A Mathematical Model for Metal Hydride Electrodes

A Mathematical Model for Metal Hydride Electrodes Minna Viitanen

Helsinki University of Technology, Fuel Cell Project, SF-20520 Turku, Finland

ABSTRACT

A mathematical model describing the discharge process of a metal hydride electrode is developed. The model is used to predict the polarization behavior of a metal hydride electrode which can be used as an anode in different types of electrochemical cells. The electrode studied in the model is cylindrical, consisting of spherical metal hydride particles. The mode] takes into account the ohmic losses in the electrolyte and in the metal hydride powder, the diffusion of hydrogen in metal, and the charge transfer reaction at the surface of metal hydride particles. The model is used to analyze how changes in parameters affect the performance of the electrode.

Metal hydride electrodes are used as anodes in electrochemical cells, e.g., in nickel-hydride batteries} -3 LaNi5 and LaNi4Cu are promising materials for use in recharge- able batteries because of their rapid and reversible storage of large quantities of hydrogen} 4 The disadvantage of LaNi~ alloys is their high cost; this has led to studies of low cost alloys in which the rare-earth metal is replaced by less expensive mischmetal (Mm)Y

Metal hydrides are used also as the storage material for hydrogen gas for use in a hydrogen-fueled battery system or as heat pumps. The main problem in the hydride beds is the formation/consumption of heat due to the hydrogen adsorption/deserption and, consequently, the transfer of heat. Numerous experimental and modeling studies have been conducted to describe and to optimize the heat transfer characteristics of the metal hydride bed] -n The kinetics of the adsorptien/desorption of hydrogen gas also has been studied and modeled by many authors. I~

Although many authors have presented different models concerning the behavior of metal hydride, there are few models which describe the metal hydride as an electrode. Since many different factors contribute to the performance of the electrode; since there also are interactions among these factors, the investigation of these phenomena by experimental test is troublesome. Due to the interactions of the phenomena it may be difficult to identify the contribu- tion of a specific factor to the performance of the electrode. The effect of different factors on the performance of the electrode can be determined with a mathematical model.

The mathematical model was formulated to describe and to analyze the polarization behavior of the electrode under various conditions, in terms of physically measurable properties such as exchange current density, diffusion coefficient of hydrogen in the metal, particle size, void fraction, and electrode size. The model takes into account the ionic ohmic losses in the electrolyte, the electronic ohmic losses in the electrode material, the diffusion of hydrogen in metal hydride and the charge transfer reaction at the surface of the metal hydride particles.

A schematic representation of the metal hydride electrode is shown in Fig. 1. The electrode is cylindrical. The inner diameter is denoted by rl and the outer diameter by r2. The height of the electrode is h0. The hydride metal is in the form of powder, and hence in the model, the electrode consists of spherical metal particles which are in contact with each other to provide the electrical connection to the current collector. The void volume of the electrode is filled with electrolyte. The current collector is placed at distance rl. Current density is the current per surface area of the current collector, which is 2~rlho.

The possible reactions taking place at the surface of the metal particles are ~'~4

MH~ ---> xM-H~d~ [1]

xM-H~d ~ + x O H - ~ x e +xH~O [2]

Reaction 1 is the dehydridization reaction, where the hydrogen is released from the hydride phase, resulting in the formation of adsorbed hydrogen atoms, M-Had,. Reaction 2 is the electrochemical reaction in which the current is gen-

936

erated when the adsorbed hydrogen reacts with the hydroxide ions. Thus, the total reaction at the anode is

MH= + xOH- -~ xe + xH20 [3]

During operation, the hydrogen at the surface of the metal particles reacts with the hydroxide ions. The decrease in the hydrogen content at the surface induces the diffusion of the hydrogen in the metal towards the surface. Hence the surface concentration is determined by the reaction rate and by the diffusion rate of the hydrogen. The electrons generated in the reaction are conducted through the metal powder towards the current collector. The electron current determines the potential gradient in the material. The hydroxide ions participating in the reaction also move towards the current collector, reacting as they move along the path. This ionic current determines the potential gradient in the electrolyte.

Potentials in the Electrolyte and Electrode Material According to reaction 2, one hydroxide ion generates one

electron. Hence, the change in ionic current is equal to the change in electronic current, with a different sign, i.e.

~jOH -- ~J~ i.e., jOH + j~ = constant = 112wrlh0 [4] aX ~X '

where jOH andje are ionic and electronic current (A); respec- tively. Since both the negative ions and electrons are mov- ing towards the current collector, which is in the negative direction of the x-axis, the currents have a positive value. Ij is the total current density (A/m 2) at the radius r I at the current collector cylinder, and h0 is the height of the cylinder.

