Mathematical simulation of heat-and-mass transfer processes in “metal hydride-hydrogen-gas impurities” systems

hf. J. Il~drop (

Pergamon

1998

Energy,

Vol. 23, No. 6, pp. 463.468. 1998 Association for Hydrogen Energy Elsevier Science Ltd reserved. Printed in Great Britain

International All

PII: SO36&3199(97)00098-0

rights

0360

MATHEMATICAL SIMULATION OF HEAT-AND-MASS PROCESSES IN “METAL HYDRIDE-HYDROGEN-GAS SYSTEMS YU. F. SHMAL’KO, Institute

V. I. KOLOSOV,

V. V. SOLOVEY,

L. A. KENNEDY*

3199/9X

$19.00+0.00

TRANSFER IMPURITIES”

and S. A. ZELEPOUGA*

for Problems in Machinery of National Academy of Sciences of Ukraine, 2!10 Pozharsky Str., 310046, Kharkov, Ukraine *University of Illinois at Chicago, College of Engineering. 851 South-Morgan Str., 60607-7043. Chicago, IL. USA

Abstract-A mathematical model of heat-and-mass transfer in “metal hydride-hydrogen-gas impurities” systems is presented. This model makes it possible to describe the heat-and-mass transfer taking mto account the convective transfer in the metal&hydride modules, for gas mixtures containing impurities that are inactive or low-active to the hydride-forming material. ‘(7’ 1998 International Association for Hydrogen Energy

INTRODUCTION

to be fixed ones, and hence, in as much as the volume of the MH crystal lattice depends on the hydrogen content, the MH porosity changes during the hydriding process; the gas phase consists of N gaseous products of reac-

The wide use of intermetallic hydrogen sorbents in energy conversion systems poses the problem of correct prediction

of their

main

operating

properties

during

the

whole period of operation in real conditions. The methods of modeling the heat-and-mass transfer processes in metal hydride-hydrogen systems [l&10] do not account for the real conditions, namely primarily the presence of various gaseous admixtures in the hydrogen being processed. This makes it impossible both to design the

elements

of real

energy

conversion

systems

with

tions

perature

a

scale

uf the “MH-gas”

system

being

assume

Me

is a metal

or an alloy.

The

other

the pores,

Darcy’s

law.

hereafter

that

the

kinetic

reactions

obey

x = 1 -exp(--kr)“,

the

(2)

where x is the conversion degree, r is time, II In this case, the hydriding reaction follows

I the

sim-

plified mechanism (1). x == six,,. where .Y is a current hydrogen concentration and x,( is the maximum hydrogen concentration in the p-phase of MH. For the mechanism

or

where

inside

Avraami-Eropheev law with the effective values of the norder reactions (as a rule close to unity) and the k-velocity constant obtained from the experimental data:

(1)

Me H, + ,yv/2H22 MeH r-,

and

The second asssumption makes it necessary to note the following. The true kinetics of the MH-forming (decomposition), as a rule, is unknown, therefore we shall

con-

it MeH,

on the walls

model;

one and obeys

sidered essentially exceeds the dimensions of a single MH particle; 2. in the system there takes place a reversible heterogeneous reaction of the following kind: Me+s/2H2

place

the regime of the gas flow in the MH layer is a viscous

sufficient degree of accuracy and to predict their reliability. We propose a model which eliminates this drawback. When forming the mathematical mbdel of the processes taking place in a fine-dispersion metal hydride (MH) layer through which the hydrogen-containing gas is filtered we will consider that: 1. the spatial

taking

and their concentrations correspond to the state of thermodynamic equilibrium: the medium “MH-gas” during the heat-and-mass transfer process is characterized by a single-tem-

equation lowing: responds

(Ia) components

(1 a) to be close

to the reality,

x = (x-x,)/(x,~-.Y,), to an ultimate

hydrogen

concentration

solid solution. Regarding the fourth assumption

in the gas phase do not react with the MH; 3. the geometric boundaries of the system are considered 463

