy of the metal hydride reactor. Based on Eqs. " -− (8) and (9), the left term can be transformed as , T[ +  T - dU⁄d] + T[ U," + # 

" . The effects of the enthalpy change caused by the inflow/outflow hydrogen, the heat transfer T U#  between the ambient environment and the metal hydride reactor, and the reaction heat on the metal hydride hydrogen storage system are taken into consideration in the right terms of Eq. (12). The heat

" ∆P0/= ψ ⁄ or − # " ∆P0/= . , and it is positive during source term is written as #    absorption and negative during desorption. The energy conservation equation of this system is transformed as: 6

, T[ +  T -

6 !

" [T[ − T U + ∆P0/= ⁄ ] (13) = " ,TD UV − T^ U- + WX Y ,UX − U- − #  

2.4. Equation of state for hydrogen gas Gaseous hydrogen in the metal hydride reactor is considered as the ideal gas:  = 9 ab ⁄cU

(14)

where 9 is the hydrogen pressure; ab is the volume for the gas phase in the metal hydride reactor; R is the universal gas constant, it is set as 8.314 J/mol/K.

3. Analytical solutions of the thermodynamic model The analytical solutions can be solved from the above thermodynamic model for the metal hydride hydrogen storage system. The mass flow rate can be set as constant through a mass flow controller in the experiment. When the mass flow rate is fixed, the reaction rate of metal hydride is almost constant [26]. Therefore, we can assume that the associated mass source terms in the mass balance equations of the metal hydride and hydrogen are constant. The variable reaction rate has been considered into the lumped parameter model in the following section, whose value is proved to be approximately constant. In this section, the reaction rate is assumed as constant in order to solve the analytical solutions for the lumped temperature of metal hydride reactor. The mass source term is set as constant. The analytical solutions are discussed during hydriding, dormancy and dehydriding processes in various conditions, where the inflow/outflow temperature UV is considered to be constant or variable in the analytical solutions. 3.1. Constant inflow/outflow temperature during hydriding/dehydriding processes The inflow/outflow temperature UV is assumed to be constant. Considering the mass flow rate and the

mass source term are set as constants, Eqs. (8) and (9) can be transformed as  =  + " -] and  =  +  − # " ], respectively. So Eq. (13) becomes: ," + #  

" T[ − # " T -]@ ? T[ +  T +  T + ," T[ + #  

" TD UV + WX Y UX −

K"% ∆J4/> Z %

dU d]

=

" T[ − # " T − U,WX Y + " T[ + #  

" T[ − # " T , Eq. (15) is transformed as: Divided by " T[ + #    T[ +  T +  T dU d + ]e = " T[ − # " T d] " T[ + #  

7

(15)

" % Qf 6g S0hi& 6h 2

j"% ∆k4/>  26,0h i&S l% 

" QR 2K% " Q& " % QR SK%  

16

" QR 2K% " Q& " % QR SK%  

" T[ − # " T -, The characteristic time ] ∗ is assumed as , T[ +  T +  T -.," T[ + #  

" ." , α = WX Y ⁄," T[ - and U0/= = − o∆P0/= ⁄, T[ - . Extracting n = TD ⁄T[ ， o = #     " T[ in the right term of Eq. (16), the energy equation becomes:

] ∗ + ]

6 !

= [nUV + αUX + U0/= − U Cα + 1 + o −

Defining U ∗ = ,nUV + αUX + U0/= -.Cα + 1 + o −

rQ& QR

rQ& QR

" T[ − # " T H]" T[ .," T[ + #  

H , reducing " T[ and 1 + o −

rQ& QR

(17)

in the

right term of Eq. (17), then the above equation is simplified as: ] ∗ + ]

6 !

= d1 +

s

tu& uR

OSr2

e U ∗ − U

(18)

Solving Eq. (18), using v = ]⁄] ∗ , we obtain 6 ∗ 26

6 ∗ 26G

O

x oT yzt{ s uR

OS

= COSwH

(19)

or U = U ∗ − U ∗ − U ⁄1 + v

oTs

}OSs.COSr2

uR

H~

(20)

3.2. Variable inflow/outflow temperature during hydriding/dehydriding processes The inflow/outflow temperature UV is set as a variable in this section, so the energy balance equation is presented as: " T[ − # " T -]@ ? T[ +  T +  T + ," T[ + #  

dU = d] "

