scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A Mathematical Model of a Lead‐Acid Cell Discharge, Rest, and Charge

01 Dec 1987-Journal of The Electrochemical Society (The Electrochemical Society)-Vol. 134, Iss: 12, pp 2953-2960
TL;DR: In this paper, a mathematical model of a lead-acid cell is presented which includes the modeling of porous electrodes and various physical phenomena in detail, including the dynamic behavior of the acid concentration, the porosity of the electrodes, and the state of charge of the cell during discharge, rest, and charge.
Abstract: A mathematical model of a lead-acid cell is presented which includes the modeling of porous electrodes and various physical phenomena in detail. The model is used to study the dynamic behavior of the acid concentration, the porosity o f the electrodes, and the state of charge of the cell during discharge, rest, and charge. The dependence of the performance of the cell on electrode thicknesses and operating temperature is also investigated. The lead-acid system is used in the largest n u m b e r of secondary bat ter ies manufac tu red in the world. The most important market remains the car battery for starting, l ighting, and igni t ion, with approximate ly 50 • 106 uni t s sold per year in the U.S.A. (1). Other appl icat ions are in emergency power supplies, load-leveling, and more recently for ins t ruments , radio, and other electrical apparatus. The design and improvemen t of these batteries are mostly done by trial-and-error. This t radi t ional approach, which consis ts of experimenta l cell bui ld-ups and extensive testing, is costly and t ime consuming . Fur thermore , resul ts from such tests provide only global in format ion and do not provide insight into the governing phenomena. It is advantageous to develop a mathematical model of the cell which would allow one to gain a better unders tand ing of the cause and effect re la t ionships and the p h e n o m e n a involved, and suggest directions for improvements . Complement ing experimental testing with mathematical model ing is a cost effective approach to the developm e n t and design of batteries. Test ing is still needed to verify predictions of the model and to uncover physical p h e n o m e n a that may not have been inc luded in the model. But with the help of this mathemat ica l tool, ex*Electrochemical Society Active Member. **:Electrochemical Society Student Member. ~Present address: Department of Chemical Engineering, Texas AM see http://www.ecsdl.org/terms_use.jsp 2954 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 December 1987 Region 1, positive electrode.-poros i t y v a r i a t i o n Oe 1 [ MWebso4 Ot 2F PPbSO4 O h m ' s law in s o l u t i o n MWebo2 I Oi~ PPb02 ] 0X 0 [6] is Od)~ RT 0 in (cf) + ( 1 2 t ~ 0 [ 7 ] eexlK 0x F 0x O h m ' s law in s o l i d i2 exml a~l _ I = 0 [8] -{[ O'Pb02 0X mate r i a l b a l a n c e OC Fig. 1. One-dimensional macro-homogeneous model of a lead-acid cell 6 gions : a l ead g r id c u r r e n t c o l l e c t o r at x = 0, w h i c h is at t h e c e n t e r o f t h e p o s i t i v e e l e c t r o d e , t h e p o s i t i v e (PbO2) e l e c t r o d e ( r eg ion 1), t he p o s i t i v e e l e c t r o d e / r e s e r v o i r int e r f ace , t he r e s e r v o i r ( reg ion 2), t he r e s e r v o i r / s e p a r a t o r in te r face , t he separa to r ( region 3), t he s epa ra to r /nega t ive e l e c t r o d e i n t e r f ace , t he n e g a t i v e (Pb) e l e c t r o d e ( reg ion 4), and t h e c e n t e r of t h e n e g a t i v e e l e c t r o d e w h e r e ano the r gr id is located. Deta i ls of the g e o m e t r y are i gno red and the who le cel l is r ega rded as a h o m o g e n e o u s macros cop ic en t i ty w i th d i s t r i bu t ed quan t i t i e s in t h e d i r ec t i on p e r p e n d i c u l a r to t he grid. An e x t e n s i v e d i s cus s ion o f ave r a g e q u a n t i t i e s u s e d in t h e d e v e l . o p m e n t of t h e m o d e l has b e e n g iven by D u n n i n g (13) and T r a i n h a m (14). Addi t iona l ly , i s o t h e r m a l c o n d i t i o n s are a s s u m e d here . T h e e l ec t ro ly t e is c o n c e n t r a t e d H2SO4 w h i c h is c o n s i d e r e d to be a b i n a r y e l e c t r o l y t e t ha t d i s s o c i a t e s in to H + and HSO4in H~O. The e l ec t rode reac t ions d u r i n g d i s cha rge a r e PbO2(s) + HSO4 + 3H + + 2e-

