How batteries discharge: A simple model
W. M. Saslowa兲
Department of Physics, Texas A&M University, College Station, Texas 77843-4242
共Received 7 June 2007; accepted 22 November 2007兲
A typical battery is a set of nominally identical voltaic cells in series and/or parallel. We consider
the discharge of a single voltaic cell. As the cell discharges due to current-carrying chemical
reactions, the densities of the chemical components decrease, which leads to an increase in the
internal resistance of the voltaic cell and, upon discharge, a decrease in its terminal voltage and
current. A simple model yields behavior similar to what is observed, although accurate battery
models are more complex. © 2008 American Association of Physics Teachers.
关DOI: 10.1119/1.2826658兴
I. INTRODUCTION
Most introductory physics textbooks give an oversimplified, black-box discussion of voltaic cells, with the internal
electrical resistance r placed outside of the cell, and the electromotive force 共emf兲 E treated artificially as a volume pump
within the cell’s electrolyte, bounded by the two
electrodes.1–3 On going around a discharging circuit the voltage profile in this model has an unphysical linear rise across
the volume of the cell and an unphysically-located linear fall
in voltage across the externally placed “internal” resistance.
The unnatural spatial separation of the voltages associated
with this incorrect model of the emf and the internal resistance is necessary in order to distinguish these voltages, because in one dimension they both vary linearly with the spatial coordinate.
A reasonably correct treatment of a discharging cell has a
linear decrease in voltage within the electrolyte and two voltage jumps at the electrode-electrolyte interfaces. Hence we
do not need to separate the emf of the voltaic cell from its
internal resistance; they have intrinsically different spatial
dependences. One of the major defects of the black-box
model is its inability to explain why a AAA cell and a D cell
have the same emf. In particular, the volume pump model
suggests that the D cell emf should vastly exceed the AAA
cell emf, in contradiction to common experience.
Although the physics of voltaic cells seems to have become lost to the physics community, voltaic cells were
treated correctly in physics texts as late as the 1940’s.4,5 As
has been discussed,6 a realistic model of a voltaic cell must
incorporate the fact that 共to use an archaism兲 the “seat of
emf” of a voltaic cell, which leads to voltage jumps at the
two electrode-electrolyte interfaces, is due to chargetransferring chemical reactions at the two electrodeelectrolyte interfaces.7–9 The phrase “two surface pumps and
an internal resistance” conveniently and reasonably accurately describes a voltaic cell.6
Previous work accounted for surface resistances at the
electrode-electrolyte interfaces, where the electric currentproducing chemical reactions occur. These additional resistances lead to three sources of internal resistance associated
with the voltaic cell: the electrolyte and the electrodeelectrolyte interfaces.6 For a lead-acid cell, the effect of the
electrode-electrolyte interfaces is negligible; a typical leadacid cell is effectively discharged when the sulfuric acid has
become depleted to the point where the terminal voltage of
the cell falls to ⬇1.7 V, below which the devices that it
drives normally will not operate.10
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Am. J. Phys. 76 共3兲, March 2008
http://aapt.org/ajp
The present work shows how the increase in internal resistance due to discharge leads to a qualitative and semiquantitative explanation of how a battery 共properly, a single voltaic cell within the battery兲 becomes ineffective. For the
lead-acid cell the internal resistance r is taken to have a fixed
part rs from the electrode-electrolyte interfaces and a part re
from the electrolyte, which is inversely proportional to the
amount 共“charge” Q兲 of sulfuric acid in the cell. This mass
decrease in proportion to the amount of electrical discharge
is a version of Faraday’s law of electrolysis. The cell emf E
is taken to be fixed, although by Nernst’s equation it is somewhat affected by the state of the electrolyte.11
This model has the property that when the external load
resistor RL dominates the internal resistance r, the current is
nearly constant and causes a nearly uniform 共in time兲 increase in the internal resistance, which causes the current to
have a slow linear decrease with time. On the other hand,
when r dominates 共for example, because the sulfuric acid has
become nearly depleted兲, the current I decreases exponentially with time. There is an intermediate regime where the
battery discharges rapidly; the crossover is characterized by a
change in the sign of the curvature of the terminal voltage
⌬VT versus time, at which time the battery is effectively
useless. All this behavior is in qualitative agreement with
experiment.
