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).
Downloaded by UNIV OF CALIFORNIA SANTA BARBARA on May 5, 2016 | http://pubs.acs.org Publication Date: January 1, 1969 | doi: 10.1021/ba-1965-0047.ch006
RECEIVED February 17, 1964.
Young and Linden; Fuel Cell Systems Advances in Chemistry; American Chemical Society: Washington, DC, 1969.