Ohm's law gives the relation between the current and the potential

JoH(X, t) = --KL(2~xho) ( ~ ) [5]

Fig. 1. Metal hydride electrode consisting of spherical metal particles and electrolyte.

J. Electrochem. Soc., Vol. 140, No. 4, April 1993 �9 The Electrochemical Society, Inc.

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J. Electrochem. Soc., Vol. 140, No. 4, April 1993 �9 The Electrochemical Society, Inc. 937

where 6L (V) is the potential in the electrolyte, ~L (1/f~m) is the effective ionic conductivity of the electrolyte, and 2~xho (m ~) is the surface area of the cylinder at distance x. Likewise, for the electronic current we get

j~(x, t) = -KM(2Zrxho) ( ~ ) [6]

where ~M (V) is the potential in the electrode material, and ~ (1/f~m) is the effective electronic conductivity of the metal powder.

The potentials 6~ and d#M are a function of distance x and time t. The dependence on time is because the potentials depend on the reaction rate, which again depends on the concentration of hydrogen on the metal surface, which is natural ly a function of time.

Taking the derivative of Eq. 5 and 6, we get

OJoE(X' t) - --~(2~hO) ~x (X OqbL(X' t) ) [7] Ox Ox

and

or t) = (x o, (X,ox t) ) [8]

The potential distributions in the electrolyte and in the metal hydride are illustrated in Fig. 2.

The value of the constant Ct depends on the meehameal connections to the current collector in the metal hydride electrode. The value of the constant Cro~ depends on the reaction taking place in the reference electrode and also on the mechanical connection, which is needed to measure the potential. These constants do not change during the operation.

The electrode potential vs. reference electrode is defined as ~lo = Crof -- r where Crof is the potential of the reference electrode, and ~ is the potential of the wire connected to the current collector of the electrode. The electrode potential as a function of x is defined by

n(x) = C1 + C~of + eL(x) - CMo [9]

Adding term CM(X) -- CM(X) to the right side of Eq. 9, and defining the potential drop in metal powder by

r i m ( X ) ---- ~)M(X) - - (~M O [10]

Eq. 9 yields

6n(X) - qbM(x) = n(x) - rim(X) - (C, 1 + C~J [11]

The relation is needed in the next Section, when we consider the current-potential equation, which describes the electrochemical reaction. 2

ref

I 1

~ r

I I

r I r 2 Fig. 2. Potential distributions in the electrode material, ~M, and in

the electrolyte, 6,. The measurable electrode potential is ~10 = ~r.f --

In general, owing to the high conductivity of metal the potential drop in metal powder, ~n, is close to zero. How- ever, during the cycling of the electrode the conductivity of the-metal powder might decrease considerably, for exam- pie, due to the oxidation of the surface of the powder or due to the weakened contact between the powder particles.

Although the electrode potential is a function of x, it can be measured only at r2 by placing the reference electrode in the immediate vicinity of the outer surface of the electrode. Thus the measurable electrode potential vs. reference electrode is

no = ~(r2) [12]

Taking the derivatives of Eq. 9 and 10, we see that

\(O'q(x't)~:(Od~L(x't)~and(~)Ox ] \ Ox ] =\~](O6M(x't)~ [13]

Substi tuting these in Eq. 7 and 8 we obtain

OjoH(X't)o~-- EL(2~rho) ~ x 0 ( ~ )

and

[14]

OJe(X't)---KM(2"rrho) 0 ( 0~1;(~ 't)) Ox ~ x [15]

The electrode potentials ~l and ~lm cannot be calculated from these quations since the left side of these equations is still unknown. Thus we have to derive an expression for the left side, for the term Oje(x, t)/Ox, which depicts the current generated across the distance dx. After this, OJoH(X, t)/OX is also known from Eq. 4.