Summary (1 min read)

1F( wPbso4 WPbo2)

  • Equations [12] and [16] satisfy the requirement that the flux of a species i, Ni, and the superficial current density, i2, are continuous across the interface.
  • The electrode solid phase potential is zero (Eq. [21] ) because there is no electrode solid phase in the reservoir.
  • Where EQUATION Equation [24] satisfies the condition that the fluxes of the species are continuous.
  • Equation [26] indicates that the superficial current density in the solution is equal to the applied current density since no charge transfer reaction occurs in either the reservoir or the separator.

Numerical Procedure

  • The governing equations were put into finite difference forms and solved using the numerical technique of Newman (18, 19) .
  • Each region was evenly divided, but node point spacings were different between regions.
  • The finite difference approximations of the derivatives for an internal mesh point can be written as.

Results and Discussion

  • The authors examine some simulated results and their implications.
  • Parameters not referenced are arbitrary but reasonable quantities.
  • The discharge cutoff was chosen to be 1.55V.
  • Implicitly, the exchange current density in the kinetic expression is also a strong function of temperature.
  • As mentioned earlier, the exchange current density was adjusted for the two temperatures so that the calculated cell voltages agreed relatively well with experimental observations.

Table I. Parameters used in the calculations

  • Cell voltage profiles during discharge are given in Fig. 2 .
  • Figure 5 shows the cell voltage profiles during charge at the two temperatures.
  • Note that the concentration of the electrolyte has not totally relaxed at the end of the lh rest.
  • The same kinetic parameters were used for the positive and negative electrodes in the simulations.
  • The solid phase conductivity does not have much influence on the electrode behavior when an electrode is thin.

Conclusions

  • This model predicts profiles of acid concentration, overpotential, porosity, reaction rate, and electrode capacity as functions of time and temperature.
  • The model can be used to evaluate the effects of electrode porosity, electrode thickness, various separators, acid reservoir volume, and operating temperature on the performance of the cell (voltage, power, and cold cranking amperage).
  • A negative electrode with 0.65 initial porosity instead of 0.53 also improves the discharge time by 7s.
  • A series of lead alloy specimens comprising binary Pb-Li, Pb-Sn, Pb-Sb, and quinary Pb-(A1, Mg, Sn, Li) alloys were electrochemically tested.
  • Homogeneous or single-phase alloys corroded uniformly and recorded similar behavior to pure lead while the richer alloys suffered severe intergranular corrosion and were anomalous.

Did you find this useful? Give us your feedback

Content maybe subject to copyright    Report

University of South Carolina University of South Carolina
Scholar Commons Scholar Commons
Faculty Publications Chemical Engineering, Department of
1987
A Mathematical Model of a Lead-Acid Cell: Discharge, Rest, and A Mathematical Model of a Lead-Acid Cell: Discharge, Rest, and
Charge Charge
Hiram Gu
T. V. Nguyen
Ralph E. White
University of South Carolina - Columbia
, white@cec.sc.edu
Follow this and additional works at: https://scholarcommons.sc.edu/eche_facpub
Part of the Chemical Engineering Commons
Publication Info Publication Info
Journal of the Electrochemical Society
, 1987, pages 2953-2960.
© The Electrochemical Society, Inc. 1987. All rights reserved. Except as provided under U.S. copyright law,
this work may not be reproduced, resold, distributed, or modiBed without the express permission of The
Electrochemical Society (ECS). The archival version of this work was published in the Journal of the
Electrochemical Society.
http://www.electrochem.org/
DOI: 10.1149/1.2100322
http://dx.doi.org/10.1149/1.2100322
This Article is brought to you by the Chemical Engineering, Department of at Scholar Commons. It has been
accepted for inclusion in Faculty Publications by an authorized administrator of Scholar Commons. For more
information, please contact digres@mailbox.sc.edu.

