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Chemical Engineering, Department of
1-1-1986
A Mathematical Model of a Zn/Br2 Cell on
Charge
M J. Mader
Texas A & M University - College Station
Ralph E. White
University of South Carolina - Columbia, white@cec.sc.edu
Follow this and additional works at: http://scholarcommons.sc.edu/eche_facpub
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Journal of the Electrochemical Society, 1986, pages 1297-1307.
© The Electrochemical Society, Inc. 1986. All rights reserved. Except as provided under U.S. copyright law, this work may not be
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Publisher's Version: http://dx.doi.org/10.1149/1.2108857
DOI: 10.1149/1.2108857
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A Mathematical Model of a Zn/Br Cell on Charge
M. J. Mader and R. E. White*
Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843
ABSTRACT
A mathematical model of a parallel plate electrochemical cell with a separator and a homogeneous bulk reaction is
presented. The model is based on the Zn]Br,~ redox couple and can be used as an aid for the design of an efficient
rechargeable storage battery. It is shown that four independent variables exist for the system at a fixed temperature:
the effective separator thickness, the residence time, the channel width, and the potential driving force. Performance
criteria of interest for the Zn]Br.2 battery are defined. Predictions of performance during the charging process are presented. It is shown that the cell performance improves as the effective thickness of the separator is increased, despite
the associated greater cell resistance. It is also shown that a change in the residence time has little effect on cell performance.
Several companies, including E x x o n and Energy Research Corporation (ERC), are pursuing the development
of a zinc/bromine (Zn]Br2) battery by building and testing
various designs (1). A comprehensive mathematical
model which can reasonably predict the performance of a
Zn]Br.~ b a t t e r y c o u l d be of great use to these companies as
a tool for d e s i g n i n g a more efficient system. Such a
model would reduce the need to build and test dozens of
designs.
A typical Zn/Br2 flow battery consists of stacks of alectrochemical cells in which reversible reactions occur at
the electrodes. An aqueous electrolyte Solution containing t h e reacting species is circulated through each cell
in a stack and stored in external tanks. To charge the battery, a current or potential is applied to the stack while
the fluid is fed to each cell from the external tanks. Energy is stored in the form of Zn and Br~ when zinc ions
(Zn '-'*) in solution are plated a s solid Zn on the cathode
and bromide ions (Br-) react within a porous layer on the
anodes to form Br2, a liquid which is soluble in the aqueous solution. However, any Br.~ that reaches the Zn electrode reacts electrochemically, thereby causing a loss of
charge. To help prevent this undesirable reaction, either a
separator or a complexing salt which causes an inert, insoluble Br2-rich phase to be formed, or both, are included.
An additional reaction which occurs, in the bulk of the
electrolytic solution, is the partial complexation of Br2
and B r - to tri-bromide ions (Brs-). The battery is discharged by circulating the stored electrolyte through the
cell stack with the terminal leads of the cell stack connected to an appropriate load. The reverse of the electrochemical reactions occur at each electrode and energy is
released.
In the work presented here, an attempt has been made
to include as many of these features as possible into a
comprehensive two-dimensional Zn/Br~ cell model. The
goal is to create a cell model from which a greater understanding of the physical p h e n o m e n a affecting cell performance can be obtained and which gives reasonable
performance predictions. The model presented, therefore,
is derived to the greatest extent possible from scientific
first principles, with a de-emphasis on empirical relationships such as mass transfer coefficients. The scope of
this work does not a l l o w all the f e a t u r e s of a typical
Zn]Br~ cell design to be included. Only a charge mode for
a single cell in the stack is considered. The electrochemical energy storage reactions producing Br.~ and Zn are included, but the Br2 electrode is modeled without a porous
layer. The undesirable Br., reaction competing with the
Zn reaction at the cathode is included in the model. A porous separator placed between the two electrodes is in the
model, but the second, Br2-rich phase has not been considered. Finally, the complexation of Br2 and B r - to Br:~has been included in the model. In addition, a o n e - d i m e n sional model, called a one-step model, which requires significantly less computation time, yet compares very well
to the two-dimensional solution, has been developed.
*Electrochemical Society Active Member.
Analysis of the mode] provides a set of independently
adjustable parameters which are shown to control the
performance of the system. To study the effect of physical p h e n o m e n a on cell performance, performance criteria
are defined and the independently adjustabl e parameters
are varied. Results of the variations lead to an understanding of the complex interaction of kinetic, thermody:
namic, and transport properties within the cell.
The Zn/Br2 cell model presented is the first model developed which predicts product conversion, energy efficiency, and other quantities of practical interest to a cell
designer, for a separated cell in which multiple electrode
reactions occur. White et aL (2) developed a mathematical
model o f a parallel plate electrochemical reactor (PPER)
which Can predict similar quantities of interest but does
not include a separator. Their work is the most desirable
from which to develop a Zn]Br2 cell model, because they
start from first principles and include many of the same
features present in a typical Zn/Br=, cell. Lee and Selman
(3, 4) developed a comprehensive model of a Zn]Br~ cell
but their work does not address at all the objectives desired in this work. Van Zee et al. (5) developed a Zn]Br~
battery model which determines an energy efficiency for
the entire battery but cannot predict product conversions.
Fedkiw and Watts (6) developed a comprehensive model
of the iron]chromium (Fe/Cr) redox battery, which, unfortunately, cannot be used for the Zn]Br2 battery presented
here.
White et al. (2) developed a mathematical model of an
electrochemical reactor which, with a few modifications,
is useful in developing a model of a Zn/Br~ cell. The
model of White et al. (2) consists of parallel plate electrodes and an electrolyte in laminar flow between them.
Their model uses a differential mass balance in two dimensions and can handle multiple electrode reactions, as
well as homogeneous reactions in the electrolyte bulk.
Additionally, their model can be used to predict current
efficiencies and conversions per pass of each species in
solution. The model of White et al. (2) is extended here to
treat the Zn]Br~ battery.
Lee and Selman (3, 4) modeled the Zn]Br2 system with a
separator, with the particular interest of studying dendrite growth due to uneven plating of Zn. They consider
the Zn]Br~ cell as a complete unit. However, their model
does not address the goals described above for the following reasons. First of all, they consider cell performance
only at fixed, typical states of charge and do not vary design p a r a m e t e r s , such as electrode length or separator
thickness. Second, Lee a n d Selman propose corrosion of
Zn as a possible reaction during charge, while it is felt
here that corrosion could n o t occur on charge due to cathodic protection. Finally, Lee and Selman characterize
their separator with parameters such as porosity and a
mass-transfer coefficient which are difficult to determine experimentally. Van Zee (7) has shown that the
MacMullin number, Nm, is an easily determined characteristic of a porous separator which, with the separator
thickness, can completely describe its transport
properties.
1297
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J. Electrochem. Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
1298
Van Zee et al. (5) d e v e l o p e d a s i m p l e ZtYBr2 battery
m o d e l in w h i c h t h e y e s t i m a t e cell resistances and total
e n e r g y r e q u i r e m e n t s to run the battery. T h e y h a v e f o u n d
that by v a r y i n g the separator t h i c k n e s s the energy efficiency may be i n c r e a s e d by at least 1%, w h i c h is a considerable e n e r g y savings for an often-cycled battery. Their
mode] is ]trotted, h o w e v e r , b e c a u s e it does not i n c l u d e the
capability of p r e d i c t i n g p r o d u c t conversion. Also, their
m o d e l does not i n c l u d e the effect of fluid flow rate on
cell p e r f o r m a n c e , s u c h as h o w the flow rate affects the
c o u l o m b i c losses caused by the r e d u c t i o n of b r o m i n e at
the zinc e l e c t r o d e on charge.
F e d k i w and Watts (6) h a v e s i m u l a t e d an e l e c t r o c h e m i cal cell w h i c h i n c l u d e s a s e p a r a t o r and m u l t i p l e electrode
reactions, b u t t h e y focus on the Fe/Cr r e d o x system
w h i c h uses p o r o u s p a c k e d b e d electrodes. Their m o d e l
i n c l u d e s studies at various states 0f charge, and t h e y follow c o n c e n t r a t i o n changes as well as local current densities. B e c a u s e t h e y study the Fe/Cr s y s t e m and h a v e d o n e
only a small a m o u n t of w o r k v a r y i n g d e s i g n parameters,
their w o r k is not helpful in u n d e r s t a n d i n g the Zn/Br~ system. Also, t h e y h a v e preferred to study p o w e r requirem e n t s and c u r r e n t flow rather t h a n c o n c e n t r a t i o n
changes, w h i c h are of interest in this work.
x
(1) Br ~ 1 / 2
f
s c h e m a t i c of a single cell for t h e
Zn/Br~ battery is p i c t u r e d in Fig. 1. It features a parallel
plate reactor w i t h a porous separator. Electrolyte is
p u m p e d to e a c h side of the cell f r o m separate tanks. T h e
a q u e o u s electrolytic solution consisting of the species
Na', B r - , Br2, Br:~-, and Zn ~ flows l a m i n a r l y in the channels on b o t h sides of t h e separator. It is a s s u m e d that species diffuse and m i g r a t e t h r o u g h t h e separator. H o w e v e r ,
to simplify the analysis, it is a s s u m e d that the p r e s s u r e
drop per unit l e n g t h is identical on b o t h sides of the cell,
t h e r e b y p r e v e n t i n g c o n v e c t i v e flow t h r o u g h the separator. This a s s u m p t i o n is a c h i e v e d by setting SA = Sc and
Assumptions.--A
Vavg :
Vavg,A :
Vavg,C.
The l e n g t h (L) and w i d t h (W) of b o t h electrodes are ass u m e d to be large relative to the gap b e t w e e n the electrodes (S). N g u y e n et al. (8), in an e x t e n s i o n of the m o d e l
of White et al. (2), d e t e r m i n e d that if the aspect ratio (a =
S / L ) is less t h a n 0.2, t h e n ~he axial diffusion and migration flux t e r m s m a y be neglected. It is a s s u m e d that the
mobility of ionic species follows t h e N e r n s t - E i n s t e i n
e q u a t i o n (u~ = D~/RT) and that the B u t l e r - V o l m e r e q u a t i o n
can be u s e d to d e s c r i b e the e l e c t r o c h e m i c a l reactions. In
addition, p s e u d o - s t e a d y - s t a t e c o n d i t i o n s are a s s u m e d to
exist. That is, it i's a s s u m e d that b e c a u s e of a low conversion per pass, t h ~ c o n c e n t r a t i o n of the feed to the cells,
within the external, storage tanks, c h a n g e s v e r y slowly.
