- Email: [email protected]
A polarization model for protonic solid electrolyte fuel cells
0360 3199/87 $3.(K) + 0.00 Pergamon Journals Ltd. © 1987 International Association for Hydrogen Energy.
hit. J. Hydrogen Energy. Vol. 12, No. 3, pp. 151-157. 1987. Printed in Great Britain.
A P O L A R I Z A T I O N M O D E L FOR PROTONIC SOLID E L E C T R O L Y T E FUEL CELLS J. D. CANADAY,T. A.
A. K. KUR1AKOSEand A. AHMAD
WHEAT,
Mineral Processing Laboratory, Energy, Mines & Resources Canada, Ottawa, Ontario, Canada
(Received for publication 9 November 1986) Abstract--A model is developed for predicting the polarization effects of protonic solid electrolyte fuel cells. It is found that concentration polarization can be eliminated by controlling the electrode morphological parameters, porosity and thickness. Consideration of ohmic polarization showed that for an electrolyte of thickness 0.1 cm, the peak power density was increased from 0.0166 to 0.106 watts cm 2 when the conductivity was increased from 0.01 to 0.1 (ohm-cm) -t. For these conductivities an increase in peak power density from 0.106 to 0.716 watts cm 2 was calculated with an electrolyte of thickness 0.01 cm. Implications for fuel cell design are discussed. It is shown that thin-film, thick-film, and microelectronic technologies will be useful in reducing concentration and ohmic polarization. Finally, it is shown that after concentration and ohmic effects have been minimized, activation polarization at the three-phase line of the electrode-electrolyte interface will remain as the rate-limiting process of the fuel cell. Exchange current densities for this process are likely to be =10 3 A cm 2.
NOMENCLATURE A B D F H
L R T W J Jm
Jl. JL ,a
JL,c Jo Jo ,a
Jo.c P
P1 ,P2
/°cell
V Vcell
r~ v,.
Ve
electrolyte cross-sectional area pre-exponential term in Arrhenius equation Fick's Law diffusion coefficient Faraday's constant activation energy for ionic motion in electrolyte electrolyte thickness gas constant temperature activation energy for Arrhenius Law reactions at the electrode-electrolyte interface electrical current density molar flux density limiting current density limiting current density at the anode limiting current density at the cathode exchange current density anode exchange current density cathode exchange current density ideal gas pressure partial pressure o f i d e a l g a s components A, B, C, D with stoichiometric coefficients a, b, c, d partial pressures of reactant gas at the outer surface of the electrode and at the electrode-electrolyte interface respectively fuel cell power density equilibrium potential of the fuel cell operating voltage of the fuel cell anode polarization cathode polarization
V° Veon~ Va,cooe Va,ac t
Vc,eo.c Vc,act Vp AG AG ° AH ° AS ° K
dc/dx dp/dx n n~, nc p r t a 6 c5L
151
O o o0 0o
electrolyte polarization equilibrium potential under standard conditions concentration polarization concentration polarization at the anode activation polarization at the anode concentration polarization at the cathode activation polarization at the cathode total polarization free energy change free energy change under standard conditions standard enthalphy standard entropy equilibrium constant molar concentration gradient partial pressure gradient of reactant gas n u m b e r of electron transfers n u m b e r of electron transfers at the anode n u m b e r of electron transfers at the cathode electrode porosity electrode pore radius electrode tortuosity transfer coefficient electrode thickness electrode thickness corresponding to the limiting current density electrical resitivity of electrolyte electrical conductivity of electrolyte pre-exponential conductivity term fractional surface coverage of adsorbed atoms
152
J. D. CANADAY, T. A. WHEAT, A. K. KURIAKOSE and A. AHMAD INTRODUCTION
Fuel cells have been constructed with a variety of electrolytes including liquid H3PO4, molten carbonate and solid ZrO2 [1]. Recent investigations of solid fl"-A1203 [2] and Nasicon [3, 4] systems have shown that the electrolyte ionic conductivity, which is due to the transport of positively charged species, is comparable to that of the ZrO2 type electrolyte in which ionic charge transport is produced by 02-. These conductivities for the /Y'-A1203 and ZrO2 electrolytes occur at =300°C and -1000°C respectively. This means that the cationic cell can be operated at a higher thermodynamic efficiency [5] than is the case for the ZrO2 cell. The corrosion and electrolyte circulation problems which occur with liquid-electrolyte fuels can also be avoided. Furthermore, the high-temperature materials problems encountered with ZrO2 will not arise. These potential advantages of the protonic solid electrolyte cell over the fuel cells using liquid electrolytes or ZrO 2 suggest the possibility of increasing the cell lifetime. In order for the cationic solid electrolytes to be used in fuel cells, it is necessary that the cations be exchanged for the hydronium, H30 +, species [6]. Rate-limiting mechanisms of the zirconia fuel cell have been analysed and compared. Steele [7] confirmed that, for selected investigations, the choice of experimental conditions and electrode morphology often controls whether the rate is associated with the gas phase mass transfer (concentration polarization) or with the adsorption and subsequent charge transfer process (activation polarization). Similarly, Gut et al. [8] noted that the rate-limiting step may be due to changes in temperature, oxygen pressure and electrode morphology.
From equation (4), this constant may be evaluated by the relation K = exp ( - A G / R T )
(6)
AG ° = AH ° - TAS °
(7)
where The terms AH ° and AS ° represent the standard enthalpy and entropy. The electromotive force (or open circuit equilibrium voltage) is related to the changes in free energy by the equations V - -AG nF
(8)
and v ~ -
-AG°
(9)
nF where n is the number of electron transfers in the reaction and F is Faraday's constant, 96 485 coulombs/ mole. Equations (2), (5), (8) and (9) can be combined to give the electromotive force as a function of gas partial pressure in the form V = ~ - I n L p--v~-~]
For the hydrogen-oxygen fuel cell, in which the reaction is: H 2 (g) + '/202 (g) --~ H20 (g)
RT
Electromotive force of fuel cells For an ideal gas undergoing the reaction aA+ bB~cC+dD
(1)
(11)
the EMF is given by:
F
PV2oJ
V=~-ff In ~PH2 ~ J D E V E L O P M E N T OF THE M O D E L
(10)
(12)
At 300°C and 1 atmosphere, AG ° = -216.4 kJ mole -1 [5] so that from equation (9), V~ = 1.121 volts. It will be assumed here that the fuel cell is operated near atmospheric pressure which means that V=Vo.
