COUPLING AND EFFICIENCY OF FUELCELLSAND MEMBRANE DESALINATION
3801
The Degree of Coupling and Efficiency of Fuel Cells and Membrane Desalination Processes
by S. R. CapIan* (Received M a y 7, 1966)
Polyner Department, Weizmann Institute of Science, Rehovoth, Israel
The efficiency of energy conversion in desalination processes and fuel cells depends on the degree of coupling of the flows concerned. Methods for determining the degree of coupling and hence the maximum efficiency of these systems are outlined.
The efficient operation of desalination processes and fuel cells is currently a matter of considerable importance, and consequently it seems worthwhile clarifying some of the considerations involved in assessing efficiency in these systems. Kedem and Caplan have recently shown that the phenomenological description of two coupled flows leads to a definition of their “degree of coupling.”’ This dimensionless parameter q is readily measurable and determines uniquely the maximum efficiency of energy conversion. I n general the efficiency q (which is defined by means of the entropy production function) depends both on q and on the conditions of operation. The extension of these concepts to multiple-flow systems, such as biological membranes, has been discussed by Caplan.2 The degree of coupling between any pair of flows is related to an over-all degree of coupling between sets of flows, which has a particularly simple form when one driving flow or energy input is present. I n this case the maximum efficiency of energy conversion is again uniquely determined. An analysis, in these terms, of membrane desalination processes and fuel cells with side reactions and leaks will be outlined below. For isothermal systems we may conveniently begin with the dissipation function 9. If n irreversible processes take place
n Ji
=
C LijXj j= 1
=
C RijJj j= 1
n
Xi
and that the phenomenological coefficients obey Onsager symmetry. Two dimensionless coefficients are defined as Jij
L d G G
=
(3) Til
=
-R i j/ h
i &
As a consequence of symmetry, l i j = l j i and ri, = rjt. It has been shown that Zv is a normalized force ratio, while rlj is a normalized flow ratio.2 The latter quantity can be envisaged as the extent to which the ith flow is “dragged” by the j t h when all other flows are stopped and the force conjugate to the ith flow is zero, and is accordingly defined as the degree of coupling between the processes i and j. If there are only two terms in 9 r12
= 112 =
q
(4)
(1)
The summation in eq. 1 must always contain a t least one positive term, which may be regarded as the energy input rate. If the nth term represents this spontaneous process, it is seen that
where J , and X i refer to conjugate flows and forces and diS/dt is the rate of entropy production- It is assumed that the flows and forces are linearly related
* Biophysical Laboratory, Harvard Medical School, Boston 15, Mass. (1) 0. Kedem and S. R. Caplan, Trans. Faraday SOC.,61,1897(1965). ( 2 ) S. R. Caplan, J. Theoret. Biol., in press.
n
= T(diS/dt) =
1
JiXt
20
Volume 69, Number 11 A’ovembeT 1965
S. R. CAPLAN
3802
n-1
=
-E JiXiIJnXn ill
and the condit’ionfor energy conversion is 0 When n = 2, one finds the relations’
(5)
6q6
1.
Thus, p and qmax may be determined, in the case of a linear two-flow system with constant driving force, by the ratio of fuel consumption rates in the two stationary states corresponding to q = 0. The first of these, when J i = 0, was termed for convenience “static head” (it is sometimes known as the stationary state of first order); the second, when X z = 0, was termed “level flow.” If n > 2, the over-all degree of coupling, although not in general independent of the forces or the flows, reaches a maximum value in a particular series of stationary statese2 These “maximum coupling states” are states of minimal entropy production under conditions of energy conversion and include as limits the states of static head (all J , = 0) and level flow (all X i = 0 ) . The latter two states may be used to measure p and qmaxby eq. 6 and 7. In the case n = 3
q2 = ( d
+ + 2 ~ 1 3 r 2 d / ( l - ~12) r232
(8)
