STUDY OF TWO-DIMENSIONAL HEAT AND MASS TRANSFER DURING.pdf

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hr. J. Hydrogen Energy, Vol. 20, No. I I, pp. 881-891, 1995Copyright @ International Association for Hydrogen EnergyElse&r Science Ltd03663199(94)001154 Printed in Great B ritain. All rights reserved0361X3199/95 $9.50 + 0.00

STUDY OF TWO-DIMENSIONAL HEAT AND MASS TRANSFER DURINGDESORPTION IN A METAL-HYDROGEN REACTORA. JEMNI and S. BEN NASRALLAH

Ecole Nationale dIng tnieurs de Monastir , Route de Kairouan, 5000 Monastir, Tunisia

(Received for publication 16 November 1994)Abstract-A study of two-dimensional dynamic heat and mass transfer in a metal-hydrogen reactor during desorptionis presented in this paper. A mathematical model has been established snd solved numerically by the method offinite domains. The numerical simulation is used to present the time-space evolution of the temperature, the pressureand the hydride density in the reactor and to determine the sensitivity to some parameters (reactor geometry, outletpressure, emperature of heating fluid and heat conductivity). This simulation allows us to study the effect of neglectingthe term related to heat transport by convection in the model.

CP4EdhHOHf&sH/MkmMPP,%RReSTtV

NOMENCLATURESpecific heat (J kg- K-l)Particle diameter (m)Activation energy (J mol- )Conductance between hydride bed and heatingfluid (W mm2 K-l)Conductance between outlet face of the reactorand exterior medium (W m- K-l)Reactor height (m)Heat coefficient exchange between solid and gas(Wm- K-l)Hydrogen-to-metal atomic ratioPermeability (m)Hydrogen mass desorbed (kg mm3 s-l)Molecular weight (kg mol.- )Pressure (Pa)Prandtl numberUniversal gas constant (J mol- Km )Reactor ray (m)Reynolds numberSolid-gas exchange area (m mm3)Temperature (K)Time (s)Gas velocity (m s-)

Greek lettersAH Reaction heat of formation (J kg-A r Thickness of control volume (m)At Time increment (s)AZ Thickness o f control volume (m)& PorosityA Thermal conductivity (W m-l K-P Dynamic viscosity (kg m- 1 s- )

P Density (kg mm3)V Kinetic viscosity (m2 s- )w Aver.aging volume (m3)Subscriptse effectiveeq equilibriumf heating fluidg gasge gas effectivem, n spatial indexse solid effectiveS solidss saturatedSuperscriptsg gaseous phasei time indexS solid phaseaverage volumefluctuation

INTRODUCTIONThe metal-hydrogen reactor can be used in many instal-lations (hydrogen accumulator, heat accumulator, com-

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pressor, heat pumps and refrigerators, heat engines, etc).A knowledge of heat and mass transfer in a metal-hydrogen reactor during the absorption and desorptionof hydrogen is very important for their reactor optimiz-ation. Several attempts have been made to analyze thehydride behavior dur ing the sorption phenomena [l-13].Among these studies there are a some studies which haveinvestigated the desorption phenomena [ll-131. The881


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882 HEAT AND MASS TRANSFER

Fig. 1. Metal-hydrogen

one-dimensional model used by Mayer et al. [l l] andthe two-dimensional model presented by Sun and Deng[12] and Pons [13] assume that gas and solid tempera-tures are equal and that gas pressure is constant in thereactor. In this paper we report on a theoretical study o fthe two-dimensional unsteady heat and mass transferduring the hydrogen desorption within a metal-hydrogenreactor. The model used in the present paper takes intoaccount the effect of the difference between the solid andgas temperatures, as well as that of the variation of thegas pressure.We first present the set of equation which govern heatand mass transfer in the reactor during the desorption.The system of equations was resolved numerically by thefinite domain method. The numerical simulation givesthe time-space evolution of different state variables(temperature, pressure, hydride density) in the reactorand determines heat and mass transfer sensitivity to someof these parameters (outlet pressure, temperature ofheating fluid, effective thermal conductivity of the solidand reactor geometry).