the change in free energy is given by [9-12]: Types of polarization AG = AG ° + R T In LW---7~J
(2)
Here, AG ° is the flee energy change under standard conditions and the logarithmic term contains the partial pressures of the gases. At equilibrium
The operation voltage produced by a fuel cell, Vc~, is reduced from its equilibrium value, V, by polarization mechanisms which occur at the anode, Va, the cathode, Vo and within the electrolyte, Vo, according to the relation (7):
AG = O
Vce, = V - Ilia[- v c - v~
(3)
so that: AG ° = - R T In K
(4)
where the equilibrium constant is K -
U c P~D P~A p b
(5)
(13)
These processes are numerous, complex and interacting which means that precise models are impossible. However, one can identify, in an approximate manner, the rate-limiting process in each of the fuel cell components and suggest means of improving the overall performance of the device. At each electrode, concentration polarization and
153
PROTONIC SOLID ELECTROLYTE FUEL CELLS D dP p
Load resistunce
Anod
Etectrotyte
Oxidant 02
Fuel H,
Here, R is the gas constant, T is the temperature, and dP/dx is the partial pressure gradient of the reactant gas across the thickness of the electrode. For a porous structure, the diffusion coefficient will involve considerations [15] of the self-diffusion of a single gas species, interdiffusion of a reactant gas in an inert carrier gas, or Knudsen flow. A non-porous structure will be characterized by the bulk diffusion coefficient. The electrical current density is given by
H~
S = nFJm
HsO ÷
activation polarization may be found so that the total anode and cathode polarization will be of the form Va = Va ..... -}- Va,act
(14)
Vc = V~. . . . . + Vc, ,ct
(15)
and Figure 1 shows schematically the general locations of the polarization processes. Concentration polarization occurs in the bulk portion of the electrodes, regions 1 and 5. Activation polarization is found at the interfaces between the electrodes and the electrolyte, regions 2 and 4. Ohmic polarization results from ionic migration within the electrolyte, region 3. The electrical circuit is completed by the external load resistor, region 6. The combined polarization effects of the electrodes and the electrolyte may be defined as Vp = Va + Vc + Vc
(16)
so that the fuel cell voltage is given by
Vp
Peel! = JVceu
J = nF D PI-P2 p RT 6 t
(18)
JL = nF --D-DP1 P R T 61_ t
Models for concentration polarization of the electrodes have been based on Fick's first law of diffusion (19)
where Jr, is the molar flux density, D is the diffusion coefficient, and dC/dx is the molar concentration gradient. For an ideal gas flowing through an electrode in which the morphology is characterized by a porosity, p, and a tortuosity, t, the molar flux is given by [13, 14]
(23)
taking the ratio of equations (22) and (23) gives
J _ P1-P2 OL JL
Pj
(24)
b
Assuming that the diffusion length is independent of partial pressure, i.e. 6L = 6, then:
When the fuel cell is operating under the closed circuit condition and producing a cui-rent, the system is not at equilibrium. However, the Nernst equation may be applied to the steady state condition in the form: RT
Vconc = ~-~ In
(26)
which together with equation (25) relates (5) the concentration polarization to the limiting current density:
T In Vco,~ = R n--F
Concentration polarization
(22)
where P1 and P2 are the reactant species partial pressures at the outer surface of the electrode and the electrode-electrolyte interface; the term 6 is the electrode thickness. When the pressure at the electrochemical reaction site located near this interface is negligibly small then the limiting current density is
(17)
The power density is the product of the fuel cell current density, J, and the cell voltage
Jm = D dC/dx
(21)
so that from equation (20)
Fig. 1. Schematic diagram of a protonically conducting hydrogen/oxygen fuel cell.
Vco,, = V -
(20)
Jm - R T d x t
JI~ l
(27)
where n = 2 for hydrogen at the anode, and n = 4 for oxygen at the cathode. The limiting current density is evaulated at each electrode for the various diffusion mechanisms.
A ctivation polarization Charge transfer and catalytic processes involving a
154
J. D. CANADAY, T. A. WHEAT, A. K. KURIAKOSE andA. AHMAD
single step reaction are usually described [16] by the Butler-Volmer equation [-~xnaFV] _ e x p IX-oOncFV_)~
J= J0expL---k -vj
U
(28)
where a represents the transfer coefficient; the number of electrons involved in charge transfer reactions at the anode and cathode is given by na and no. The term J0 is the exchange current density which follows the Arrhenius relation [16]: J0 = B exp (-W/RT)
(29)
where B is a constant, and Wis the activation energy for the reaction occurring at each electrode. The electrode reaction site for the charge transfer process is generally thought to occur for the 0 2. conducting ZrO2 cell [17, 18, 19] at the three-phase line in which the electrolyte, electrode, and reactant gas species are in mutual contact. For protonic electrolytes, very little understanding of this mechanism has been obtained [14]. The limiting current density for activation polarization has been shown [20] to be of the form jL =
(30)
J0
1 - 00
where 00 is the fractional surface coverage of adsorbed oxygen atoms. The anode and cathode activation polarization terms are found from equation (28) to be Va,act = ~
In
(31)
and
RT
Vc,act - 4 F ( l _ a 9 In
~oxl
(32)
Ohmic polarization Within the electrolyte, electrical conductivity results from the migration of mobile ionic species, and the ohmic polarization can be described by the relation
Ve = JAR
(33)
where A is the cross-sectional area of the electrolyte and the resistance, R, is given by
R-
LO
a = -T- exp (H/RT)
V~ = oLJ
(35)
The pre-exponential term, o0, depends on dopant concentration. The activation energy for ionic motion is
(36)
Thus, the voltage drop can be reduced either by lowering the electrolyte resistivity or by reducing its thickness.