" T[ + # " T − WX Y -U + WX Y UX − K% ∆J4/> ," TD − " T[ − #   Z " T[ − # " T , then Eq. (21) becomes: Divided by " T[ + #   d

 T[ +  T +  T dU + ]e = " " d] " T[ + # T[ − # T 





" QR SK% " Q& 20hi& -6S0h i& 6h2 , " % Qf 2 " % QR 2K%   " QR 2K% " Q& " % QR SK%  

j"% ∆k4/>  l% 

%

(21)

(22)

The characteristic time ] ∗ , the dimensionless heat transfer coefficient , the absorption/desorption

contributed temperature U0/= , the specific heat ratio n and the ratio of hydrogen mass source term to 8

mass flow rate o are defined in the former section. Extracting " T[ in the right term of Eq. (22), then the above equation becomes: ] ∗ + ]

6 !

" T[ − # " T = [n − 1 − o + oT ⁄T[ − αU + αUX + U0/= ]" T[ ⁄," T[ + #  

(23) Reducing " T[ and 1 + o − ] ∗ + ]

6 !

= d1 +

s2€

tu& uR

OSr2

rQ& QR

, Eq. (23) is simplified as:

e U ∗ − U

(24)

where the characteristic temperature U ∗ = ,UX + U0/= -.Cα − n + 1 + o −

rQ& QR

H. Defining v = ]⁄] ∗ ,

the solution of Eq. (24) is obtained as 6 ∗ 26

6 ∗ 26G

=C

O

OSw

tu OSs2€.COSr2 u & H

H

R

or U = U ∗ − U ∗ − U ⁄1 + v

(25) tu& H] uR

[OSs2€.CrSO2

(26)

3.3. Temperature during dormancy process " and # " , are neglected The mass flow rate " is set as 0, and the source terms, including #  during dormancy. The energy balance equation, which only takes the heat transfer between metal hydride reactor and ambient environment into consideration, is written as: , T[ +  T +  T - dU⁄d] = WX Y ,UX − U-

(27)

We assume that the characteristic time ] ∗ = , T[ +  T +  T -.,WX Y - , and the characteristic temperature U ∗ =UX . Then Eq. (27) becomes,

dU⁄d] = U ∗ − U⁄] ∗

(28)

U = U ∗ − U ∗ − U exp −v

(29)

Solving Eq. (28), using v = ]⁄] ∗ , we get:

4. Application of analytical solutions The thermodynamic model for the metal hydride hydrogen storage system consists of the mass conservation equation, the reaction kinetics equation, the equilibrium pressure equation, the energy conservation equation, and the equation of state for the ideal gas. The analytical solution is solved from the thermodynamic model, its results are calculated by the compiled program on the Matlab software platform. The lumped parameter numerical model is developed on the Matlab/Simulink software platform. The variable-step type and the ode 45 (Dormand-Prince) solver are used in the 9

simulation process. The max step size is 0.1 and the relative tolerance is 1e-3. The lumped parameter model can call the program written for the analytical solution to compare their results directly. The analytical solutions can be considered as a reference to validate the numerical model. In this section, the numerical solutions of the lumped parameter model with constant mass source term are compared with the analytical solutions. Various analytical solutions using different inflow/outflow temperature are discussed. The analytical solution in the present work is compared with the reduced model proposed in the reference. 4.1. Description of hydrogen storage system Based on a metal hydride reactor compacted with LaNi5, the above analytical solutions are used to validate a numerical model. The numerical model is the lumped parameter thermodynamic model or the zero-dimensional model, which is developed on the Matlab/Simulink platform. There are three stages in a hydriding-dehydriding cycle, including absorption, dormancy and desorption processes. The metal hydride reactor is cooled by circulating water during absorption, and heated in the dormancy and desorption stages. The lumped temperature of the metal hydride reactor during dormancy increases since the reactor is heated by the circulating water, and a relatively high temperature in metal hydride reactor is good for the dehydriding reaction. Talaganis et al. have simulated absorption-desorption cyclic processes and presented detailed parameters in their cases. As presented in the previous section, the solid materials in the reactor generally include the alloy and the metal hydride. At the beginning of our case, the mass of metal hydride is 27.375kg, the mass of alloy is 82.3kg. When the whole of the metal hydride decomposes as the alloy and the gaseous hydrogen, the total mass of alloy is 109.3kg. The parameters used in our model are obtained from Ref. [26], as shown in Table 2. 4.2. Validation of numerical model by analytical solution A lumped parameter model can be solved numerically and sometimes analytically. The analytical solution can be considered as a simple benchmark to validate the lumped parameter numerical model. " in each stage is set as constant in the For simplification, the hydrogen mass source term #  numerical model and the analytical solution. The mass flow rate used in the present work refers to that in Case 2 from Ref. [26] in order to maintain a hydrogen production of 2kgH2/h. The time of an absorption-desorption cycle in Case 2 [26] is 3600s. The hydrogen is produced during desorption (about 720s), and the mass outflow rate is about 1.389g/s in the dehydriding process. The absolute values of the mass flow rate and the hydrogen mass source term are both assumed as 1.389 g/s during 10