FORMULATION OF HEAT ANDMASS TRANSFERThe cylindrical reactor considered in this paper ex-changes heat through lateral and base areas at a constant

temperature and a constant flow rate heating flu id (Fig.1).The reactor is composed of a solid phase (metal-hydride)and a gaseous phase (hydrogen), so it is a discontinuousmedium. The equations which govern heat and masstransfer in the reactor media are obtained by changingthe scale of description in space [14]. We pass from amicroscopic view, in which the averaging volume w issmall compared with the pores, to a macroscopic view,in which the averaging volume is large with respect tothe pores. This scale changing permits conversion fromthe real discontinuous medium to a fictious continuousequivalent medium. Each macroscopic term is obtainedby averaging the microscopic one. We define the averageof some microscopic function 4 as :

where 4i is a quantity associated with the i phase. Wealso define the intrinsic average over a phase i as:

where wi is the volume occupied by phase i in the totalaveraging volume w.The macroscopic differential equations areobtained by taking the average of microscopic equationsover the averaging volume w and using closing assump-tions. The microscopic equations are the mass, the energyand the momentum equations balance in each phase andat the interface. These equations are obtained by usingthermodynamic and mechanic laws of continuous media.Several simplifying assumptions are made in order toobtain a closed set of governing equations at the macro-scopic scale:(1) the viscous dissipation and compression work andnegligible;(2) the gas phase is ideal from the thermodynamic viewpoint;(3) the dispersion term and the tortuousity term canbe modeled as diffusive fluxes.The equations governing heat and mass transfer in ametal-hydrogen reactor are [lo]:

Energy equationsWhen the transfers are two-dimensional and dependon the r-and z-axes, the energy equations become:

for the fluid

- H,,( T; - T:)S + mc,,( T; - Tz) (I)for the solid

+ H8J T; - T:)

x S - m(AH + CpsT; - C,,T;). (2)


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A. JEMNI and S. BEN NASRALLAH 883Momentum equation for the fluid

The gas velocity can be expressed by Darcys law:

Hydrogen mass balanceFor the gas: the mass conservation equation of thehydrogen is:

XJ:)F:at + div(pi V,) := -m. (4)Assuming that the hydrogen is an ideal gas and consider-ing Darcys law, the mass conservation equation of thehydrogen becomes

1

k 1 avg r 3

For the solid: the mass conservation equation of the solidbecomes:

(6)Initial conditions

Initially, the temperature, the pressure and the hydridedensity in the reactor are assumed to be constants:T; = T; = To, Pi = PO, p* = po.

Boundary conditionsTaking into account the symmetry about the z-axis,we deduce that

T (2, 0) = 0; 2 (2, 0) = 0; F (2, 0) = 0. (7)

The wall is impervious and therefore:

F (z, fo = 0; 2 (H, r) = 0. (8)The outlet reactor pressure is assumed to be constantQz = 0, r) = PO.The heat flux continuity through the lateral area andbase area (r = R and z = H) allow us to write thefollowing equations:

Pi a(T:)~ (z, R) = h(T; - T,);w ar

a (H, r) = h(Ti - T,),-%e aZfor the solid

-) a(T)~ (z, R) = h(Tz - T,);se ar-), a(T)se x W, 4 = h(TS - T), (10)

where h is the conductance between hydride bed and heatingfluid and Tf is the temperature of the heating fluid.The heat exchange at the outlet reactor and the flowexisting above a porous surface are more complex be-cause, on the one hand, the outlet face of the reactor incontact with the gas can release a natural convectionmovement which is analogous to the one observed on ahorizontal plate-the metallic edge at the outlet of thereactor in fluences this movement; on the other hand, theconvective fl,~x of the gas at the reactor outlet disturbsthe effect of l.he natural convection movement. In orderto resolve this problem, we introduce a heat coefficientexchange h, between the outlet face of the reactorand the extl:rior medium; so at the outlet, the gastemperature and the solid temperature are related by theequations:

i,, 2 (0, r) = h,(Ti - 7 ;.);

4, 2 (0, r) = h,(Tz - Tf),Reaction kinetic

The expression of the hydrogen mass desorbed per unittime and unit volume, m, given for LaNi,-hydrogendesorption in Ref. [ 111, is:

m = C, exp

where E, is the activation energy for the diffus ion of thehydrogen atoms through the hydride phase and C, is aconstant. Based on the work of Suda and Kobayas [l],E, and C, were equal to 16,473 J mol- and 9.57 s-lrespectively. The expression for the hydrogen equilibriumpressure P,, was deduced by smoothing of the experi-mental results obtained by Uchida et al. [15]:


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N-l

n+ln

n-l

21 1 2 m-l m m+l M-l M

Fig. 2. Numerical grid

-1193.2852@+ 479.9432(;~}

x ,x,(-3323.884(+-&)) x 105. (13)