The term -Oil(x, t)/Ox (A/m) is the current generated across the distance dx. Dividing this by the cross-sectional area of the element, we get the current generated per volume, i.e, -Oj~(x, t)lax �9 1/(2~rxho) (A/m3). The number of spherical metal particles in a volume unit is denoted by N (l/m3). Thus the current generated per spherical metal particle is -Oil(x, t)/Ox. 1/(N2~rxho) (A/particle). Dividing this by Faraday's constant F and the number of electrons in- volved in the reaction n, we get the flux out of the surface of the spherical metal particle Fo (mol/s/particle)

_ (aj~(x, t)~ 1 1 Fo(x, t )= \ ~ ] x 2~hoNnF [16]

The terra aj~(x, t)/Ox in Eq. 16 can be solved, and substiimt- ing i t in Eq. 15 we obtain

Fo(x, t) 1 a (Onm(X,X)) =~'xax x Ox [17]

The term NFo [mol/(m3s)] is the reaction rate. Using Eq. 4 we get the equation for the electrode potential

NFo(x,t) = ~L 1 a ( ~ ) X OX X [18]

The equations for the electrode potential ~l(x, t) and for ~lm(X, t) are given by Eq. 17 and 18. Next we consider the boundary conditions for these equations.

Boundary conditions.--It appears from Eq. 4 that the sum of ionic and electronic currents is constant, i.e., jr + Jo~ = constant. At distance rl, where the current collector is placed, the ionic current is zero, since ions cannot be transferred through the current collector. Thus Je represents the total current at distance rl, i.e., Je = (2"srlh0) �9 Using Ohm's law, the boundary condition for ~1~ at rl is obtained

(O~m(r,, t)~ z, o x / - ~m [191

Since the ionic current at rl is zero, according to Ohm's law, the gradient of potential ~1 is zero also. Thus the boundary condition for electrode potential ~ at distance rl is

(O~l(r~, _ 0 [20] t)~ Ox / -



938 J. Electrochem. Soc., Vol. 140, No. 4, April 1993 �9 The Electrochemical Society, Inc.

At distance r2 the situation is the opposite; the electronic current is zero, and the total current is equal to the ionic current. The total current is constant throughout the electrode, but since the electrode is cylindrical, the Cross-sectional area increases as distance x increases. Hence the current density decreases as distance x increases. This must be taken into account when we derive the boundary condition at distance r2.

The total Current is 2~rrlhoI1, where I1 is current density at distance rl. Dividing this by the surface area at distance r2 we get the current density at r2, which is I" = I~ �9 rl/r~. Ac- cording to Ohm's law, the boundary condition is

r2~L [21]

The electronic current at r2 is zero, analogous to the ionic current at rl. Thus we obtain

O'qm(r2, Ox t)] =0 [22]

We now have all four boundary conditions required to solve the potentials ~] and ~]~ from Eq. 18 and 17. It appears from Eq. 10 that

~ ( r i ) = 0 [23]

since at distance r~, the potential +M(x) is equal to SMO- Thus Eq. 23 can be used as a boundary condition instead of Eq. 22.

Current-Voltage Characteristics The local current density generated at the surface of the

spherical metal particle is denoted by i (A/m2). The local current density, i, means the current across the surface of the spherical metal particle. The current density i is dependent on the concentration of hydrogen at the surface of the spherical metal particles, c~(x, t), and on the potential difference between the electrode material and the electrolyte according to the equation

( ~zF t))) i(x, t) = io c : exp \ ~ , - (~Sa(X , t) --4)M(X, [24]

where i~ is the exchange current density (A/m2), c ~ is the initial surface concentration of hydrogen (mol/m3), ~ is the transfer coefficient, z is the stoichiometric number, R is the gas constant (J/kmol), and T is the temperature (K).

Substituting Eq. 11 for the term 4~L -- 4)M in Eq. 24 we obtain

i(x, t) i o ~ e x p / c t z F " " t))) [25] = ~ ('O[X, t ) -- "Gin(X, Cs

where io = io exp [-c~zF/(RT) �9 (C~ + C,~f)]

Dividing i by nF, we get the flux across the surface of the sphere, i (x, t ) /nF (mol/m2s). Further, multiplying this by (l/m), which is the surface area of the electrode per unit volume, and dividing by N(1/m3), the number of spheres per unit volume, we obtain the flux across the surface of the spherical metal particle in unit time (tool/s)

[26] Fo(X, t) - ~ ( ~ / ) N

The local current density generated, i(x, t), now can be solved from this equation and substituted for Eq. 25, giving

NFo(:C, t )=n~ iO ~ e z ~ p [~-/azF'',~],x, t)--'qm(X , t))) [27]

NFo(X, t) is the reaction rate (mol/m~s), giving the flux generated per unit volume per unit time. The reaction rate, NFo, appears on the left side of Eq. (17) and (18), from which the potentials ~] and ~]~, are calculated.

