Modeling and optimization of multi-tubular metal hydride beds for efficient hydrogen storage

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 0

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Modeling and optimization of multi-tubular metal hydridebeds for efficient hydrogen storage

Constantinos A. Krokos a,b, Dragan Nikolic a, Eustathios S. Kikkinides a,*,Michael C. Georgiadis c, Athanasios K. Stubos b

a Department of Mechanical Engineering, University of Western Macedonia, Bakola & Salvera Str., 50100 Kozani, Greeceb National Center of Scientific Research DEMOKRITOS, Institute of Nuclear Technology and Radiation Protection,

15310 Ag. Paraskevi Attikis, Athens, Greecec Department of Engineering Informatics and Telecommunications, University of Western Macedonia, Karamanli & Ligeris Str.,

50100 Kozani, Greece

a r t i c l e i n f o

Article history:

Received 24 July 2009

Received in revised form

9 September 2009

Accepted 9 September 2009

Available online 2 October 2009

Keywords:

Metal hydrides

Hydrogen storage

Multi-tubular reactors

Numerical simulations

* Corresponding author.E-mail address: [email protected] (E.S. Kikk

1 Based on data from Honda FCX Clarity.0360-3199/$ – see front matter ª 2009 Profesdoi:10.1016/j.ijhydene.2009.09.021

a b s t r a c t

This work presents a novel systematic approach for the optimal design of a multi-tubular

metal hydride tank, containing up to nine tubular metal hydride reactors, used for

hydrogen storage. The tank is designed to store enough amount of hydrogen for 25 km

range1, for a fuel cell vehicle. A detailed 3D Cartesian, mathematical model is developed

and validated against a 2D cylindrical developed by Kikkinides et al. [1]. The objective is to

find the optimal process design so as to increase the overall thermal efficiency, and thus

minimize the storage time. Optimization results indicate that almost 90% improvement of

the storage time can be achieved, over the case where the tank is not optimized and for

a minimum storage capacity of 99% of the maximum value.

ª 2009 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

1. Introduction technologies do not meet all the requirements of future fuel

Hydrogen is shown to be the future fuel from the point of view

of human fuel evolution. The fuel evolution experienced the

history from coal through petroleum to natural gas following

the direction of increasing the content of hydrogen, therefore,

it must finally reach the destination of pure hydrogen [1–3].

Automotive industry is slowly moving towards this direction

by developing fuel cell vehicles, which have the benefit of zero

emissions. However, before introducing the fuel cell vehicle in

the market, several issues including the hydrogen storage

need to be addressed. Currently, high pressure tanks and

liquid hydrogen tanks are used for road tests, but both

inides).

sor T. Nejat Veziroglu. Pu

cell vehicles [3]. Storage of hydrogen in a pressurized cylinder

is not likely to be applied in the future due to low density, high

pressurization costs and safety considerations [2]. Liquid

hydrogen could be applied if the unit cost becomes compa-

rable with gasoline, yet inevitable boiling-off liquid might be

in concern [2]. Metal hydride materials seem to exhibit good

storage capabilities and constitute a promising direction

towards the hydrogen economy [4–6]. This can be attributed to

the reduced operating pressure compared to compressed gas

technology, thus ensuring less weight and increased safety [1].

The current work is motivated by the desire to design a metal

hydride tank, with small charge–discharge duration germane

blished by Elsevier Ltd. All rights reserved.

Fig. 1 – Circle drawn with the 8-way symmetry on pixelized

surface.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 0 9129

to automotive use. The tank is designed to store enough

amount of hydrogen for 25 km range for a fuel cell vehicle

similar to Honda’s FCX Clarity. It is possible to increase the

range by increasing the number of reactor modules.

The performance of hydrogen absorption to and desorp-

tion from metal hydride depends on the heat transfer, mass

transfer and kinetic-reactions of the metal hydride bed [7].

Hence, there have been numerous experimental and theo-

retical investigations on several aspects of metal hydride

storage systems. Muthukumar et al. [8] analyzed the effect of

various operating conditions in metal hydride based on

hydrogen storage device. MacDonald and Rowe [9] analyzed

dynamic metal hydride storage systems by a combination of

experimental tests and simulations. Several parameters that

play a significant role on precise modeling results were

investigated. Jemni and Narsallah [10] presented a theoretical

Fig. 2 – The five regions for the case of the reactor with 4

metal hydride beds.

study of the mass and heat transfer dynamics in a metal

hydride reactor. Some of the assumptions used in their work

were validated in a subsequent contribution [11]. Jemni et al.

[12] presented an experimental approach to determine the

reaction kinetics equilibrium conditions and transport prop-

erties in LaNi5–H2 system. Nakagawa et al [13] presented a 2D

model for the transient heat and mass transfer within a metal

hydride bed. Mat and Kaplan [14] developed a mathematical

model to describe hydrogen absorption in a porous

lanthanum metal bed. The model takes into account the

complex mass and heat transfer and reaction kinetics. Model

prediction results were found to be in a good agreement with

experimental data. Demircan et al. [15] examined experi-

mentally the hydrogen absorption in two LaNi5H2 reactors.

