Fuel Cell Systems

reduced to an elliptic integral of the first kind when α/β attains one of the following values: V 2 , 1, 2. In practical cases, α/β usually will n...
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6 Theory of Polarization of Porous Electrodes

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KAREL MICKA J. Heyrovsky's

Polarographic

Institute, Prague,

Czechoslovakia

A theory of polarization of porous electrodes with constant current is given under the assumption

that concentration changes

of de-

polarizer and supporting electrolyte are negligible. No restrictions are made as to the pore geometry, specific resistances of electrode and electrolyte, and polarization of the solid-liquid interface, the electrode being regarded as a macrohomogeneous

superposition of two con-

tinua, a solid and a liquid one. Some special cases are discussed briefly—namely, small or large specific electrode resistance, small

and

large polarization. It is shown that the same theory can be applied under reasonable

as-

sumptions to the case where the depolarizer is in the gaseous phase, i. e. to a gaseous porous electrode. Some implications as to the structure of the latter are presented.

y h e problem of polarization of porous electrodes with relation to the resistance of the electrode material was solved first by Coleman ( 1 ) i n the case of cylindrical cathodes of Leclanché elements. In his differential equation, a supposition is implicitly included that the faradayic current, u, is directly proportional to polarization of manganese dioxide particles. Therefore, Coleman's expression for faradayic current as a function of distance from electrode surface is substantially in accord with that of Euler and Nonnenmacher (3) who assumed a linear polarization curve of manganese dioxide electrode. Daniel-Bek (2) was the first to deduce fundamental differential equations in the form which is used nowadays. H e gave the solution for two limiting cases—that the faradayic current is an exponential or linear function of polarization. Finally, Newman and Tobias (6) solved the differential equations under the assumption that the 73 Young and Linden; Fuel Cell Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

FUEL CELL SYSTEMS

74

faradayic current is an exponential function of polarization and their results are substantially i n accord with those of Daniel-Bek. The latter assumption is valid only when the thickness of the electrode (the "pore length") is small, the electrolyte conductivity high, and the current large. These restrictions are severe enough, and it is therefore desirable to get a more general solution which would be applicable i n any case. Such a solution can indeed be obtained i n a relatively simple way.

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Mathematics L e t us consider a porous electrode flooded with electrolyte and backed by a metallic conductor of negligible resistivity (Figure 1). Experi­ mental conditions are chosen such that the depolarizer and supporting electrolyte concentrations within the pores can be considered as con­ stant. This applies (for a limited time interval) when the electrode ma­ terial itself undergoes electrode reaction or when the electrolyte is forced to flow sufficiently rapidly through the pores. Also, a physical model of a gaseous porous electrode was proposed (7) for which the following theory is applicable (4).

Ε

χ

It is unnecessary to make any assumptions concerning the structure of the porous electrode except that this structure is sufficiently fine to be regarded as macrohomogeneous. Thus, the electrode may be described as a superposition of two continua—a solid and a liquid one ( 6 ) . A more detailed analysis of this idea is given i n another paper ( 5 ) . F o r potential, φι, of the electrode material, O h m s law holds:

Young and Linden; Fuel Cell Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

6.

MICKA

75

Polarization of Porous Electrodes

(1)

— = —pill dx

where pi is the specific resistance of the porous electrode matrix and h is the electronic current density corresponding to unit area of the electrode cross section perpendicular to the x-axis. A n analogous equation holds for the potential,



dx where p is the resistance of the electrolyte contained i n one cubic centi­ meter of the electrode-that is, the specific resistance of the "electrolyte matrix —and i is the ionic current density corresponding again to unit area of the electrode section. Finally according to Daniel-Bek (2) we may write the following equation for the faradayic current density, D , on the inner pore surface: 2

,,

2

ψ

dx

= SD

(3)

where S is the active surface area of a cubic centimeter of the electrodethat is, the area on which the electrode reaction takes place. L e t us choose for D the following function of polarization ( overvoltage ), E:

= i (^exp

D

βηΞΕ

0

-anFE\ - exp -^r)

. (4) iA

in which Ε = φ — φ fulfills the condition that Ε = 0 when D = 0; i is the exchange current density for reaction R e d ?± O x + > equal to nFk° [Ox] [Red]", k° being the standard rate constant of the electrode reac­ tion proper. Other symbols have their usual meaning. Boundary conditions for Equations 1 to 3 are: χ

2

0

ne

e

e

e

χ = 0: h = /, x = L:

h = 0, φ* = 0,

(5) (6)

and the conservation law of current ii + h = /.