hfaximum coupling states are characterized by the following two mutually dependent conditions: the J t are in the proportions (relative to one another) which they assume at level flow, and the X t are in the proportions they assume at static head. There is no other series of stationary states in which the relative proportions of both the J i and the X t remain fixed. I n general phenomenological coefficients are functions of the parameters of state and are not constant. However, it is possible to determine p over a narrow range of maximum coupling states by means of the relation2
bination) may also be evaluated by eq. 9 and 10. The degree of coupling in such circumstances is less than p. Membrane Desalination Processes It is useful to examine the over-all degree of coupling in a clear-cut physical system not unrelated to biology, For the system chosen we can express q2 in terms of the practical phenomenological coefficients of Kedem and Katchalsky,s proeiding the membrane separates aqueous solutions of a single salt (and no other permeable solute is present). Electrodes reversible to one of the ions are immersed in the solutions. We shall use the practical coefficients without further explanation. The three flows involved are (1) volume flow, (2) salt flow, and (3) current flow. The dissipation function is 9 = J,(Ap
Any system in which the dissipation function has been reduced to two terms by maintaining all driven processes but the ith at static head or level flow (in any comThe Journal of Physical Chemistry
(11)
where A p and AT are mechanical and osmotic pressure differences, and cs is the average concentration of the permeable salt. For the sake of simplicity, we define the following dimensionless parameters, all of which can be expressed in terms of lt, or rij only
- K / K ‘ = P 2 ~ / L ,=, 11S2 w = w ’ / w - 1 = cs(l - u)2L,/w T = T I ’ / T ~ - 1 = cS(l - u)PvIzIF/T1 K = 1
(12)
where the coefficients involved are: K , K ’ , electrical conductance under different restrictions; w, w’, salt , transpermeability under different restrictions; T ~ T~‘, port number under different restrictions (of ion not participating in electrode reaction) ; p, eIectroosmotic permeability; u, reflection coefficient ; L,, filtration coefficient. Three cases may now be considered Driving process J,(Ap
- AT) 81’ = (K
+ W)/(1 + W )
(13)
Driving process J s A r S / c s
“’ = {l where any flow i has been chosen among the “driven” flows, and the partial derivatives, forces, and flows refer to some maximum coupling state within the range. A t static head this reduces to
- A T ) + J s A ~ s /+~ Is E
Driving process I E
T2(1 - K) W ( T 2 2KT K)
+
qa2 = K ( T *
+
+ W)/ (T2
+ KW)
(15)
However, it is not particularly useful to regard J , . ( A p - A T ) as an energy source. In practical salt filtration procedures (usually referred to as “hypefiltration” ~s posior “reverse osmosis”) the term J s A ~ s /remains tive in the dissipation function. One is really con(3) 0.Kedem and A. Katchalsky, Trans. Faraday SOC.,59, 1918, 1931 (1963).
COUPLING AND EFFICIENCY OF FUELCELLSAND MEMBRANE DESALINATION
3803
~
Table I: Over-all Degree of Coupling and Maximum Efficiency of Energy Conversion (%) for a Typical Synthetic Ion-Exchange Membranes between Ag-AgC1 Electrodes Hyperfiltration Either charge Either charge
Mean concn. of KC1, M
5 . 5 X 10-3 2 . 2 X 10-2
51’
3.68 4.02
.
--Concentration Cation exch.
?mas
91.’
qmax
Fa8
Vmax
Fz’
nmax
0.94 1.03
6.35 4.43
1.64 1.13
99.6 95.5
88.1 65.1
0.05 5.12
0.01 1.31
cerned with the mechanical energy invested, in which case it is advantageous to transform eq. 11 into the form
CP = JvAp
+ JDAT, + I E
(16)
which assumes that impermeable solutes do not contribute significantly to the total osmotic pressure difference. The diffusional flow J D is given to a close approximation by
JD
J ~ / c, Jv
(17)
W’ = csa2L,/w
(18)
we may consider a further case based on eq. 16
gla2 = ( K
,
Electrodialysis-Cation exch. Anion exch. Ga’ ~max Gap ~ma
99.6 95.5
88.1 65.1
3.64 8.13
x
0.93 2.12
given in Table I. Since neither transport number nor conductivity are now involved explicitly in the degree of coupling, uncharged membranes may offer advantages. Loeb and Manjikian4 have described the performance of a desalination cell incorporating cellulose acetate membranes. From their data, an efficiency q = - J D A T , / J , A ~ of 7.8% is calculated for their conditions of operation. This is equivalent to a value of rlZ2not less than 25%.
Fuel Cells
If we now define an additional parameter
Driving process J,Ap
cellAnion exch.