Total polarization Substitution of equations (27), (31), (32) and (36) into equation (13) relates the fuel cell voltage to all of the polarization processes considered Vcell = V -
In
"
In
2F
Rr --
--
4F
FJ c] Inl-
-F-'~
RT ---In
F ]
(37)
[_Juc-J_J 4F
-pLJ Equation (37) shows that limiting current densities occur in the terms for the concentration and activation polarization at the anode and cathode. It therefore becomes necessary to determine reasonable estimates for these quantities. Table 1 shows the limiting current densities for the four types of concentration polarization [15] in which Knudsen diffusion is considered for pores of radii 10 -6 and 10 -5 cm. The temperature and pressures were assumed to be 300°C and one atmosphere. The bulk electrode thickness was taken to be 10 pm, and the ratio of porosity to tortuosity was assumed to be 0.1. Also shown in Table 1 are the estimates of limiting current densities at the anode and cathode which arise because of activation polarization. The limiting current density for the case of zirconia solid electrolyte fuel cells has been investigated [20] and was found to have the form given by equation (30). It was determined that 0.05 Jo=3 mA cm -2 so that as a first approximation, Jc --~ 1 mAcro -e. For ceria-based electrolytes with porous electrodes, the limiting current density [19] was seen to be of the Table 1. Limiting current densities for concentration and activation polarization mechanisms Limiting current densities (A cm 2) Anode Cathode
(34)
A in which 0 and L are is the resistivity and thickness of the electrolyte. It can also be noted that 0 = o - a , and the conductivity, o, has been shown [21] to generally obey the Arrhenius law O0
represented by H. The ohmic polarization voltage may also be written as
Concentration polarization Self diffusion lnterdiffusion Knudsen diffusion r = 10 6 r = 10-s Bulk diffusion
Activation polarization
82.0 82.9
22.2 44.3
67.1 67.1 =10 I
33.7 33.7 =10-4
210 3
~10-3
PROTONIC SOLID ELECTROLYTE FUEL CELLS
155
1.2
012
I 0
--
08
-
-
Ol
i 0
0"=01
+
O-
=
O08
001
06--
006
O4
004 "%
O2
002
-%
o
l I06
10-5
Eo ,n
-3 I0
q
~0
TO 2
~
d [ampslcm
lO-I
I
10
2}
Fig. 2. Cell voltage and power density for an electrolyte thickness of 0.1 cm. same order of magnitude as the exchange-current density. Depending on the electrode morphology, the electrode process would be dominated by JL or by Jo. Thus, from these two studies, the limiting current density for oxygen-ion electrolytes is estimated to be on the order of 10 -~ A cm -2. Data regarding limiting current density and exchange current density for electrodes on protonically conducting solid electrolytes are unavailable. A recent study [16] has considered the exchange current density to be that of the rate-limiting step. An overall fuel-cell efficiency of 66% was calculated for an assumed exchange current density of 1 mAcm -2. This value will, therefore, be assumed as the limiting current density at the anode of a protonic-electrolyte fuel cell for activation polarization.
1.2
(~'t%
1.0 -
" ~
RESULTS AND DISCUSSION OF MODEL CALCULATIONS Equation (37) has been plotted as a function of current density in Figs 2 and 3 for electrolyte thicknesses of 0.1 and 0.01 cm. ; in each case electrolyte conductivities of 0.01 and 0.1 (ohm cm) -t were considered. The anode and cathode limiting current densities associated with an interdiffusion of hydrogen and oxygen respectively, 82.9 and 44.3 Acm -2, were taken from Table 1. The open circuit voltage of 1.121 volts is nearly maintained to a current density of 10-3 A cm -2. At this value of J, the maximum ohmic polarization voltage, occurring for o = 0.01 (ohm cm) - l and L = 0.1 cm, is 10-° volts. Beyond a current density of 10 -3 Acm 2, the
112 --
1.0
0.4
02
\ + " Logo 10 -6
I0 -5
10 `4
I0 5
i0-2
I0 I
I
I0
d ( O m p s / c m ~)
Fig. 3. Cell voltage and power density for an electrolyte thickness of 0.01 cm.
156
J. D. CANADAY, T. A. WHEAT, A. K. KURIAKOSE and A. AHMAD
value assumed for the anodic and cathodic exchange current densities for activation polarization, the fuel cell voltage is reduced as result of ohmic and activation polarization. Concentration polarization resulting from self diffusion, interdiffusion, or Knudsen diffusion of reactant gas species does not reduce the cell voltage for electrodes with 10% porosity and 10/~m thickness. Bulk diffusion of reactant gas species through a non-porous anode or cathode with limiting current densities of 10 -1 or 10 4 Acm -2 will result in a discontinuous drop in the cell voltage to a value of zero at these current densities. The fuel cell power density as a function of current density, which is calculated from equation (18), is also shown in Figs 2 and 3. The effects of reducing the electrolyte thickness and reducing its conductivity are readily apparent. With an electrolyte thickness of 0.1 cm, the maximum power densities for conductivities of 0.1 and 0.01 (ohm cm) -1 are 0.106 and 0.0166 watts cm -2. Reducing the electrolyte thickness to 0.01 cm results, for conductivities of 0.1 and 0.01 (ohm cm)- 1, in maximum power densities of 0.716 and 0.106 watts cm -2. These calculations also assume an exchange current density of 10 -3 Acm -2. Increasing the electrode thickness from 10/~m to 100 /~m (with a porosity of 10%) will lower the limiting current densities arising from self diffusion to 8.2 and 2.2 Acm -2 at the anode and cathode. These limits are still in excess of the maximum power produced for an electrolyte of thickness 0.1 cm. When the electrolyte thickness in reduced to 0.01 cm, the current density corresponding to the maximum power density and the limiting current density for self diffusion at the anode are approximately the same, 2.2 A c m -2. Results of the model are in qualitative agreement with those found for the oxygen electrode of O - 2 conducting ZrO2-type cells [14]. Here, it was determined that the electrode reaction rates were independent of electrode thickness, for thickness in the range of 0.05 to 10/~m, if the electrodes had good porosity. It may thus be inferred from equation (23) that the limiting current densities arising from the diffusion of the reactant gas through the electrodes in these experiments were greatly in excess of the measured current densities. It would also appear that the electrode porosity was greater than 10%. Therefore, the concentration polarization due to these electrodes, as given by equations (27) and (37) would have been negligible.