On charge, t h e reactions that are a s s u m e d to take place
at the electrodes are
B r - ~ 1/2 Br2 + e -
(anode, reaction 1)
1/2 Br, + e - ~ B r -
(cathode, r e a c t i o n 2)
1/2 Z n ~* + e - --> 1/2 Zn
(cathode, reaction 3)
R e a c t i o n 1 and
e n e r g y storage.
reaction w h i c h
Corrosion of Zn
3 are the desired reactions w h i c h lead to
R e a c t i o n 2 is an u n d e s i r e d self-discharge
leads to inefficiency and energy losses.
by Br~ d u r i n g c h a r g e a c c o r d i n g to
Zn + Br~ -~ ZnBr2
has b e e n p r o p o s e d by Lee and S e l m a n (3, 4), b u t d u r i n g
charge, the zinc is cathodically protected, and thus corrosion is at least negligible and p r o b a b l y nonexistent. D u e
to the large voltaic driving force, h y d r o g e n e v o l u t i o n at
the c a t h o d e m i g h t be proposed, but this reaction has a
small e x c h a n g e c u r r e n t density on zinc electrodes. In addition, the solution is a s s u m e d to h a v e a large e n o u g h p H
that there is no a p p r e c i a b l e c o n c e n t r a t i o n of h y d r o g e n
ion.
Br2+e
lJnl(X)
C. . . . . . . . . . . . . .
U,~= 1.087V
$
. C,(X,y)
,~(x,y)
Cu.ea.C ~
vav~,c ,
~! A
C....... ( x = L )
I
Sc
cLa'g'c(x=L)
L
(2) 1/2 Br2 + e- - - B r - U~ = 1.087 V
(3) 1/2 Zn 2+ + e - - 1 / 2 Zn LI3e = -0.763 V
V=
Fig. 1. Schematic of Qn individual cell in o Zn/Br~ bottery
A h o m o g e n e o u s c o m p l e x a t i o n reaction p r o d u c i n g Br:~Br2 + B r - m Br3-
(bulk, r e a c t i o n 4)
occurs in the solution bulk. It is a s s u m e d this reaction is
fast and, thus, t h e c o n c e n t r a t i o n s of the three Species are
in e q u i l i b r i u m t h r o u g h o u t the cell, a c c o r d i n g to the equil i b r i u m c o n s t a n t g i v e n by E i g e n and K u s t i n (9)
Model Development
The a s s u m p t i o n s n e e d e d to d e v e l o p the m o d e l are presented first, followed by the g o v e r n i n g e q u a t i o n s and
b o u n d a r y conditions.
July 1986
K,q -
cB,3-
- 17M - ' = 17,000 (mol/cm3) -1
[1]
CBr_ -- CBr2
Governing e q u a t i o n s . - - T h e cell in Fig. 1 is d i v i d e d into
three sections, two electrolyte flow c h a n n e l s of w i d t h SA,
and a separator of w i d t h Ss. The g o v e r n i n g e q u a t i o n s for
each section, as d e v e l o p e d in t h e P P E R m o d e l of White et
al. (2), i n c l u d e the etectroneutrality c o n d i t i o n and a
steady-state material balance e q u a t i o n for each ' species.
The e q u a t i o n s m a y be used to solve for t h e c o n c e n t r a t i o n
(ci) and potential ((a) distributions as f u n c t i o n s of the axial
(x) and radial (y) position.
The e l e c t r o n e u t r a l i t y c o n d i t i o n
[2]
zici = 0
i
ensures that t h e r e is no b u i l d u p of positive or n e g a t i v e
charge a n y w h e r e in the electrolytic solution. F o r steady
state, t h e material balance e q u a t i o n s for each species are
- V - Ni + R~ = 0
(i = B r - , Br~, Br:~-)
V 9 Ni = 0
(i = Na*, Zn '-'~)
[3]
[4]
where
Di
Ni = -DiVci - z i - ~
FciVcP + vci
(flow channels)
[5]
and
Die
N~ = -Dj.eV q - zi ~
FciVq~ (separator)
[6]
In the separator, an effective diffusivity, Di.e, is required,
given by Van Zee (7) as
Di
D~,e -
Nm
[7]
w h e r e Nm is the MacMullin n u m b e r . T h e MacMullin
n u m b e r is a characteristic of a p o r o u s separator, e q u i v a lent to the ratio of the separator's t o r t u o s i t y to its porosity. H o w e v e r , u n l i k e porosity or tortuosity, it is easily det e r m i n e d and not a p r o p e r t y of solution strength, as
s h o w n by Van Zee (7). The MacMullin n u m b e r is given by
a resistivity ratio
N~-
p
Po
[8]
w h e r e po is the resistivity of a solution w i t h o u t a separa-
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A MATHEMATICAL
Vol. 133, No. 7
tor in its midst, and p is the resistivity of separator and solution together 9
The v e l o c i t y distribution w i t h i n the flow c h a n n e l s is
a s s u m e d to be w e l l - d e v e l o p e d l a m i n a r flow as g i v e n b y
(10)
V~ = 0;
Vx = 6Vavg
SA
[9]
SA ~
The material b a l a n c e e q u a t i o n s g i v e n by Eq. [3] can be
simplified by a s s u m i n g that reaction [4] is at e q u i l i b r i u m
so that Rsr- = R ~ r ~ = - - R B r : ~ - . F o r t h e species B r - and Br:~-,
Eq. [3] for e a c h can be a d d e d to get
( - V 9 N ~ - + R~r-) + ( - V ' N~r:~- + R~r:~-) = 0
[10]
0
[11]
V 9 Ns~.~ + V 9 Nsr:~- = 0
[12]
9 NBr:~-
=
L i k e w i s e for species Br~ and Br:~-
S i n c e the n u m b e r of e q u a t i o n s has b e e n r e d u c e d by one,
t h e e q u i l i b r i u m e x p r e s s i o n of Eq. [1] is n e e d e d to describe t h e s y s t e m fully. Therefore, Eq. [1], [2], [4], [11], and
[12] c o m p r i s e t h e set of g o v e r n i n g e q u a t i o n s n e e d e d to
s o l v e the s y s t e m for the u n k n o w n variables, c~ and (I).
As p o s t u l a t e d by White et al. (2) and later d e m o n s t r a t e d
by N g u y e n et al. (8), w h e n the a s p e c t ratio (a ~ S / L ) is
small, t h e n the diffusion and m i g r a t i o n t e r m s of the flux
e x p r e s s i o n in t h e a x i a l , or flow, d i r e c t i o n are negligible
c o m p a r e d to t h e radial or n o r m a l diffusion and m i g r a t i o n
terms. Thus, u p o n c o n v e r t i n g to t h e d i m e n s i o n l e s s variables
Initial and b o u n d a r y conditions.--To c o m p l e t e the syst e m of e q u a t i o n s , t h e initial and b o u n d a r y c o n d i t i o n s
m u s t be specified. Conditions are specified at t h e cell
entrance, the electrodes, and the electrolyte/separator interface.
The initial c o n d i t i o n s at the cell e n t r a n c e and b o u n d a r y
conditions at the electrodes are t h e s a m e as specified by
White et al. (2). At the cell entrance, t h e feed concentrations are k n o w n , and the e l e c t r o n e u t r a l i t y c o n d i t i o n m u s t
hold
at ~ = 0
0<y<SA
[15]
(SA+Ss)<y<S
2Sv~v~
at V = 0
S'-'
(V 9 Ni)
+
0'-' 0i
Zi F
0 H'-'
RT L
[
0i
cT-'qb
00i
(H' - H'~) ~ -
0
RT
[21]
n~F
at H = 1 (cathode)
- -
n:~F
[22]
= Nni
(0i,o) pij
-
exp \ RT
H
i
Hj
( - a c J F H0}
(6~.~)"'~ e x p \ R T
[23]
Hj = V~ - qbo~ - Uj,ref
[24]
In Eq. [24], V~ is t h e e l e c t r o d e potential (either V~ or V~),
q)o~ is the potential of the solution at the surface of that
electrode (either qbo~ or ePo~), and Uj.,~ is t h e local opencircuit potential of reaction j at t h e r e f e r e n c e concentrations. The open-circuit potential is a t h e r m o d y n a m i c par a m e t e r of the solution g i v e n by
H---:;
ziF [
Nni
si2in2
- = Nni
[18]
and for the s e p a r a t o r as
eCi,refDi,-- (V - N i ) = -
-
w h e r e 0~.o is a d i m e n s i o n l e s s local surface c o n c e n t r a t i o n
and Vj is the reaction overpotential, defined as
0 ~'-'
00i0__~_
0(I)0H~]+ 3 ~DI~ P e -
n~F
and
the d i v e r g e n c e of t h e flux of species i in Eq. [4], [11], and
[12] m a y be r e w r i t t e n for the flow c h a n n e l s as
ci.r~Di
Sil~nl
(anode)
inj = ioj.ref
[14]
Pe - - -
T h e b o u n d a r y c o n d i t i o n s at t h e e l e c t r o d e s are that t h e
e l e c t r o n e u t r a l i t y c o n d i t i o n h o l d s and that t h e rate of cons u m p t i o n or p r o d u c t i o n of a species by e l e c t r o c h e m i c a l
reaction at the e l e c t r o d e is e q u a l to the n e t n o r m a l flux of
t h e species t o w a r d or away f r o m t h e electrode. T h a t is
[17]
H = y/S
Oi = ci/ci.~
[20]
i
[16]
[13]
Y - (SA + Ss)
Sc
~ z~c~.~fO~.f~,d= 0
0j = 0L~,a;
w h e r e i,j is the c u r r e n t d e n s i t y for reaction j, s~ is the
s t o i c h i o m e t r i c coefficient for e a c h species in the electroc h e m i c a l reaction, nj is the n u m b e r of electrons transferred in t h e reaction, and N,~ is t h e net n o r m a l flux of
species i, in w h i c h t h e derivatives are defined to be in t h e
positive y direction. It should be p o i n t e d out that the
signs u s e d for N,~ in Eq. [21] and [22] are consistent w i t h
the c o n v e n t i o n that positive c u r r e n t leaves the a n o d e to
enter the electrolytic solution, and n e g a t i v e current f r o m
the solution enters the cathode.