Kaplan Y. [16] studied experimentally the effect of heat

transfer mechanisms in three different geometric design

configurations of MH reactors under various charging pres-

sures. It was concluded that the charging of the MH reactors is

primarily heat transfer depended and the reactor with better

cooling exhibits the fastest charging characteristics.

Aldas et al. [17] and Mat et al. [18], presented an integrated

model of heat and mass transfer, reaction kinetics and fluid

dynamics in a hydride bed. An important conclusion of this

work is that the flow conditions do not significantly affect the

amount of hydrogen absorbed. Askri et al. [19] extended the

previous studies [10–12], by investigating the effect of radiate

heat transfer in LaNi5H2 and MgH2 hydride beds. Simulation

results showed that radiation effects are negligible on the

sorption process in the LaNi5H2 bed but they play an important

role in the case of MgH2 hydride bed. The effects of heat transfer

on the sorption kinetics of complex hydride reacting systems

werestudiedbyLozanoetal. [20]. Itwasfoundthat thesizeofthe

hydride bed as well as the container has a significant effect on

the hydrogenation kinetics of NaAlH4 system. Recently Chung

and Ho [21] studied the effect of the expansion volume on top of

a metal hydride reactor using detailed simulation models. It was

found that the expansion volume has a retardation effect on the

reaction rates due to an increase in the thermal resistance to the

heat transfer from the heat exchanger. Moreover, heat convec-

tion is no longer negligible as in similar studies with hydride

reactors without expansion volume.

The systematic optimization of hydrogen storage in

a metal hydride bed was investigated by Kikkinides et al. [1],

where two novel cooling options were investigated by intro-

ducing additional heat exchangers at a concentric inner tube

and an annular ring inside the tank. Askri et al. [22] investi-

gated numerically the impact of the tank wall thermal mass

on the hydride process and it was found that there is no

significant effect on the hydrogen storage time. In the same

work the dynamic behaviour inside four metal hydride tank

designs was studied. The design that includes a concentric

heat exchanger tube equipped with fins and filled with flowing

cooling fluid presents the shortest storage duration.

A lumped-parameter numerical model of a metal hydride

hydrogen-storage tank was developed and validated against

experimental results by Gambini et al. [23]. The model was

used as a tool to predict the performance of metal hydride

hydrogen storage devices operating in Fuel Cell based energy

systems. Brown et al. [24] following the original work of Mayer

et al. [25] developed a simplified dynamic model to be applied

Fig. 3 – Boundaries between metal hydride (region 2) and

cooling fluid (region 1) formed on a pixelized surface.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 09130

to integrated hydride tank fuel-cell systems in order to

develop control systems and strategies. Chaise et al. [26]

suggested a criterion which allows to quantify the error which

is made when the fluid flow is neglected. Although the crite-

rion was successfully applied on a co-milled magnesium with

transient metals, characterized as P. De Rango et al. [27] and

Huot et al. [28] proposed, there are no known successful

implementations on LaNi5 based hydrides.

All these studies have shown that the charge as well as

discharge time of a metal hydride storage tank strongly depend

on the successful heat transfer from/to the hydride bed.

Numerous attempts to improve the effective thermal conduc-

tivity of metal hydride beds can be found in the literature,

including: insertion of aluminium foam [29], integration of

copper wire net structure [30], packing in a multilayer waved

sheet structure, microencapsulated metal hydride compact [31]

and compacting metal hydride powder with an expanded

graphite [32]. Mohan et al. [33] suggested a novel complex LaNi5hydrogen storage device with embedded, in a recurring motif,

filters and cooling tubes. Simulation results showed the

importance of bed thickness as a major parameter controlling

the rate of hydrogen absorption at all locations within the bed.

Mellouli et al. [34] described a novel design of a metal hydride

storage vessel with a spiral heat exchanger and they have

shown, experimentally, that the charge and discharge time are

significantly reduced. Recently the same group [7] developed

a 2D model in order to study various designs of their suggested

metal hydride storage vessel. The model was validated against

experimental results and was used to test three vessel config-

urations. Simulation results revealed that the configuration

with the most effective heat management exhibits the smallest

charge–discharge duration.

In general, configuration of metal hydride tanks for

hydrogen storage fall into the following categories:

1. A cylindrical tube filled with metal hydride with heat

exchangers placed at the outer surface and at different

positions inside the tank including the center and the

annulus at different positions, in the form of a spiral.

2. Multiple heat transfer tubes embedded in a cylindrical

metal hydride tank.

3. Multiple metal hydride reactors embedded in a cylindrical

tank containing the cooling fluid.

Case 1, has been considered and modeled in great detail as

the particular configuration results in straightforward

geometrical representations that can be easily simulated in

cylindrical coordinates under conditions of axial symmetry

reducing the original 3D problem into a 2D one.

Case 2, is far more complex and can be solved in cylindrical

coordinates only under specific conditions [33].