The total current density/1 at radius r~ can be repre- sented with the aid of the reaction rate N F o. Integrating the current generated per uni t volume, i.e., 7i(x, t), over the volume of the metal hydride electrode, we get the total current, and by dividing this by the surface area of the

current collector, the total current density/1 at radius r~ is obtained. Thus

I _ ~ 1 ( ",/i (x, t )dv [281 1 - 2~rrlh ~ �9 j~

where dv is a volume element. The volume element is dv = 21rxhodX. Hence, using Eq. 26 we obtain

Ii =n--F'rl f~i xNF~ t )dx [29]

For the correct solution this equation natural ly must be valid. This equation can be used to check the accuracy of the solution calculated.

Release of Hydrogen from Metal Particles When the hydrogen concentration c~(x, t) at the surface

of the spherical particles is known, we can solve the potentials ~] and ~]m from Eq. 17 and 18 with the aid of Eq. 27, which gives the flux NFo.

In the following set of equations we derive an equation for c (r, t), from which the value of cs(x, t) can be solved. The metal particles are assumed to be spherical. The radius of the spheres is ro. For the diffusion of hydrogen inside the particle we can write

Oc(r, t) D 0 (r 2 Oc(r, t)~ Ot - r 2 o r \ Or / [30]

where c(r, t) is the concentration of hydrogen in metal (tool/m3), and D is the diffusion coefficient of hydrogen (mZ/s). The boundary condition in the middle of the sphere, where r = 0, is on account of the symmetry

oc(O, t) _ o [31] Or

The boundary condition at the surface of the sphere is related to the flux out of the surface, which is further related to the reaction rate. The flux Fo is the total amount of reacting hydrogen coming out of the surface of the sphere in the unit time. Hence we can write

Fo(X, t ) = - 4 ~ r ~ D (Oc(ro, t)~ \ Or / [32]

This boundary condition shows that, since the flux Fo is a function of the x-coordinate, the concentration c is also a function of the x-coordinate. Thus from now on we use the notation c(r, x, t) for the concentration of hydrogen in the metal hydride. The initial condition is that the concentration of hydrogen is constant throughout the sphere; thus

c(r~ x~ O ) -- Cinitia 1 [33]

With Eq. 30-33 we can calculate the concentration of hydrogen in the metal and also the surface concentration

cs(x, t) = c(ro, x, t) [34]

which is needed in Eq. 25 to calculate the local current density generated.

We also can write another expression for the total current density,/1, besides the Eq. 29, where it was calculated as an integral of the reaction rate NFo. If we multiply Eq. 32 by the number of spheres in a unit volume N, we get an expression for the reaction rate, which can be substituted for Eq. 29. Also, if we notice that N = ~/(4~rr02), we get

I1 nF~D [r2 0c (r0, x, - rl "JroX" ~ t ) d x [35]

which can be used for checking the accuracy of the solution.

Summary of the Equations The electrode potential ~](x, t)

NFo(x, t) = - ~ . x Ox x [18]




The boundary conditions for ~(x, t)

( ~ ) _ rill [21] r2KL

The potential ~m(X, t)

EM 1 a ( a ~ ( x , t ) ) [17] Nfo(x, t ) : ~ . ~ x Ox

The boundary conditions for ~m(X, t)

Clam(re, t ) ) I~ [19] OX -- K% ~m(rt) = 0 [23]

The reaction rate NFo(x, t) (mol/(m3s))

N F o ( x , t ) = n ~ i o ~ e x p { ~zF - t)]) [27] \ RT [~(x, t) ~m(X,

The concentration of hydrogen in metal c(r, x, t)

Oe(r ,x , t ) D a ( a c ( r , x , t ) ) at - r 2 Or r2 Or [30]

The boundary and initial conditions for c(r, x, t)

Oc(O, x, t) _ 0 [31] 0r

Fo(x, t )=-4~rr~D {Oc(r~_x, t) ) [32] \ Or

e(r~ x~ 0) = Cimtm 1 [33]

The surface concentration of hydrogen c~(x, t)

c~(x, t) = c(ro, x, t) [34]

Numerical Solution of the Equations The model equations are solved numerically. Equation

30, hydrogen diffusion, is solved by using the implicit finite difference method. The potential Eq. 17 and 18 also are solved by the finite difference method. The equations of the model are complex since the hydrogen diffusion Eq. 30 in- volves the potentials 3 and ~1~, and the potential equations involve the hydrogen concentration. However, the dif- ficulty in solving these equations is because only the gradi- ents at r~ and r2 of the potential ~1 are known. This problem can be avoided by using an extra boundary condition, which is an estimated value of 3 at rl. However, this means that when the solution is calculated, the gradient at r2 is incorrect if the estimated value of ~q at r~ is wrong. Thus new estimates must be made for 3(r~) until the correct gradient at r2 is obtained. This is called the shooting method.