The n o r m a l c o m p o n e n t of t h e current density for reaction j, inj, is g i v e n by t h e B u t l e r - V o l m e r equation, as g i v e n
by White et at. (11)
= x/L
Y
1299
n u m b e r is defined h e r e in t e r m s of S (Eq. [17]) instead of
SA, w h i c h w o u l d be m o r e a p p e a l i n g physically; h o w e v e r ,
since no d i m e n s i o n l e s s analysis is p r e s e n t e d in this paper, the definition of P e is arbitrary.
w h i c h r e d u c e s to a g o v e r n i n g e q u a t i o n consisting only of
flux t e r m s
V - N~- + V
MODEL
0~q)
__00i 04PJ
[19]
O~ OH'-' + OH 077
N o t e t h a t t w o d i m e n s i o n l e s s radial variables are u s e d
in Eq. [18]. The v a r i a b l e H is m a d e d i m e n s i o n l e s s by the
total e l e c t r o d e gap S, and is u s e d to define the step size
in any section w h e n c o n v e r t i n g the differential g o v e r n i n g
e q u a t i o n s to finite difference form. T h e variable H' is
m a d e d i m e n s i o n l e s s by the flow c h a n n e l w i d t h SA, and is
n e e d e d to e x p r e s s position w i t h i n the flow channels,
since the local e l e c t r o l y t e v e l o c i t y d e p e n d s on the relative
position w i t h i n t h e channel. N o t e also that the P e c l e t
Uj.r~f= Uj ~
U~E8 -
njF
~ sijln
RT
+ nl~eF
i
(ci, f)
( ci.,~: ~
~ si.i~EIn \ do /
[25]
w i t h the s u b s c r i p t R E referring to an arbitrary reference
electrode. V~, ~Poe, UJ~, and U ~ e are all defined relative to a
reference electrode.
The b o u n d a r y c o n d i t i o n s at the separator/electrolyte interface are that the electroneutrality c o n d i t i o n holds (as it
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1300
J. E l e c t r o c h e m . Soc.: E L E C T R O C H E M I C A L
SCIENCE
d o e s e v e r y w h e r e in t h e reactor) a n d t h a t t h e n o r m a l flux
o f e a c h s p e c i e s is c o n t i n u o u s a c r o s s t h e interface. T h a t is,
at t h e a n o l y t e / s e p a r a t o r i n t e r f a c e
at y = SA
J u l y 1986
AND TECHNOLOGY
Table I. Fixed parameter values for the Zn/Br~ cell
Kinetic and thermodynamic (T = 298.15 K, V~ = 0.0V)
[26]
N n i , a = Nni,s
Ujo I)
ioj, refa
E x p a n d i n g w i t h t h e n o r m a l flux t e r m s in Eq. [5] a n d [6]
a n d c o n v e r t i n g to d i m e n s i o n l e s s q u a n t i t i e s , this b o u n d ary c o n d i t i o n b e c o m e s
_ (O0, I
z~F
1
Oe)
(00i~
ziF
1
OC i
C i --
[28]
In d i m e n s i o n l e s s c o o r d i n a t e s , w i t h c~.~r = c~.f~, Eq. [28]
becomes
~Oi
@
Oi- 1
- 1
a~j
c~j
nj
Vj, refc
(V)
(V)
Stoichiometry and electrochemical reaction orders
Ci,foed
L
(A/cm-')
1
0.0031
0.5
0.5
1
1.087
1.783
2
0.0031
0.5
0.5
1
1.087
1.783
3
1.00
0.5
0.5
1
-0.763
0.0
Reaction 4 (Homogeneous, bulk) K~,,'~ = 17000 (moYcm:~)-~
/00~
w h e r e 00JB~ a n d 0*/B~ are e v a l u a t e d at t h e interface,
b u t w h o l l y w i t h i n t h e r e g i o n of t h e i r r e s p e c t i v e s u b s c r i p t .
N o t e t h a t Di h a s b e e n c a n c e l e d f r o m this e q u a t i o n , b u t
t h e M a c M u l l i n n u m b e r , N~, d o e s n o t cancel. A t t h e
s e p a r a t o r / c a t h o l y t e interface, t h e b o u n d a r y c o n d i t i o n is
similar.
In a n e f f o r t to c u t d o w n on c a l c u l a t i o n time, a s i m p l e
m o d e l w a s d e v e l o p e d in w h i c h only o n e s t e p w a s t a k e n in
t h e flow d i r e c t i o n [see M a d e r et al. (12)]. This m o d e l ass u m e s t h a t i,j is a c o n s t a n t a l o n g t h e l e n g t h o f t h e r e a c t o r
a n d t h e r e f o r e n o t a f u n c t i o n o f ~ as in t h e p r e v i o u s ease.
Similarly, c~ a n d q) b e c o m e f u n c t i o n s o f ~ only. In this
way, t h e cell acts m u c h like a C S T R in t h e flow direction.
A t a g i v e n v a l u e o f 7, t h e r e is only a s t e p c h a n g e in conc e n t r a t i o n o f a s p e c i e s f r o m its f e e d v a l u e to its exit value,
instead of a gradual concentration gradient d o w n the
l e n g t h of t h e r e a c t o r as in t h e m o r e c o m p l i c a t e d model.
(To d i s t i n g u i s h t h e t w o m o d e l s , t h e s i m p l e o n e will b e
called t h e o n e - s t e p m o d e l a n d t h e o t h e r t h e c o n t i n u o u s
model.)
In t h e o n e - s t e p m o d e l , a finite d i f f e r e n c e e x p r e s s i o n
o v e r t h e l e n g t h o f t h e r e a c t o r r e p l a c e s t h e axial c o n c e n t r a t i o n g r a d i e n t in t h e c o n v e c t i o n t e r m of t h e c o n t i n u o u s
model
Ox
Reaction O)
[29]
All o t h e r e q u a t i o n s f r o m t h e c o n t i n u o u s m o d e l r e m a i n
t h e s a m e in t h e o n e - s t e p m o d e l .
B e c a u s e it e s s e n t i a l l y b e c o m e s a o n e - d i m e n s i o n a l
m o d e l , t h e o n e - s t e p m o d e l is a less a c c u r a t e s o l u t i o n o f
t h e e l e c t r o c h e m i c a l cell t h a n t h e t w o - d i m e n s i o n a l c o n t i n u o u s m o d e l . H o w e v e r , t h e o n e - s t e p m o d e l d o e s retain t h e
c o n c e p t s o f radial g r a d i e n t s a c r o s s t h e cell a n d t h e possibility o f m u l t i p l e e l e c t r o d e r e a c t i o n s o c c u r r i n g at e a c h
e l e c t r o d e . In a d d i t i o n , t h e o n e - s t e p m o d e l r e q u i r e s less
c o m p u t e r t i m e to solve. T h e a c c u r a c y of t h e o n e - s t e p
m o d e l relative to t h e c o n t i n u o u s m o d e l e s s e n t i a l l y dep e n d s on t h e c o n v e r s i o n p e r p a s s o f t h e s y s t e m . T h e ass u m p t i o n t h a t t h e c o n c e n t r a t i o n in t h e flow d i r e c t i o n is
u n i f o r m is r e a s o n a b l e for a s y s t e m only if t h e c o n v e r s i o n
per pass is small, as is often the case for a ZrffBr2 cell.
Parameters.--The parameters used in the model consist
of the fixed parameters of the cell and the independently
adjustable parameters. The fixed parameter values used
in the Zn/Brx cell model are given in Table I. The system
was set up to simulate as closely as possible the design of
Exxon (5, 13, 14) from which the most design parameters
were available. The feed composition at initial conditions,
0~.f~a, is assumed to be the same in both channels, for convenience. To determine the concentration of each species
in the feed, it is first assumed that the solution is composed of Na*, Br-, Br~, and Zn ~ at the reference concentrations listed in Table I, and that Br- and Br., are uncomplexed. Then, an equilibrium calculation is done using
t h e t r i - b r o m i d e c o m p l e x a t i o n r e a c t i o n a n d K~,~ to d e t e r -
Species (i)
Na
Br- ~
Br~
Zn'-" f
Br:,-
Reactions [1] and [2]
0= 1, 2)
s~
PiJ
qiJ
0
1
- 1/2
0
0
0
1
0
0
0
0
0
1/2
0
0
Reaction [3] (j - 3)
s~j
p~j
qi~
0
0
0
-1/2
0
0
0
0
0
0
0
0
0
1/2
0
Transport and reference concentrations
Species (i)
zi
D,~
(cm'-Ts)x 10~
Na ~
BrBr~
Zn ~"
Br:,
1
-1
0
2
-1
1.334
2.084
1.310
0.754
1.310
ci. ,.c~
(moYcm :~)• 10:~
1.00
3.00
0.05
1.00
0.10
0i. ~,~.~h
1.000
0.983
0.0203
1.000
0.510
" Chosen to duplicate current densities reported in Ref. (13).
"SeeRef. (15).
~ The reference electrode is reaction [3] at the reference
conditions.
'~See Ref. (9).
Limiting reactant at Br~ electrode.
f Limiting reactant at Zn electrode.
See Ref. (16-18); D~,.:~- assumed equal to D,,..,.
h At initial charging conditions, the feed to each channel is assumed to be the same, for convenience.
m i n e t h e a d j u s t e d c o n c e n t r a t i o n s of B r - a n d Br~, a n d t h e
c o r r e s p o n d i n g e q u i l i b r i u m c o n c e n t r a t i o n o f Br:~-. A f t e r
all o t h e r p a r a m e t e r s w e r e set, t h e e x c h a n g e c u r r e n t d e n s i ties i,,j.r~f w e r e p i c k e d b y trial a n d e r r o r to give c u r r e n t
d e n s i t i e s s i m i l a r to t h a t r e p o r t e d b y E x x o n (13). W h e n
c o m p a r e d to v a l u e s in t h e l i t e r a t u r e (19, 20), t h e s e ioj.,.r
w e r e w i t h i n an o r d e r o f m a g n i t u d e .