II. INTERNAL RESISTANCE
To provide a self-contained discussion we present in more
detail than in Ref. 6 a derivation of the net internal resistance
of a voltaic cell in terms of the electrolyte resistance and the
two electrode-electrolyte-interface resistances. We neglect
nonfaradaic 共or noncharge-transferring—e.g., ordinary兲
chemical reactions at the electrode-electrolyte interfaces,
which are responsible for the slow discharge of voltaic cells
that are on an open circuit.12 We will assume that the discharge is slow, corresponding to the use of the headlights of
a car. We will also neglect the rearrangement of ions that can
occur within a voltaic cell.13 If such rearrangement occurs,
both the electric field and diffusion contribute to the motion
of ions within the cell, and the problem becomes considerably more complex and unsuitable for discussion at the introductory level.
We treat the two electrode-electrolyte interfaces as supporting chemical reactions that provide emfs E1 ⬍ E2, which
tend to drive current out of the cell, so that for the cell of Fig.
1 the net rightward emf E is, by Volta’s law for electrodes,
© 2008 American Association of Physics Teachers
218
Fig. 1. Model of a voltaic cell. The asymmetry indicates that the electrodes
are distinct, with the larger electrode having the larger emf. ⌬V2 = V2T − V2e,
⌬V1 = V1T − V1e, ⌬Ve = V2e − V1e.
E = E2 − E1 .
共1兲
When the cell discharges, positive current I will leave the
cell by electrode 2 and enter the cell by electrode 1. The
values of E1, E2, and thus E1 are determined by the cell’s
chemistry.
Let the electrodes be characterized by surface resistances
r1 and r2 and the electrolyte by the resistance re. The voltages of the terminals are V2T and V1T, and the electrolyte
voltages adjacent to the respective terminals are V2e and V1e.
We define
⌬V2 = V2T − V2e,
⌬V1 = V1T − V1e,
⌬Ve = V2e − V1e .
共2兲
We use these definitions of the voltage differences and the
convention that positive current is to the right and apply
Ohm’s law to each electrode and to the electrolyte. The result
is
I=
E2 − ⌬V2
r2
,
I=−
E1 − ⌬V1
r1
,
⌬Ve
I=−
.
re
共3兲
共The signs reflect the definitions of voltage difference and of
positive current flow being to the right. Thus E2 ⬎ 0 drives
current to the right, E1 ⬎ 0 drives current to the left, and a
higher electrolyte voltage on the right drives current to the
left.兲
With these definitions the terminal voltage is given by
⌬VT = V2T − V1T = ⌬V2 + ⌬Ve − ⌬V1
=共E2 − Ir2兲 − Ire − 共E1 + Ir1兲 = E − Ir,
共4a兲
共4b兲
where
r = r1 + r2 + re .
共5兲
We rewrite Eq. 共4b兲 to obtain the equation for the system as
a whole,
I=
E − ⌬VT
r
.
共6兲
Note that
re
⌬Ve = − Ire = − 共E − ⌬VT兲 .
r
共7兲
For I ⬎ 0, the cell discharges, and the voltage across the electrolyte is opposite the emf, as can be seen in Fig. 2共a兲.
As an example we take r1 = r2 = 0.01 ⍀ and re = 0.08 ⍀, so
r = 0.1 ⍀; in this case the electrolyte dominates the internal
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Am. J. Phys., Vol. 76, No. 3, March 2008
Fig. 2. The voltage profile for a voltaic cell. The numbers refer to voltage
changes on moving across each circuit element from left to right for the
model discussed in the text. 共a兲 Open circuit. 共b兲 Closed circuit with resistance R.
resistance. We also take E2 = 1.4 V and E1 = −0.6 V 共the negative sign means that the reaction at electrode 1 drives current
in the same direction as does electrode 2兲, so E = 2.0 V. 共A
well-charged single cell of a lead-acid battery, with geometrical plate area of 100 cm3 has an emf of about 2.15 V
and an internal resistance on the order of 0.01 ⍀.14兲 The
similar example depicted in Fig. 9 of Ref. 6 took r1 = r2 = 0,
so that the voltage drop across the electrodes was unaffected
by the current flow.
For an open circuit for which I = 0, Eq. 共3兲 yields ⌬Ve
= −0.0 V, ⌬V2 = 1.4 V, and ⌬V1 = −0.6 V, so that ⌬VT = 1.4
− 0.0+ 0.6= 2.0 V. With an external resistor R = 0.4 ⍀, for
which I = ⌬V / R holds, and the current is I = E / 共r + R兲 = 4 A.