Case 3, has not been considered yet from a modeling point

of view since the mathematical problem in its full complexity

requires detailed 3D CFD simulations in the domain that

contains the cooling fluid coupled with the simultaneous

simulation of hydrogen storage at each metal hydride reactor.

Evidently the development of the modeling equations in

cylindrical coordinates is now very tedious as the axi-

symmetric conditions no longer hold. In such cases it is often

preferred to develop the modeling equations in 3D Cartesian

coordinates. This way the formation of the equations is

straightforward and can be easily expanded to include any

number of MH reactors, however one must deal with the

accurate representation of the curved surfaces in Cartesian

coordinates as well as the proper implementation of the

relevant boundary conditions at fluid-reactor interfaces.

This work presents for the first time the development of

a detailed 3D Cartesian mathematical model to describe

hydrogen storage in multi-reactor MH configurations. As

a first step a constant axial velocity for the cooling fluid is

assumed, an assumption that is normally valid under condi-

tions of full turbulence. The modeling equations have been

developed in 3D Cartesian coordinates with special care in the

implementation of boundary conditions at the fluid-reactor

interfaces. The mathematical model has been implemented

and validated in the gPROMS modeling environment [35] and

found to be in excellent agreement with the 2D cylindrical

model developed by Kikkinides et al [1]. The developed model

provides significant insight into the problem of minimizing

the storage time by manipulating the parameters that affect

the thermal efficiency of the reactor. The first studied

parameter is the cooling fluid velocity. Secondly, the effect of

several different number of beds, containing each time the

same amount of metal hydride alloy is examined. Finally

various positions of the metal hydride tubes inside the reactor

are studied. Optimization results indicate that significant

improvements on the total storage time can be achieved

under a safe and economically attractive operation, using

simple reactor geometries.

2. Mathematical model

The purpose of developing a 3D Cartesian model is to

accurately study the heat transfer efficiency of multi-

tubular metal hydride reactors. Several barriers were found

during the development of the model. The two most

important among them were the cylindrical geometry

formation on a Cartesian plane and the heat transfer at the

boundaries between the metal hydride tubes and the cooling

medium.

Fig. 4 – The base-case metal hydride reactor consistent of

one metal hydride bed (metal hydride tube) surrounded by

a cooling medium. MH stands for metal hydride and CF for

cooling fluid. Lx, Ly, Lz are the lengths at the X, Y, Z axis

respectively.

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2.1. The transformation of a cylindrical problem toa Cartesian

The problem of forming a circle on a Cartesian plane is well

known in the computer graphics area, where a circle must be

formed on a monitor which is divided in several pixels. An 8-

way symmetry is used to form a circle on an X–Y plane. By

reproducing this circle towards the Z axis for n times

a cylinder is formed. A circle drawn with the 8-way symmetry

algorithm is shown in Fig. 1. The major advantage of the 3D

Cartesian model against the 2D cylindrical models found in

the literature is that any design option can be easily modeled.

Therefore the position of the metal hydride tubes in the

reactor could be exactly located just by defining the X and Y

coordinates of the center of the tubes.

2.1.1. Geometrical mapsSince the tubular reactor geometry is developed on a 3D

Cartesian coordinate system, the mathematical expressions

that describe the physicochemical phenomena that occur at

each region (e.q. metal hydride, cooling medium etc) must be

defined. Therefore the metal hydride reactor is divided into

five regions as shown in Fig. 2:

1. Metal hydride (MH)

2. Cooling fluid (CF)

3. Surroundings (S)

4. Boundaries MH/CF

5. Boundaries S/CF

At each region different heat, mass and momentum

balance equations apply. To switch on and off the relevant

terms in the equations geometrical maps are used as follows:

� Metal hydride (MH) – xyMapMH(x,y)

� Cooling Fluid (CF) – xyMapCF(x,y)

� Surroundings (S) – xyMapS(x,y)

� Boundaries MH/CF – xyMapBoundsMHCF(x,y)

� Boundaries S/CF – xyMapBoundsSCF(x,y).

Each map is a binary function that takes the value of 1 if

a point belongs in a certain region and 0 otherwise. The

geometrical maps are calculated by an in-house software

developed in Cþþ and used as a foreign object in gPROMS

[35]. Each equation is therefore expressed in the following

form:

xyMapSðx; yÞ fSurrounding equationsgþxyMapBoundsSCFðx; yÞ fSurrounding

=Cooling fluid boundary equationsgþxyMapCFðx; yÞ fCooling fluid equationsgþxyMapBoundsMHCFðx; yÞ fMetal hydride

=Cooling fluid boundary equationsgþxyMapMHðx; yÞ fMetal hydride equationsg ¼ 0 (1)

x ˛ (0, 1), y ˛ (0, 1), z ˛ (0, 1).