The m o d e l i n p u t v a r i a b l e p a r a m e t e r s are t h e l e n g t h o f
t h e e l e c t r o d e (L), t h e a v e r a g e v e l o c i t y o f t h e e l e c t r o l y t e
(v,,v~), t h e flow c h a n n e l w i d t h (SA), t h e t h i c k n e s s o f t h e
s e p a r a t o r (Ss), t h e MacMullin n u m b e r (Nm), a n d t h e app l i e d cell p o t e n t i a l (E~e~ = V~ - Vr N o t e t h a t t h e total
e l e c t r o d e gap (S) is a linear c o m b i n a t i o n of t h e flow c h a n nel w i d t h s a n d t h e s e p a r a t o r t h i c k n e s s , a n d is t h e r e f o r e
n o t an i n p u t variable, s i n c e it is c o m p l e t e l y s p e c i f i e d b y
t h e o t h e r w i d t h s . U p o n r e v i e w i n g t h e g o v e r n i n g equations and b o u n d a r y conditions and analyzing the model
results, it is f o u n d t h a t only four i n d e p e n d e n t l y adjustable p a r a m e t e r s e x i s t in t h e m o d e l , a l t h o u g h six i n p u t variables m a y be specified. A m e a n i n g f u l set of i n d e p e n d e n t
p a r a m e t e r s c o n s i s t s of t h e r e s i d e n c e t i m e o f t h e electrolyte in t h e cell, L/v,v~ ( w h i c h a p p e a r s as a ratio in t h e t e r m
Pea), t h e flow c h a n n e l w i d t h , Sa ( w h i c h a p p e a r s a l o n e as
t h e d e n o m i n a t o r in t h e d e f i n i t i o n o f 7'), t h e effective separator t h i c k n e s s , NmSs ( w h i c h a p p e a r s as a p r o d u c t in t h e
d e n o m i n a t o r o f t h e finite d i f f e r e n c e f o r m o f Eq. [27], as
s h o w n in Ref. (21)), a n d t h e cell p o t e n t i a l , Er (since V~
a n d Vr are relative to t h e s a m e r e f e r e n c e potential). V a n
Z e e (7) s h o w e d in a s e p a r a t o r f l o w - t h r o u g h e x p e r i m e n t
t h a t t h e p r o d u c t NmSs is i m p o r t a n t as a g r o u p , a n d n o t Nm
or Ss s e p a r a t e l y . I n t h e case o f t h e ZrdBr., cell, t h e m o d e l
i n d i c a t e s t h a t NmSs is also an i n d e p e n d e n t v a r i a b l e for
flow by a s e p a r a t o r .
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A MATHEMATICAL
VoL 133, No. 7
In this study, e a c h of t h e i n d i v i d u a l parameters, L, V~g,
SA, Ss, Nm, and Ece u a r e set at v a l u e s to be e x p e c t e d in a
typical Zn/Br~ battery. The p a r a m e t e r L/Vav ~ iS v a r i e d by
c h a n g i n g either L or Va,~, or both, b e t w e e n runs. T h e
length, L, is v a r i e d b e t w e e n 20 and 30 c m in t h i s study,
w h i l e v ~ is v a r i e d b e t w e e n 2.0 and 1.5 cm/s. T h e c h a n n e l
flow w i d t h (SA) is assigned values b e t w e e n 0.005 and 0.10
cm. E i t h e r Nm or Ss is v a r i e d to alter N~Ss, w i t h Nm varying b e t w e e n 1.5 and 10.0, and Ss b e t w e e n 0.02 and 0.12
cm. F o r Er Vc is set equal to 0 V for c o n v e n i e n c e , and V~
is v a r i e d b e t w e e n 1.85 and 2.025V.
Solution Technique and Material Balance Closure
H a v i n g set the fixed and variable parameters, the syst e m e q u a t i o n s can be solved for the c o n c e n t r a t i o n and potential distributions by using an i m p l i c i t s t e p p i n g techn i q u e in t h e axial direction (for the t w o - d i m e n s i o n a l
model) and N e w m a n ' s t e c h n i q u e (16) in the radial direction. This solution p r o c e d u r e is d i s c u s s e d by White et al.
(2) and M a d e r et al. (12). To e n s u r e that the average current densities p r e d i c t e d by t h e m o d e ] are consistent w i t h
the p r e d i c t e d a v e r a g e exit c o n c e n t r a t i o n s of each species,
a material b a l a n c e closure is d o n e on each species. At
steady state, the m o l a r (or mass) rate of c o n s u m p t i o n or
p r o d u c t i o n of a species by e l e c t r o c h e m i c a l and h o m o g e neous reaction w i t h i n the cell m u s t e q u a l the net m o l a r
(or mass) flow rate of that species t h r o u g h the reactor
from e n t r a n c e to exit. A q u a n t i t a t i v e e v a l u a t i o n of the
material balance closure s t a t e m e n t was d o n e for each
case s t u d i e d and t h e m o d e l p r e d i c t i o n s w e r e f o u n d to be
c o n s i s t e n t e a c h time. F u r t h e r i n f o r m a t i o n on this test for
c o n s i s t e n c y can be f o u n d by referring to M a d e r I21).
MODEL
1301
the species w h i c h m u s t be stored to c h a r g e the battery.
F o r the Zn/Br2 cell to charge, Br~ m u s t be p r o d u c e d at t h e
a n o d e and stored in the an01yte and Zn m u s t be plated at
the cathode. S i n c e the e l e c t r o d e material is often one of
the largest costs in the battery, it is i m p o r t a n t to charac=
terize t h e p r o d u c t i o n rate in t e r m s of t h e surface area of
electrode required.
B r o m i n e is stored in the anolyte in two forms, either as
dissolved l i q u i d Br2 or as part of t h e ionic c o m p l e x Br:~-.
Br~ is p r o d u c e d at the a n o d e d u r i n g charge by reaction of
B r - . Most of the Br2 p r o d u c e d at the a n o d e is c o m p l e x e d
w i t h excess B r - to create Br:c. S o m e of the b r o m i n e ,
b o t h as liquid Br~ a n d as the c o m p l e x e d Br:~-, m a y diffuse
out of the anolyte into the separator, w h i l e s o m e m a y enter the anolyte f r o m t h e separator, d e p e n d i n g on t h e concentration and potential gradients. H o w e v e r , only the brom i n e w h i c h exits t h e reactor in the anolyte, as either Br..,
or Br:~-, is c o n s i d e r e d stored. The Br~ p r o d u c t i o n rate per
area of a n o d e is, therefore, defined as t h e rate of gain of
Br2 in either form, in the anolyte alone, w h i c h is formulated as the p r o d u c t of the average flow rate and the
c h a n g e in average c o n c e n t r a t i o n f r o m e n t r a n c e to exit,
per area of anode. Thus
PB,',2=
[(CBr2,avg.A 2- CBr3--,avg,A).r
= L
.__ (CBr2,feed,A 2- CBr3_ feed,A)]
v~v~SAW
[31]
LW
w h e r e Ci,avg,A(X) is the local average radial c o n c e n t r a t i o n
in the anolyte section. Taking into a c c o u n t the l a m i n a r
flow of the electrolyte, this average c o n c e n t r a t i o n is [see
Mader et al. (12)]
Results and Discussion
O n c e the c o n s i s t e n c y of the solution t e c h n i q u e is verified, the c o n c e n t r a t i o n and potential distributions can be
u s e d to analyze the Zn/Br2 cell model. First. p e r f o r m a n c e
criteria of specific interest to this cell are derived. Then,
N~Ss is verified as an i n d e p e n d e n t parameter. Next. the
t w o - d i m e n s i o n a l m o d e l and one-step m o d e l p r e d i c t i o n s
are c o m p a r e d . E x a m p l e s of p e r f o r m a n c e predictions that
can be o b t a i n e d f r o m the Zn/Br._, cell m o d e l at initial
c h a r g i n g c o n d i t i o n s are t h e n presented. Finally, t h e perf o r m a n c e of the cell as a f u n c t i o n of the state of charge of
t h e s y s t e m is c o n s i d e r e d for t h r e e v a l u e s of t h e effective
separator thickness, N~S~, and three v a l u e s of the residence time.
Performance criteria.~o deve}op a m e a n i n g f u l set of
p e r f o r m a n c e criteria for t h e Zn/Br~ cell, the desirable
characteristics of a r e c h a r g e a b l e battery s y s t e m s h o u l d be
considered. First, it is i m p o r t a n t to k e e p the cell current
at a relatively low level to p r e v e n t d e n d r i t i c g r o w t h of
zinc d u r i n g charging. Second, it is i m p o r t a n t that a rec h a r g e a b l e battery can be c h a r g e d quickly.-A third desirable characteristic, d u r i n g the c h a r g i n g process, is a low
a m o u n t of e n e r g y c o n s u m p t i o n relative to the a m o u n t of
p r o d u c t stored. Finally, it is i m p o r t a n t to h a v e a h i g h energy efficiency, indicating the e n e r g y c o n s u m e d has
b e e n put to useful work.
To m o n i t o r the cell current, the average current density
at an electrode, ia~, is a useful q u a n t i t y . It is defined as
the s u m of t h e a v e r a g e c u r r e n t densities for each reaction
at an e l e c t r o d e
i~v~ = ~ i,j ....
[30]
w h e r e i ~ , ~ is an a v e r a g e of the local c u r r e n t densities,
i.j(x), o v e r the l e n g t h of the electrode. The v a l u e of inj.~v~is
s o l v e d for directly in the one-step model.
The a v e r a g e c u r r e n t density m a y also be defined as the
total cell current, I, per surface area of electrode, LW, so
that for electrodes of e q u a l surface area, the v a l u e of i ~
m u s t be t h e s a m e at b o t h electrodes.
A s e c o n d desirable characteristic is t h e capability to
c h a r g e quickly. In other words, for a g i v e n area of electrode, it is i m p o r t a n t to a c h i e v e a h i g h p r o d u c t i o n rate of
c~...... (x) = ~
6 fs~ y
, S~
Y~)c~(x,y)dy
[32]
A similar definition m a y be written for t h e local radial
average c o n c e n t r a t i o n in the catholyte section. Storage of
Zn is a c c o m p l i s h e d solely by p l a t i n g Zn ~" ions on the
cathode. The p r o d u c t i o n rate of Zn, then, can be f o u n d
from the rate of the zinc reaction, i,~:~,~v~,as
Pzn- (Sz'"-+':~)(in:~'~v~)
[33]
n:~F
A third characteristic w h i c h is desirable in a battery,
d u r i n g the c h a r g i n g process, is a low rate of energy cons u m p t i o n r e l a t i v e to the rate of storage of t h e product.
The rate of e n e r g y c o n s u m p t i o n is the p r o d u c t of the applied voltage and t h e total cell current. T h e charging rate
is e q u i v a l e n t to t h e rate of p r o d u c t i o n , P~LW. So, in general, t h e e n e r g y c o n s u m p t i o n per m o l e of p r o d u c t stored
can be defined as
~oi --
(Ecell)(iav~LW)
PiLW
[34]
N o t e that t h e e l e c t r o d e area cancels out. S u b s t i t u t i n g Eq.