we take

the fol-

where X, = y and this corin the c(-

it should be noted

464

Y U. F. SHMAL’KO

that the composition of the gaseous products includes L elements (e.g. H, C, N, 0, Ar) and Q molecular components (e.g. H,, O,, H?O, CO, CO,, NZ, NH,, CHJ. The fifth assumption is true in the case where the relaxation times of the processes of heat-and-mass transfer in the gas and solid phases are small compared to the characteristic time of heat-and-mass transfer. ,Since the process relaxation times are less as the sizes of pores grow smaller, then the fifth assumption is true if the mean statistical sizes of the pores are sufficiently small. According to the accepted assumptions and following [8, 1 l-141, let us formulate a system of equations describing the processes of heat-and-mass transfer in a porous reacting MH medium. To obtain the conservation equation let us introduce, following [13, 141, the notions of partial density and the component volume fraction, as well as some other notions of the mechanics of two-phase reacting porous media. The partial (effective) density of the i-th component will be called the limit of the ratio of the mass m, of the i-th component in the volume V and the value of this volume:

ef c/l.

2

/=I x(-)

By using equation (6) the relation metric fraction of the i-th component centration can be found:

EQUATIONS

(’

’

pot, = p,2’,,$ =

P!

between the voluand its mass con-

OF CONTINUITY COMPONENTS

OF

To derive the continuity equations, separate out an elementary volume AV in the porous body (Fig. 1). The metal hydride mass in the separated volume is equal to (p’&A V at the moment of time z and to (P~)~+~~A V at the moment of time z+Ar. Hence the changing of the MH mass to the accuracy of the third-order infinitesimal is equal to (apy/&)AtAV. Since the MH mass changing is due to only the reactions of formation or decomposition of the MH, then the MH mass conservation equation takes the form:

ad

Further, indices 1,2 will be used for denoting the thermodynamic parameters of the metal hydride and gas phase, respectively. The effective density of a multiphase multicomponent medium will be called the sum of all the partial densities: p” = i py. ,= I Hence, the mass concentrations equal to: c, = pplp’

and

of the components

aZ+R,=O

(4)

Here R, is the mass rate of formation or decomposition of the MH. Equations (7) is in essence the equation of chemical kinetics of formation or decomposition of the MH. Therefore, it can be rewritten in an alternative form by using the mass concentration of hydrogen (x) in the metal hydride [8]:

are

R, = pyax/asT,

63)

axjaz = + (x,, ~ x,)Kexp (- KT). ,i, c, = 1’

The volumetric fraction of the i-th component t’; will be called the ratio of the partial density of the i-th component and its true density:

Here x, and xfi are the hydrogen concentrations in the MH on the boundaries of the x- and P-phases, respectively. The dependence of the reaction rate equation (8) on the hydrogen pressure (PH) and temperature (T’) is taken into account by the equation:

vi = PPIP,. (5) Here p, = my (m,/ V,) is the true density of the i-th component and Vi is the volume occupied by this component. From this definition it follows that

v-2 is the volumetric fraction of the gas phase-metal hydride porocity (6). Further, consider the MH porosity to be active, i.e. all the pores can freely communicate with one another and with the surface bounding the MH layer. From the definition of values p and v there follow the next relationships:

u2

Fig.

1. Elementary

volume

A V.

HEAT-AND-MASS

K = K,(P,/P,-

TRANSFER

l]exp(-E,/RT).

(9)

Here P, is the equilibrium hydrogen pressure, Es, is the activation energy and R is the gas constant. The equilibrium dissociation pressure PC =,f(x,T) is defined according to experimental data. Let us derive the continuity equation for the i-th component ofthe gasphase. At the moment of time z the mass of thej-th component in the elementary volume is equal to (p&,A V, and at the moment of time r + AT:

1

a(Pz,%)

-+ s

AV.

7

Hence, the increment of the mass of thej-th component in the elementary volume during the time interval AT to the accuracy of third-order infinitesimals is equal to [CJ(p2,~32)/dsr]ArA V. This increment in particular is due to the inflow of the mass pz,v,(a~,+ W,,],,ArAu,Au, in the direction of axis u, through the left-hand boundary of the elementary volume caused by the convective flow with the mean mass velocity W, and the molecular diffusion ( W,, is the mean diffusion velocity of thej-th component). At the same time the mass: ~z,d+r',

+ W,,)I,,,+

Al*,AzAu,Au,

=

P>,v~(I.I~, + W,,)I,,

+ W,,)].,]Au, +

a(a ~+div(cp,,w+rJ,)=

R,,+R*,,

shall be held within the elementary volume BP’ in the direction of axis u,. Similarly, the masses: _( - & [p2,C,2(~~,Z+ W,2j]A7Au,AuzAuJ and 2

c J,= 0; ,=: I PZ,=PZ;

,+,=R,;

,= I

+ &[P,o;

3

(%+

w,,)l

= R,,+R,,,j

Let us use the notations

=

l,...,N.