Vol. 134, No. 12
EFFICIENT VANADIUM REDOX FLOW CELL 2953
The independent sizing of the redox system's power
and storage capacity makes it particularly attractive for
long term electricity storage in remote areas where solar
arrays or wind generators have been installed. An at-
tractive feature of the vanadium redox battery is that
since there is no solution contamination due to cross-
mixing, after the original capital investment, there
would be negligible running and maintenance costs.
Since the cell can be completely discharged without any
deterioration in performance, it would be ideally suited
for large scale energy storage in load leveling applica-
tions.
A larger scale five-cell battery unit is currently under
construction so that the system performance can be fur-
ther tested before scaling up to a 1 kW unit.
Acknowledgments
Support for this project was provided under the Na-
tional Energy Research Development and Demonstra-
tion Programme which is administered by the Australian
Commonwealth Department of Resources and Energy.
The authors are grateful to Dr. Miron Rychcik for useful
discussions.
Manuscript submitted Sept. 2, 1986; revised manu-
script received Dec. 16, 1986.
The University of New South Wales assisted in meeting
the publication costs of this article.
REFERENCES
1. L. H. Thaller, NASA TMX-71540, National Aeronautics
and Space Administration, U.S. Department of En-
ergy, 1974; and U.S. Pat. 3,996,064 (1974).
2. L. H. Thaller, NASA TMM-79143, National Aeronautics
and Space Administration, U.S. Department of En-
ergy, 1979.
3. Redox Flow Cell Development and Demonstration
Project, NASA TM-79067, National Aeronautics and
Space Administration, U.S. Dept. of Energy, 1979.
4. H. H. Hagedorn and L. H. Thaller, NASA TM-81464,
National Aeronautics and Space Administration,
U.S. Department of Energy, 1980.
5. D. A. Johnson and M. A. Reid, NASA TM-82913, Na-
tional Aeronautics and Space Administration, U.S.
Department of Energy, 1982.
6. K. Nozaki, H. Kaneko, A. Negishi, and T. Ozawa, in
"Proceedings of Meeting" pp. 1641-1646, 18th IECEC
(1983).
7. R. F. Gahn, N. H. Hagedorn, and J. A. Johnson, NASA
TM87034, National Aeronautics and Space Adminis-
tration, U.S. Department of Energy, 1985.
8. M. Skyllas-Kazacos, M. Rychcik, and R. Robins,
Patent Applications filed.
9. M. Skyllas-Kazacos, M. Rychcik, R. Robins, A. Fane,
and M. Green, This Journal, 133, 1057 (1986).
10. R. Rychcik and M. Skyllas-Kazacos, J. Power Sources,
19, 45 (1986).
11. D.-G. Oei, J. Appl. Electrochem., 15, 231 (1985).
A Mathematical Model of a Lead-Acid Cell
Discharge, Rest, and Charge
Hiram Gu* and T. V. Nguyen, **'1
Electrochemistry Department, General Motors Research Laboratories, Warren, Michigan 48090-9055
R. E. White*
Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
ABSTRACT
A mathematical model of a lead-acid cell is presented which includes the modeling of porous electrodes and various
physical phenomena in detail. The model is used to study the dynamic behavior of the acid concentration, the porosity of