+ W’)/(l + W’) (19)
As a concrete example, the typical synthetic ionexchange membrane described in detail by Kedem and Katchalsky3 is evaluated in Table I. Two average salt concentrations within the range of their model have been chosen. The calculation of p 2 values offers a good check on the internal consistency of a set of assumed or measured coefficients, apart from setting an upper limit to q and providing a basis for comparison. The concentration cell (also referred to as “reverse electrodialysis”) is included in Table I for comparison with the desalination processes. The asymmetry of coupling in both reverse and normal electrodialysis with respect to the membrane charge, for electrodes reversible to a given ion, disappears if one takes as the working unit a pair of membranes of opposite fixed charges. The efficiency of this particular membrane in converting mechanical energy to electrical and osmotic energy is extremely low. Note that in hyperfiltration the over-all degree of coupling, whichever way it is calculated, is independent of the transport number in the membrane. Actually, only the conversion of mechanical energy to osmotic energy is of importance, and the current is always zero, no electrodes being inserted. It is readily shown that under this restriction K vanishes in eq. 13 and 19, considerably reducing the figures
In a previous discussion‘ it was pointed out that the degree of coupling in a fuel cell operating without side reactions and leaks is unity. The appearance of side reactions (Le., incomplete oxidations) may partially decouple the system2 and gives rise to additional terms in the dissipation function. The considerations set out above are then applicable. Consider a working unit, in a stationary state, operating on a gaseous fuel. The unit includes the cell and any recycling arrangements and is provided with feed lines and exhaust lines. The dissipation function for this system, if we suppose that only one side reaction takes place, is 0 = IE
+ AZJ+
Asus
+ J r b r + JoApo
(20)
where A , A , and ZJ,v, represent the affinities and velocities, respectively, of the main and side reactions, while Jr, J Oand Apf, Apo represent exhaust flows and chemical potential differences (across the unit) of fuel and oxygen. The affinities here are calculated by using the chemical potentials of the reactants on the feed side and the products on the exhaust side (if the feed lines contain products, this will simply contribute further flow terms to 0). The last two terms arise owing to incomplete utilization of the fuel; this is characteristic of fuel cell technology since at high degrees of conversion appreciable polarization occurs. The reaction velocities are calculated on the basis of fuel actually consumed. Within the cell the vectorial flows Jt and J O (4) 9. Loeb and 8. Manjikian, “Brackish Water Desalination by an Osmotic Membrane,” University of California, Department of Engineering Report No. 63-37, 1963.
Volume 69, Number I 1 Nowmber 1966
S. R. CAPLAN
3804
take place in isotropic spaces adjacent to the electrodes and, hence, cannot be coupled to the scalar chemical reactions. It is evident that they also cannot be coupled to the current, so that the sum of the last two terms is separately positively definite, and we can divide CP into two parts
a‘
=
a‘‘
IE
+ Av + Asv8
= JrApr
+ JoApo
(21) (22)
For a variety of fuel cells operated a t constant affinity, the linear region of the voltage-current curve is remarkably broad, frequently extending to open circuit.6 For such cells eq. 10 may be used to evaluate the degree of coupling. For convenience it is rewritten below; static head in this case corresponds, of course, to open circuit q 2 = [l
@’’represents an input of energy which is completely wasted and, hence, decreases the efficiency (although the fuel may subsequently be recovered by operating cells in cascade). However, the efficiency of the unit as an energy converter under given conditions of fuel utilization depends on a’. Note that successive interactions in the cell, forming short-lived intermediates (such as hydrogen peroxide) which do not appear in the exhaust gases, contribute no terms to a’. In order to evaluate the degree of coupling and the efficiency, we make the linear transformation
CP’ = I E
+ Aut + ( A , - A)v,
(23)
+
where ut = v us denotes the total fuel actually consumed. The third term in eq. 23 now represents a back reaction; ie., v, is the velocity and ( A , - A ) the affinity of a hypothetical process in the exhaust gases whereby the products of the main reaction (usually carbon dioxide and water) are reduced to the products of the side reaction, releasing oxygen into the feed. This affinity is clearly negative if not zero, and it seems reasonable to assume that it is very small in comparison to A since otherwise in the presence of catalyst complete oxidation of the fuel would occur. If this is the case, the back reaction may be considered to be a t level flow, and the efficiency is given by
-
(bv/dE) vOc/EoC A
I-’
Equation 25 requires the measurement of fuel consumption a t several values of the load resistance. Once q2 has been determined, the efficiency is known‘ for any potential difference across the terminals 7 =
1 - (E/Eoc) (Eoc/Eq2)- 1
If the linear region of the voltage-current curve does not extend to open circuit, an extrapolated value for Eo’ may be used in eq. 26, q2 having been determined either by using extrapolated values for both voc and E”’ in eq. 25 or by means of the following form of eq. 9
Equation 27 also applies over any small region of a nonlinear characteristic. It will be observed that the degree of coupling q here is actually an I*,.
(24)
Acknowledgments. The author wishes to express his indebtedness to Drs. Ora Kedem and Donald C. Mikulecky for their stimulating criticism. This investigation was supported by P.H.S. Research Grants GM-09432-01 and GM-09432-02 from the National Institute of General Medical Sciences, Public Health Service.
where the output is compared to the total input of chemical energy on the basis of complete oxidation of the fuel consumed.
(5) See, for examples, G . J. Young, Ed., “Fuel Cells,’’ Reinhold Publishing Corp., New York, N. Y . , 1960; W. Mitchell, Ed., “Fuel Cells,” Academic Presa Inc., New York, N. Y., 1963.
7 =
-IE/Avt
The Journal of Physical Chemistry