(10)

J,, = p>,W,,, & = pz,W,,,

N.

(11)

,$

R, = R,;

(12) we obtain whole:

the continuity aa) 7 s

equation,for

the gas phase as a

+ div (t~?,w) = R, + R,.

(13)

Let us introduce the mass concentration of the gaseous components c,j = pJ/pz into equation (11). As a result we obtain: 2

+wdiv(c2,)

1

+div(tJ,)

= R,,+ R,, acP2 c 21 7 + div(tp,w)

Using equation to the form: tpz 2 [

(13) we transform

+wdiv(c,,)

+div(tJ,)

1

the obtained equation

= R,,+R?,-cZ,(R,+

R:),

I ,j= l,...,

N-l.

(14)

Hence, instead of N equation (11) we can use N- 1 (equations (11) and (13)) or N- 1 (equation (14)) and the relationship: j,

shall be held within the elementary volume in the direction of axes u2 and uj. The change of the mass of the ,j-th component is connected also with the running of different chemical reactions. Denote by R2j and R, j the mass rates of formation (disappearance) of the j-th component as a result of different homogeneous and heterogeneous chemical reactions. Then the equation of mass balance of the gas phase component takes the form:

.j=l,...,

Here values w and J, are vectors. The values of J, are determined with the help of the Stefan-Maxwell relationship [1 l-141. If we take the sum of equation (11)) by ,j from 1 to N, then with account of the equations: .v

AtAu,Au,

flows out through the right-hand boundary of the elementary volume in the direction u,. Hence, the mass equal to:

465

J,, = pz,Wj, which are the projections of the density of the diffusive flow of the j-th component in the direction of the axes u,, u2, uj, as well as the fact that u2 = t. Let us rewrite equation (10) in the divergence form:

EP2

+ &&1(., 1

PROCESSES

(‘2,= 1

(15)

to be calculated on the basis of the law of mass action. The use of methods of linear algebra and chemical kinetics determines the invariants of chemical reactions, i.e. the linear combinations of the concentrations of components which do not change in the course of one or another complex of chemical reactions. To the invariants of chemical reactions there can belong, for instance, the concentration of atoms. The use of the fact that the number of elements does not change in the course of chemical conversions allows simplification of the continuity equations. The fourth assumption about the chemical equilibrium gas flow in the pores allows simplification of the conservation equations for the components. A flow is considered to be in equilibrium if during the residence time of the liquid particle in the

YU. F. SHMAL’KO

466

vicinity of the flow around body, the composition of this particle instantly becomes adjusted to the local value of the particle temperature. From the definition of chemical equilibrium flows it follows that the convective and diffusive transfer practically does not influence the composition of the liquid particle and the concentrations of the components of the liquid particle are a conservation system for separate elements. Indeed, since the element in chemical reactions of the gas phase is not formed and does not disappear, then:

Here ,M,R2,/M, is the mass rate of transition of element 1 into the composition of substancej due to all the chemical reactions, aj is the number of atoms of element I in the composition of thej-th component, M, is the mass of the atom of element I and M, is the molecular mass of thejth component. Hence, by multiplying equation (11) by a&,/M, and by summing with respect to j from 1 to N we obtain the continuity equation for the elements:

et ul.

where A, is the symbol of the independent component, and a,, is the reaction stoichiometric coefficient. The law of mass action which represents the condition of chemical equilibrium of this reaction has the form: P,=

,ii,flp

/t$

(

q=l,_..,

Q

1

(21)

where P, and P, are partial pressures and e is the equilibrium constant by the partial pressures. Hence, besides the L equations for equilibrium flows, we obtain Q independent relationships equation (21) for determining the equilibrium concentrations. EQUATION OF CONSERVATION PHASE IMPULSE (EQUATIONS

OF THE GAS OF MOTION)

Taking into account the sixth assumption, let us use the equation of motion suggested in work [4]: k, (22) w = -IrgradP. Here p is the gas dynamic viscosity coefficient and k, is the permeability of the porous medium. EQUATION