2.1.2. Heat transfer at the boundariesA Cylindrical model produces distinct regional separation

contrary to the Cartesian one. In the Cartesian model, cells

comprising both metal hydride and cooling fluid regions exist,

as shown in Fig. 3. Thermal conductivity values at the nodes of

each cell are defined by Eq. (2).

k ¼

8<:

k1; Adjacent cells are of region 1k2; Adjacent cells are of region 22k1k2k1þk2

; Adjacent cell are of different regions(2)

If two adjacent cells are of the same region, the thermal

conductivity of that region is assigned. If the adjacent cells

belong to different regions a combined thermal conductivity is

derived using the equality of thermal flux and temperature.

The thermal conductivity values are also calculated by the

developed foreign object as shown in Eq. (3)

kþx ¼2k1k2

k1 þ k2(3a)

k�x ¼ k1 (3b)

kþy ¼ k1 (3c)

k�y ¼2k1k2

k1 þ k2(3d)

The central finite difference scheme is applied for the heat

transfer equation at the boundary MH/CF. Thus, the heat

transfer equation takes the form of Eq. (4)

k

�v2Tvx2þ v2T

vy2þ v2T

vz2

�z�

kþx ðx; yÞðTxþ1 � TxÞ þ k�x ðx; yÞ

ðTx�1 � TxÞ�.

Dx2 þ�

kþy ðx; yÞ�Tyþ1 � Ty

�þ k�y ðx; yÞ�

Ty�1 � Ty

��.Dy2 þ ðkðx; yÞðTzþ1 � 2Tz þ Tz�1ÞÞ=Dz2 ð4Þ

In the present study no special treatment is required for

the boundary CF/S, due to the assumption of infinite heat

transfer between cooling fluid and surroundings. Neverthe-

less inclusion of an additional heat transfer coefficient

between CF and S is straightforward and can be easily

included.

Table 1 – List of basic model input parameters.

Parameter Units Value

Lz m 0.2

e dim/less 0.5

u0 m s�1 0.2

DH J mol�1 �29879

Cps J kg�1 K�1 429.1

Cpg J kg�1 K�1 14.3� 103

Cpf J kg�1 K�1 4184

K0 m2 1� 10�8

P0 Pa 1� 106

T0 K 290

m Pa s 8� 10�6

lg W m�1 K�1 0.12

ls W m�1 K�1 1.20

lf W m�1 K�1 0.60

h W m�2 K�1 1650

rs,0 kg m�3 4140

rf kg m�3 1000

a

b

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2.2. Mathematical modeling of heat and mass transfer

The base case of a metal hydride reactor with one metal

hydride tube (bed), as shown in Fig. 4, is used for the presen-

tation of the mathematical model. The bed is considered to be

filled with a metal hydride alloy (namely LaNi5), surrounded by

cooling medium that flows with constant velocity towards the

a

b

Fig. 5 – Time evolution of the total hydrogen mass

absorbed for two cooling fluid velocities (a) 0 m/s, (b) 1 m/s,

for both Cartesian and Cylindrical models.

Fig. 6 – Time evolution of the average temperature in the

metal hydride beds for two cooling fluid velocities (a) 0 m/s,

(b) 1 m/s, for both Cartesian and Cylindrical models.

Z axis. The developed mathematical model is based in the

following assumptions:

� The ideal gas law holds in the gas phase.

� The solid and fluid temperatures are essentially equal.

� The axial pressure drop in the bed depends linearly on the

superficial velocity through Darcy’s Law.

� Infinite heat transfer between cooling fluid and

surroundings.

� Radial flow is neglected.

The latter assumption is justified from previous studies on

LaNi5 MHT, where the effect of convection is found to be

negligible compared to that of conduction of the Energy

balance [7,11]. Since in the present work hydrogen is assumed

to be introduced axially in the MHT, the axial velocity term is

taken into account (and the respective convection term). This

is also justified from previous studies which revealed that at

very short times and for special types of pressure charging,

the axial convection term may not be completely negligible.

On the other hand, the radial velocity term has been always

negligible in magnitude for the LaNi5 system and thus it was

not included in the model equation. This leads to significant

reduction in computational requirements, an important issue

as the number of MHT increases.

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2.2.1. Mass balance of hydrogen in the gas phase

vbrvsþ 1

e

vðbrbuzÞvbz þW0

vbrs

vs¼ 0; (5)

where s, is the dimensionless time, s¼ t$Lz/u0, (Lz is the

modular reactor length, u0 is the reference velocity), bz stands

for the dimensionless axial distance, normalized by the total

bed length Lz ðbz ¼ z=LzÞ, buz is the axial superficial velocity,br ¼ r=r0 is the density of hydrogen in the gas phase, r0 being the

density at the bed inlet ðbz ¼ 0Þ, and brs is the solid (metal)

density of the alloy. W0 is a dimensionless parameter defined in

Section 2.2.7, and e is the void fraction of the bed.