[31] for the p r o d u c t i o n rate of Br.z into Eq. [34] a n d
rearranging yields
(Er
r176 =
iav~
SA
ACBr~
[35]
w h e r e Ac,~r,~ = (CBr2,avg,A 2- C ~ r . ~ - - , a v ~ , A ) . r = L - - (CBr2,feed,A -~
cBr:~-.feeoA). The e n e r g y c o n s u m p t i o n per m o l e of Zn stored
is defined by using Eq. [33] for Pz,, giving
~Oz,( n'~F)
(~1
[36]
A final characteristic desirable in a r e c h a r g e a b l e battery is a h i g h e n e r g y efficiency, a q u a n t i t y related to the
e n e r g y c o n s u m p t i o n term. The e n e r g y efficiency is defined as the p r o d u c t of the c o u l o m b i c efficiency and t h e
voltaic efficiency
eT = ecev
[37]
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1302
J. E l e c t r o c h e m . Soc.: E L E C T R O C H E M I C A L
T h e c o u l o m b i c efficiency is a m e a s u r e of t h e f r a c t i o n of
c u r r e n t p a s s e d w h i c h gives t h e d e s i r e d reaction. A t t h e
a n o d e , t h e Br~ r e a c t i o n r u n s u n c o n t e s t e d so t h a t t h e
c o u l o m b i c efficiency is unity. T h e c o u l o m b i c efficiency
of m o r e i n t e r e s t is t h e c u r r e n t efficiency at t h e c a t h o d e ,
w h i c h is d e f i n e d as
[in3.~d
ec - - -
T h e v o l t a i c efficiency o n c h a r g e is d e f i n e d as t h e ratio
of t h e t h e o r e t i c a l v o l t a g e r e q u i r e d for c h a r g i n g , to t h e actual a p p l i e d voltage. T h e t h e o r e t i c a l r e q u i r e d v o l t a g e is
g i v e n b y t h e o p e n - c i r c u i t p o t e n t i a l o f t h e cell b a s e d o n
t h e r e f e r e n c e c o n d i t i o n s . T h a t is, it is t h e d i f f e r e n c e of t h e
half-cell p o t e n t i a l s f o r t h e d e s i r e d r e a c t i o n s : T h e r e f o r e ,
t h e v o l t a i c e f f i c i e n c y o n c h a r g e is g i v e n b y
Ul,re f -
U3,re f
[39]
Ece|l
T h e a p p l i e d cell p o t e n t i a l , EteH, i n c l u d e s t h e t w o c a u s e s
of voltaic inefficiency, t h e o v e r p o t e n t i a l s , ~j,. d e f i n e d
earlier in c o n j u n c t i o n w i t h t h e B u t l e r - V o l m e r e q u a t i o n ,
a n d t h e p o t e n t i a l d r o p a c r o s s t h e cell d u e to o h m i c resista n c e of t h e s e p a r a t o r a n d t h e electrolyte. T h e p o t e n t i a l
d r o p across t h e cell, or I R drop, is d e f i n e d as
IR drop = qbo~ - epor
[40]
where r and dPoc are the potentials of the electrolytic solution at the Br2 electrode and Zn electrode, respectively.
Substituting the definitions of coulombie and voltaic
efficiencies into Eq. [37], the total energy efficiency of
t h e cell o n c h a r g e m a y b e r e w r i t t e n as
eT :
[41]
(i.~)(E~,,)
w h i c h is t h e f r a c t i o n of e n e r g y a p p l i e d to t h e cell t h a t
l e a d s to u s e f u l s t o r a g e of energy. N o t e t h a t t h i s d e f i n i t i o n
of t o t a l e n e r g y efficiency d o e s n o t t a k e i n t o a c c o u n t energy losses e x t e r n a l to t h e c h a r g i n g p r o c e s s , s u c h as t h e
e n e r g y r e q u i r e d to p u m p t h e e l e c t r o l y t e or t h e h e a t losses
c a u s e d b y friction. A p r o c e d u r e for e s t i m a t i n g s u c h addit i o n a l l o s s e s is g i v e n b y V a n Z e e et al. (5).
T h e e n e r g y efficiency as d e f i n e d in Eq. [41] m a y b e
u s e d to r e d e f i n e t h e e n e r g y c o n s u m p t i o n p e r m o l e of Zn.
S o l v i n g Eq. [41] for Er
s u b s t i t u t i n g i n t o Eq. [36] a n d
r e a r r a n g i n g gives
OJz, =
[42]
ET
w h i c h i n d i c a t e s t h a t t h e e n e r g y c o n s u m p t i o n p e r m o l e of
Z n is a f u n c t i o n o f only eT a n d fixed s y s t e m p a r a m e t e r s .
N o t e p a r t i c u l a r l y t h a t OJz, is a m i n i m u m w h e n t h e e n e r g y
efficiency is 100%. U s i n g t h e v a l u e s i n T a b l e I for t h e
c o n s t a n t s i n Eq. [42] a n d a s s u m i n g 100% efficiency, t h e
J u l y 1986
AND TECHNOLOGY
Table II. Demonstration that the effective separator
thickness is an independent parameter
Input parameters:
L/v.~g =
15s, E ~ . = 1.9V, SA = S c = 0 . 0 6 5 cm
N~
Ss
(cm)
N~Ss
(cm)
S = SA +
Ss + Sc
(cm)
e~
i~
(mA/cm=')
2.0
3.0
6.0
0.06
0.04
0.02
0.12
0.12
0.12
0.19
0.17
0.15
0.6322
0.6322
0.6322
20.54
20.54
20.54
[38]
iavg
ev -
SCIENCE
m i n i m u m p o s s i b l e e n e r g y c o n s u m p t i o n p e r m o l e of Z n
p r o d u c e d is 344.1 k J / m o l Zn. A s i m i l a r c a l c u l a t i o n for Br2
yields t h e s a m e v a l u e of m i n i m u m e n e r g y c o n s u m p t i o n
of 344.1 k J / m o l Br2, s i n c e at 100% e f f i c i e n c y 1 tool of Z n
is p r o d u c e d for e v e r y m o l e of Br2 stored.
Effective separator thickness.---A s i m p l e v e r i f i c a t i o n
t h a t t h e p r o d u c t NmSs is a n i n d e p e n d e n t p a r a m e t e r in a
cell w i t h a s e p a r a t o r , a n d n o t Nm or Ss s e p a r a t e l y , is g i v e n
i n T a b l e II. W h i l e k e e p i n g L/Vavg, Ecen, a n d SA c o n s t a n t ,
t h e r e a c t o r p e r f o r m a n c e is t h e s a m e w h e n Nm a n d Ss a r e
a d j u s t e d , as l o n g as t h e effective s e p a r a t o r t h i c k n e s s ,
NmSs, is h e l d c o n s t a n t . N o t e t h a t T a b l e II also s h o w s t h a t
t h e t o t a l cell gap, S, m a y b e a d j u s t e d w i t h o u t a f f e c t i n g t h e
r e a c t o r p e r f o r m a n c e , t h u s v e r i f y i n g t h a t S is n o t a n i n d e p e n d e n t p a r a m e t e r for a s e p a r a t e d cell.
Comparison of the one-step and the continuous
models.--A c o m p a r i s o n of p e r f o r m a n c e criteria for t h e
Zn/Br2 cell as p r e d i c t e d b y t h e o n e , s t e p a n d b y t h e cont i n u o u s m o d e l s is p r e s e n t e d i n T a b l e III. T h e c o n t i n u o u s
m o d e l for t h e Zn/Br~ cell e m p l o y s a v a r i e t y of axial s t e p
sizes in o r d e r to s t e p d o w n t h e r e a c t o r as q u i c k l y as possib l e a n d still r e t a i n a c c u r a c y to t h r e e digits i n s o l v i n g for ci
a n d ~. To a c h i e v e t h i s a c c u r a c y , s t e p sizes o n t h e o r d e r o f
5~ = 10 -~ are r e q u i r e d at t h e e n t r a n c e d u e to t h e l a r g e
d r i v i n g force i n s t a n t a n e o u s l y a p p l i e d , w h i l e in t h e last
60% of t h e r e a c t o r l e n g t h , step sizes of h~ = 0.01 are acc e p t a b l e . T h e a g r e e m e n t b e t w e e n t h e o n e - s t e p a n d cont i n u o u s m o d e l s is as g o o d as 1.3%, for I R d r o p i n t h e c a s e
shown. The worst agreement between the two techniques
is for ~T, at 5.6% in t h e c a s e s h o w n . T h e a g r e e m e n t is v e r y
g o o d for t h e Zn/Br~ cell b e c a u s e t h e f r a c t i o n a l c o n v e r s i o n
of r e a c t a n t is small, a b o u t 1.3% for B r - a n d 0.8% for Z n ~
w h e n SA = 0.065 cm. As t h e c h a n n e l w i d t h , SA, gets
smaller, t h e c o n v e r s i o n i n c r e a s e s d u e to a s m a l l e r r e a c t o r
volume. Accordingly, the agreement between models
w o r s e n s , to a b o u t 10% for eT w h e n SA = 0.02 cm, at w h i c h
t h e c o n v e r s i o n for B r - is a b o u t 4.1% a n d for Z n 2 is a b o u t
2.6%. B e c a u s e t h e a g r e e m e n t is g o o d a n d t h e d i f f e r e n c e i n
c o m p u t a t i o n t i m e is so g r e a t (up to 100 t i m e s f a s t e r for
t h e o n e step), t h e o n e - s t e p m o d e l is u s e d i n all s u b s e q u e n t w o r k to p r e d i c t t h e p e r f o r m a n c e of t h e cell.
Performance predictions at the initial state of
charge.--The Zn/Br~ cell m o d e l c a n b e u s e d to p r e d i c t cell
Table III. Comparison of the one step to the continuous model for the Zn/Br2 cell
Input parameters: L/v,v~ = 15s, Ete. = 1.9V
SA = 0.065 cm, N~Ss = 0.18 cm
One step
iav~
IR drop
PB~
Pzn
EI,
Computation
time on
CDC CYBER
170/825
(mA/cm ~)
(moYcm~s) • 10~
(mol/cm~s) x 10~
19.69
15.146
10.307
6.724
0.619
(CPU s)
97.0
(mV)
Continuous
20.02
15.347
10.495
6.471
0.586
% Difference
(from
continuous)
1.6
1.3
1.8
3.9
5.6
10,080.
Downloaded 06 Jun 2011 to 129.252.106.103. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp
Vol. 133, No. 7
40.0
.
.
.
.
A MATHEMATICAL MODEL
I
.
.
.
.
.
.
]
.
.
.
.
I
.
.
.
240
.
. . . .
,
. . . .
,
.
.
1303
.
.
.
.
,
. . . .
,
. . . .
,
. . . .
r
30.0
r
35.0
25,0
/
30.0
20.0
%
25.0
A
E
20.0
x
<
150
E
~E
00.