The shooting method is used at each time step, but natu- rally the correct values of ~(r~) of the successive time steps are close to each other, and hence less calculation is needed than at the first step.

Experimental Determination of Parameters Ionic conduc t i v i t y . - -The effective ionic conductivity EL

is determined by measuring the potential drop across a layer of metal hydride powder. The measurement system is shown in Fig. 3.

The current flows through the metal hydride powder, which is inside a tube as shown in Fig. 3. There is a porous membrane at both ends of the tube to keep the powder inside. The potential drop, AV, across the powder is determined by measuring potential with a reference electrode at both ends of the tube.

The effective conductivity of the porous membrane, ~ , is known. At current density, I, the potential drop across the membrane is A V m = I �9 Im/~, where l~ is the total thickness of membrane. Hence, the potential drop across the powder is AVMH = AV - AV~ and the effective conductivity is

EL = I . I/AVMH [36]

tube

metal powder

membrane

Fig. 3. Determination of effective ionic conductivity.

where ! is thickness of the layer of the metal hydride powder.

The effective conductivity of the electrolyte was measured using several different void fractions t3 and the relation

EL = ~. EL [37]

where KL, the conductivity of the electrolyte, is roughly valid and is used in the metal hydride electrode model.

De La Rue and Tobias I~ have suggested the use of the Bruggeman equation, EL = KL ' (1 -- ~)3/2, for predicting the effective conductivity of electrolytes in the presence of spherical nonconductive particles. They experimented with glass beads of different sizes and in the range of 1 < 13 < 0.6. A good correlation was found between their experiments and the Bruggeman equation. The circumstances in the metal hydride electrode are, however, somewhat different from that of the experiments of De La Rue and Tobias. In the metal hydride electrode the void fraction is smaller, ~ < 0.5, the metal particles most probably are not spherical in practice, and the metal particles are conduc- tive. Thus, the Bruggeman equation might not be a good approximation for a metal hydride electrode, although it is evident that the relation given by Eq. 37 is a rough approximation.

Other parameter s . - - The parameters i0 and az are determined by measuring the polarization of the metal hydride electrode with a very small current, I, and while the hydrogen concentration in the metal hydride powder is still close to the initial value. In these circumstances it can be assumed that the potential drop in material ~lm = 0, which is also true due to the high conductivity of the metal hydride electrode, and also due to the surface concentration of hydrogen c(ro) = Cln,tial. From Eq. 25 we get

RT RT . ln(i0~V) [38] ~l = a ~ " i n (I) - a ~

When the electrode potential ~1 is plotted against In (I), a linear function is obtained, which gradient gives a value for ~z, and the intercept with the In (/)-axis gives a value for i0, when the surface area per unit volume ~ and the volume of the electrode V are known.

The diffusion coefficient of hydrogen in metal hydride is obtained from the literature. Lebsanft et al. 16 investigated the proton diffusion in several metal hydrides with quasi- elastic neutron scattering. For the diffusion coefficient of hydrogen in LaNisH6 they obtained at room temperature a value of 6.1 �9 10 -13 m2/s. Fischer et al. n determined the diffusion coefficient by neutron scattering, and the result obtained at room temperature was 1.2 �9 10 -1~ mf/s. Thus the diffusion coefficients differ somewhat, but the average value at room temperature is 3.6 - 10 -12 mf/s. TM The value used in the model is 1. 10 -12 mf/s.

Consideration of Different Rate-Determining Factors of the Problem

Diffusion of hydrogen in metal p a r t i c l e s . - - N e x t we consider & special case. The reaction is assumed to be very fast, so fast that the surface concentration of hydrogen is zero, meaning that whenever a mole of hydrogen reaches the surface, it immediately reacts according to reaction 2.



940 J. Etectrochem. Soc., Vol. 140, No. 4, April 1993 �9 The Electrochemical Society, Inc.

Table I. Parameters used in the calculations.

Parameter Value

D 1 . 1 0 -I2 m2/s Cinitia I 125,000 mol/m a

r 1 2 mm r2 6 mm ho 50 mm

Equations for this special case are the diffusion Eq. 30 and the boundary condition 3 i. The other boundary condition is now, instead of 32 ..

c(r0, x, t) = 0 [46]

W h e n concentra t ion is so lved as a func t ion of t ime, from Eq. 35 we can calculate the total current density as a function of time.