(7)

F o r an anodic current, I > 0, Ε > 0, and D > 0; for a cathodic current, inverse relations apply. The problem defined b y Equations 1 to 7 can be reduced to a single differential equation. Subtracting Equation 2 from 1 yields dE — = P2*2 — Plh = (pi + p2)H — Pi/. dx

Young and Linden; Fuel Cell Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

(8)

76

FUEL CELL SYSTEMS

When differentiating the latter formula and combining the result with Equations 3 and 4, we get cPE = W(pi +

/ βηΞΕ -anFE\ ) ^exp — - exp ) x

P2

/nN

(9)

Let us now introduce the following parameters: u = anFE/RT = aE, λ = \/^2aioS( + p ), h = 2/a\p . 2

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Pi

2

(10)

Equation 9 then becomes 2λ ~

= exp ^ ^ - exp ( - « ) .

2

(11)

Multiplying Equation 11 b y du/dx and integrating, we obtain

{iJ

X2

l

=

{i )

exp

u

+

e

x

p

{

-

u

)

-

c

( 1 2 )

where the integration constant C > 0 is defined b y the condition that when du/dx — 0, then the absolute value of dimensionless polarization ]ti|, attains its minimum \u \: m

C = ~ exp (-Um) + exp ( ~ 0 · β V* /

03)

The solution of Equation 12 may be written in the form du *(* "

= λ J

/β \

(14) +

exp ( —«)

— G

in which s denotes the sign of du/dx and x is the value of χ for which u= w . The integral on the right-hand side of Equation 14 cannot, in general, be expressed by elementary or other known functions; however, it can be reduced to an elliptic integral of the first kind when α/β attains one of the following values: V , 1, 2. In practical cases, α/β usually w i l l not be appreciably different from unity. Instead of Equation 9, we may there­ fore write an approximate one: m

w

2

(PE — « 2ioS(pi + p ) smhaE dx

(9')

2

1

in which a = finF/RT when Ε > 0, and a = anF/RT Thus, instead of Equation 14 we obtain s(x ~ xm)

=

λ

—p-

r

\2 Ju

u

m

-,

when Ε < 0.

du

— \ c o s h u — cosh u

(14') m

Young and Linden; Fuel Cell Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

6.

MICK A

Polarization of Porous Electrodes

77

B y substitution cos ψ = (sinh 3^Wm)/sinh 3^«, k = 1/cosh Y^Um

(15)

Equation 14' takes the form of the final solution (16)

\x - x \ = UF{k,t) m

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in which F(fc, ψ) denotes the elliptic integral of the first kind with the modulus k and amplitude «'·

\1 — k s i n ψ

ο

2

2

Formally, Equation 16 is analogous to that which Winsel (8) derived for the case of pi = 0. The faradayic current as a function of distance, x, from the metallic conductor is given b y D/D

m

= c o s - V λ/l - k sin ψ 2

2

(18)

χ

ψ being defined b y Equation 16 as sin φ = sn (^^-,

*),

09)

where sn denotes Jacobi's elliptic function. Further D stands for 2i sinh «,„, so that jD„,| represents the minimum absolute value of D . A n important measurable quantity is the potential,

Literature Cited (1)

Coleman, J. J.,

Trans. Electrochem. Soc.

90, 545 (1946).

(2) Daniel-Bek, V. S., Zhur. Fiz, Khim. 22, 697 (1948).

Young and Linden; Fuel Cell Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1969.

82

(3) (4) (5) (6) (7) (8)

FUEL CELL SYSTEMS

Euler, J., Nonnenmacher, W., Electrochim. Acta 2, 268 (1960). Micka, K., Collection Czech. Chem. Commun., in press. Ibid. 29, 1998 (1964). Newman, J. S., Tobias, Ch. W., J. Electrochem. Soc. 109, 1183 (1962). Pshenichnikov, A . G., Dokl. Akad. Nauk SSSR 148, 1121 (1963). Winsel, Α., Ζ. Elektrochem. 66, 287 (1962).

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RECEIVED February 17, 1964.

Young and Linden; Fuel Cell Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1969.