hydriding and dehydriding. The ratio of the hydrogen mass source term to the mass flow rate is assumed as −1. The hydriding/dehydriding reaction is ignored during dormancy. The mass flow rate is a positive quantity and the hydrogen mass source term is a negative quantity during hydriding. On the contrary, the mass flow rate is a negative quantity and the hydrogen mass source term is a positive quantity during dehydriding. The inflow temperature is assumed as constant while the outflow temperature is variable. Fig.1 shows the comparisons between the Matlab/Simulink numerical solutions and the analytical solutions, including the variations of the lumped temperature, the pressure, the mass source term of metal hydride, the mass of metal hydride, the equilibrium pressure and the mass of gaseous hydrogen, in order to validate the lumped parameter numerical model. In general, the numerical solutions of the lumped parameter model with constant mass source term agree well with the analytical solutions. The lumped temperature of the metal hydride reactor increases during hydriding and decreases during dehydriding. The temperature increases during dormancy since the reactor is heated, which is good for the dehydriding reaction. The metal hydride mass increases during hydriding and decreases during dehydriding. The sum of the mass flow rate and the hydrogen mass source is 0, and the mass of gaseous hydrogen in the metal hydride reactor is a fixed value. The variation curve of pressure is as same as that of lumped temperature, which is according to the equation of state in the assumed conditions. The equilibrium pressure is presented in Fig.1e, which is calculated by Eq. (11). The equilibrium pressure is related to the lumped temperature and the mass of metal hydride, and the effect of the lumped temperature on the equilibrium pressure is more obvious than that of the mass of metal hydride. The equilibrium pressure shows the same trend as the lumped temperature. 4.3. Comparison among different analytical solutions The analytical solutions using different inflow/outflow temperature are discussed. Then, the deduced model proposed in Ref. [26] is compared with the above analytical solutions in this section. The mass flow rate in the analytical solutions is assumed to be constant, the case using a variable mass flow rate is not considered in this work. Any of hydrogen inflow and outflow temperatures can be assumed as constant or variable, which results in analytical solutions under four different inflow/outflow conditions. The detailed information on the combinations is presented in Table 3. Fig.2a shows the comparison of lumped temperature among the analytical solutions with different inflow/outflow temperature. When the inflow temperature is set as constant, the lumped temperature is about 0.7K lower than that of others during hydriding. The lumped temperature in Case 2 and Case 3, where the outflow temperature is set as constant, is about 0.7K higher than the others during 11

dehydriding. The effect of the inflow/outflow temperature on the lumped temperature is not obvious due to a large amount of reaction heat. A reduced model has been introduced and its usefulness has been confirmed in Ref. [26], where the hydrogen specific heat capacity and the enthalpy change caused by the hydrogen inflow are ignored. Based on these assumptions, the energy balance equation Eq. (12) becomes: " . d T U⁄d] = WX Y ,UX − U- − ∆P0/= #  

(30)

Solving Eq.(30), we obtain: U = U ∗ + U − U ∗ exp?− WX Y ]⁄ Tƒ @

" .,WX Y  -. where the characteristic time U ∗ is equal to UX − ∆P0/= #  

(31)

Based on the same parameters and conditions, we compare the lumped temperature of the analytical solutions with that of the reduced model results. Fig.2b shows that the lumped temperature of analytical solutions is about 5.5 K higher than that of the reduced model during hydriding. The results calculated by the reduced model is about 6.5 K higher than the other during dehydriding. The analytical solutions in the present work show more accurate than the reduced model.