NUMERICAL RESOLUTIONThe system of equations presented is solved numeri-cally by the method of finite differences based on thenotion of control domain as described by Patankar [ 161.The advantage of this method is to ensure the fluxconservation, and thus to avoid the generation of para-sitical sources. The method consists of defining a grid ofpoints P,,, within the calculated domain and then buildsaround each point a control domain (n, m). In our case,we have considered a regular mesh within the integrateddomain and we have used 35 x 25 nodes. Figure 2 showsthe mesh used in the numerical resolution. The point P,,,is located in the center of the control domain. The value

of the physical scalar 4 at P,,,, and at time t + At willbe denoted &,,. The equations are integrated on thiscontrol domaib and on the interval of time [t, t + At].At the boundary limits of the reactor, the equationsare made discrete by integrating over half of the controldomain by taking into account the boundary conditions.At the corner we have used the quarter of control domain.In order to bring the resulting integral equations backto algebraic equations tying together the solution valuesto the nodes on the grid, we make the following hypothe-ses:(1) the fluxes are constant on the face of the controldoman that is perpendicular to them;

(2) the accumulated terms and the source terms canbe approximated by their averages on the control domaincontructed around P,,,.In order to ensure the stability of the numerical model,we used an implicit scheme and we supposed that the

values ofconvected quantities along the face of the controldomain are equal to their values at the grid point situatedin the upstream (upwind scheme).The first derivatives, which are evaluated on the controldomain faces, are approximated by:w

0i+l

az = ccl,, - 43n+(l/ZLm AZConsidering these assumptions, the form of the result-ing algebraic equations becomes:

+ A,&+,:,, + A,&+:,, + A,. (14)The resulting system of algebraic equations is solvednumerically by the iterative line-by-line method scanning.The choice of this method is justified by its rapidity ofconvergence compared with the point-by-point method.Scanning along the z-axis, we set:

where Z = A,~~~~,, 1 + A,$:,_ 1 + A,.This resolution method consists in the evaluation, usingthe predicted solution, of A,, A,, A, and Z coefficents.Then, the tridiagonal resulting system of equations issolved by the standard Gaussian elimination method. Ifthe difference between the calculated and estimatedsolutions is small, the convergence to the solution isachieved, or else we repeat the procedure of evaluationof the coefficients using the solutions which have alreadybeen calculated until convergence.RESULTS AND DISCUSSION

During the theoretical simulation, the volume of theconsidered reactor is 235.6 cm3, and it is heated by aheating fluid at constant temperature Tf = 313 K. Theoutlet hydrogen pressure is P, = 1.5 x 10 Pa. Thereactor is filled with LaNi,. Initially the solid temperatureis 290 K. The permeability, the porosity, the effectivethermal conductivity of solid and the effective thermalconductivity of the gas are, respectively, lo- rn, 0.5, 1.2W m-l K- and 0.12 W mm Km.The heat exchange coefficient between solid and hy-drogen is given by [ 171:H,, = 2 (2 + l.lP,3 Re0.6),

P(16)


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A. JEMNl and S. BEN NASRAL.LAH 885t = 300 s

t= 60s

Fig. 3. Time-space evolut ion of temperature (R = 0.05 m, H = 0.03 m, T = 313 K, T, = 290 K, P, = 1.5 x lo5 Pa, K = lo-*E = 0.5, i,se= I.2 W m- Km. 4, = 0.12 W mm K-l).m2,


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HEAT AND MASS TRANSFER

t = 1500 5:

38, i-1 t = 9000 Y

I = 5400 Y

Fig. 4. Time-space evolution of solid and gas temperature dilTerence(R = 0.05 m, H = 0.03m, T = 3 13 K, T, = 290 K, P, = 1.5 x lo5Pa, K = 10-s m*, E = 0.5, I.,, = 1.2 W m- K-l, I.,, = 0.12 W m- K-l).


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A. JEMNI and S. BEN NASRALLAH 887

t = IOROO s

Fig, 5. Time-space evolution of hydride metal density CR = 0.05 my H = 0.03 m, Tf = 313 K, To = 290 K, P, = 1.5 X 10 Pa,K = 10-s mZ,e=o.5,;.,,= 1.2~m~ K-,i.,, =0.12 W mm K-).