nologies can be made on the basis of film thickness and technique. The film-thickness criteria is somewhat arbitrary, but the value of 1/,m (10 -4 cm) may be considered as an upper limit to the thickness of a thin film and as a lower limit to the thickness of a thick film. Evaporation sputtering, and ion implatation are common methods of producing thin films. Silk-screen printing and plasma spray are used to fabricate thick films. Thin film and thick film technologies have been used to construct a number of electrochemical devices including batteries, gas sensors, and fuel cells. For example, sputtering and silk-screen printing have been used to produce /j/fi"-alumina and N A S I C O N solid electrolytes for hydrogen sensors [23]. Sputtering has also been employed in preparing a /J-alumina electrolyte for a solid-state secondary battery [24]. Fuel cell electrodes have been produced by the plasma spraying of perovskite powders [25].
CONCLUSIONS
IMPLICATIONS O F M O D E L RESULTS F O R F U E L CELL D E S I G N
A fuel cell model has been developed in which concentration polarization within the electrodes was based on gas diffusion laws. Ohmic polarization was assumed to hold for the electrolyte, and activation polarization at the electrode-electrolyte interface was characterized by the Butler-Volmer equation. Based on the results of the model calculations, concentration polarization will not limit fuel cell performance for electrode thicknesses of 10 -5 cm and a porosity of 10%. Self diffusion, interdiffusion and Knudsen diffusion of reactant gas species will each result in similar values of limiting current densities at the anode and cathode. Bulk diffusion of the reactant gases through non-porous electrodes, however, will yield negligible power densities. Thus, fundamental electrode morphological parameters will be thickness and porosity. Ohmic polarization will be decreased by reduction in electrolyte resistivity and/or thickness. The reduction of concentration and ohmic polarization can be implemented by the use of thin-film/thickfilm and microelectronic technologies. This will result in a device with solid-state components. Following the successful employment of these technologies to minimize concentration and ohmic polarization by optimizing the electrode morphology and electrolyte resistance, activation polarization at the electrode-electrolyte interface will remain as the limiting mechanism. Reducing the effect of the process will necessitate the development of catalysts with lower exchange current densities.
The electrolytes and electrodes considered in this model have thicknesses of 0.01-0.1 cm and 0.001-0.01 cm respectively. Thus, thin film/thick film and microelectronic technologies will be appropriate for their construction. Reviews of these methods for electrolyte fabrication have recently been given [22, 23]. Distinctions between thin-film and thick-film tech-
REFERENCES 1. H. Ehringer, P. Zegers and H. Hoyaux (eds), Advanced batteries and fuel cells, Commission of the European Communities, Luxembourg (1983). 2. A. Ahmad, T. A. Wheat, A. K. Kuriakose and B. Kindl, J. Can. Cerarn. Soc. 52, 1 (1983).
PROTONIC SOLID ELECTROLYTE FUEL CELLS 3. A. K. Kuriakose, T. A. Wheat and A. Ahmad. In T. A. Wheat, A. Ahmad and A. K. Kuriakose (eds), Progress in Solid Electrolytes, Energy, Mines and Resources, Canada (1983). 4. A. K. Kuriakose, T. A. Wheat, A. Ahmad, and J. Dirocco, J. Am. Ceram. Soc., 179 (1983). 5. A. McDougaU, Fuel cells, Wiley, New York (1976). 6. J. Jensen. In J. Jensen and M. Kleitz (eds), Solid State Protonic Conductors, Odense University Press, Odense, Denmark (1982). 7. B. C. H. Steele. In M. Kleitz and J. Dupuy (eds), Electrode Processes In Solid State lonics, D. Reidel, Dordrecht, Holland (1976). 8. T. M. Gur, I. D. Raistrick and R. A. Huggins, J. Electrochem Soc. 127, 2620 (1980). 9. A. R. Denaro, Elementary Electrochemistry, Butterworths, London (1965) 10. J. R. W. Warn, Concise Chemical Thermodynamics, Van Nostrand Reinhold, London (1969). 11. D. T. Sawyer and J. R. Roberts Jr, Experimental Electrochemistry For Chemists', Wiley, New York (1974). 12. H. Iwahara, H. Uchida and N. Maeda, Solid State lonics, 11,109 (1983). 13. S. Pizzini, M. Bianchi, P. Colombo and S. Torchio, J. Appl. Electrochem. 3, 153 (1973). 14. E. J. L. Schouler, M. L. S. Birchall and J. S. Lundsgaard. In J. Jensen, M. Kleitz (eds), Solid State Protonic
15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
157
Conductors 1, Odense University Press, Odense, Denmark (1982). J. D. Canaday, T. A. Wheat, A. Ahmad and A. K. Kuriakose, J. Appl. Electrochem., submitted for publication. M. F. Bell, M. Sayer, P. S. Nicholson and K. Yamashita, Presented at the Hydrogen Conference, Toronto (1984). D. Y. Wang and A. S. Nowick, J. Eleetrochem. Soc. 128, 55 (1981). D. Braunshtein, D. S. Tannhauser and I. Riess, J. Electrochem. Soc. 128, 82 (1981). M. J. Verkerk, M. W. J. Hammink and A. J. Burggraaf, J. Electrochem. Soc. 130, 70 (1983). T. H. Etsell and S. N. Flengas, J. Electrochem Soc. 118, 1890 (1971). D. K. Hohnke. In Vashishta, Mundy and Shenoy (eds), Fast Ion Transport In Solids, Elsevier-North Holland, (1979). G. Velasco, J. Ph. Schnell, N. Nouailles and M. Croset, Solid State Protonic Conductors I, Odense University Press, Odense, Denmark (1982). Ph. Colomban, H. Perthius and G. Velasco, Solid State Protonic Conductors II, Odense University Press, Odense, Denmark (1983). F. Kanamaru, Technocrat, 18, 50 (1985). H. W. King, G. J. Murphy and J. Wojtowicz, J. Can. Ceram. Soc., in press (1985).
hit. J. Hydrogen Energy. Vol. 12, No. 3, pp. 151-157. 1987. Printed in Great Britain.