5. Parametric study by numerical solution The lumped parameter numerical model with constant source term is validated by the analytical solutions in the previous section. Then, the validated lumped parameter numerical model is used for parametric study in this section. The variable source term is easily and widely used in numerical models to simulate the hydriding-dehydriding cycle. The effect of variable mass source term on the numerical results of the lumped parameter model is discussed. The simulation results of the lumped parameter numerical model with variable mass source term are compared with those with constant mass source term. The effect of the mass flow rate on the metal hydride system is studied to exam the upper limit of the analytical solutions. Matlab/Simulink is used for numerical solutions. 5.1. Effect of variable mass source term The effect of variable mass source term on the numerical solutions of the lumped parameters model is studied. The variable metal hydride source terms during hydriding and dehydriding can be calculated

by Eq. (10a) and Eq. (10b). The mass source term is set as 0 when the hydrogen pressure 9 is between 9:;0 and 9:;= . The variable mass source term means the variable reaction rate. The

numerical solutions using the variable mass source term are compared with the numerical solutions using the constant mass source term. 12

Fig.3 shows the effects of the variable mass source term on the numerical solutions, including the lumped temperature, the pressure, the mass source term of metal hydride, the mass of metal hydride, the equilibrium pressure and the mass of gaseous hydrogen. As for the lumped temperature, the metal hydride mass and the equilibrium pressure, the results of the numerical model using variable mass term agree well with the other in Fig.3a, Fig.3d and Fig.3e. Fig.3c shows that the mass source term of metal hydride during hydriding and dehydriding can be considered roughly as constant, which confirms that the constant mass source term used in the analytical solution gives reasonable approximated results. The value of variable mass source term is below that of constant mass source term due to a relative lower temperature at the beginning of the hydriding process. As for the pressure and the gaseous hydrogen mass, the difference between two numerical solutions is obvious. The pressure and the gaseous hydrogen mass in the metal hydride reactor are related to the

source term. The ratio of the hydrogen mass source term to the mass flow rate is assumed as −1 in the numerical model with the constant mass source, which makes the mass of gaseous hydrogen in the metal hydride reactor to be constant. Therefore, the pressure calculated by the model with constant source term is approximately constant. As for the numerical model with variable source term, the " is a negative quantity after about 731s during dormancy in metal hydride mass source term #

" is a positive quantity and the gaseous hydrogen Fig.3c. Thus, the hydrogen mass source term # 

mass increases in that process, which causes the increase of the hydrogen pressure 9 during the

later dormancy in Fig.3b. Due to the similar reason, the hydrogen pressure 9 decreases during the earlier dormancy. Considering the gaseous hydrogen mass in the reactor is a relatively smaller quantity, the effect of gaseous hydrogen mass on the numerical results is not significant. In general, the numerical model with a variable source term is more accurate than the other to predict the hydriding-dehydriding cycle process. 5.2. Effect of mass flow rate Based on the Matlab/Simulink numerical model using the variable source term, the effect of the mass inflow rate on the metal hydride system is taken into consideration. The absolute value of the mass inflow rate is equal to that of the mass outflow rate. Four cases with different mass flow rates are discussed in this section, and the absolute values of the mass flow rate are presented in Fig.4. The

reaction fraction X is defined as the ratio of the metal hydride mass  to the saturated solid

material mass ƒƒ0! . The reaction fraction X is about 100% when the hydrogen storage capacity of the metal hydride system approaches the limited capacity. 13

Fig.4 shows the comparison of the numerical solutions among four cases with different mass flow rates, including the lumped temperature, the mass source term of metal hydride and the reaction fraction. Fig.4a shows the lumped temperature during hydriding generally rises with the increase of mass inflow rate, and the lumped temperature during dehydriding decreases with the increase of mass " during hydriding and dehydriding can outflow rate. Fig.4b shows the metal hydride source term #

be considered roughly as constant in the cases with a relatively small mass flow rate. Fig.4c shows the maximum reaction fractions in four cases are about 64.1%, 77.2%, 90.3% and 100%. It takes less time to approach the limited capacity by taking a relatively larger mass flow rate.

" decreases when As for the case with the maximum mass flow rate, the metal hydride source term # the reaction fraction is beyond about 90% during hydriding, and the lumped temperature reduces due to the decrease of reaction heat and the heat transfer from the metal hydride bed to the cooling water. The analytical solution is solved from the thermodynamic model based on the assumption that both the mass flow rate and the mass source term are constant. The metal hydride source term or the reaction rate during hydriding is almost constant when the reaction fraction is below about 90% during hydriding, as shown in Fig.4b and Fig.4c. When the hydriding reaction is close to the saturation condition, the metal hydride source term or the reaction rate decreases and cannot be considered roughly as constant. In this article, we exactly solved the model to give analytical solutions under different conditions. The model used for analytical solutions is an approximate model when the hydrogen storage capacity of the metal hydride system is above about 90% of its limited capacity.