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HEAT AND MASS TRANSFER

O.10t = 9000 s t = 10800 s

Fig. 6.Time-space evolution of pressure (P - PO) (R = 0.05 m, H = 0.03m, T, = 313 K, To = 290 K, f, = 1.5 x 10 Pa, K = 1~m2, 6 = 0.5, A,, = 1.2 W m-l K-l, I,, = 0.12 W m- K-l).


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A. JEMNI and S. BEN NASRALLAH 889where d, is the particle diameter, P, is the Prandtl numberand Re is the Reynolds number.The results of numerical simulation are presented as athree-dimension curve giving the time-space evolutionduring the desorption of the temperature, the hydridebed density and the pressure. Taking into account thesymmetry about the z-axis, the figures are plotted foronly half of the cylinderFigure 3 shows the temperature distribution of fluidwithin the reactor after 60, 300, 1800, 5400, 9000 and10,800 s. Keeping in mind that the dehydriding reactionwas endothermic, the temperature inside the reactor firs tdecreases, then increases. This is because of the decreasein reaction velocity. Against the wall, the temperature isgreater than in the interior of the reactor, due to theexternal heating fluid. Afte r a substantial period of time,the remaining quantity of hydrogen in the reactor be-comes too small. Consequently, the heat needed for thehydrogen dissocation tends to zero, so the problem comesdown to one of stationary heat conduction inside an inertporous medium.The time-space evolution of the solid temperaturewithin the reactor is similar to the time-space evolutionof gas temperature; however, the difference between thesolid and the gas temperature is important for a shorttime, then decreases with time (Fig. 4) . This difference ismore important near the impervious wall, because thethermal characteristics of the two phases are very differ-ent. The results show that there is no thermal equilibriumin the whole reactor.Fig 5 shows that the mass desorbed is higher near thewall where the temperature is high. This is because thedissocia tion reaction velocity increases with temperature.When time is long enough, the hydride density inside areactor tends to a constant.

c PO = 0.25 x 10e5 a--- PO = 0.50 x 10.PaPO = 1.50 x 10m5 a

Fig. 7. Influence of the hydrogen outlet pressure on the totalmass desorbed.

15 r

- Tf=313K- Tf=343Kv Tf=373K

Fig. 8. Influence of the heating fluid temperature on the totalmass desorbed.

Figure 6 shows the over-pressure inside the reactorwhich is due to the hydrogen desorption. This over-pressure is more noticeable near the heated partition sincethe desorption reaction velocity is high. When time islong, pressure increases within the reactor and tends tothe outlet pressure.The results of the numerical simulation , for the selecteddimensions (R = 5 cm and H = 3 cm), show that the heatand the mass transfers depend on r and z, so neglectingthe two-dim.ensional effect can generate an importanterrorSensitivity to the outlet pressure

From Fig. 7 the effect of the outlet hydrogen pressureon the hydrogen mass desorbed can be seen. We noticethat the desorption reaction is faster when the outletpressure is small.Sensitivity to the temperature of the fluid heating

The evolution of the total reactor hydrogen massdesorbed with time for different temperatures of the fluidheating Tr ~Fig. 8) shows that the reaction velocityincreases with Tr.

Sensitivity to the effective thermal conductivity of the solidThe heat needed for hydrogen desorption is transferredfrom the heal.ed fluid to the inside o f the reactor essentiallyby a conducion process. Thus it is of interest to explorethe effect of thermal conductivity. Figure 9 shows thatthe time required to release hydrogen stored is small forhigh values of the effective thermal conductivity. Accord-ing to this, increasing the effective conductivity of thesolid permits an acceleration of the desorption reaction


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15r t=3600s

- hse = 1.2 W / (Kin)- Ase=SW/(Km)- hse=lOW/(Km)

Time (s)Fig. 9. Influence of the effective thermal conductivity of the solidon the total mass desorbed.

velocity. Several methods have been suggested for in-creasing this conductivity ; for example, addition of wireor metal powder of high thermal diffusivity. Choi andMills [S] have studied this effect on hydrogen absorption.They showed that increasing the thermal conductivity upto a value of about 4 W K- m-i gives a substantialimprovement in the rate of hydrogen absorption, whereasan increase above a value of about 5 W K- m-l yieldslittle fu rther improvement. In the case of desorption, wenote that this result remains valid.