20.0
=*
19.0
.3
e
eL-
15.0
10.0
PBr
5,0
SA = 0.065 c m
NmSs = 0.18 cm
Er
L/v~,g = 15 s
5.0
= 1.g V
\
SA = 0 . 0 6 5 c m
Pz,
N m S s (crn)
0.0
.
.
.
.
I
180
,
,
,
1.86
~
,
,
,
L
1.90
.
.
.
.
L
1.95
.
.
.
.
200
205
Ece. ( V )
Fig. 2. The effect of applied cell potential on the production rate
per area of electrode, of Br2 and Zn, at initial charging conditions.
performance at a specific state of charge as a function of
any of the i n d e p e n d e n t parameters. Figures 2-5 show
some of the performance predictions that can be made at
the initial state of charge corresponding to the feed concentrations listed in Table I.
For instance, the predicted effect of E~H on PB,.~ and Pz,,
at initial charging conditions, is shown in Fig. 2. Note that
4500
.
.
.
:
.
.
.
~
9
~
.
.
.
~
.
,
.
Fig. 4. The effect of effective separator thickness on the average
current density and potential drop across the cell, at initial charging
conditions.
the rate at which Brz is stored is significantly different
from the rate at which Zn is stored. It is important that
P,~r~ is not greater than Pzn throughout all states of charge,
since the amount of product (Br~ and Zn) and of reactant
(Br- and Zn ~) would become greatly mismatched
quickly, causing battery failure. This problem is resolved
when considering performance under charging conditions. Figure 2 also suggests that the battery can be
0.70
.
. . . .
[
. . . .
L
N~Ss = 0.18 cm
L/Vavg = 15 S
E~e,, = 1.9 V
425.0
0.65
400:0
~-
060
3
375.0
0.55
350.0
theoretical minimum
energy consumption
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
SA = 0 . 0 6 5 c m
Eceil :
3 2 5 . 0
.
0.00
.
.
.
0.02
i
0.04
r
I
I
J
005
I
I
0.08
'
'
]
'
0.10
SA (cm)
Fig, 3. The effect of channel width on the energy consumption per
mole of Br,: stored, at initial charging conditions.
0.50
. . . .
I
00
. . . .
1.g V
~ .
01
.
.
.
~
0.2
. . . .
03
N m S s
I
04
. . . .
I
0.5
. . . .
L
06
. . . .
0.7
(Ore)
Fig. 5. The effect of effective separator thickness on the total energy efficiency, at initial charging conditions (ev = 0.9384).
Downloaded 06 Jun 2011 to 129.252.106.103. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp
J. Electrochem. Soc.: E L E C T R O C H E M I C A L
1304
charged about as rapidly as desired, by increasing the applied cell potential. However, since the amount of current
passed will also increase with Ece~, the m a x i m u m allowable value of current density to avoid dendrite growth
must be carefully considered.
In Fig. 3, the predicted effect of SA on the energy consumption per mole of Br2, ~Br2, i s illustrated at initial
charging conditions. The dashed line indicates the theoretical m i n i m u m value of ~B,2, 344.1 kJ/mol of Br2. At
values of SA less than 0.02 cm, the energy consumption
per mole of Br~ Stored rises unacceptably high, while for
values of SA greater than 0.05 cm, ~B~2levels out to about
363. kJ/mol of Br2. Figure 3 shows that, at the initial state
of charge, the energy consumption per mole of Br2 is dramatically and adversely affected by narrow flow channels. It suggests that designing a cell with channel widths
of less than 0.02 c m is not advisable.
Figure 4 illustrates the predicted effect of NmSs On the
IR drop and o n iavg , at initial charging conditions. Figure 4
shows that the potential drop increases as NmSs increases,
as expected, since the separator contributes to the overall
cell o h m i c resistance. Figure 4 also shows that the average current density drops with increasing NmSs since the
cell mass-transfer resistance is increased by the thicker
separator.
Figure 5 demonstrates the predicted effect of NmSs on
the total energy efficiency at initial charging conditions.
Because Er is fixed at 1.9V, the voltaic efficiency is
fixed at ev = 1.783/1.9 = 0.9384. Thus, Fig. 5 is actually a
measure of the change of coulombic efficiency with effective separator thickness. At initial charging conditions,
the relationship between e~ and NmSs is nearly linear, with
the efficiency surprisingly dropping as the separator resistance increases. This performance suggests that, at
least at the state of charge considered, the increased mass
transfer and ohmic resistance of t h e separator unfortunately influences the cell performance more strongly
than does the beneficial aspect of preventing Br2 diffusion from the anode to the cathode. This trend is reversed
as charging continues, as discussed below.
Performance predictions at various states of
charge.--Figures 6-11, in which the state of charge is followed for three values of NmSs (Fig. 6-8) and three values
of L / v ~ (Fig. 9-11), resolve the problems found of mismatched production rates and poorer performance as the
effective separator thickness is increased at the initial
state of charge.
Although pseudo-steady state is assumed in the model,
the concentration and potential distributions in the cell
do change over long periods of time, and therefore the
state of charge and cell performance also change. Bellows
of E x x o n (13) h as determined that a convenient parameter for following the state of charge of a ZrgBr~ battery is
the amount of Zn ~+ which has been consumed and deposited as Zn. The percentage of Zn ~* plated as solid Zn,
0.9
~-
,
i
9 h
9 i
. . . .
,
. . . .
r
. . . .
i
July 1986
SCIENCE AND TECHNOLOGY
12.0
[
~.
N~Ss=
0.06 cm
L/v~g
= 15 s
Ece, = 1 , 9 V
~
e.o
:"
m- 4.o
"'-..
9)
9
2.0
.......
Br2
Zn
o.oo.,,,,,~o,
018cm
~ 0.12 cm
cm
,,,Lo,,,,4o,,,,/o.o,
' /,.o'"'~,,.o"
1,
% Z n ~+ P l a t e d
Fig. 7. The effect of effective separator thickness on production
rote per area of electrode, under charging conditions.
then, can be defined as the difference between the original amount of Zn ~§ introduced as feed to the system and
the amount of Zn 2§ currently entering the cell. The initial
feed concentration of Zn 2. introduced to both channels of
the cell is Czn2+,rer,while the amount currently entering the
system is Czn2+.feed.A in the anolyte and Czn2+.feed.C in the
catholyte. Therefore, the percentage of Zn ~§ plated relative to the start-up concentration of Zn ~§ is
%Zn '-'§ plated = 100%
x
]
SA(CZn2+,ref -- Czn2+ feed,A) 4- Sc(Czn2+ ref -- Czn2+,feed,C)
- s ~
+ S~(czn'~,,oa
"
[433
or, when SA = Sc.
%Zn 2+ plated = 100% x [1.0 - 0.5(ez~§
+ eZn2+.r~d.c)]
[44]
Bellows (13) claims that when approximately 70% of the
Zn 2+ ions are plated, the system is fully charged. Because
of the large amount of computer t i m e required to follow
the state of charge, studies were limited to approximately
25% of Zn '-'" plated.
In order to follow the concentration distribution as a
function of the state of charge, it is necessary to assume
that the exit average concentration of each species in
each channel, at a known state of charge, will be the feed
concentration of each species for that particular channel
at some later state of charge. That is, it is reasonable to assume that since the composition of anolyte and catholyte
is assumed to change slowly under pseudo-steady-state
conditions, then the composition leaving the cell at any
state of charge will eventually be duplicat"ed in the storage tanks and thus in the feed to the anolyte and catholyte channels. Also, it is assumed that the anolyte is
9
L/v..~, = 1 5 s
~oo.o
I \
/,,,./
SA = 0 . 0 6 5 c r n
/ ~
0.7
iloo.o
E~., = 1 . 9 V
N~Ss = 0.06 cm
06
,- .....
0.5
B,,
/
700.0
04
/>o12cm
,'J"
03
0.2 0.0
50
100
150
20 0
% Zn 2+ Plated
25o
30.0
350
Fig. 6. The effect of effective separator thickness on energy efficiency, under charging conditions (ev = 0.93841.
seoo
oo
5.0
100
150
20.0
% Z n 2+ P l a t e d
2B.O
30.0
35.0
Fig. 8. The effect of effective seporator thickness on energy consumption, under charging conditions.
Downloaded 06 Jun 2011 to 129.252.106.103. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp
Vol. 133, No. 7
A MATHEMATICAL MODEL
stored in a tank separate from the catholyte tank, so that
the feed composition of each channel changes independently. Although the assumptions that the process is under
pseudo-steady-state conditions and that the exit composition at one state of charge is the feed composition at a future state of charge are valid only for very low conversion,
these assumptions at least present a rough estimate of the
performance possible as a function of the state of charge.
Cases studied under charging conditions include three
effective thicknesses of the separator and three settings of
residence time, under otherwise fixed conditions. For all
cases, the start-up composition of the electrolyte is the
same as that used to study initial charging conditions.
The applied cell potential and channel are also fixed,
with Er = 1.9V and SA = 0.065 cm. For the cases in which
the effective separator thickness is varied, L/vavg is fixed
at 15s and the three values of NmSs studied are 0.06, 0.12,
and 0.18 cm. For the cases in which the residence time is
varied, N~Ss = 0.18 cm and the three values of L/v~v~ studied are 10, 15, and 20s. These cases were chosen as representative of typical cell designs. The performance of the
cells with different separators under charging conditions
is shown in Fig. 6-8 and the performance of the cells at
different residence times is shown in Fig. 9-11.
The predicted effect of NmSs on CT u n d e r charging conditions is demonstrated in Fig. 6. For each of the three
cases shown, e~ improves through the initial charging process, levels off at a m a x i m u m efficiency, and then drops
steadily as the cell is charged further. At initial charging
conditions, up to about 2% of Zn x* storage, the efficiency
is best for the cell with the thinnest separator (0.06 cm)
and worst for the cell using the thickest separator (0.18
cm). This behavior is due to the differences in cell mass
transfer and ohmic resistance for the three separators. As
the charging process continues, however, e~ becomes opt i mu m for the cell with the thickest separator and the
lowest for the cell with the thinnest separator considered,
as shown in Fig. 6. In addition, the discrepancy between
cell designs b eco m e s more p r o n o u n c e d as the amount of
charge progresses. Figure 6 shows, therefore, that the separator overcomes the disadvantage of an increased cell resistance when the effective thickness of the separator is
increased, by minimizing the amount of Br~ which diffuses from the anolyte to the catholyte. Therefore, a
thicker separator is necessary to i m p r o v e the cell energy
efficiency, especially when the cell is to be driven to a
high state of charge. For inStance, if it is desired to keep
eT above 0.5 t h r o u g h a state of charge of 30% Zn ~ plated,
then NmSs must be greater than 0.18 cm.