2.2.2. Pressure drop equationThe steady state momentum balance of laminar gas flow

through a packed bed can be expressed by Darcy’s law.

buz ¼ �KzvbPvbz: (6)

where Kz is the dimensionless permeability value of the bed, bPis the dimensionless pressure ðbP ¼ P=P0Þ and P0 is the inlet

pressure. For the derivation of Eq. (6), it is assumed that the

transient and inertial velocity terms are negligible, according

a

b

Fig. 7 – Cooling fluid velocity effect on storage (a) and

cooling (b) duration, T0 [ 290 K.

to previous related studies on adsorption dynamics in fixed

beds [36].

2.2.3. Energy balance for the metal hydride bed

vðbrqÞvsþ Le

vðbrsqÞvs

� g

g� 1vbPvsþ 1

e

vðbrbuzqÞvbz �

�1

Pet

�V2q

�W0bvbrs

vs¼ 0; (7)

where q¼ T/T0 is the dimensionless gas temperature, T0 being

the temperature at the bed inlet. Le is the dimensionless Lewis

number expressing the ratio of the metal hydride heat

capacity over the gas heat capacity. Pet is the dimensionless

thermal Peclet number for the metal hydride bed.

2.2.4. Mass balance for the metal hydrideWe use the following kinetic scheme that has been successfully

employed for the case of LaNi5–H2 system [5,10–12,14,17–19]:

ð1� eÞvrs

vs¼ Catresexpð�ea=qÞln

� bPbPeq

��brsat � brs

�; (8)

where bPeq is the dimensionless equilibrium pressure, brsat is

the ‘‘saturated’’ bed density after complete reaction of the

alloy with H2 and ea is the dimensionless activation energy of

the reaction kinetics process.

2.2.5. Definition of equilibrium pressureFor the case of LaNi5–H2 system the equilibrium pressure is

defined as follows:

bPeq ¼ fðH=MÞ � exp

�DHR

�1

T0q� 1

Tref

�; (9)

where the function fðH=MÞ is the equilibrium pressure at the

reference temperature Tref. This is normally determined by

fitting the experimental data to a polynomial of nth degree. For

the case of LaNi5 available polynomial fits of 5th and 9th

degree are given in [10,12], respectively.

2.2.6. Energy balance for the cooling fluidThe cooling medium is assumed to flow at a constant axial

velocity (z-direction). The energy balance for the cooling fluid

is defined as follows:

vbTf

vs� 1

PetfV2 bTf þ

uf

u0

vbTf

vbz ¼ 0; (10)

where bTf is the dimensionless cooling fluid temperature and

uf being the cooling medium velocity. Petf is the dimensionless

thermal Peclet number for the cooling fluid.

2.2.7. Other auxiliary expressionsThe dimensionless form of the model introduces several

dimensionless parameters, which are defined below.

W0 ¼�

1� e

e

�$rs;0

r0

; (11)

Le ¼W0$Cps

Cpg(12)

b ¼ �DHT0$Cpg

; (13)

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Lam ¼ e$lg þ ð1� eÞ$ls; (14)

Kz ¼K0$P0

m$u0$Lz; (15)

ea ¼Ea

Rg$T0; (16)

tres ¼Lz

u0; (17)

Pet;x ¼e$Cpg$r0$T0$u0$Lx

Lam; Pet;y ¼

e$Cpg$r0$T0$u0$Ly

Lam;

Pet;z ¼e$Cpg$r0$T0$u0$Lz

Lam; (18)

Petf;x ¼rf$Cpf$u0$Lx

lf; Petf;y ¼

rf$Cpf$u0$Ly

lf; Petf;z ¼

rf$Cpf$u0$Lz

lf:

(19)

Note that the operator V2, become, in the dimensionless

form:

V2C ¼ Lz

Lx

v2C

vbx2 þLz

Ly

v2C

vby2 þv2C

vbz2 : (20)

where bx and by are the dimensionless distances at the X and Y

direction respectively, normalized by the total bed length Lz,

ðbx ¼ x=Lz; by ¼ y=LzÞ. It should be noted that the aspect ratio

Lz/Lx and Lz/Ly does not appear in the 2nd power in Eq. (20)

because of the way the dimensionless time, s and the Peclet

numbers, Pet, Petf have been defined.

2.2.8. Boundary and initial conditionsThe boundary conditions between the metal hydride bed and

cooling fluid, and between cooling fluid and surroundings are

included in the model as described previously. The remaining

necessary boundary and initial conditions to complement the

model equations are given bellow.

At the modular reactor inlet ðbz ¼ 0Þ:Dirichlet boundary conditions:

br ¼ 1 (21a)

q ¼ 0 (21b)

bP ¼ 1 (21c)

bTf ¼ 0 (21d)

At the modular reactor outlet ðbz ¼ 1Þ:

vbrvbz ¼ 0 (22a)

vq

vbz ¼ 0 (22b)

vbP ¼ 0 (22c)

vbTf

vbz ¼ 0 (22e)

Initial conditions ðs ¼ 0Þ

The metal hydride bed is initially assumed to contain no

hydrogen and to be at a temperature equal to the inlet

temperature. The cooling medium is initially assumed to flow

with constant velocity uf towards the z-direction and to be at

a temperature equal to ambient temperature.