A P O L A R I Z A T I O N M O D E L FOR PROTONIC SOLID E L E C T R O L Y T E FUEL CELLS J. D. CANADAY,T. A.
A. K. KUR1AKOSEand A. AHMAD
WHEAT,
Mineral Processing Laboratory, Energy, Mines & Resources Canada, Ottawa, Ontario, Canada
(Received for publication 9 November 1986) Abstract--A model is developed for predicting the polarization effects of protonic solid electrolyte fuel cells. It is found that concentration polarization can be eliminated by controlling the electrode morphological parameters, porosity and thickness. Consideration of ohmic polarization showed that for an electrolyte of thickness 0.1 cm, the peak power density was increased from 0.0166 to 0.106 watts cm 2 when the conductivity was increased from 0.01 to 0.1 (ohm-cm) -t. For these conductivities an increase in peak power density from 0.106 to 0.716 watts cm 2 was calculated with an electrolyte of thickness 0.01 cm. Implications for fuel cell design are discussed. It is shown that thin-film, thick-film, and microelectronic technologies will be useful in reducing concentration and ohmic polarization. Finally, it is shown that after concentration and ohmic effects have been minimized, activation polarization at the three-phase line of the electrode-electrolyte interface will remain as the rate-limiting process of the fuel cell. Exchange current densities for this process are likely to be =10 3 A cm 2.
NOMENCLATURE A B D F H
L R T W J Jm
Jl. JL ,a
JL,c Jo Jo ,a
Jo.c P
P1 ,P2
/°cell
V Vcell
r~ v,.
Ve
electrolyte cross-sectional area pre-exponential term in Arrhenius equation Fick's Law diffusion coefficient Faraday's constant activation energy for ionic motion in electrolyte electrolyte thickness gas constant temperature activation energy for Arrhenius Law reactions at the electrode-electrolyte interface electrical current density molar flux density limiting current density limiting current density at the anode limiting current density at the cathode exchange current density anode exchange current density cathode exchange current density ideal gas pressure partial pressure o f i d e a l g a s components A, B, C, D with stoichiometric coefficients a, b, c, d partial pressures of reactant gas at the outer surface of the electrode and at the electrode-electrolyte interface respectively fuel cell power density equilibrium potential of the fuel cell operating voltage of the fuel cell anode polarization cathode polarization
V° Veon~ Va,cooe Va,ac t
Vc,eo.c Vc,act Vp AG AG ° AH ° AS ° K
dc/dx dp/dx n n~, nc p r t a 6 c5L
151
O o o0 0o
electrolyte polarization equilibrium potential under standard conditions concentration polarization concentration polarization at the anode activation polarization at the anode concentration polarization at the cathode activation polarization at the cathode total polarization free energy change free energy change under standard conditions standard enthalphy standard entropy equilibrium constant molar concentration gradient partial pressure gradient of reactant gas n u m b e r of electron transfers n u m b e r of electron transfers at the anode n u m b e r of electron transfers at the cathode electrode porosity electrode pore radius electrode tortuosity transfer coefficient electrode thickness electrode thickness corresponding to the limiting current density electrical resitivity of electrolyte electrical conductivity of electrolyte pre-exponential conductivity term fractional surface coverage of adsorbed atoms
152
J. D. CANADAY, T. A. WHEAT, A. K. KURIAKOSE and A. AHMAD INTRODUCTION
Fuel cells have been constructed with a variety of electrolytes including liquid H3PO4, molten carbonate and solid ZrO2 [1]. Recent investigations of solid fl"-A1203 [2] and Nasicon [3, 4] systems have shown that the electrolyte ionic conductivity, which is due to the transport of positively charged species, is comparable to that of the ZrO2 type electrolyte in which ionic charge transport is produced by 02-. These conductivities for the /Y'-A1203 and ZrO2 electrolytes occur at =300°C and -1000°C respectively. This means that the cationic cell can be operated at a higher thermodynamic efficiency [5] than is the case for the ZrO2 cell. The corrosion and electrolyte circulation problems which occur with liquid-electrolyte fuels can also be avoided. Furthermore, the high-temperature materials problems encountered with ZrO2 will not arise. These potential advantages of the protonic solid electrolyte cell over the fuel cells using liquid electrolytes or ZrO 2 suggest the possibility of increasing the cell lifetime. In order for the cationic solid electrolytes to be used in fuel cells, it is necessary that the cations be exchanged for the hydronium, H30 +, species [6]. Rate-limiting mechanisms of the zirconia fuel cell have been analysed and compared. Steele [7] confirmed that, for selected investigations, the choice of experimental conditions and electrode morphology often controls whether the rate is associated with the gas phase mass transfer (concentration polarization) or with the adsorption and subsequent charge transfer process (activation polarization). Similarly, Gut et al. [8] noted that the rate-limiting step may be due to changes in temperature, oxygen pressure and electrode morphology.
From equation (4), this constant may be evaluated by the relation K = exp ( - A G / R T )
(6)
AG ° = AH ° - TAS °
(7)
where The terms AH ° and AS ° represent the standard enthalpy and entropy. The electromotive force (or open circuit equilibrium voltage) is related to the changes in free energy by the equations V - -AG nF
(8)
and v ~ -
-AG°
(9)
nF where n is the number of electron transfers in the reaction and F is Faraday's constant, 96 485 coulombs/ mole. Equations (2), (5), (8) and (9) can be combined to give the electromotive force as a function of gas partial pressure in the form V = ~ - I n L p--v~-~]
For the hydrogen-oxygen fuel cell, in which the reaction is: H 2 (g) + '/202 (g) --~ H20 (g)
RT
Electromotive force of fuel cells For an ideal gas undergoing the reaction aA+ bB~cC+dD
(1)
(11)
the EMF is given by:
F
PV2oJ
V=~-ff In ~PH2 ~ J D E V E L O P M E N T OF THE M O D E L
(10)
(12)
At 300°C and 1 atmosphere, AG ° = -216.4 kJ mole -1 [5] so that from equation (9), V~ = 1.121 volts. It will be assumed here that the fuel cell is operated near atmospheric pressure which means that V=Vo.