Then ⌬V = IR = 1.6 V. Moreover, ⌬Ve = −0.32 V, ⌬V2
= 1.36 V, and ⌬V1 = −0.56 V, so that ⌬VT = 1.36− 0.32
+ 0.56= 1.6 V.
If we compare the current-carrying closed circuit and
noncurrent-carrying open circuit cases, we see that the voltage drop across the electrode-electrolyte interfaces is not the
ideal value of the associated emf, but differs by an amount
proportional to the current. In principle, the surface resistance can be determined by measuring the voltage difference
between a given electrode and the electrolyte near that electrode, and determining its dependence on the current flow.
The voltage profile for this circuit is plotted in Fig. 2 for
an open and a closed circuit. The voltaic cell is to the left,
and to the right is either an air gap 关Fig. 2共a兲兴 or a resistor
关Fig. 2共b兲兴.
More realistic models for a voltaic cell include the fact
that for large currents the current crossing an interface is a
nonlinear function of the voltage difference across that interW. M. Saslow
219
face. Including this effect makes the equations much more
complex and is unnecessary for small currents.
III. CELL PROPERTIES DEPEND ON THE STATE
OF CHARGE
In a lead-acid cell the electrode-electrolyte interfaces are
associated with the sulfuric acid-containing pores of lead and
lead oxide.15–19 When the cell is well charged, the pores are
large with a large surface area; when the cell is poorly
charged, the pores are small with a small surface area. The
reason for the change in pore size is that under discharge
共charge兲, chemical reactions plate lead sulfate onto the lead
and lead-oxide electrode, thus decreasing 共increasing兲 the
area that has become sulfated 共desulfated兲. The change in the
pore size is not the dominant effect on the internal resistance,
because the fractional utilization of the interface area does
not decrease below 0.7–0.8 unless there is passification 共that
is, under normal reverse current the area can no longer be
recharged兲 by formation of sulfate crystals on the surface.20
The most important effect of discharge is to change the electrolyte resistance.
We therefore take the internal resistance r to be the sum of
a small and 共nearly兲 fixed part rs = r1 + r2 associated with the
electrode-electrolyte interfaces and a part re that varies with
the charge Q of the cell. This variation of r is not difficult to
obtain. Recall that in the Drude theory of conductivity, which
explains the difference between conductors and insulators,
the conductivity ␴ is proportional to the carrier density n. In
the present case the conductivity of the electrolyte also is
proportional to n. Because the sulfuric acid is the limiting
component, its density n is proportional to the charge Q of
the cell,
共8兲
Q = 2en⍀,
where each atom of sulfuric acid provides a net ionic charge
2e 共e from the rightward motion of H+ and e from the leftward motion of HSO−4 兲 and ⍀ is the volume of the electrolyte. Hence, because the resistance varies inversely with the
conductivity, we have re = r0Q0 / Q, where the initial electrolyte resistance and charge are r0 and Q0. Then
r = rs + r0
Q0
,
Q
rs = r1 + r2 .
共9兲
We repeat that voltaic cells are not really ohmic devices,
and that the electrode-electrolyte interface resistance depends
nonlinearly on the voltage difference between the electrode
and electrolyte.15–20 In addition, both the electrolyte density
and the active electrode area require treatment as continua
共that is, they depend on position兲, especially for high rates of
discharge, where ionic motion in the bulk may not be able to
keep up with the needs of the chemical reactions at the
interfaces.15–20 We neglect such niceties here to obtain a
model that is manageable and contains the basic elements to
explain how batteries discharge.
IV. INCREASE OF ELECTRODE RESISTANCE
WITH DISCHARGE
The charge of a cell decreases in proportion to its integrated discharge:
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Am. J. Phys., Vol. 76, No. 3, March 2008
dQ
= − I.
dt
共10兲
Equation 共10兲 relates the current I to the rate of change of the
matter in the cell, expressed in units of charge. It incorporates the ideas implicit in Faraday’s law of electrolysis.
Equation 共10兲 explicitly assumes that the discharge rate is
sufficiently slow that concentration-driven diffusion and
electric field-driven drift can maintain the ionic densities to
be nearly uniform within the electrolyte. For high discharge
rates, this assumption fails.