The response of P ~ and Pz, to the three cells of various
effective separator thickness, as charging progresses, is
shown in Fig. 7. Again, for the lowest states of charge, the
cell with the separator of width 0.06 cm shows the best
performance, having the highest production rate for either Br._, or Zn, but as charging continues, the production
rate drops to the lowest of the three cases considered.
oB~
. . . .
,
. . . .
,
9
,
9
-
~
IO,O
:
-
,
,
1305
.
.
,
.
.
.
,
9
.
Llv,vg = 10 s
,
9
,.
.
NmSs=O.18cm
~ ' ~
SA = 0.065 cm
S
= 1.g V
Er
20;" ~ .
x
"N
s,o
I
60
Br~
.......
4"0
0'0
I
Zn
r
i
,
~ 0
'
I
k
'
I;.0
t
'
t
% Zn 2+
I
6 0
a
'
,
I
;.0
I
t
,
t
Z~.O
Plated
Fig. 10. The effect of residence time on production rote per area of
electrode, under charging conditions.
Likewise, the thickest separator, at 0.18 cm, has the
slowest production rate at initial charging conditions, but
shows the best performance at the highest states of
charge.
At initial charging conditions, Pzn is significantly less
than PB,2, but as the state of charge increases to the moderate to high range, Zn is deposited faster than Br., is
stored, for each of the cells considered. This behavior is
important to the success of the Zn/Br~ battery. Because
the battery is a closed system, it is necessary that in a
complete cycle of charge and discharge, Br2 is not produced in excess of the amount of Zn deposited, and vice
versa. Figure 7 shows that although Br,~ is initially produced faster than Zn, fortunately the relative rates reverse so that through a full charging process, the amounts
of each gtored species will be approximately equivalent.
Integration of P,,~ and Pz, over the length of charge will
indicate whether Br2 or Zn ='~is lost in each cycle, allowing
predietion of the n u m b e r of cycles until the concentrations are badly mismatched.
In Fig. 8, the energy consumption based on both Br2
and Zn is illustrated as charging progresses, for the three
values of NmSs. As the cell is charged, (o~,~.~increases quite
dramatically and is particularly sensitive to the thickness
of the separator. At an initial state of charge, the value of
(o,,= is identical for every value of NmSs, but as the state of
charge progresses past 20% of the Zn'-'" ions deposited, the
energy required to produee and store a mole of Br~ when
NmSs is 0.06 cm is more than double that required when
NmSs is 0.18 am. The value of ~oz, is likewise sensitive to
the effective thickness of the separator.
As with the production rates, it is important that the energy consumed to produce and store each of Br~ and Zn
be equivalent for a full cycle of the battery. Figure 8 demonstrates that over a large'amount of charging, the energy
consumption per mole of Zn is about the same as the consumption per mole of Br~.
,
rose
0.75
10s
z
20 S
0.70
A
5 s
5500
15s
"
,
,
~
~ 2 . 0 S
2
los
~- ~00.0
15S
",',,,
20
~",,','.
Br2
- ......
~
~
;
]
"
,
'
;
:
:
:
"
Zn
000
N~Ss=
0.18 cm
.......
"~'~
. . . . . . . . . . . . . . . . . . . . . . . .
SA = 0.065 cm
E~a, = 1.9 V
o65
E~,
o.~o
00
o
NmSs= 0.15 cm
N = : ; - O0:'086::rn
400C
Ioo% Zn 2+ Platedl~~
20.o
~5o
. . . .
3500
o o
'
6.0
. . . .
r
. . . .
1oo
% Zn2§
Fig. 9. The effect of residence time on energy efficiency, under
charging conditions (ev = 0.9384).
i
la o
,
,
,
= 1.9 V
i. . . . .
2oo
2s.o
Plated
Fig. 11. The effect of residence time on energy consumption, under
charging conditions.
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1306
J. Electrochem. Soc.: E L E C T R O C H E M I C A L
Consideration of Fig. 6-8 as a w h o l e s h o w s that of the
three separators considered, the t h i c k e s t one, with NmSs =
0.18 cm, is by far the best. This t h i c k n e s s gives the
h i g h e s t efficiency and p r o d u c t i o n rates of Br2 and Zn per
area of electrode, o v e r the range of c h a r g e considered,
and p r o v i d e s a dramatically l o w e r e n e r g y c o n s u m p t i o n
per m o l e of b o t h Br2 and Zn. In addition, w h e n N,,Ss has a
v a l u e of 0.18 cm. Fig. 6-8 suggest that to a c h i e v e the best
battery, c h a r g i n g should be b e g u n a p p r o x i m a t e l y w h e n
the status of the battery is at a state of charge of 5% Zn ~
plated, rather t h a n at a fully d i s c h a r g e d state, and that the
battery s h o u l d be charged up only to a 20% deposit of
Zn ~* ions. Within this range of charge, er, Pz,, and C0z,, all
r e a c h o p t i m u m v a l u e s and PB~ and ~B~ deteriorate, but at
a c c e p t a b l e rates. On the other hand, at states of charge
l o w e r than 5% Zn 2+ plated, ~t, Pz,, and ~z~ exist at decidedly n o n d e s i r a b l e values, while at states of charge h i g h e r
t h a n a 20% d e p o s i t of Zn 2. ions, p e r f o r m a n c e criteria begin to deviate f r o m t h e i r o p t i m u m values rapidly.
In Fig. 9-11, a c h a n g e in r e s i d e n c e t i m e on cell performance u n d e r c h a r g i n g c o n d i t i o n s is s h o w n to h a v e little effeet. F i g u r e 9 s h o w s that v a r y i n g the r e s i d e n c e t i m e bet w e e n values of 10, 15, and 20s has only a slight effect on
eT, and the d i v e r g e n c e that occurs is m o s t p r o n o u n c e d at
initial charging states or w h e n the battery is charged for
e x t e n d e d periods of time.
In Fig. 10, again t h e r e is v e r y little v a r i a n c e in production rates of either Br.~ or Z n as t h e r e s i d e n c e t i m e is
changed. P e r h a p s m o s t r e m a r k a b l e is that at the h i g h e s t
states of charge illustrated (above 15% d e p o s i t i o n of Zn-'~),
the p r o d u c t i o n rate of Brz per surface area of a n o d e is
e q u i v a l e n t for all t h r e e r e s i d e n c e times. N o t e that Pz,,
starts out at an initial state of charge v e r y low relative to
the rate of storage of Br~. Pz, rises slightly and t h e n drops
off slower t h a n PB~, w i t h Zn e v e n t u a l l y b e i n g p r o d u c e d at
a rate h i g h e r t h a n Br~ is stored. S u c h b e h a v i o r again is desired, b e c a u s e o v e r a cycle of charge and discharge, it is
i m p o r t a n t that n e i t h e r excess Br.2 nor excess Zn be
produced.
In Fig. 11, t h r o u g h all states of charge, ~0,~r~ d i v e r g e s
very little due to different values of L/v~v~. On the o t h e r
hand, ~z, is s h o w n to be u n a f f e c t e d by L/V~v~ only w h e n
the e n e r g y c o n s u m p t i o n is at its m o s t o p t i m u m , b e t w e e n
a 5% and 10% d e p o s i t i o n of Zn-". A r e s i d e n c e t i m e of 20s
p r o v i d e s the m i n i m u m value of ~OZnfor all states of charge.
C o n s i d e r a t i o n of Fig. 9-11 shows that a change of resid e n c e t i m e has little effect on overall cell performance,
w h e t h e r efficiency, p r o d u c t i o n rate, or energy c o n s u m p tion. This is especially true relative to the change in perf o r m a n c e u n d e r g o n e , as charging progresses, w h e n NmSs
is varied. The o p t i m u m v a l u e of L/v~v~, a m o n g the t h r e e
values considered, is the largest time, 20s. The t r e n d s
s h o w n suggest that l e n g t h e n i n g the r e s i d e n c e t i m e bey o n d 20s will further e n h a n c e the p r e d i c t e d performance,
If the c h a r g i n g process is restricted b e t w e e n 5 and 20% of
Zn ~" plated as d i s c u s s e d earlier, h o w e v e r , t h e n Fig. 9-11
indicate that the v a l u e of L/v~v~ u s e d has virtually no effeet on eT, P~, or ~o~.
Conclusions
The Zn/Br~ cell m o d e l p r e s e n t e d is a useful tool to aid
in the design of a c o m p l e t e battery system. Its ability to
predict a c c u r a t e l y the p e r f o r m a n c e of cells within batteries currently u n d e r d e v e l o p m e n t by E x x o n , ERC, and
others is u n t e s t e d b e c a u s e data are v e r y limited for proprietary reasons. H o w e v e r , it is a versatile m o d e l w h i c h
can be e x t e n d e d easily or modified to fit a s s u m p t i o n s
not used here. In addition, t h e m o d e l can be u s e d in conj u n c t i o n w i t h a p a r a m e t e r e s t i m a t i o n t e c h n i q u e and exp e r i m e n t a l results from a w o r k i n g cell to d e t e r m i n e unk n o w n s y s t e m constants s u c h as diffusivities, D~,
e x c h a n g e c u r r e n t densities, io~.,,f, and reaction transfer coefficients, a~, and a~j.
Acknowledgment
This w o r k was s u p p o r t e d in part by the C e n t e r for Energy and Minerals R e s e a r c h at Texas A & M University.
SCIENCE
AND TECHNOLOGY
July 1986
M a n u s c r i p t r e c e i v e d J u l y 19, 1985.