the change in free energy is given by [9-12]: Types of polarization AG = AG ° + R T In LW---7~J
(2)
Here, AG ° is the flee energy change under standard conditions and the logarithmic term contains the partial pressures of the gases. At equilibrium
The operation voltage produced by a fuel cell, Vc~, is reduced from its equilibrium value, V, by polarization mechanisms which occur at the anode, Va, the cathode, Vo and within the electrolyte, Vo, according to the relation (7):
AG = O
Vce, = V - Ilia[- v c - v~
(3)
so that: AG ° = - R T In K
(4)
where the equilibrium constant is K -
U c P~D P~A p b
(5)
(13)
These processes are numerous, complex and interacting which means that precise models are impossible. However, one can identify, in an approximate manner, the rate-limiting process in each of the fuel cell components and suggest means of improving the overall performance of the device. At each electrode, concentration polarization and
153
PROTONIC SOLID ELECTROLYTE FUEL CELLS D dP p
Load resistunce
Anod
Etectrotyte
Oxidant 02
Fuel H,
Here, R is the gas constant, T is the temperature, and dP/dx is the partial pressure gradient of the reactant gas across the thickness of the electrode. For a porous structure, the diffusion coefficient will involve considerations [15] of the self-diffusion of a single gas species, interdiffusion of a reactant gas in an inert carrier gas, or Knudsen flow. A non-porous structure will be characterized by the bulk diffusion coefficient. The electrical current density is given by
H~
S = nFJm
HsO ÷
activation polarization may be found so that the total anode and cathode polarization will be of the form Va = Va ..... -}- Va,act
(14)
Vc = V~. . . . . + Vc, ,ct
(15)
and Figure 1 shows schematically the general locations of the polarization processes. Concentration polarization occurs in the bulk portion of the electrodes, regions 1 and 5. Activation polarization is found at the interfaces between the electrodes and the electrolyte, regions 2 and 4. Ohmic polarization results from ionic migration within the electrolyte, region 3. The electrical circuit is completed by the external load resistor, region 6. The combined polarization effects of the electrodes and the electrolyte may be defined as Vp = Va + Vc + Vc
(16)
so that the fuel cell voltage is given by
Vp
Peel! = JVceu
J = nF D PI-P2 p RT 6 t
(18)
JL = nF --D-DP1 P R T 61_ t
Models for concentration polarization of the electrodes have been based on Fick's first law of diffusion (19)
where Jr, is the molar flux density, D is the diffusion coefficient, and dC/dx is the molar concentration gradient. For an ideal gas flowing through an electrode in which the morphology is characterized by a porosity, p, and a tortuosity, t, the molar flux is given by [13, 14]
(23)
taking the ratio of equations (22) and (23) gives
J _ P1-P2 OL JL
Pj
(24)
b
Assuming that the diffusion length is independent of partial pressure, i.e. 6L = 6, then:
When the fuel cell is operating under the closed circuit condition and producing a cui-rent, the system is not at equilibrium. However, the Nernst equation may be applied to the steady state condition in the form: RT
Vconc = ~-~ In
(26)
which together with equation (25) relates (5) the concentration polarization to the limiting current density:
T In Vco,~ = R n--F
Concentration polarization
(22)
where P1 and P2 are the reactant species partial pressures at the outer surface of the electrode and the electrode-electrolyte interface; the term 6 is the electrode thickness. When the pressure at the electrochemical reaction site located near this interface is negligibly small then the limiting current density is
(17)
The power density is the product of the fuel cell current density, J, and the cell voltage
Jm = D dC/dx
(21)
so that from equation (20)
Fig. 1. Schematic diagram of a protonically conducting hydrogen/oxygen fuel cell.
Vco,, = V -
(20)
Jm - R T d x t
JI~ l
(27)
where n = 2 for hydrogen at the anode, and n = 4 for oxygen at the cathode. The limiting current density is evaulated at each electrode for the various diffusion mechanisms.
A ctivation polarization Charge transfer and catalytic processes involving a
154
J. D. CANADAY, T. A. WHEAT, A. K. KURIAKOSE andA. AHMAD
single step reaction are usually described [16] by the Butler-Volmer equation [-~xnaFV] _ e x p IX-oOncFV_)~
J= J0expL---k -vj
U
(28)
where a represents the transfer coefficient; the number of electrons involved in charge transfer reactions at the anode and cathode is given by na and no. The term J0 is the exchange current density which follows the Arrhenius relation [16]: J0 = B exp (-W/RT)
(29)
where B is a constant, and Wis the activation energy for the reaction occurring at each electrode. The electrode reaction site for the charge transfer process is generally thought to occur for the 0 2. conducting ZrO2 cell [17, 18, 19] at the three-phase line in which the electrolyte, electrode, and reactant gas species are in mutual contact. For protonic electrolytes, very little understanding of this mechanism has been obtained [14]. The limiting current density for activation polarization has been shown [20] to be of the form jL =
(30)
J0
1 - 00
where 00 is the fractional surface coverage of adsorbed oxygen atoms. The anode and cathode activation polarization terms are found from equation (28) to be Va,act = ~
In
(31)
and
RT
Vc,act - 4 F ( l _ a 9 In
~oxl
(32)
Ohmic polarization Within the electrolyte, electrical conductivity results from the migration of mobile ionic species, and the ohmic polarization can be described by the relation
Ve = JAR
(33)
where A is the cross-sectional area of the electrolyte and the resistance, R, is given by
R-
LO
a = -T- exp (H/RT)
V~ = oLJ
(35)
The pre-exponential term, o0, depends on dopant concentration. The activation energy for ionic motion is
(36)
Thus, the voltage drop can be reduced either by lowering the electrolyte resistivity or by reducing its thickness.