A l t h o u g h this s impl i f ied case does not cons ider the potent ia l s in the metal hydride electrode, it gives a good idea of the value of the maximum current density we can get from the e lectrode w i t h spec i f ied d i m e n s i o n s and part ic le size, since the only l imiting factor here is the diffusion of hydrogen.

Table I shows the parameters used in the following calculations. The results of ca lcu la t ions are presented in Fig. 4. Compar i sons have been m a d e a lso to see the ef fect of the part ic le s ize on the current density. Figure 4 s h o w s that b y decreas ing the part ic le s ize an increased current dens i ty is obtained. This is an obv ious result , s ince wheri w e decrease

C u r r e n l D e n s i t y ( A / c m 2 )

6

5

4

3

2

r i

X X

X X X X

X X X X X

r0=] m m 1 4- rO=08 mm

-~ tO=05 m m

[ ] to=03 rnm

X rO=O.I mm

X X X X x

[]

[] D

50 100 ]50 200 250 300 350 400 450 500 550 600

Time (seconds)

Fig. 4. The maximum current density obtained in circumstances where the only limiting factor is the diffusion of hydrogen inthe metal particle.

Cur ren l Dens i ty ( A / c m 2 J 0

[ ~ ro=l.0 mm

5 -4-- rO=05 mm

re=0.1 mm

i 2 ~

0 , ~ i i i

0 1O 20 30 40 50 60 70 80 90 100

Initial Concenl ra l ion (% ol max)

Fig. 5. The total current density with different hydrogen concentra- tions expressed in percentages of the maximum concentration, thus, as a function of discharge degree.

vo l t age vs Zn [V] - 0 3 6 [

]

- 0 3E i

b " 0 3

0 4 -4-- b = 0 4

-->g:- b = 0 5

-El- r = ]e-6 m

0 4 2 ~ r : lee-6 m

- ~ - r = 50e-6 m

- 0 4 4

0 ,46 i i i I i

0 0.5 1 15 2 2 5 3

cu r ren t [A]

Fig. 6. Potential vs. total current; the effect of void fraction and particle size when ohmic losses are neglected.

v o l t a g e vs Zn [V] - 0 8 9

m 0 4 1

\ \ - - ~ b = 0 4 - 0 . 4 3 " ~ ---t4- b ; 0 5

- 0 45

- 0 . 4 7 I I I I r

0 0 5 1 1.5 2 2 .5 3

current [A]

Fig. 7. Potentials vs. total current; the effect of void fraction.

the d iameter of the meta l part ic les , the surface area per uni t v o l u m e increases .

We also can demons tra te the effect of the in i t ia l c o n cen- trat ion of the surface on the tota l current density. This is s h o w n in Fig. 5, w h e r e the in i t ia l concentra t ion is e x - pressed in percentages of the m a x i m u m concentrat ion .

Diffusion and reaction of hydrogen.- -In this case w e as- sume that the e f fect ive conduct iv i t i e s of the l iquid and the meta l hydr ide are very high; thus the po ten t ia l and the loca l current are cons tant throughout the e lectrode, i.e., ~ = ~(t) and i = i(t). In these c i rcumstances the gradient of hydrogen is determined by the local current density according to equation

( a c ) 1 i ( t ) [40] - D . ~ r=,0=~-~ �9

This is the b o u n d a r y condi t ion that w e use in this spec ia l case. The loca l current dens i ty can be e x p r e s s e d by the tota l current as follows

i(t) = I _ [411 ~ v

Table II. The parameters used in the calculations.

Parameter Value

D 1 . 1 0 ]2m2/s z0 2 . 10 -2:~ A/m 2 uz -3.21 c,,, 125,000 mol/m :~

0.5 r. 125 ~m rl 2 mm r~ 6 mm h0 50 mm




voltage vs Zn [V]

-0 4!~

Table III. The parameters used to calculate the general solution

Parameter Value

- 0 4 2 ~z

-0 .44

- 0 4 6

0 4 8

- 0 5

-0 52 0

100000 1/{Ohm'm)

--~-- 10000 1/(Ohm'm)

--,14-- 1000 1/[Ohm'm)

- B - 100 1/(Ohm'm)

55 1/(Ohm'm)

25 1/(Ohrwm)

5 1/(Ohm*m)

i r r r r

0.5 1 15 2 2 5 3

cu r ren t [A]

Fig. 8. Potential vs. total current; the effect of the conductivity of the electrolyte.

v o l t a g e vs Zn [V]

- 0 . 3 6 " , + r = l e - 6 m

--F- r - IOe-6 m k ~ 0 38 ~ , ~ --,14- r = 50e-6 m

- 0 4

-0.42

-0.46

- 0 4 8 I , I I ,

0 0 5 1 15 2 2 5

current [A]

Fig. 9. Potential vs. total current; the effect of the particle size.

where I is total current (A), ~/is surface area of the metal powder per unit volume (mf/m~), and V is the volume of the electrode (m~).