L I S T OF S Y M B O L S
c~
c o n c e n t r a t i o n of species i, moYcm 3
Ci,avg.A average c o n c e n t r a t i o n o f species i in the anolyte
channel, moYcm 3
el,feed feed c o n c e n t r a t i o n of species i, moYcm '~
C~.ref
r e f e r e n c e c o n c e n t r a t i o n of species i, moYcm ~
diffusion coefficient of species i, cm~/s
D~
effective diffusion coefficient of species i in the
D~.~
separator, em~/s
d e n s i t y of p u r e solvent, k g / c m 3
do
diffusion coefficient of limiting reactant, cm2/s
DR
Eo,,I
applied cell potential (= V~ - Vo), V
F
F a r a d a y ' s constant, 96,487 C/tool
I
total cell current, m A
iav~
average c u r r e n t density at an e l e c t r o d e (= I/LW),
mA/cm-'
n o r m a l c o m p o n e n t of c u r r e n t d e n s i t y d u e to reaci,j
tion j, mA/cm-'
inJ'avg a v e r a g e n o r m a l current d e n s i t y due to reaction j,
mAYem ~
.
e x c h a n g e c u r r e n t density of r e a c t i o n j, m A / c m ~
Z~
I R drop
potential drop across the cell (= qSoa - ahoy), V
K~
e q u i l i b r i u m c o n s t a n t for t r i - b r o m i d e reaction, cm3/
mol
L
e l e c t r o d e length, c m
L/Vavg r e s i d e n c e t i m e of the reactor, s
flux v e c t o r of species i, moYcm~-s
N~
n u m b e r of electrons passed in r e a c t i o n j
nj
MacMullin n u m b e r in the separator
N~
NmSs effective separator thickness, cm
n o r m a l c o m p o n e n t of the flux (y- or q-direction)
Nnl
of species i, mol/cm~-s
p~
p r o d u c t i o n rate of species i p e r e l e c t r o d e area,mol/
em2-s
p~j
anodic reaction order of species i in reaction j
P e c l e t n u m b e r (= 2Sv~JDR)
Pe
c a t h o d i c reaction o r d e r of species i in reaction j
q~
gas law constant, 8.314 J / m o l - K
R
h o m o g e n e o u s reaction rate, mol/cm3-s
R~
total e l e c t r o d e gap, cm
S
SA
anolyte c h a n n e l width, c m
Sc
catholyte c h a n n e l width, c m
s~j
s t o i c h i o m e t r i c coefficient of species i in reaction j
Ss
w i d t h of t h e separator, c m
temperature, K
T
standard half-cell potential, V
U~~
Uj,ref open-circuit potential of reaction j based on the
r e f e r e n c e concentrations, V
electrolyte v e l o c i t y vector, cm/s
vy~
a n o d e potential, V
V~v~
average v e l o c i t y of the electrolyte, cm/s
Vr
c a t h o d e potential, V
V~
potential of electrode, V
v.~.
velocity c o m p o n e n t of the electrolyte in the x-direction, cm/s
b r e a d t h of t h e electrode, cm
W
axial coordinate, c m
x
n o r m a l coordinate, cm
Y
charge n u m b e r of species i
z~
Greek
aspect ratio, S/L
anodic transfer coefficient for reaction j
a~
c a t h o d i c transfer coefficient for r e a c t i o n j
Q~eJ
Ac,~ difference in average c o n c e n t r a t i o n of b r o m i n e carrying species (Br~ and Br:,-) from e n t r a n c e to
exit, m o l / c m :~
ec
c o u l o m b i c efficiency
eT
total cell e n e r g y efficiency
ev
voltaic efficiency
d i m e n s i o n l e s s axial c o o r d i n a t e (x/L)
d i m e n s i o n l e s s n o r m a l c o o r d i n a t e (y/S)
~j
o v e r p o t e n t i a l at e l e c t r o d e surface (V~ - q~o~- U~,r~f),
V
7'
d i m e n s i o n l e s s n o r m a l coordinate, specific to a
flow c h a n n e l
0~
d i m e n s i o n l e s s c o n c e n t r a t i o n of species i (cj/cj,~f)
0~.~,ed d i m e n s i o n l e s s feed c o n c e n t r a t i o n of species i
0i,o
d i m e n s i o n l e s s c o n c e n t r a t i o n of species i at the
e l e c t r o d e surface
p
resistivity of separator and its s u r r o u n d i n g electrolytic solution, ~ - c m
po
resistivity of the p u r e electrolytic solution, ~ - c m
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Vol. 133, No. 7
A MATHEMATICAL MODEL
r
CPoa
@oc
@oe
~i
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
solution potential, V
solution potential at the anode, V
solution potential at the cathode, V
solution potential at electrode, V
energy consumption per mole of product stored,
kJ/mol
REFERENCES
J. L. Chamberlin, in "Proceedings of the EPRI/LBL
Workshop on the Electrochemistry of Zinc/Halogen
Batteries," Palo Alto, CA, Nov. 30-Dec. 1, 1983, Vol.
I, p. 2-15 (1984).
R. E. White, M. Bath, and M. Raible, This Journal, 130,
1037 (1983).
J. Lee and J. R. Selman, ibid., 129, 1670 (1982).
J. Lee, P h . D . Dissertation, Illinois Institute of Technology, Chicago, IL (1981).
J. Van Zee, R. E. White, P. Grimes, and R. Bellows, in
"Electrochemical Cell Design," R. E. White, Editor,
p. 293, P l e n u m Press, New York (1984).
P. S. Fedkiw and R. W. Watts, This Journal, 131, 701
(1984).
J. Van Zee, Ph.D. Dissertation, Texas A&M University, College Station, TX (1984).
T. Nguyen, C. Walton, R. E. White, and J. Van Zee,
This Journal, Submitted for publication.
M. Eigen and K. Kustin, J. Am. Chem. Soc., 84, 1355
(1962).
D. J. Pickett, "Electrochemical Reactor Design," 2nd
ed., Elsevier Scientific Publishing Co., New York
(1979).
R. E. White, S. E. Lorimer, and R. Darby, This Jour-
1307
nal, 130, 1123 (1983).
12. M. Mader, C. Walton, and R. E. White, ibid., 133, 1124
(1986).
13. R. Bellows, in "Proceedings of the EPRI/LBL Workshop on the Electrochemistry of Zinc/Halogen Batteries," Palo Alto, CA, Nov. 30-Dec. 1, 1983, Vol. I,
p. 4-3 (1984).
14. R. Bellows, P. Grimes, H. Einstein, E. Kantner, P.
Malachesky, and K. Newby, IEEE Trans. Veh. Technol., vt-32, 26 (1983).
15. "CRC Handbook of Chemistry and Physics," 60th ed.,
CRC Press, Inc., Boca Raton, FL (1979).
16. J. S. Newman, "Electrochemical Systems," Prentice
Hall, Inc., Englewood Cliffs, NJ (1973).
17. R. E. White, in "Proceedings of the EPRI/LBL Workshop on the Electrochemistry of Zinc/Halogen Batteries," Palo Alto, CA, Nov. 30-Dec. 1, 1983, VoL I,
p. 5-65 (1984).
18. R. Selman, in "Proceedings of the EPRI/LBL Workshop on the Electrochemistry of Zinc/Halogen
Batteries," Palo Alto, CA, Nov. 30:Dec. 1, 1983, Vol.
II, p. 10-63 (1984).
19. P. Grimes, in "Proceedings of the EPRI/LBL Workshop on the Electrochemistry of Zinc/Halogen
Batteries," Palo Alto, CA, Nov. 30-Dec. 1, 1983, Vol.
II, p. 7-25 (1984).
20. U. Landau, in "Proceedings of the EPRI/LBL Workshop on the Electrochemistry of Zinc/Halogen
Batteries," Palo Alto, CA, Nov. 30-Dec. 1, 1983, Vol.
II, p. 10-75 (1984).
21. M. Mader, M. S. Thesis, Texas A&M University, College Station, TX (1985).
The Lithium Surface Film in the Li/S02 Cell
K. M. Abraham* and S. M. Chaudhri
EIC Laboratories, Incorporated, Norwood, Massachusetts 02062
ABSTRACT
Evidence is presented which suggests that the Li surface film in the Li/SO~ cell comprises a complex mixture of
products which include Li~S and several Li sulfur-oxy compounds. Elemental analysis, IR spectral, and XPS data indicate that the Li sulfur-oxy compounds may be Li~S~O4, Li~SO:~, Li~S,O,~, and Li=,S~O~.
Lithium batteries, particularly those containing liquid
electrolytes, owe their stability to protective films on the
Li anode surface. These films, formed by spontaneous reaction of Li with the electrolyte solution, have been
identified to affect such battery properties as voltage delay (1), anodic polarization (2), self-discharge rates (3),
rechargeability (4), and safety (1, 5).
One of the most extensively studied interphases between the Li anode and a liquid electrolyte is the LiC1
film in the Li/SOCl~ cell (1, 3). Studies of the morphological and electrical properties of this film have led to the
recognition of it as a solid electrolyte interphase (SE!).
Such an interphase, while allowing the migration of LF
across it, behaves as an electronic insulator preventing
self-discharge and serves as a barrier to further reactions
between the Li anode and the electrolyte. It is now accepted that a SEI or a conceptually similar film exists in
all liquid electrolyte Li batteries and the film is formed
instantaneously upon contact of the Li anode with the
electrolyte. This compact SEI is believed to be the "protective" film. The latter is often covered by a coarse porous film whose thickness is determined
by, among
other
factors, the length of exposure of the Li electrode to the
electrolyte. The chemical composition and physical properties of Li surface films have also been studied in
PC/LiC104 (6), THF/LiAsF,, (7), and Li~S,, (8).
It has been believed all along that the Li surface film in
the Li/SO~ cell solely comprised Li._,S~O4 (1). However,
recent studies have begun to indicate a more complex
composition for the film. Auger spectroscopic results of
Nebesny et al. (9) of fresh Li surfaces after exposure to
* Electrochemical Society Active Member.
low levels of SO~ indicated the presence of Li~S and LifO.
X-ray photoelectron spectroscopic (XPS) studies of similar Li surfaces by Hoenigman and Keil (10) showed the
presence of Li sulfur-oxy compounds in addition to Lifo
and Li~S. The latter authors also found only Li._,Oand Li~S
at low levels of SO~ exposure. With longer SO~ exposure,
however, they observed the evolution of XPS peaks at
168.8, 166, and probably 163.2 eV. Considering that only
1.3 eV separates the 163.2 eV peak from the Li~S main
peak at 161.9 eV, the former could be the 2P1/2 transition
of the sulfide. The sulfur 2P1/2 9 3/2 doublet separation,
evaluated from the spectra of a n u m b e r of Li sulfur-oxy
compounds, is - 1 eV. The peak at 168.8 eV was assigned
to Li~SO:~.No assignment was given to the peak at 166 eV.
Recently we reported (11) on the composition of the Li
surface film found on the anodes of partially discharged
and stored Li/SO~ cells. Infrared (IR) spectral and XPS
analyses of the film indicated the presence of several Li
sulfur-oxy compounds including Li~S~O4 and Li~S,Q.
Lately, we have examined by SEM, IR, and XPS, the Li
surfaces of a large n u m b e r of L i / S Q cells after they have
been subjected to various regimes of discharge, such as
partial and full discharge at low to moderate rates, high
current pulse discharge, and forced overdischarge. The
results of these studies have led us to conclude that the Li
surface film in the Li/SO._, cell comprises a complex mixture of products including several Li sulfur-oxy compounds and Li~S. Our results are reported in this paper.
Experimental
All the experiments involving manipulation of air sensitive materials were carried out inside a Vacuum Atmo-
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