Total polarization Substitution of equations (27), (31), (32) and (36) into equation (13) relates the fuel cell voltage to all of the polarization processes considered Vcell = V -
In
"
In
2F
Rr --
--
4F
FJ c] Inl-
-F-'~
RT ---In
F ]
(37)
[_Juc-J_J 4F
-pLJ Equation (37) shows that limiting current densities occur in the terms for the concentration and activation polarization at the anode and cathode. It therefore becomes necessary to determine reasonable estimates for these quantities. Table 1 shows the limiting current densities for the four types of concentration polarization [15] in which Knudsen diffusion is considered for pores of radii 10 -6 and 10 -5 cm. The temperature and pressures were assumed to be 300°C and one atmosphere. The bulk electrode thickness was taken to be 10 pm, and the ratio of porosity to tortuosity was assumed to be 0.1. Also shown in Table 1 are the estimates of limiting current densities at the anode and cathode which arise because of activation polarization. The limiting current density for the case of zirconia solid electrolyte fuel cells has been investigated [20] and was found to have the form given by equation (30). It was determined that 0.05 Jo=3 mA cm -2 so that as a first approximation, Jc --~ 1 mAcro -e. For ceria-based electrolytes with porous electrodes, the limiting current density [19] was seen to be of the Table 1. Limiting current densities for concentration and activation polarization mechanisms Limiting current densities (A cm 2) Anode Cathode
(34)
A in which 0 and L are is the resistivity and thickness of the electrolyte. It can also be noted that 0 = o - a , and the conductivity, o, has been shown [21] to generally obey the Arrhenius law O0
represented by H. The ohmic polarization voltage may also be written as
Concentration polarization Self diffusion lnterdiffusion Knudsen diffusion r = 10 6 r = 10-s Bulk diffusion
Activation polarization
82.0 82.9
22.2 44.3
67.1 67.1 =10 I
33.7 33.7 =10-4
210 3
~10-3
PROTONIC SOLID ELECTROLYTE FUEL CELLS
155
1.2
012
I 0
--
08
-
-
Ol
i 0
0"=01
+
O-
=
O08
001
06--
006
O4
004 "%
O2
002
-%
o
l I06
10-5
Eo ,n
-3 I0
q
~0
TO 2
~
d [ampslcm
lO-I
I
10
2}
Fig. 2. Cell voltage and power density for an electrolyte thickness of 0.1 cm. same order of magnitude as the exchange-current density. Depending on the electrode morphology, the electrode process would be dominated by JL or by Jo. Thus, from these two studies, the limiting current density for oxygen-ion electrolytes is estimated to be on the order of 10 -~ A cm -2. Data regarding limiting current density and exchange current density for electrodes on protonically conducting solid electrolytes are unavailable. A recent study [16] has considered the exchange current density to be that of the rate-limiting step. An overall fuel-cell efficiency of 66% was calculated for an assumed exchange current density of 1 mAcm -2. This value will, therefore, be assumed as the limiting current density at the anode of a protonic-electrolyte fuel cell for activation polarization.
1.2
(~'t%
1.0 -
" ~
RESULTS AND DISCUSSION OF MODEL CALCULATIONS Equation (37) has been plotted as a function of current density in Figs 2 and 3 for electrolyte thicknesses of 0.1 and 0.01 cm. ; in each case electrolyte conductivities of 0.01 and 0.1 (ohm cm) -t were considered. The anode and cathode limiting current densities associated with an interdiffusion of hydrogen and oxygen respectively, 82.9 and 44.3 Acm -2, were taken from Table 1. The open circuit voltage of 1.121 volts is nearly maintained to a current density of 10-3 A cm -2. At this value of J, the maximum ohmic polarization voltage, occurring for o = 0.01 (ohm cm) - l and L = 0.1 cm, is 10-° volts. Beyond a current density of 10 -3 Acm 2, the
112 --
1.0
0.4
02
\ + " Logo 10 -6
I0 -5
10 `4
I0 5
i0-2
I0 I
I
I0
d ( O m p s / c m ~)
Fig. 3. Cell voltage and power density for an electrolyte thickness of 0.01 cm.
156
J. D. CANADAY, T. A. WHEAT, A. K. KURIAKOSE and A. AHMAD
value assumed for the anodic and cathodic exchange current densities for activation polarization, the fuel cell voltage is reduced as result of ohmic and activation polarization. Concentration polarization resulting from self diffusion, interdiffusion, or Knudsen diffusion of reactant gas species does not reduce the cell voltage for electrodes with 10% porosity and 10/~m thickness. Bulk diffusion of reactant gas species through a non-porous anode or cathode with limiting current densities of 10 -1 or 10 4 Acm -2 will result in a discontinuous drop in the cell voltage to a value of zero at these current densities. The fuel cell power density as a function of current density, which is calculated from equation (18), is also shown in Figs 2 and 3. The effects of reducing the electrolyte thickness and reducing its conductivity are readily apparent. With an electrolyte thickness of 0.1 cm, the maximum power densities for conductivities of 0.1 and 0.01 (ohm cm) -1 are 0.106 and 0.0166 watts cm -2. Reducing the electrolyte thickness to 0.01 cm results, for conductivities of 0.1 and 0.01 (ohm cm)- 1, in maximum power densities of 0.716 and 0.106 watts cm -2. These calculations also assume an exchange current density of 10 -3 Acm -2. Increasing the electrode thickness from 10/~m to 100 /~m (with a porosity of 10%) will lower the limiting current densities arising from self diffusion to 8.2 and 2.2 Acm -2 at the anode and cathode. These limits are still in excess of the maximum power produced for an electrolyte of thickness 0.1 cm. When the electrolyte thickness in reduced to 0.01 cm, the current density corresponding to the maximum power density and the limiting current density for self diffusion at the anode are approximately the same, 2.2 A c m -2. Results of the model are in qualitative agreement with those found for the oxygen electrode of O - 2 conducting ZrO2-type cells [14]. Here, it was determined that the electrode reaction rates were independent of electrode thickness, for thickness in the range of 0.05 to 10/~m, if the electrodes had good porosity. It may thus be inferred from equation (23) that the limiting current densities arising from the diffusion of the reactant gas through the electrodes in these experiments were greatly in excess of the measured current densities. It would also appear that the electrode porosity was greater than 10%. Therefore, the concentration polarization due to these electrodes, as given by equations (27) and (37) would have been negligible.