6. Conclusions The analytical solutions of lumped temperature for the hydrogen storage system using metal hydride are obtained from the thermodynamic model in various process, including hydriding, dormancy and dehydriding processes. The analytical solutions are applied to validate the lumped parameter numerical model with a constant source term. Further, the analytical solution can be used as a reference for more detailed two-dimensional or three-dimensional models. Four cases under constant/variable inflow/outflow temperature conditions are analytically studied, the results show that the effect the variable inflow/outflow temperature condition on the lumped temperature in metal hydride system is not obvious compared to the constant inflow/outflow temperature condition. 14

The analytical solution is compared with the result of the reduced model proposed in the reference, the comparison shows that the analytical solution is more accurate than the reduced model. The variable mass source term is taken into consideration in the validated lumped parameter numerical model. The numerical model with variable source term shows better performance than that with constant source term. The effect of mass flow rate on the metal hydride hydrogen storage system is discussed, the results show that the metal hydride source term during hydriding can be considered roughly as constant, which is true when the hydrogen storage capacity of the metal hydride system is not beyond about 90% of its limited capacity.

Acknowledgments We wish to thank the financial supports from the National Natural Science Foundation of China for the project No. 51476120, from the State Administration of Foreign Experts Affairs of China for the 111 Project (No. B17034). Mr. Liang Tong also thanks the support from the China Scholarship Council (CSC) and the Fonds de Recherche du Québec - Nature et Technologies (FRQNT) for the PBEEE fellowship (No. 203790).

Nomenclature Y TD

surface area for heat transfer, m2

" # 

specific heat capacity at constant pressure

kg/s

" #

of gaseous hydrogen, J/kg/K T[

]∗

specific heat capacity at constant volume

U0/=

of gaseous hydrogen, J/kg/K T

specific heat capacity of solid material,

L…



U∗

UX

plateau slope coefficient

ab

mass, kg net mass flow rate, kg/s

M

molecular weight, kg/mol

9

UV

activation energy, J/mol

"

equilibrium pressure, MPa

R

universal gas constant, J/mol/K

characteristic time, s absorption/desorption contribution

characteristic temperature, K inflow/outflow temperature, K ambient temperature, K volume for gaseous phase in the reactor,

3

m

∆P ∆#

reference pressure, MPa

9:;

mass source term for metal hydride, kg/s

temperature, K

J/kg/K „

mass source term for gaseous hydrogen,

enthalpy of reaction, J/mol entropy of reaction, J/mol/K

Greek letters  15

dimensionless heat transfer coefficient, 1

WX o

overall heat transfer coefficient, W/m2/K

g

gaseous phase

ratio of hydrogen mass source term to mass

H

hydrogen absorbed in metal hydride

H2

gaseous hydrogen

stoichiometric coefficient

in

inflow

ratio of specific heat, 1

M

alloy

MH

metal hydride

max

maximum

out

outflow

s

solid phase

sat

saturation

flow rate, 1 ψ n

Superscripts and subscripts a

absorption process

d

desorption process

eq

equilibrium

References [1] Nastasi B, Lo Basso G. Hydrogen to link heat and electricity in the transition towards future Smart Energy Systems. Energy 2016; 110:5-22. [2] Principi G, Agresti F, Maddalena A, Lo Russo S. The problem of solid state hydrogen storage. Energy 2009; 34:2087-91. [3] Gkanas EI, Khzouz M, Panagakos G, Statheros T, Mihalakakou G, Siasos GI, Skodras G, Makridis SS. Hydrogenation behavior in rectangular metal hydride tanks under effective heat management processes for green building applications. Energy 2018; 142:518-30 [4] Alijanpour Sheshpoli M, Mousavi Ajarostaghi SS, Delavar MA. Waste heat recovery from a 1180 kW proton exchange membrane fuel cell (PEMFC) system by Recuperative organic Rankine cycle (RORC). Energy 2018; 157:353-66. [5] Paskevicius M, Sheppard DA, Williamson K, Buckley CE. Metal hydride thermal heat storage prototype for concentrating solar thermal power. Energy 2015; 88:469-77. [6] Yang FS, Wu Z, Liu SZ, Zhang Y, Wang G, Zhang ZX, Wang YQ. Theoretical formulation and performance analysis of a novel hydride heat Pump(HHP) integrated heat recovery system. Energy 2018; 163:208-20. [7] Laurencelle F, Goyette J. Simulation of heat transfer in a metal hydride reactor with aluminium foam. Int J Hydrogen Energy 2007; 32:2957-64. [8] Tsai ML, Tang TS. On the selection of metal foam volume fraction for hydriding time minimization of metal hydride reactors. Int J Hydrogen Energy 2010; 35:11052-63. [9] Bennett C, Eastwick C, Walker G. Effect of a varying effective thermal conductivity term on heat conduction through a physical model of a hydride bed. Int J Hydrogen Energy 2013; 38:1692-701.