A. Discharge through a fixed load RL
We now connect an external load resistor RL to the cell.
With no other source of emf in the circuit, the cell produces
a current through the load given by
I=
⌬VT
.
RL
共11兲
Equations 共6兲 and 共11兲 lead to I = E / 共r + RL兲. We use Eq. 共9兲
to rewrite Eq. 共11兲 as
I=
E
r + RL
=
E
Q
,
R Q + Q0r0/R
R ⬅ RL + rs .
共12兲
The quantity r0 / R must be small for a voltaic cell to be
effective in driving a load for a significant amount of time.
The combination of Eqs. 共10兲 and 共12兲 yields
E
Q/Q0
Q
dQ
Q0
=−
,
=−
dt
R Q + Q0r0/R
␶ Q/Q0 + r0/R
共13兲
with ␶ = RQ0 / E. Here ␶ is the characteristic discharge time
for a circuit of emf E, charge Q0, and resistance R. Equation
共13兲 has the properties that for a well-charged cell, where
Q / Q0 Ⰷ r0 / R, the decay is linear in time, whereas when the
cell has become depleted, so Q / Q0 Ⰶ r0 / R, the decay is exponential. Equation 共13兲 can be solved numerically to find
Q / Q0 as a function of t / ␶. It can also be solved analytically,
leading to mixed linear and logarithmic behavior:
冉
冊
Q r0
Q
t
=1−
+ ln
,
␶
Q0 R
Q0
共14兲
which can be used to find Q / Q0 as a function of t / ␶.
The dimensionless terminal voltage is given by
⌫⬅
Q/Q0
⌬VT RL
,
=
E
R Q/Q0 + r0/R
共15兲
where RL / R is slightly smaller than unity. Comparison with
Eq. 共13兲 shows that ⌫ is proportional to dQ / dt.
We expect that for short times ⌫ has a downward curvature as it approaches the regime where the cell becomes ineffective, and that for times long enough that the cell has
become useless ⌫ has a positive curvature. Therefore, the
time at which d2⌫ / dt2 = 0 indicates the crossover in behavior.
We thus consider
Q/Q0
1
d共Q/Q0兲
d⌫ RL d
RL
=
=
dt
R dt Q/Q0 + r0/R R 共Q/Q0 + r0/R兲2 dt
共16兲
W. M. Saslow
220
Fig. 3. The normalized charge Q / Q0 versus the normalized time t / ␶
= tE / RQ0 for R / r0 = 1 , 10, 100. See Eq. 共13兲. Legend: R / r0 = 1 共dot-dashed
line兲, R / r0 = 10 共dashed line兲, R / r0 = 100 共solid line兲.
Q/Q0
1 RL
=−
␶ R 共Q/Q0 + r0/R兲3
共17兲
Fig. 5. The normalized terminal voltage ⌫ = ⌬VT / E versus the normalized
discharge 共1 − Q / Q0兲 for R / r0 = 1 , 10, 100 and RL / R = 0.95. See Eq. 共20兲. The
legend is the same as in Fig. 3.
and
d2⌫ 1 RLr0 共Q/Q0兲共r0/R − 2Q/Q0兲
=
.
dt2 ␶2 R2
共Q/Q0 + r0/R兲5
共18兲
Thus the curvature is zero for Q = Qc, where QcR = Q0r0 / 2.
Alternatively, note that Q / Q0 = ⌫共r0 / R兲 / 共RL / R − ⌫兲 and
Q / Q0 + r0 / R = 共r0 / R兲 / 共R / RL − ⌫兲, which leads to
1 R2
d⌫
⌫共RL/R − ⌫兲2 .
=−
dt
␶ R Lr 0
共19兲
Equation 共19兲 can be solved numerically to give ⌫共t兲. In this
case it is helpful to use ⌫0 = RL / 共R + r0兲, which for the relevant case of large R / r0 is slightly less than RL / R.