the electrodes, and the state of charge of the cell during discharge, rest, and charge. The dependence of the performance of
the cell on electrode thicknesses and operating temperature is also investigated.
The lead-acid system is used in the largest number of
secondary batteries manufactured in the world. The
most important market remains the car battery for start-
ing, lighting, and ignition, with approximately 50 106
units sold per year in the U.S.A. (1). Other applications
are in emergency power supplies, load-leveling, and
more recently for instruments, radio, and other electrical
apparatus. The design and improvement of these
batteries are mostly done by trial-and-error.
This traditional approach, which consists of experi-
mental cell build-ups and extensive testing, is costly and
time consuming. Furthermore, results from such tests
provide only global information and do not provide in-
sight into the governing phenomena. It is advantageous
to develop a mathematical model of the cell which would
allow one to gain a better understanding of the cause and
effect relationships and the phenomena involved, and
suggest directions for improvements.
Complementing experimental testing with mathemat-
ical modeling is a cost effective approach to the develop-
ment and design of batteries. Testing is still needed to
verify predictions of the model and to uncover physical
phenomena that may not have been included in the
model. But with the help of this mathematical tool, ex-
*Electrochemical Society Active Member.
**:Electrochemical Society Student Member.
~Present address: Department of Chemical Engineering, Texas
A&M University, College Station, Texas 77843-3122.
tensive experimental testing is no longer needed. Great
savings in material, labor, and time can be realized in the
development of a new battery.
The first use of mathematical models to describe the
behavior of the lead-acid system was applied to the po-
rous positive electrode (PbO2) by Stein (2, 3) and Euler
(4), with further improvements made by Simonsson (5-7),
Micka and Rousar (8), Gidaspow and Baker (9), and oth-
ers. A good review of the development in the theory of
flooded porous electrodes prior to 1975 has been pro-
vided by Newman and Tiedemann (10). Recently, Tiede-
mann and Newman (11) and Sunu (12) applied Newman's
theory to the development of a.complete cell model de-
scribing the discharge behavior of the lead-acid battery
system. However, a model for predicting the cell behav-
ior during charge and rest, as well as the effects of cy-
cling, is not available. To assist designers and engineers
in the further development of the lead-acid batteries
with improved performance and cycle life, a detailed
mathematical model of a lead-acid cell is presented that
can be used to predict the dynamic behavior of the cell
not only during discharge, but also during charge, rest,
and cycling.
Model Development
A schematic for the lead-acid cell is shown in Fig. 1.
The cell consists of the following boundaries and re-
Downloaded 13 Jun 2011 to 129.252.106.227. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