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[10] Mazzucco A, Rokni M. Generalized computational model for high pressure metal hydrides with variable thermal properties. Int J Hydrogen Energy 2015; 40:11470-7. [11] Chung CA, Ho CJ. Thermal–fluid behavior of the hydriding and dehydriding processes in a metal hydride hydrogen storage canister. Int J Hydrogen Energy 2009;34:4351-64. [12] Bao ZW, Yang FS, Wu Z, Cao XX, Zhang ZX. Simulation studies on heat and mass transfer in high-temperature magnesium hydride reactors. Applied Energy 2013;112:1181-9. [13] Kikkinides ES, Georgiadis MC, Stubos AK. Dynamic modeling and optimization of hydrogen storage in metal hydride beds. Energy 2006; 31:2428-46. [14] Raju M, Kumar S. System simulation modeling and heat transfer in sodium alanate based hydrogen storage systems. Int J Hydrogen Energy 2011; 36:1578-91. [15] Raju M, Kumar S. Optimization of heat exchanger designs in metal hydride based hydrogen storage systems. Int J Hydrogen Energy 2012; 37:2767-78. [16] Nyamsi SN, Yang FS, Zhang ZX. An optimization study on the finned tube heat exchanger used in hydride hydrogen storage system - analytical method and numerical simulation. Int J Hydrogen Energy 2012;37:16078-92. [17] Bhouri M, Goyette J, Hardy BJ, Anton DL. Honeycomb metallic structure for improving heat exchange in hydrogen storage system. Int J Hydrogen Energy 2011;36:6723-38. [18] Wu Z, Yang FS, Zhang ZX, Bao ZW. Magnesium based metal hydride reactor incorporating helical coil heat exchanger: Simulation study and optimal design. Applied Energy 2014;130:712-22. [19] Kyoung S, Ferekh S, Gwak G, Jo A, Ju H. Three-dimensional modeling and simulation of hydrogen desorption in metal hydride hydrogen storage vessels. Int J Hydrogen Energy 2015; 40:14322-30. [20] Ferekh S, Gwak G, Kyoung S, Kang HG, Chang MH, Yun, SH, Oh YH, Kim W, Kim D, Hong T, Ju H. Numerical comparison of heat-fin- and metal-foam-based hydrogen storage beds during hydrogen charging process. Int J Hydrogen Energy 2015; 40:14540-50. [21] Mellouli S, Askri F, Abhilash E, Ben Nasrallah S. Impact of using a heat transfer fluid pipe in a metal hydride-phase change material tank. Applied Thermal Engineering 2017; 113:554-65. [22] Muthukumar P, Prakash Maiya M, Srinivasa Murthy S. Experiments on a metal hydride-based hydrogen storage device. Int J Hydrogen Energy 2005;30:1569-81. [23] Gambini M, Manno M, Vellini M. Numerical analysis and performance assessment of metal hydride-based hydrogen storage systems. Int J Hydrogen Energy 2008;33:6178-87. [24] Raju M, Ortmann JP, Kumar S. System simulation model for high-pressure metal hydride hydrogen storage systems. Int J Hydrogen Energy 2010; 35:8742-54.