nologies can be made on the basis of film thickness and technique. The film-thickness criteria is somewhat arbitrary, but the value of 1/,m (10 -4 cm) may be considered as an upper limit to the thickness of a thin film and as a lower limit to the thickness of a thick film. Evaporation sputtering, and ion implatation are common methods of producing thin films. Silk-screen printing and plasma spray are used to fabricate thick films. Thin film and thick film technologies have been used to construct a number of electrochemical devices including batteries, gas sensors, and fuel cells. For example, sputtering and silk-screen printing have been used to produce /j/fi"-alumina and N A S I C O N solid electrolytes for hydrogen sensors [23]. Sputtering has also been employed in preparing a /J-alumina electrolyte for a solid-state secondary battery [24]. Fuel cell electrodes have been produced by the plasma spraying of perovskite powders [25].
CONCLUSIONS
IMPLICATIONS O F M O D E L RESULTS F O R F U E L CELL D E S I G N
A fuel cell model has been developed in which concentration polarization within the electrodes was based on gas diffusion laws. Ohmic polarization was assumed to hold for the electrolyte, and activation polarization at the electrode-electrolyte interface was characterized by the Butler-Volmer equation. Based on the results of the model calculations, concentration polarization will not limit fuel cell performance for electrode thicknesses of 10 -5 cm and a porosity of 10%. Self diffusion, interdiffusion and Knudsen diffusion of reactant gas species will each result in similar values of limiting current densities at the anode and cathode. Bulk diffusion of the reactant gases through non-porous electrodes, however, will yield negligible power densities. Thus, fundamental electrode morphological parameters will be thickness and porosity. Ohmic polarization will be decreased by reduction in electrolyte resistivity and/or thickness. The reduction of concentration and ohmic polarization can be implemented by the use of thin-film/thickfilm and microelectronic technologies. This will result in a device with solid-state components. Following the successful employment of these technologies to minimize concentration and ohmic polarization by optimizing the electrode morphology and electrolyte resistance, activation polarization at the electrode-electrolyte interface will remain as the limiting mechanism. Reducing the effect of the process will necessitate the development of catalysts with lower exchange current densities.
The electrolytes and electrodes considered in this model have thicknesses of 0.01-0.1 cm and 0.001-0.01 cm respectively. Thus, thin film/thick film and microelectronic technologies will be appropriate for their construction. Reviews of these methods for electrolyte fabrication have recently been given [22, 23]. Distinctions between thin-film and thick-film tech-
REFERENCES 1. H. Ehringer, P. Zegers and H. Hoyaux (eds), Advanced batteries and fuel cells, Commission of the European Communities, Luxembourg (1983). 2. A. Ahmad, T. A. Wheat, A. K. Kuriakose and B. Kindl, J. Can. Cerarn. Soc. 52, 1 (1983).
PROTONIC SOLID ELECTROLYTE FUEL CELLS 3. A. K. Kuriakose, T. A. Wheat and A. Ahmad. In T. A. Wheat, A. Ahmad and A. K. Kuriakose (eds), Progress in Solid Electrolytes, Energy, Mines and Resources, Canada (1983). 4. A. K. Kuriakose, T. A. Wheat, A. Ahmad, and J. Dirocco, J. Am. Ceram. Soc., 179 (1983). 5. A. McDougaU, Fuel cells, Wiley, New York (1976). 6. J. Jensen. In J. Jensen and M. Kleitz (eds), Solid State Protonic Conductors, Odense University Press, Odense, Denmark (1982). 7. B. C. H. Steele. In M. Kleitz and J. Dupuy (eds), Electrode Processes In Solid State lonics, D. Reidel, Dordrecht, Holland (1976). 8. T. M. Gur, I. D. Raistrick and R. A. Huggins, J. Electrochem Soc. 127, 2620 (1980). 9. A. R. Denaro, Elementary Electrochemistry, Butterworths, London (1965) 10. J. R. W. Warn, Concise Chemical Thermodynamics, Van Nostrand Reinhold, London (1969). 11. D. T. Sawyer and J. R. Roberts Jr, Experimental Electrochemistry For Chemists', Wiley, New York (1974). 12. H. Iwahara, H. Uchida and N. Maeda, Solid State lonics, 11,109 (1983). 13. S. Pizzini, M. Bianchi, P. Colombo and S. Torchio, J. Appl. Electrochem. 3, 153 (1973). 14. E. J. L. Schouler, M. L. S. Birchall and J. S. Lundsgaard. In J. Jensen, M. Kleitz (eds), Solid State Protonic
15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
157
Conductors 1, Odense University Press, Odense, Denmark (1982). J. D. Canaday, T. A. Wheat, A. Ahmad and A. K. Kuriakose, J. Appl. Electrochem., submitted for publication. M. F. Bell, M. Sayer, P. S. Nicholson and K. Yamashita, Presented at the Hydrogen Conference, Toronto (1984). D. Y. Wang and A. S. Nowick, J. Eleetrochem. Soc. 128, 55 (1981). D. Braunshtein, D. S. Tannhauser and I. Riess, J. Electrochem. Soc. 128, 82 (1981). M. J. Verkerk, M. W. J. Hammink and A. J. Burggraaf, J. Electrochem. Soc. 130, 70 (1983). T. H. Etsell and S. N. Flengas, J. Electrochem Soc. 118, 1890 (1971). D. K. Hohnke. In Vashishta, Mundy and Shenoy (eds), Fast Ion Transport In Solids, Elsevier-North Holland, (1979). G. Velasco, J. Ph. Schnell, N. Nouailles and M. Croset, Solid State Protonic Conductors I, Odense University Press, Odense, Denmark (1982). Ph. Colomban, H. Perthius and G. Velasco, Solid State Protonic Conductors II, Odense University Press, Odense, Denmark (1983). F. Kanamaru, Technocrat, 18, 50 (1985). H. W. King, G. J. Murphy and J. Wojtowicz, J. Can. Ceram. Soc., in press (1985).