2954
J. Electrochem. Soc.:
ELECTROCHEMICAL SCIENCE AND TECHNOLOGY
December 1987
Region 1, positive electrode.--
porosity variation--
Oe
1 [ MWebso4
Ot
2F PPbSO4
Ohm's law in solution--
MWebo2 I Oi~
PPb02 ] 0X
0 [6]
is Od)~ RT
0 in (cf)
--+ -- (1- 2t~ -- -0 [7]
eexlK 0x F 0x
Ohm's law in solid--
i2
exml
a~l
_
I
=
0 [8]
-- {[ O'Pb02 0X
material balance--
OC
Fig. 1. One-dimensional macro-homogeneous model of a lead-acid cell 6
--
gions: a lead grid current collector at x = 0, which is at
the center of the positive electrode, the positive (PbO2)
electrode (region 1), the positive electrode/reservoir in-
terface, the reservoir (region 2), the reservoir/separator
interface, the separator (region 3), the separator/negative
electrode interface, the negative (Pb) electrode (region
4), and the center of the negative electrode where an-
other grid is located. Details of the geometry are ignored
and the whole cell is regarded as a homogeneous macro-
scopic entity with distributed quantities in the direction
perpendicular to the grid. An extensive discussion of av-
erage quantities used in the devel.opment of the model
has been given by Dunning (13) and Trainham (14). Addi-
tionally, isothermal conditions are assumed here. The
electrolyte is concentrated H2SO4 which is considered to
be a binary electrolyte that dissociates into H + and
HSO4- in H~O.
The electrode reactions during discharge
are
PbO2(s) + HSO4 + 3H + + 2e-
and
Pbls) + HSO(
discharge
) PbSO4(~) + 2H20
(positive electrode)
discharge
) PbSO4(~) + 2e- + H +
(negative electrode)
There are five explicit unknowns in the model: acid
concentration (c), electrode porosity (6), superficial cur-
rent density in the electrolyte (i2), potential in the solid
phase (qbl), and potential in the electrolyte ((b2). The gov-
erning equations and boundary conditions are presented
next.
Center of the positive electrode, x = O.--
Oc
- 0 [1]
ox
o6
- 0 [2]
Ox
i2 = 0 [3]
(bl =
0
[4]
O62
= 0 [5]
OX
Equations [1], [2], and [5] are in accordance with the as-
sumption of symmetry. Equation [3] states that, at the
center of the positive electrode, all the current is in the
current collector and none is in the electrolyte. In Eq. [4],
as a convenient choice, the solid phase potential, d)l, is
designated to be 0 V at this boundary. Without this refer-
ence potential, a particular solution cannot be obtained.
1F( wPbso4 WPbo2)
PPbS04
0t 2F PPb02
(3
2t~ + 2V ] i2 ac O ( Oc )
-- -- o --
E exl D
Ox Ox Ox
- . ~ 9 (1-cYo)+~ ox
electrode kinetics--for discharge and rest
~ -- 6ol
0i2
ama• ~ C~-e~ erna• Co1
0X
[ RT (~)1 - ~D2 --
P-l} o
exp [~- (q), - r - =
and for charge
0i2 * (C tYI( ~--~o1 )~1 e .... --E
--OX -- amaxll~ \ C-~ef ] 6maxl -- 6ol ~maxl -- ~ol
where
[9]
[10a]
exp [ c~IF Uebo~)] -
t RT (~1- ~b2-
[ -CtelF "'
Upb02)] } 0
exp [~ (r - r - =
[lOb]
Upb02 ~ Upbo20- Upb 8 [11]
Equation [6] describes the change in porosity with time
due to the conversion of the active material in the elec-
trode reaction. Equation [7] is a modified Ohm's law for
the electrolyte which states that the current in the elec-
trolyte is driven by the electric potential and chemical
potential gradients. Equation [8] is Ohm's law applied to
the solid matrix. Equation [9] states that the electrolyte
concentration at any point in space changes with time
because of the electrode reaction, diffusion, and migra-
tion. Equations [10a] and [10b] are kinetic expressions
for the electrode reaction. Equation [10b] includes a fac-
tor to account for the depletion of lead sulfate as lead di-
oxide is being formed. For convenience, the dependence
of the overpotential on the acid concentration is neg-
lected, and a concentration independent lead electrode
is used as a reference electrode in Eq. [10a], [10b], and
[11].
Interface between region 1 and 2.--
~C
region
OC
region
e .... [12]
~X ~ aX 2
0e 1 ( MWebso4
MWpboz)OiZ=o[13]
Ot
2F
PPbS04 '. PPbO2 0X
Downloaded 13 Jun 2011 to 129.252.106.227. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