Using Eq. 30 and boundary conditions 31 and 40, and as the total current is known, the concentration of hydrogen can be calculated. The potential then can be calculated using Eq. 25, since ~ = 0.

Figure 6 shows the potentials of electrodes which have different void fractions (~ = 0.3, 0.4, 0.5) and different particle sizes (r0 = i, I0, 50 ~m). It appears from Fig. 6 that, when the void fraction is decreased, the potential drop decreases. This can be explained by an increase of surface area of the m e t a l p o w d e r , a n d h e n c e a d e c r e a s e of t h e loca l c u r r e n t dens i ty . W h e n t h e p a r t i c l e s ize is d e c r e a s e d , t h e p o t e n t i a l d r o p d e c r e a s e s m o r e m a r k e d l y . T h e d e c r e a s e i n p o t e n t i a l

vo l t age vs Zn [V]

o4i[- ! : : - | ~ experimental ] _0 4 3 / _ - .... . . . . ~ calculated J

/

- 0 . 47 - ~

-049 ~" ~ ^ .

- 0 5 1

- 0 . 5 3

- 0 55 i i , ~ i i i 0 0.2 0.4 0.6 0,8 1 12 14

cur reiqt [A]

Fig. 10. Experimental and calculated polarization curves.

1.6

D 1 �9 10 -12 m~/s i o 2 , 10 23 A /m 2 c~z -3 .21 cin~ 125,000 mol /m 3

0.5 r0 50 ~m rl 2 m m r 6 m m

_h0 50 m m EM (1 -- ~)- 1 �9 l0 s 1/s ~L ~ . 55 i/~m

Table IV. The parameters used in calculations.

Parameter Value

D 1 10 -12 mf/s i0 2 ." 10 -28 A / m 2 c~z -3 .21 cm~ 125,000 mol /m 3

0.5 ro 13 txm r 1 1.5 m m r 2 4 m m h 0 50 m m EM (1 -- ~) - 1 �9 l0 s 1/~lcm K L ~ " 55 i/flcm

d r o p a l so c a n b e e x p l a i n e d b y t h e i n c r e a s e of t h e s u r f a c e a rea .

General Solution of the Problem T h e i n f l u e n c e o f t h e a m o u n t of m e t a l h y d r i d e p o w d e r o n

t h e p e r f o r m a n c e of t h e e l e c t r o d e is s t u d i e d b y c a l c u l a t i n g t h e p o t e n t i a l w i t h d i f f e r e n t v a l u e s of v o i d f r a c t i o n 13. T h e p a r a m e t e r s u s e d i n t h e c a l c u l a t i o n s a r e p r e s e n t e d i n T a b l e III. F i g u r e 7 s h o w s p o t e n t i a l s v s . t o t a l c u r r e n t w h e n t h e v o i d f r a c t i o n is b e t w e e n 0.3 a n d 0.6. I t a p p e a r s t h a t t h e i n f l u e n c e of t h e v o i d f r a c t i o n is v e r y s m a l l . T h e d e c r e a s e of t h e v o i d f r a c t i o n i n c r e a s e s t h e s u r f a c e a rea , a n d i t d e - c r e a s e s t h e e f f e c t i v e c o n d u c t i v i t y of l i q u i d . H o w e v e r , t h e effects of these two factors are opposite; the increase of the suz:faee area decreases the local current density and there- fore decreases the losses of the reaction, and the decrease of the effective conductivity increases the ohmic losses.

The decrease of the void fraction decreases the losses, and this is true also at small current densities, since the ohmic losses are small But at large current densities the situation is reversed, since the ohmic losses increase so much that the benefit of the easier reaction is lost, and thus better performance is achieved at smaller effective density of the metal powder.

The effect of the conductivity of the electrolyte is shown in Fig. 8, where the conductivity changes from 5 i/~m to i00,000 lfllm. As shown in Fig. 8, at a total current of 3 A, the losses due to the finite conductivity of the electrolyte are about 20 mV, when a 7M KOH solution with a conductivity of 55 i/~m is used.

Particle size also has a great effect on the performance of the electrode. As shown in Fig. 9, when the radius of the particle is decreased from i00 p~m to 1 p~m, the decrease in l o s s is a b o u t 40 mV.