OF ENERGY

CONSERVATION

According to [l l-141 and the accepted assumptions, the energy conservation equation has the form: The gas component of the system “metal hydride-gas” obeys the law of state for an idea1 gas:

cp,pyE

= div (lgradT)-c&w

grad T+ R,Al# + i &,A#,. ,= I

where Z is the compressibility factor. Let us rewrite equation (17) in terms of the partial pressure of the j-th component, taking into account that the diffusive transfer is negligibly small as compared to the convective one [4]:

1= l,...,L

Here cpl is the heat capacity at constant pressure, A@ is the heat of MH formation, A@, is the heats of formation of gas phase components and 1 is the effective thermal conductivity of the metal hydride layer. For the system of conservation equations to be a closed one it is necessary to write down the thermal equation of state:

(19)

Consider a gas mixture consisting of N different individual substances, and being in the equilibrium state. Let us choose a minima1 number L of substances taken from N substances with the help of which we may uniquely write down the reactions of formation of the remaining Q = N-L substances. It is most simple to choose atoms as L substances, though this is not obligatory. For the sake of simplicity in the future we shall call any chosen L substances as independent components (index I) and the Q remaining substances as dependent ones (index q). The reaction of formation of the dependent component B, may be presented in the form:

Bq= 2 Q,A,, I= I

(23)

(20)

(24) and the normalizing

equation: P=?P,

(25)

,=I

As a result, we obtain a system of N+ 5 equations, for determining the N+5 unknowns x, pz, w, P,, P and T (equations (8), (19) and (21)-(25)). In the genera1 case, the thermodynamical and transfer coefficients of the metal hydride-hydrogen system depend on the amount of sorbed hydrogen, and temperature and pressure. In the given mode1 we use relationships obtained from the results of experimental investigations 1151.

HEAT-AND-MASS

EDGE

TRANSFER

CONDITIONS

Specific solving of the equation system requires setting some initial and boundary conditions. A typical operating cycle of an energy-technological metal hydride system (ETMHS) is evident to include four characteristic time areas: an induction period of desorption (t,,J, a desorption period (7J, an induction period of sorption (7,,) and a sorption period (t,). Separation of processes taking place in the system into induction and main ones permits defining their duration. It allows the duration of main processes (sorption and desorption) to be chosen on the basis of a required mean hydrogen supply (reception) rate for the time period. Then drawing an operating cyclogram for a metal hydride plant with continuous hydrogen supply, calculation of the necessary number of successive generators--sorbers becomes possible. Next formulate an edge setting to calculate ETMHS operating cycle parameters for each characteristic period, i.e. we can set edge conditions, or simple conditions, in an initial time period and on the surface that restricts the metal hydride layer. The initial conditions for any characteristic period are the same, but the output (end) parameters for a previous process are input (initial) parameters for the next one. So in an initial time (7 = 0):

PROCESSES

461

sure (P,, = P,,), is assumed to be impermeable without any heat exchange with the ambient medium, i.e.: q = 0,

8 P/&z = 0, ap,jall

P”(U,,w4

p, = P,“(Ul,w4 .Y = x( To, P,,)

(2’3

P2 = PZV,,,P”)


period

of desorption

The surface area S,, when hydrogen pressure on boundary (S,) is less than the operating desorption pres-

T=

T(7)

or q = q(z) or q = a( T,? - TJ,

dP/&r = 0. I aP,/Zn = 0.

(28)

where r is a heat-exchange coefficient, T,y,, is a temperature of a MH on a surface SZ, T,,, is a temperature of a heat carrier. Desorption

period

When the pressure on the surface area S, is equal to = P,, ,, the heat exchange with the ambient medium is realized with the hydrogen flow, i.e.: P,

P,, , = P,,, =

const, q = pzwAh2.

(29)

Here Ah, is gas enthalpy. The correlations of equation (29) are supplemented with continuity conditions for partial pressures and temperatures: (30)

where the subscript 1~’shows the gas parameters on the boundary of S, with the ambient medium and the subscripts shows the MH parameters on the boundary of S, with the MH. The area Sz is the heat supply area and has boundary conditions according to the correlations of equation (28). Induction

period

qf sorption

The surface area S, when the hydrogen pressure on the boundary (S,) exceeds an operating sorption pressure (P, = P,,,,) is assumed to be impermeable without any heat exchange with the ambient medium. The boundary conditions for this area correspond to the correlations of equation (27). The surface area SZ is considered as impermeable and gives up its heat. The boundary conditions correspond to the correlations of equation (28). Sorption

Fig. 2. A plan of dividing a surface S that restricts a MH layer into areas S, and SZ.