Vol. 134, No. 12
MATHEMATICAL MODEL OF A LEAD-ACID CELL 2955
i2 = I [14] e = %p [25]
~r i2 = I [26]
o~ = 0 [15]
Ox
~ $~ = 0 [27]
~ext 0520x ~g~o~ I - 0520x ~egio, 2 [16]
Equations [12] and [16] satisfy the requirement that the
flux of a species i, Ni, and the superficial current density,
i2, are continuous across the interface. In other words,
Ni,region 1 =
Ni,region 2 and
i2,regio n 1 = i2,region
2; with i = +, -, or
o. But since [Ref. (15)]
V |= ~--~ CiviY i
[17]
i
and
N i = civ i
[18]
with Vi assumed constant, the requirement that the flux
of species i is continuous implies that the volume aver-
age velocity is also continuous,
i.e., v~" = vs'.
Equation [13] describes the porosity variation with
time and Eq. [14] states that all the current at the inter-
face is in the electrolyte phase. Finally, Eq. [15] states
that the electrode solid phase potential gradient is equal
to zero at the interface because the electrode solid phase
ends there.
Region 2, reservoir.-
porosity-
Ohm's law in solution--
i~ 0r RT
+--
K
0x F
solid phase potential--
= 1 [19]
In (c3O
-- (1 - 2t~ - 0 [20]
0x
~, = 0 [21]
material balance--
0C 1 [( MWpbso 4 MWpbo2 )
Ot
2F PPbSO4 PPbO2
2t~
0c _ D 9-cc =0
(3
o J OX OX 2
current in solution--
[22]
i~ = I [23]
Equation [19] states that the reservoir is filled with
electrolyte. Equation [20] is Ohm's law applied to the
electrolyte written with respect to Pb reference elec-
trode. The electrode solid phase potential is zero (Eq.
[21]) because there is no electrode solid phase in the res-
ervoir. Equation [22] is the material balance for the elec-
trolyte in the reservoir. Finally, Eq. [23] states that all the
applied current is flowing through the solution phase
since there is no electrode solid phase in the reservoir.
Interface between region 2 and region 3.--
0c
region .C~u.~. region
D
-- CV2 g = D~sep exa
-- CV3 I
OX z 3
where
[24]
V2 m
1 1( MWpbso4 MWpbo2
2F PPDSO4 PPbO2
- (3 - 2t~ + 2VolI
1 [( MWpbso4
v3" = 2F PPbSO4
MWpb ) +(1- PPb 2t~ I
[0
(~2
RT
0 In (cf) ]
Ox
F (1 - 2t~ 0x r~g~on
=[e~opex30~)2
RT
01n(cf) ] [28]
0--x-- - e~Ptx~ F (1 - 2t~ 0~ ~gio~ 3
Equation [24] satisfies the condition that the fluxes of the
species are continuous. As shown by the expressions for
v2" and v3" below Eq. [24], v2" is not equal to v3". v2" is
equal to Vl" as stated earlier and v3" is equal to v4" which
is the volume average velocity due to migration and
changes in the structure of the lead electrode
(i.e.,
region
4). The implicit requirement that v2" = v3" is not met here.
This inconsistency would be removed in a two-dimen-
sional model which included the fact that the level of the
acid changes during charge and discharge. Previous
workers (11, 16, 17) have avoided this problem by assum-
ing that the reservoir is well mixed and, consequently,
separates the convective flows out of (or into) the porous
electrodes. Since the magnitudes of the volume average
velocities are typically small, this inconsistency is ig-
nored in the model presented here.
Equation [25] sets the porosity at this interface to be
that of the separator porosity, which is a constant value.
Equation [26] indicates that the superficial current den-
sity in the solution is equal to the applied current density
since no charge transfer reaction occurs in either the res-
ervoir or the separator. Equation [27] sets the electrode
solid phase potential to zero since there is no electrode
here. Equation [28] satisfies the condition that the cur-
rent is continuous across the interface. Unlike Eq. [16]
where the terms with the concentration gradients cancel
according to Eq. [12], the concentration gradient terms in
Eq. [28] do not cancel because they are not equal accord-
ing to Eq. [24].
Region 3, separator.--
porosity--
e = esep [29]
Ohm's lawin solution--
i2 a$2 RT
--+ -- -- (1-- 2t~ --
~sep ex3
K 0X F
0
in (cf)
- 0 [30]
Ox
solid phase potential--
material balance--
~, = 0 [31]
1 [( wpbso4 Mwpb)
oc +--
esep O~ 2F PPbSO4 {OPt)
-2t~ 1
I ac ~a D
02c
+
(1
0x -~' ~=0
current in solution--
[32]
i~ = I [33]
Equation [32] is the material balance for the electrolyte
in the separator. The second term in the equation is v3"
(Oc/Ox),
where v3" is equal to that at the interface of re-
gions 3 and 4. The porosity of the separator is fixed (Eq.
[29]), and the current density in the solution is equal to
the applied current density (Eq. [33]). The electrode solid
phase potential again is set to zero because there is no
electrodehere (Eq. [31]). Finally, Eq. [30] is Ohm's law
applied to the solution phase in the separator.
Interface between region 3 and region 4.--
~!isepeX 3
OC
~ ~ex4 C~
OX region 3 OZ region 4
[34]
Downloaded 13 Jun 2011 to 129.252.106.227. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