Sensitivity to the reactor geometryThe total hydrogen mass desorbed is plotted for eachreactor dimension as a function of time in Fig. 10. Thereactor volume, the physical characteristcs and theboundary conditions were kept constant during thesimulation. Examining this plot, we note that the totalhydrogen mass desorbed is a minimum at a ratio of thereactor height(H) to the reactor radius (R) equal to unity,so for low values of H/R, the heat and mass transfer in

the reactor are one-dimensional and depend only on z.Under these conditions, the resistance to the transfersalong the z-direction increases with H/R values, hencethe total hydrogen mass desorbed decreases. For largevalues of H/R, the transfers are also one-dimensional anddepend only on r. When H/R rises the resistance to thetransfers according to r decreases and the total massdesorbed increases. For intermediate values of H/R, thetwo-dimensional effects are not negligeable. When H/Rincreases the resistance to the transfers along the z-direction increases and that along the radial directiondecreases. These competing effects explain the existenceof the minimum. This resu lt also applies in the absorptioncase [lo].

0 I I I I5 10 15 20h/rFig. 10. nfluence of the height to the radius ratio of the reactoron the total hydrogen mass desorbed.

Efict of the term of heat transport by convectionIn order to simplify the theoretical model, the numeri-cal simulation during the desorption was realized withand without taking into account the term of heat trans-port by convection (Pi I/,C,, T:). The results for differenttemperatures of heating flu id, pressures and H/R valuesshow that the change of solid and gas temperature is lessthan 1 %, so the convection heat transport term can be

neglected.

CONCLUSIONSA mathematical two-dimensional model describing thestationary heat and mass transfer processes within ametal-hydrogen reactor has been developed and solvedfor hydrogen desorption. The model takes into accountthe effect of the difference between the solid and gastemperatures, as well as that of the variation of gas

pressure. The calculated time-space evolution of pressure,temperature and mass shows the significant influence oftwo-dimensional effects for the reactor dimensions used.According to the numerical simulation results the choiceof reactor geometry, the outlet pressure and the tempera-ture of the heating fluid are very important. These resultsalso show that the thermal equilibrium assumption is notvalid in the whole reactor, that the term of heat transportby convection can be neglected and that an increase inthe effective thermal conductivity of the solid leads tosignificant improvement of performance for the reactor.Therefore, the model can be used as a helpful tool in theoptimization of the metal-hydrogen reactor designs andperformance.


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A. JEMNI and S. BEN NASRA LLAH 891REFERENCES

1. S. Suda and N. Kobayashi, J. Less-Common Metals 73, 119(1980).2. P. D. Goodell, G. D. Sandrock and E. L. Huston, J.km-Common Metals 73, 135 (1980).3. W. Supper, M. Groll and U. Mayer, J. Less-Common Met&104, 279 (1984).4. M. Y . Song and J. Y. Lee, Int. J. Hydrogen Energy 8, 363(1983).5. M. Kawamura, S. Ono and Y. Mizuno, J. Less-CommonMetals 89, 365 (1983).6. A. Jemni, S. Ben Nasrallah, J. Lamloumi and A. PercheronGuegan, In!. Symp. on Metal-Hydrogen Systems, Uppsala,Sweden, 8812 June (1992).7. M. Ram Gopal and S. Srinivasa Murthy, Int. J. HydrogenEnergy 17, 795 (1992).8. H. Choi and A. F. Mills. Int. J. Heat Mass Transfer 33. 1281( 1990)

9. S. Wakao, M. Sekine. H. Endo, T. Ito and H. Kanazawa,J. Less-Common Metals 89, 341 (1983).10. A. Jemni and S. 9. Nasrallah, Int. J. Hydrogen Energy 20 ,.43-52 (1995).11. U. Mayer, M. Groll and W. Supper, J. Less-Common Met&131, 235 (1987).12. D. W. Sun and S. J. Deng, J. Less-Common Metals 155, 271(1989).13. M. Pons, Transferts de chaleur dans la poudre de LaNi, etieur ccuplage avec la reaction, dhydruration, Thesis,Univerrite Paris (1991).14. S. Whitaker, Advances in Heat Transfer, pp. 119-203.Academic Press, New York (1977).15. H. Uchida, K. Temao and Y. C. Huang, Int. Symp. onMetul-jYydrogen Systems, Stuttgart, Germany, 449 Septem-ber (1988).16. S. V. Patankar, Numericul Heat Trunsier Fhid Flow. Hemi-sphere/MacGraw-Hill, New York (1980).17. C. Dang Vu and B. Delcambre, Rec. Phys. Appl. 22, 487(1987).

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