Experimental Verification of the Model To verify the model, the calculated polarization curve is

compared with a measured polarization curve. It appears from Fig. i0 that close agreement is obtained between the calculated and measured polarization curves. Two parameters, i 0 and c~z, are determined and all the others are known. The parameters used to obtain the calculated curve are presented in Table IV.

Manuscript submitted May 18, 1992; revised manuscript received Dec. 21, 1992.



942 J. Electrochem. Soc., Vol. 140, No. 4, April 1993 �9 The Electrochemical Society, Inc.

C C1 Cre~

D F F0 ho i i,

::o JOH l lm n N

r r0 r l

r2

R t T V hV h V ~ Avm X

Z

Greek O~

r

LIST OF SYMBOLS concentration of hydrogen in metal, mol/m 3 potential drop due to the contacts at rl, V potential drop due to the contacts and reference at rl, V diffusion coefficient of hydrogen in metal, m2/s Faraday 's constant, 96,487 C/tool reaction rate, mol/s height of the metal hydride electrode, m local current density, A/m 2 exchange current density, A/m 2 current density at r~ electronic current, A ionic current, A thickness of powder, m thickness of membrane, m number of electrons transferred in reaction number of spherical metal particles in volume unit, 1/m 3 radia l coordinate of the sphere, m radius of spherical metal particles, m inner radius of the cylindrical metal hydride electrode, m outer radius of the cylindrical metal hydride electrode, m gas constant, 8.3143 J/(mol �9 K) time, s absolute temperature, K volume of the electrode, m s potential drop across the tube, V potent ial drop across the powder, V potent ial drop across the membrane, V radial coordinate of the cylinder, m stoichiometric number

transfer coefficient void fraction surface area in unit volume, 1/m potential in liquid, V potential in metal, V potential in metal at r 1 electrode potential , V potential drop in metal, V

00 electrode potential at r2, V KL conductivity of liquid, 1/(~m) KL effective conductivity of liquid, I/(~m) ~M effective conductivity of metal powder, i/(~m) ~m effective conductivity of membrane, 1/(~m)

REFERENCES 1. J . J .G . Willems, Phillips J. Res., 39, Suppl. no. 1 (1984). 2. T. Sakai, H. Ishikawa, K. Oguro, C. Iwakura, and H.

Yoneyama, This Journal, 134, 558 (1987). 3. T. Sakai, H. Miyamura, N. Kuriyama, A. Kato, K.

Oguro, and H. Ishikawa, ibid., 137, 795 (1990). 4. G. Bronoel, J. Sarradin, M. Bonnemay, A. Percheron,

J. C. Achard, and L. Slapbach, Int. J. Hydrogen En- ergy, 1, 251 (1976).

5. T. Sakai, K. Muta, H. Miyamura, N. Kuriyama, and H. Ishikawa, in Proceedings of the Symposium on Hy- drogen Storage Materials, Batteries, and Electro- chemistry, D. A. Corrigan and S. Srinivasan, Editors, PV 92-5, p. 59, The Electrochemical Society Soft- bound Proceedings Series, Pennington, NJ (1992).

6. A. Apostolov, N. Stanev, and P. Tcholakov, J. Less-Com- mon Met., 110, 127 (1985).

7. U. Mayer, M. Groll, and W. Supper, ibid., 131, 235 (1987).

8. D. Sun and S. Deng, ibid., 155, 271 (1989). 9. P. Dantzer and E. Orgaz, Z. Phys. Chem. N. E, 164, 1267

(1989). 10. B. Luxenburger and W. Miiller, Int. J. Hydrogen En-

ergy, 10, 305 (1985). 11. G. G. Lucas and W. L. Richards, ibid., 9, 225 (1984). 12. H. BjurstrSm and S. Suda, J. Less-Common Met., 131,

61 (1987). 13. S. H. Lim and J. Y. Lee, Int. J. Hydrogen Energy, 8, 369

(1983). 14. P. H. L. Notten and P Hokkeling, This Journal, 138,

1877 (1991). 15. R.E. De La Rue and C. W. Tobias, ibid., 1{}6,827 (1959). 16. E. Lebsanft, D. Richter, and J. M. TSpler, Z. Phys.

Chem. N.F., 116, 175 (1979). 17. P. Fischer, A. Furrer, G. Busch, and L. Schlapbach,

Hel. Phys. Acta, 50, 421 (1977). 18. J. J. G. Willems and K. H. J. Buschow, J. Less-Common

Met., 129, 13 (1987).



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A Mathematical Model for Metal Hydride Electrodes

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