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[25] Raju M, Khaitan S. Charging dynamics of metal hydride hydrogen storage bed for small wind hybrid systems. Int J Hydrogen Energy 2011; 36:10797-807. [26] Talaganis BA, Meyer GO, Aguirre PA. Modeling and simulation of absorption-desorption cyclic processes for hydrogen storage-compression using metal hydrides. Int J Hydrogen Energy 2011; 36:13621-13631. [27] Talaganis BA, Meyer GO, Oliva DG, Fuentes M, Aguirre PA. Modeling and optimal design of cyclic processes for hydrogen purification using hydride-forming metals. Int J Hydrogen Energy 2014; 39:18997-9008. [28] Corgnale C, Motyka T, Greenway S, Perez-Berrios JM, Nakano A, Ito H, Maeda T. Metal hydride bed system model for renewable source driven regenerative Fuel Cell. Int J Alloys and Compounds 2013;580:S406-9. [29] Yang JC. A thermodynamic analysis of refueling of a hydrogen tank. Int J Hydrogen Energy 2009;34:6712-21. [30] Hosseini M, Dincer I, Naterer GF, Rosen MA. Thermodynamic analysis of filling compressed gaseous hydrogen storage tanks. Int J Hydrogen Energy 2012;37:5063-71. [31] Xiao JS, Tong L, Yang TQ, Bénard P, Chahine R. Lumped parameter simulation of hydrogen storage and purification systems using metal hydrides. Int J Hydrogen Energy 2017;42:3698-707. [32] Xiao JS, Bénard P, Chahine R. Charge-discharge cycle thermodynamics for compression hydrogen storage system. Int J Hydrogen Energy 2016;41:5531-9. [33] Xiao JS, Bénard P, Chahine R. Adsorption–desorption cycle thermodynamics for adsorptive hydrogen storage system. Int J Hydrogen Energy 2016;41:6139-47.

18

360

0.245

Numerical Analytical

350

Numerical Analytical

0.240 0.235 0.230

Pressure (MPa)

Temperature (K)

340 330 320 310

0.225 0.220 0.215 0.210 0.205

300

0.200

290 0

200

400

600

800

1000

1200

1400

1600

0

400

600

a

800

1000

1200

1400

1600

b

110

Numerical Analytical

0.10

Numerical Analytical

100

0.05

0.00

-0.05

90 80 70 60 50 40 30

-0.10

20

0

200

400

600

800

1000

1200

1400

1600

Time (s)

0

200

400

600

800

1000

1200

1400

1600

Time (s)

c

d

0.0040

1.4

Numerical Analytical

Numerical Analytical

0.0035

Mass of gaseous hydrogen (kg)

1.2

Equilibrium pressure (MPa)

200

Time (s)

Mass of metal hydride (kg)

Mass source term of metal hydride (kg/s)

Time (s)

1.0

0.8

0.6

0.4

0.0030 0.0025 0.0020 0.0015 0.0010 0.0005

0.2 0.0000

0

200

400

600

800

Time (s)

1000

1200

1400

1600

0

e

200

400

600

800

Time (s)

1000

1200

1400

1600

f

Fig.1 –Validation of numerical model by analytical solutions: (a) lumped temperature, (b) pressure, (c) mass source term of metal hydride, (d) mass of metal hydride, (e) equilibrium pressure and (f) mass of gaseous hydrogen.

19

360

Case 1 Case 2 Case 3 Case 4

350

Temperature (K)

340 330 320 310 300

330

332

328

330

326

328

324

326

322

324

320 322 400 450 500 550 600 650 700 750 950

290 0

200

400

600

800

1000 1050 1100 1150 1200

1000

1200

1400

1600

Time (s)

a

360 350

Temperature (K)

340 330 320 310 300

Reduced model [19] Analytical solution

290 0

200

400

600

800

Time (s)

1000

1200

1400

1600

b Fig.2 – Temperature comparison of analytical solutions under different conditions: (a) variable/constant inflow/outflow temperature, (b) reduced model.

20

1.4

350

1.2

340

1.0

Pressure (MPa)

Temperature (K)

360

330 320 310 300

0.8 0.6 0.4 0.2

Constant source term Variable source term

290

0.0 0

200

400

600

800

1000

1200

1400

1600

0

Time (s)

400

600

a

800

1000

1200

1400

1600

b

110

Constant source term Variable source term

100

Mass of metal hydride (kg)

0.08

0.04

0.00 0.03

0.02

-0.04

0.01

0.00

-0.01

-0.08

-0.02

90 80 70 60 50 40

Constant source term Variable source term

30

-0.03 700

750

800

850

900

-0.12

20 0

200

400

600

800

1000

1200

1400

1600

0

Time (s)