3. Simulation results and discussion

The model of a hydrogen modular reactor comprises a set of

integral, partial differential and algebraic equations (IPDAEs).

It is solved numerically using gPROMS [35], a software package

for the modeling and simulation of processes combining

discrete and continuous characteristics. gPROMS allows the

direct modeling of systems described by a combination of

partial and/or ordinary differential equations combined if

necessary with algebraic ones. The solution method is based

on a two-phase method-of-lines approach. In the first phase

the spatial dimensions are discretized in terms of finite

dimensional representations and this reduces the IPDAEs into

sets of differential algebraic equations (DAEs). In the second

phase the DAEs are integrated over the time horizon of

interest using advanced techniques with stiff integrators. The

external software that develops the reactor geometry is

properly linked to the gPROMS model as a foreign object.

In this case spatial discretization is done manually using

central finite differences. The X and Y domains are discretized

using from 12 up to 40 cells/MHT diameter to describe the radial

geometry of the reactor(s) as accurately as possible. The Z

domain was discretized using only 10 points without sacrificing

the accuracy of the model due to the homogeneity of regions

through the Z-direction. All model equations have been imple-

mented in the gPROMS environment. Simulation runs in

gPROMS for the problem under question require about 20 min.

3.1. Model validation

The goal of this task is to validate the Cartesian model by

reproducing the results of the cylindrical model developed by

Kikkinides et al [1]. The base case of an external cooled reactor

with one metal hydride tube, as shown in Fig. 4, was simulated

using both models. All the data used in the simulations are

summarized in Table 1.

The first examined parameter was the temporal evolution of

the total hydrogen mass adsorbed. The results between the two

models and for two cooling fluid velocities, 0 m/s and 1 m/s, are

in excellent agreement, as shown in Fig. 5. Thus, it is concluded

that the present Cartesian model represents adequately the

cylindrical geometry of a metal hydride reactor and for this

a

b

Fig. 8 – Time-space evolution of temperature in the reactor

in the reactor. (a) Reactor with one metal hydride bed, (b)

reactor with five metal hydride beds. Axial z [ 0.5, t [ 50 s.

a

b

Fig. 9 – Number of metal hydride tubes effect on (a) storage

time and (b) cooling time, T0 [ 290 K.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 0 9135

reason it can be used in more complex geometries. A second

step was to study the temporal evolution of the average

temperature in the metal hydride bed, for two cooling fluid

velocities, 0 m/s and 1 m/s. The simulation results produced by

the two models, as shown in Fig. 6, are almost identical. Indeed

the error of the results produced by the 3D Cartesian model

against the results produced by the 2D cylindrical is less than 1%

thus indicating the accuracy of the 3D Cartesian model.

From the above results it is evident that the developed

Cartesian model can represent accurately the cylindrical

geometry of the MH storage tanks provided that we use a fine

enough resolution to describe the reactor’s cross-sectional area.

If the mesh resolution is not fine enough we may expect some

error in the treatment of the boundary layer at the powder bed/

wall interface. Nevertheless, this error is much smaller when

comparing volume averaged quantities including the average

amount of hydrogen stored in the MHT as a function of time. For

this reason extensive simulations have been performed using

a single MHT reactor to study the effect of resolution on the

average amount of hydrogen stored in the MHT. Our studies

have shown that for 15 cells/diameter, the relative error is

approximately 4.6% at small to medium storage times (70% of

maximumcapacity) dropping to 1.7% at large storage times (99%

of maximum storage). The error is still tolerable even for

12/cells/diameter becoming 7% and 2.9%, at small and large

times, respectively. The above trend remains when using more

than 1 MHT, so the impact on the reduction of storage time due

to thepresence ofmulti-tubularconfigurations isnot affectedby

errors in resolution. Obviously improvement of the resolution

can be easily achieved provided that the necessary resources of

computer memory are available.

3.2. Optimization of the thermal efficiency of the metalhydride reactor

There are two main factors affecting the storage efficiency of

a metal hydride reactor. The first factor is the type of the metal

hydride alloy used in the reactor, which defines the maximum

capacity of the tank. The second is the thermal efficiency,

which defines the time required for complete hydrogen

storage. In this case the metal hydride used in the simulations

is LaNi5 with a maximum theoretical capacity of 0.0138 kg

hydrogen/kg alloy. So, this work focused on the optimal heat

management of the reactor in order to minimize the storage

time. More specifically, first the cooling fluid velocity is

Fig. 10 – Studied configurations and the temperature profiles inside the reactor for each configuration. Axial z [ 0.5, t [ 50 s.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 09136

a

b

Fig. 11 – Configuration effect on (a) storage time and (b)

cooling time, T0 [ 290 K.

Fig. 12 – Metal hydride tanks. (a) The base-case tank, (b) the

optimized tank.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 0 9137

optimized followed by the number and positions of the metal

hydride tubes inside the reactor.