To find Q / Q0 in terms of t / ␶ we need to specify only the
parameter R / r0. For ⌫共t兲 we must specify the parameters
R / r0 and RL / R. For R / r0 = 1, 10, 100 and RL / R = 0.95 we
show Q / Q0 and ⌫ = ⌬VT / E as functions of t / ␶ in Figs. 3 and
4. Note the nearly linear decrease of Q共t兲 / Q0 to zero for
small t / ␶, especially for R / r0 = 100 共initially well-charged
cell兲, and the exponential behavior for large t / ␶, especially
for R / r0 = 1 共initially poorly charged cell兲. Also note the
rapid falloff of ⌫共t兲 for t ⬇ ␶. For larger R / r0, Q共t兲 decreases
linearly until it reaches the abscissa, and ⌫共t兲 has an even
sharper falloff.
B. Terminal voltage versus discharge
Battery properties are often considered as a function of
discharge.10 Thus we consider the normalized terminal voltage ⌫ as a function of the amount of discharge 1 − Q / Q0.
This quantity is initially zero and is unity at full discharge.
To obtain it we rewrite Eq. 共15兲 as
RL
⌫=
R
冉 冊
冉 冊
1− 1−
Q
Q0
r0
Q
1+ − 1−
R
Q0
共20兲
.
We plot ⌫ as a function of 共1 − Q / Q0兲 in Fig. 5 for the same
values of R / r0 as in Fig. 4. It is implicit that RL is constant
and that there is no other source of emf.
C. Constant current discharge
We also consider the case of constant current discharge I
= I0. This situation can be produced by applying a variable
共forward兲 emf Ev that compensates for the increased resistance as the cell discharges. Its value is immaterial for our
purposes. In this case Q = Q0 − I0t by Eq. 共10兲, so that it is
convenient to write
⌫=
冉
冊
⌬VT
Q0
I0
=1−
rs + r0
.
E
E
Q
共21兲
If we fix I to the initial current for a given R / r0, we have
from Eq. 共12兲
Fig. 4. The normalized terminal voltage ⌫ = ⌬VT / E versus the normalized
time t / ␶ = tE / RQ0 for R / r0 = 1 , 10, 100 and RL / R = 0.95. See Eq. 共19兲. The
legend is the same as in Fig. 3.
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Am. J. Phys., Vol. 76, No. 3, March 2008
1
I0
=
.
E R + r0
共22兲
Equation 共22兲 permits us to rewrite Eq. 共21兲 as
W. M. Saslow
221
thought to be lithium cells, given their non-heavy-metal nature, their high energy density 共from lithium’s high energy
chemical bond—and therefore high emf—and from its low
mass density兲, and the possibility that they may have a useful
lifetime up to fifteen years. Despite the fact that this model
has the lead-acid cell in mind, the principles we have discussed should apply to other types of voltaic cells, although
the details will differ.
ACKNOWLEDGMENTS
Fig. 6. Normalized terminal voltage ⌫ = ⌬VT / E versus normalized time t / ␶
= tE / RQ0 for R / r0 = 1, 10, 100 and RL / R = 0.95 for constant current. See Eqs.
共23兲 and 共24兲. The legend is the same as in Fig. 3.
冉
冊
1
Q0
⌫=1−
rs + r0
,
R + r0
Q
where
冋
Q = Q0 1 −
册
R t
.
R + r0 ␶
a兲
共23兲
共24兲
Equation 共23兲 is plotted as a function of t / ␶ in Fig. 6 for the
same values as in Fig. 4. Note the similarity to Fig. 5, although the corresponding curves are not identical.
V. DISCUSSION
The model has the same qualitative behavior that we expect for a voltaic cell discharging through a fixed load resistor or discharging at constant current. There is a linear fall in
the terminal voltage until the cell is nearly discharged, beyond which the terminal voltage drops exponentially. The
model is not expected to be quantitative because it does not
incorporate a number of important properties: 共1兲 the ion
densities in the electrolyte 共and even in the electrode兲 are
nonuniform, which is especially important for high currents
when the ions in the volume cannot keep up with the demands of the reactions at the electrode-electrolyte interface,
共2兲 the relations between current and voltage drop across the
electrode-electrolyte interfaces are nonlinear, and 共3兲 the
electrode resistances are not fixed, but must be treated using
a porosity 共characterized by a surface area per unit volume of
the order of an inverse pore radius兲, which depends on the
amount of discharge.17,18
Nevertheless, the model gives the reasonable qualitative
result that for short times the current and terminal voltage
fall linearly until the active area becomes so small that its
resistance dominates that of the rest of the system, in which
case the decay becomes exponential. This behavior is in
qualitative agreement with I共t兲 data on how voltaic cells
discharge.21
Our considerations have in mind the lead-acid cell, which
is normally used to start internal combustion engines for automobiles. From the point of view of electric-powered and
hybrid combustion-electric automobiles, the lead-acid cell
has been superseded by Ni-metal hydride cells. The future is
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Am. J. Phys., Vol. 76, No. 3, March 2008
The author would like to acknowledge the referees for
their thoughtful and stimulating comments. Venkat Srinivasan was an invaluable source of information about real
lead-acid batteries. Art Belmonte generously provided assistance with the figures. Konstantine Romanov provided useful
comments. The author also would like to acknowledge the
general support of the Department of Energy through Grant
No. DE-FG02-06ER46278.