(27)

Here q = i ST/an is a heat flow. The surface area S, is assumed to be impermeable with the heat supply according to the known law, i.e.:

P,,,, = P,s,> T,,, = Ts,, p=

= 0.

period

The surface area S, is assumed to be permeable with heat exchange with the ambient medium by means of the hydrogen flow, and the pressure on this boundary area is equal to sorption pressure (P,). It corresponds to relation equation (28). The correlations of equation (29) are supplemented with the continuity conditions for partial pressures and temperatures (equation (30)). The surface area Sz is considered as impermeable with heat giving up. The boundary conditions correspond to the correlations of equation (28).

468

YU.

et al.

F. SHMAL’KO

CONCLUSION The proposed model makes possible a description of heat-and-mass transfer transfer in THMS elements when working on gas mixtures that contain in addition to the main component (hydrogen) any admixtures, either nonactive or of low activity towards a hydride-forming material. The proposed boundary conditions set permits drawing an operating cyclogram for a metal hydride plant with continuous hydrogen supply and calculating the minimum required number of successive generators-sorbers. Besides a computation of the main characteristics of the elements (temperature, pressure, output), the model can serve to evaluate the degree of purity of the hydrogen being given up when it is purified from its admixtures or extracted from any hydrogen-containing mixtures.

5. 6.

7.

8.

9. 10. 11.

12.

REFERENCES 1. Fisher, P. W. and Watson, I. S., International Journal oj Hydrogen Energy S(2), 1983, 1099119. 2. El Ozery, I. A., International Journal of Hydrogen Energy, 1983, S(3), 191-198. 3. Svinarev, S. V. and Trushevskoy, S. N., ZFZh (Russian Engineering Physics Journal) S(6), 1983, 99551002. In Russian. 4. Artemenko, A. N., Vopr. atom. nauki i tekhniki (Problems in Nuclear Science and Engineering), ser. Nuclear-Hydrogen

13. 14.

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Pokver Engineering and Technology. 1987, 3, 61-63 (in Russian Luxenburger, B. and Muller, W., International Journal of Hydrogen Energy 10(5), 1983, 3055315. Tarasevich, V. L., FatceLl G. A. e.a.- Vestsi AN BSSR (Bulletin of Belorussia Academy of Science), 1988, 4, 94498 (in Belorussian). Haller, U., Untersuchung des Warme-und-Stqfltransports in Metallhvdrid-Reaktionsbetten. Fortschr.-Ber. VDI. R.6. N 214, l-2, 1988. Kolosov, V. I. and Solovey, V. V., e.a.-Vopr. atom. nauki i tekhniki (Problems in Nuclear Science and Engineering), ser. Nuclear Engineering and Technology, 1991,3,4660 (in Russian) Liventsov, V. M. and Kuznetsov, A. V., ZFZh (Engineering Physics Journal) 61(6), 199 1, 928-936 In Russian) Jemni, A. and Ben Nasrallah, S., International Journal of Hydrqqen Energy 20(l), 1995, 43-52. Alekseev, V. B. and Grishin, A. M., Physical Gas Dynamics of Reacting Media. Vysshaya Shkola Publ., Moscow, 1985 (in Russian). Sergeev, G. T., Fundamentals of Heat-and-mass Transfer in Reacting Media. Nauka i Tekhnika Publ., Moscow, 1977 (in Russian). Lykov, A. V., Heat-and-mass Transjir. Energia Publ., Moscow, 1971 (in Russian). Nigmatulin, R. I., Fundamentals of the Mechanics of Heterogeneous Media. Nauka Publ., Moscow, 1978 (in Russian). Solovey, V. V. and Artemenko, A. N., e.a.-Vopr. atom. nauki i tekhniki (Problems in Nuclear Science and Engineering), ser. Nuclear-Hydrogen Power Engineering and Technology, 1984, 3, 46-60 (in Russian)