3.2.1. Effect of cooling fluid velocitySeveral cooling fluid velocities were studied for the base case

of the modular reactor shown in Fig. 4. The effect of cooling

fluid velocity on the storage time is shown in Fig. 7(a) while

Fig. 7(b) shows the effect of cooling fluid velocity on cooling

time, which is the time required for the temperature to drop to

a certain value. It is clear that the storage time depends

strongly on the amount of hydrogen adsorbed which in this

case is measured as the percentage of the maximum theo-

retical amount. It is clear that a limit cooling fluid velocity

exists, in this case 1 m/s, beyond which neither the storage

time nor the cooling time are affected.

The exothermic absorption process increases the temper-

ature in the metal hydride bed analogous to the formation

enthalpy of the hydride (DH¼�29879 J mol�1 for LaNi5 case).

The heat generated is then transported to the bed’s boundary

by conduction. The heat transfer rate from the metal hydride

alloy to the boundary is defined by the thermal conductivity of

the alloy (lMH¼ 0.66 W m�1 K�1 for LaNi5 case). The cooling

fluid rejects the available amount of heat from the boundary

to the environment. Increasing the cooling fluid velocity

increases the heat transfer rate from the metal hydride

boundary to the environment. The problem is that if the

cooling fluid velocity is increased above 1 m/s there is no

improvement on heat transfer rate because there is no avail-

able heat to be transferred. Conduction thus becomes the

limiting factor.

Despite the existence of a limiting cooling fluid velocity the

time required for 99% hydrogen storage changes from 6910 s

when the cooling fluid velocity is 0 m/s to 4780 s when the

cooling fluid velocity is 1 m/s, indicating an improvement of

30%. Similarly the time required for the temperature inside the

metal hydride tube, to drop to T/T0¼ 1.01 changes from 8200 s

to 5590 s indicating an improvement of 31%. Note that the tank

was designed to store enough amount of hydrogen for 25 km

using the 99% of its theoretical maximum storage capacity.

It should be noted that the assumption of constant cooling

velocity is only justified at high cooling fluid velocities. For

low cooling fluid velocities on the other hand it is better to

include an appropriate heat transfer coefficient to account for

the convective boundary layer effect. This is easy to imple-

ment but for the purposes of the present work, which is to

provide the methodology and tools to perform simulations

and optimizations of hydrogen storage using multi-tube

metal hydride reactor configurations, it will not change any

of the conclusions in a substantial manner. The developed

in-house software is relatively straightforward to build, easy

to implement and maintain and gives accurate results

compared to cylindrical geometries provided that the mesh is

fine enough. Furthermore an even more accurate treatment

would be to employ a commercial software that allows for

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 09138

more suitable meshes and modeling of more complex flows

for the coolant, however, there are currently some compu-

tational difficulties and limitations in interfacing gPROMS

with such a software. Nevertheless this will be systematically

considered in a future work.

3.2.2. Effect of the number of metal hydride tubesIncreasing the cooling fluid velocity is not the panacea for the

reduction of storage time. More complicated, non-concentric

metal hydride reactor geometries should be studied to achieve

a further decrease of the storage time. The same amount of

metal hydride as the concentric reactor shown in Fig. 4, was

divided to 2, 3, 4, 5 and 9 tubes to test multi-tubular metal

hydride reactor geometries. It is essential to keep the amount

of metal hydride the same in order to ensure a constant

overall storage capacity of the tanks under consideration. The

cooling fluid velocity was also kept constant and equal to 1 m/

s for all the cases studied.

Fig. 8 shows that the division of the metal hydride alloy in

several tubes reduces the zones of the temperature gradients

that occur inside the reactor and thus improve the perfor-

mance of the storage time by reducing the time needed for

complete storage and release of hydrogen. Fig. 9 shows the

effect of the number of the metal hydride tubes on the

storage and cooling time. As the number of metal hydride

tubes increases, the time for complete storage decays expo-

nentially. More specifically the time required for 99%

hydrogen storage reduces from 4789 s, when one metal

hydride tube was used, to 2010 s when metal hydride was

divided to three metal hydride tubes, indicating an

improvement of 58%. If the number of tubes is increased to

nine then the time required for 99% storage becomes 1030 s

which represents a total improvement of 78%. Analogously

the time required for the temperature, inside the metal

hydride tubes, to drop to T/T0¼ 1.01 decreased from 5590 s,

when a single metal hydride tube was used, to 2350 s when

three metal hydride tubes were used, indicating an

improvement of 58%. When the number of tubes is increased

to nine then a further improvement of 48.9% on the cooling

time was achieved.

The improvement of cooling and storage time, when the

metal hydride is divided in several tubes is attributed to the

increment of the Surface/Volume ratio as the number of metal

hydride tubes increases. More specifically:

AV¼ 2$p$r$L

p$r2$L¼ 2

r; (23)

Note that the Surface/Volume ratio increases as the metal

hydride tube radius decreases.