400

600

c

Constant source term Variable source term

1.0

0.8

0.6

0.4

0.2

800

1000

1200

1400

1600

d

Constant source term Variable source term

0.012

Mass of gaseous hydrogen (kg)

1.2

200

Time (s)

1.4

Equilibrium pressure (MPa)

200

Time (s)

0.12

Mass source term of metal hydride (kg/s)

Constant source term Variable source term

0.010

0.008

0.006

0.004

0.002

0.000 0

200

400

600

800

Time (s)

1000

1200

1400

0

1600

e

200

400

600

800

Time (s)

1000

1200

1400

1600

f

Fig.3 – Comparison for cases using lumped parameter numerical model with constant/variable source term: (a) lumped temperature, (b) pressure, (c) mass source term of metal hydride, (d) mass of metal hydride, (e) equilibrium pressure and (f) mass of gaseous hydrogen.

21

360

Mass flow rate (kg/s) 0.6*1.389e-3 0.8*1.389e-3 1.0*1.389e-3 1.2*1.389e-3

350

Temperature (K)

340 330 320 310 300 290 0

200

400

600

800

1000

1200

1400

1600

Time (s)

a

Mass source term of metal hydride (kg/s)

0.15

Mass flow rate (kg/s) 0.6*1.389e-3 0.8*1.389e-3 1.0*1.389e-3 1.2*1.389e-3

0.10

0.05

0.00

-0.05

-0.10

-0.15 0

200

400

600

800

1000

1200

1400

1600

Time (s)

b

100

Reaction fraction (%)

90 80 70 60 50

Mass flow rate (kg/s) 0.6*1.389e-3 0.8*1.389e-3 1.0*1.389e-3 1.2*1.389e-3

40 30 20 10 0

200

400

600

800

1000

1200

1400

1600

Time (s) c Fig.4 –Comparison for cases using lumped parameter numerical model with different mass flow rates: (a) lumped temperature, (b) mass source term of metal hydride and (c) reaction fraction.

22

Table 1 - Definitions of parameters used in analytical solutions Constant inflow/outflow Variable inflow/outflow Dormancy process temperature temperature WX Y ⁄," T^ TD ⁄T[

 n " T[ − # " T - , T[ +  T + T -.,WX Y , T[ +  T + T -.," T[ + # ]∗    " # ." o U0/= − o∆P0/= ⁄, T[ ∗ ⁄ v ] ] nU + αU + U αU + U V X 0/= X 0/= UX U∗ α + 1 + o − o T ⁄T[ α − n + 1 + o − o T ⁄T[ Table 2 – Values of parameters used in Matlab/Simulink numerical model and analytical solution Parameter Value Parameter Value Parameter Value Y [m2]

4.147

WX [W/m /K]

243

ab [m3] 2

0.012

 [kg]

109.3

UX†**‡()b [K]

298

UV [K]

UXˆ:()b [K]

 [kg/mol]

 [kg/mol]

290 353 0.002 0.438

„0 [J/mol]

21170

„= [J/mol]

16420

/= [1/s]

9.6

/0 [1/s]

59.2

T [J/kg/K]

355

TD [J/kg/K] T[ [J/kg/K] U()((‡ [K]

8.314

∆#0/= [J/mol/K]

108

LD [1]

0.13

9 [MPa]

0.1

ta [s]

720

14.3e3

td [s]

720

10.3e3

tdormancy [s]

180

298

9()((‡ [MPa]

R [J/mol/K]

0.2

∆P0 [J/mol]

∆P= [J/mol]

30478 30800

Table 3 - Description of four cases with different inflow/outflow temperature Case Absorption stage Desorption stage 1 2 3 4

Constant inflow temperature, UV = /‰Š‹] Constant inflow temperature, UV = /‰Š‹] Variable inflow temperature, UV = U Variable inflow temperature, UV = U

Variable outflow temperature, UV = U Constant outflow temperature, UV = /‰Š‹] Constant inflow temperature, UV = /‰Š‹] Variable inflow temperature, UV = U

23

Highlights

Analytical solutions of thermodynamic model for metal hydride system are obtained.

The solutions are applicable for constant/variable inflow/outflow temperatures.

We exactly solved the model to give analytical solutions under different conditions.

Analytical solutions can be used as basic benchmarks to validate numerical models.

Our analytical solution is more accurate than the reduced model reported in reference.

Conflict of Interest Statements

I wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Jinsheng Xiao