2956
J. Electrochem. Soc.:
ELECTROCHEMICAL SCIENCE AND TECHNOLOGY
December 1987
O e + 1 ( MWpb) 0i2 0 [35 ]_ =
Ot
2F PPbSO4 PPb c]X
i2 = I [36]
00010x
~g~on 4 =
0
[37]
esepeX3
0(~2 eex4 0~)2 region
OX
reg[o, 3 :
OX
4
[38]
The equations for this interface were established with
the same reasoning that was used for the interface be-
tween region 1 and region 2.
Region 4, negative electrode.-
porosity variation--
0: + 1 ( . MWpb ) 0i2 0 [39
Ot
2F pvbso4 pw 0x
Ohm's law in solution--
i2 or RT
0 In (cJ)
--+-- (1-2t~ [40]
eex4K
0X F 0X
Ohm's law in solid--
04,
ig
- I~exmgo'pb- -- I = 0 [41]
Ox
material balance--
1 [( wPbso4 M pb)
oc +
Ot 2F-
PPuso4 PPb
+(1-2t~
OxO (e
ex4D
oxOC )
t - 2t~ )
Oig = 0
- (1 -
C~re)-oX
[421
~f
electrode kinetics--
for discharge and rest
0~- -- amax4?'~ \
ere f ] emax4
-- ~io4 /
{ exp [ r
RT (~b,-
r exp kl[ -aOgFRT
(~)l
-- (~2)]} = 0
[43a]
and for charge
(ct, ( ;(
0X amaxg?'~ ........
\
ref
/ ~max4 -- ~o4 ~max4 -- ~o4
[43b]
Equations [39]
through [43] are counterparts of the equa-
tions established for the positive electrode. The concep-
tual reference electrode used to measure the potential
difference is a lead electrode, the same kind as the nega-
tive electrode.
Center of the negative electrode,
x =/.--Equations [1],
[2], [3], [51, and [43a1 or [43b] apply. Since the electrode
solid phase potential was set to zero at the other bound-
ary (x = 0), a kinetic expression is used to calculate 4, at
this boundary.
Numerical Procedure
The governing equations were put into finite differ-
ence forms and solved using the numerical technique of
Newman (18, 19). The cell was divided into
NJ
node
points with J = 1 designated to be the center of the posi-
tive electrode and
J = NJ
to be the center of the negative
electrode. Each region was evenly divided, but node
point spacings were different between regions.
The finite difference approximations of the derivatives
for an internal mesh point can be written as
02C~(J) Ck(J +
1) + Ck(J - 1) - 2Ck(J)
[44]
Ox g ( ax) g
aCk(J) Ck(J +
1) -
C~(J -
1)
ox
2(5x)
and for a boundary node
0Ck(J) --Ck(J
+
2) +
4Ck(J +
1) - 3Ck(J)
[45]
[46]
ox
2(AX)
OCk(J) Ck(J -
2) -
4Ck(J -
1) + 3Ck(J)
[47]
Ox
2(hx)
where hx denotes the distance between node points and
Ck(J)
represents the k th unknown at node J.
The accuracy of the finite difference approximations of
the above derivatives is (hx) 2. Equation [46], in the for-
ward difference form, is used at J = 1; and Eq. [47], in the
backward difference form, is used at
J = NJ.
for the inter-
nal boundaries, Eq. [46] is used on the higher region
number side and Eq. [47] is used on the lower region
number side. For example, Eq. [24] written in the finite
difference form is
CI(J - 2) -
4C,(J -
1) + 3CI(J)
D - C,(J)v2"
2(hx2)
-C,(J + 2) + 4C~(J + i) - 3C,(J)
= Desep ex3 - CI(J)v3"
2(Axe)
[48]
where the electrolyte concentration c is written as "un-
known 1". The subscript on Ax refers to the region num-
ber. We note that five node points are needed to describe
the continuity of the flux at an internal boundary to
maintain
(5x) 2
accuracy. Newman's BAND(J) can be
used for only three node points (18, 19). Consequently,
since up to five node points are used at an internal
boundary, a modified version of Newman's subroutine,
called Pentadiagonal BAND(J) (20, 21), was used in-
stead. As the name implies, this subroutine allows up to
five node points to be used at any position.
For the time derivative, implicit stepping was used
OCk(J) Ck(J) -
CKk(J)
[49]
ot At
where CKk(J) refers to the value at the previous time
step,
t - At.
Initial distributions of variables can be deter-
mined by taking a small time step (10-4s,
e.g.).
Results and Discussion
In this section, we examine some simulated results
and their implications. The parameters used in the calcu-
lations are given in Table I. Parameters not referenced
are arbitrary but reasonable quantities. The same ex-
change current densities were assigned to the positive
and negative electrodes. These values were chosen so
that the calculated cell voltages agree well with mea-
sured ones.
Effect of temperature on discharge.--The
simulated
discharge behavior at 25 ~ and -18~ under a constant
current density of 340 mA/cm ~ is presented first. The dis-
charge cutoff was chosen to be 1.55V. The temperature
dependence of the kinetic expression is explicitly ex-
pressed in the exponential terms. Implicitly, the ex-
change current density in the kinetic expression is also a
strong function of temperature. As mentioned earlier,
the exchange current density was adjusted for the two
temperatures so that the calculated cell voltages agreed
relatively well with experimental observations.
Downloaded 13 Jun 2011 to 129.252.106.227. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp

Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the authors reviewed the current state of the simulation, optimization and control technologies for the stand-alone hybrid solar-wind energy systems with battery storage, and found that continued research and development effort in this area is still needed for improving the systems' performance, establishing techniques for accurately predicting their output and reliably integrating them with other renewable or conventional power generation sources.

809 citations


Cites methods from "A Mathematical Model of a Lead‐Acid..."

  • ...[52] and incorporation of the diffusion–precipitation mechanism studied by Ekdunge and Simonsson [53] in the reaction kinetics of the negative electrode, Kim and Hong [54] analyzed the discharge performance of a flooded lead–acid battery cell using mathematical modelling....

    [...]

Journal ArticleDOI
15 Apr 2003
TL;DR: In this article, two electrical models of a lead-acid battery, a short-term discharge model and a long-term integrated model, were used to investigate the system performance of a battery-supported dynamic voltage restorer (DVR).
Abstract: Two electrical models of a lead-acid battery, a short-term discharge model and a long-term integrated model, were used to investigate the system performance of a battery-supported dynamic voltage restorer (DVR). The short-term model provides a simple but effective description when the DVR compensates voltage sags over a short period. The integrated model can predict accurately the terminal voltage, state of charge, battery capacity and gassing current. It gives a good description of the battery response during both discharge and charge. Parameters of both models can be determined easily from measured battery output voltages obtained from load-step tests. Both models were successfully implemented in EMTDC/PSCAD and interfaced with the digital model of a 10 kVA DVR physical prototype. They gave a very close agreement between extensive experimental data and simulation results. Application issues such as current harmonics and microcycles during charge/discharge are discussed with respect to their impact on loss of capacity and reduced lifetime of the lead-acid battery.

109 citations

Journal ArticleDOI
TL;DR: In this paper, a survey is given of the pros and cons of the existing models for the lead-acid battery and a diffusion model is proposed to explain the occurrence of sharp lead sulfate walls observed earlier by Haebler et al. The model also leads to an expression similar to Peukert's law.

65 citations

Journal ArticleDOI
TL;DR: In this paper, free convection and stratification of the electrolyte in a lead-acid cell with porous electrodes and during recharge were studied theoretically and experimentally, and the results from the experiments were compared with numerical results obtained from the mathematical model.

56 citations

Journal ArticleDOI
TL;DR: In this article, the dynamic behavior of VRLA batteries can be predicted using theoretical cell model for basic processes, which is applied for viewing unobservable processes in battery by observable processes.

50 citations

References
More filters
Book
02 Oct 2011
TL;DR: In this paper, a simple model of a Diaphragm -type chlorine cell was used to calculate the voltage drop in a battery and showed the effect of migration/diffusion of electrolytes.
Abstract: Design and Development of Electrochemical Chlor-Alkali Cells.- A Simple Model of a Diaphragm - Type Chlorine Cell.- Design Principles for Chlorine Membrane Cells.- Hydroxyl Ion Migration, Chemical Reactions, Water Transport and Other Effects as Optimizing Parameters in Cross-, Co- and Countercurrently Operated Membrane Cells for the Chlor/Alkali Electrolysis.- Hydraulic Modelling as an Aid to Electrochemical Cell Design.- Calculating Mechanical Component Voltage Drops in Electrochemical Cells.- Electrolysis Cell Design for Ion Exchange Membrane Chlor-Alkali Process.- Experiences with a Bench-Scale Electrochemical Plant.- Economic Driving Force in Electro-Organic Synthesis.- Design of SU Modularized Electrochemical Cells.- Electrochemical Techniques for the Extraction of Heavy Metals in Industry: Concepts, Apparatus and Cost.- The Design and Application of Rotating Cylinder Electrode Technology to Continuous Production of Metal.- Shunt Current Control in Electrochemical Systems - Theoretical Analysis.- Shunt Current Control Methods in Electrochemical Systems - Applications.- A Simple Model of Exxon's Zn/Br2 Battery.- A Finite Element Model of Bipolar Plate Cells.- Changes in Overall Ohmic Resistance Due to Migration/Diffusion of Electrolytes.- Mathematical Model for Design of Battery Electrodes: Lead-Acid Cell Modelling.- Extension of Newman's Numerical Technique to Pentadiagonal Systems of Equations.

47 citations

Journal ArticleDOI
TL;DR: Comportement electrochimique d'echantillons d'alliages binaires pb-Li, Pb-Sn, and quaternaire Pb-(Al, Mg, Sn, Li) en solution a 30% H 2 SO 4.
Abstract: Comportement electrochimique d'echantillons d'alliages binaires Pb-Li, Pb-Sn, et quaternaire Pb-(Al, Mg, Sn, Li) en solution a 30% H 2 SO 4 . Comparaison avec l'alliage conventionnel Pb-4% Sb

44 citations

Journal ArticleDOI

17 citations


Additional excerpts

  • ...O62 = 0 [5] OX E q u a t i o n s [1], [2], and [5] a re in a c c o r d a n c e w i t h t h e ass u m p t i o n o f s y m m e t r y ....

    [...]

  • ...- - E q u a t i o n s [1], [2], [3], [51, and [43a1 or [43b] app ly ....

    [...]