3.2.3. Effect of metal hydride tubes position in the reactorIt is important to investigate if there is an ‘‘optimal’’ config-

uration (metal hydride tubes position in the reactor). Hence

we studied five different configurations for the case of the nine

metal hydride tubes reactor, to explore the effect of the posi-

tion of the tubes on the storage and cooling time. The five

studied configurations are shown in Fig. 10.

The effect of process configuration on cooling and

storage time is shown in Fig. 11. It is evident that the posi-

tion of the metal hydride tubes plays an important role

regarding the temperature profile inside the reactor.

Therefore the configuration with the best storage time is

configuration E, where the metal hydride tubes are placed in

such a manner so as the available space of the reactor

(where cooling fluid resides) is more efficiently utilized. So,

if they the tubes are not evenly distributed with respect to

each other and the storage container then the available

cooling fluid is not effectively utilized. For example in

configuration C the right part of the cross-sectional area is

not-well utilized as the cooling fluid in that region flows

without removing enough heat from the reactors. On the

other hand, the upper and lower regions are over-crowded

with MH reactors and as result there is not enough cooling

fluid per unit time to remove the produced heat due to

absorption of hydrogen in the neighboring MH reactors. If

the reactors touch each other we loose interfacial area of

reactor to fluid. More specifically the time required for 99%

hydrogen storage changes from 1030 s, for the configuration

A to 780 s when the configuration E was used, indicating an

improvement of 25%. Similarly the time required for the

temperature to drop to T/T0¼ 1.01 changed from 1200 s to

900 s implying an improvement of 25%.

In summary, it should be emphasized that the aim of the

present work is to provide a systematic methodology and

modeling tools to perform simulations and optimizations of

hydrogen storage using multi-tube metal hydride reactor

configurations. This has been illustrated by several cases. The

main modeling assumptions of this study relate to the use of

a constant cooling fluid velocity, the rather simple represen-

tation of the wall structure and the negligence of the radial

flow within the hydride bed. The latter has indeed been found

negligible in all cases studied. As for the remaining assump-

tions, they do not affect in any significant manner the

conclusions and principal goals of the work. While the quali-

tative tendencies found and the proposed methodology, do

not bear any dependence on these assumptions, it is true that

the actual calculated tank filling times carry an estimated

uncertainty of about 10–15% as a result of the simplifying

assumptions. On the other hand, a worth mentioning limita-

tion of the computational approach has to do with the mesh

resolution employed. As shown by extensive simulations that

were performed specifically to study the effect of resolution,

the errors induced are less than 10% in all cases, i.e. smaller

than the uncertainty imposed by the assumptions.

4. Conclusions

A 3D Cartesian mathematical model has been developed to

study complex metal hydride reactor geometries. The

Cartesian model has been validated and then used to study

the effect of three basic parameters on the storage and

cooling time of the tank. The first parameter is the cooling

fluid velocity whose optimal value found to 1 m/s. A further

step along this study was to explore the effect of the number

of the metal hydride tubes (for the same amount of metal

hydride alloy) on the storage and cooling time. Then the

effect of the number of metal hydride tubes and their posi-

tion was systematically studied. It was shown that the

configuration with nine tubes resulted in the highest

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 1 2 8 – 9 1 4 0 9139

improvement on the storage and cooling time. The justifica-

tion of this trend is due to the fact that as the number of

metal hydride tubes increases the radius of the tubes

decreases and thus the Surface/Volume ratio increases.

Finally, the optimal position of the metal hydride tubes was

also studied. It was illustrated that the tubes should be

placed in a uniform manner to utilize more effectively the

available space in which the cooling medium resides that

leading to a more effective heat management.

The base-case tank and the optimized tank, both capable

for storing enough amount of hydrogen for 25 km range, are

shown in Fig. 12. The time required for 99% hydrogen storage

changed from 6910 s, for the base-case tank with cooling fluid

velocity 0 m/s, to 780 s, when the cooling fluid velocity, the

number of metal hydride tubes and the configuration of the

tubes in the reactor were optimized, indicating a total

improvement of 88.7%. Evidently the above figures could be

further improved if more efficient metal hydrides are applied

and this is the subject for a future work.

Acknowledgements

Financial support from the European Commission under the

DIAMANTE Marie Curie ToK project of FP6 (contract No: MTKI-

CT-2005-029544) is gratefully acknowledged. A. K. Stubos and

C.A. Krokos wish to acknowledge the EU/NESSHY Integrated

Project (SES6-518271) for partial financial support.

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Nomenclature

HM: dim/less hydrogen to metal atomic ratioK: permeabilityL: lengthLe: dim/less Lewis numberP: pressurePet: dim/less thermal Peclet numberPeq: equilibrium pressure

T: temperaturetres: residence timeu: velocityW0: dim/less parameter

Greek lettersb: dim/less parametere: void fractionea: dim/less activation energyr: densityq: dim/less temperature

s: dim/less time

Subscripts0: reference or inletf: cooling fluidg: gass: solidx: axial – Xy: axial – Y

z: axial – Z

Superscripts^: dim/less