Electronic mail: wsaslow@tamu.edu
D. Halliday, R. Resnick, and K. S. Krane, Physics 共Wiley, New York,
1992兲, Vol. 2, 4th ed., Figs. 3 and 4, pp. 719–720.
2
H. D. Young and R. D. Freedman, University Physics 共Addison–Wesley,
San Francisco, 2004兲, Vol. 2, 11th ed., Fig. 25.20, p. 961.
3
R. A. Serway and J. W. Jewett, Principles of Physics 共Harcourt, Fort
Worth, 2002兲, Vol. 2, 3rd ed., Fig. 21.13, p. 775.
4
O. Blackwood, General Physics 共Wiley, New York, 1943兲, Fig. 5, p. 469.
5
O. M. Stewart, Physics: A Textbook for Colleges 共Ginn and Co., Boston,
1924兲, Figs. 270 and 271, pp. 472–473.
6
W. M. Saslow, “Voltaic cells for physicists: Two surface pumps and an
internal resistance,” Am. J. Phys. 67, 574–583 共1999兲.
7
E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course
共McGraw–Hill, New York, 1965兲, Vol. 2, Figs. 4.16–4.19, pp. 136–137.
A similar discussion does not appear in the 2nd edition. Figures 4.18 and
4.19 of the 1st edition, which give the voltage profiles for open-circuit
and current-drawing circuits with the Weston cell, are not discussed in the
text. Similar figures appear in textbooks of the early 1900s; these figures
may represent what Purcell was exposed to as a student.
8
W. M. Scott, The Physics of Electricity and Magnetism 共Wiley, New
York, 1966兲, 2nd ed., Fig. 5.5, pp. 211–213. The text gives no discussion
of the electrode emfs or internal resistance.
9
W. M. Saslow, Electricity, Magnetism, and Light 共Academic/Elsevier,
Amsterdam, 2002兲, Figs. 7.20 and 7.21, pp. 314–315.
10
V. Srinivasan 共private communication兲.
11
Properly, E = E0 + 共kBT / e兲ln共关HSO−4 兴关H+兴兲, where E0 = 2.041 V, and
关HSO−4 兴 and 关H+兴 are molar concentrations of the ions in the lead-acid
cell. For a six molar concentration of both ions and T = 298 K, we obtain
E = 2.133 V. Reduction to one molar concentration gives the slightly different value E0 = 2.041 V.
12
If nonfaradaic discharge 共that is, ordinary, noncharge-transferring chemical reactions at the electrode-electrolyte interface兲 occurs, then even
when a voltaic cell is not part of an electrical circuit, the cell can lose
energy. Nonfaradaic discharge causes unused batteries to go bad. In addition, nonfaradaic discharge causes the voltage profile within the electrolyte to develop a spatially quadratic component. See W. M. Saslow,
“Nonlinear voltage profiles and violation of local electroneutrality in ordinary surface reactions,” Phys. Rev. E 68, 051502-1–051502-5 共2003兲.
13
When steady-state faradaic discharge 共that is, charge-transferring chemical reactions at the electrode-electrolyte interface兲 occurs, the voltage
profile within the electrolyte can develop in the steady-state a spatially
quadratic component. See W. M. Saslow, “What happens when you leave
the car lights on overnight: Violation of local electroneutrality in slow,
steady discharge of the lead-acid cell,” Phys. Rev. Lett. 76, 4849–4852
共1996兲.
14
See Ref. 9, Example 8.4, p. 345.
15
G. W. Vinal, Storage Batteries 共Wiley, New York, 1955兲, 4th ed.
16
J. Newman, Electrochemical Systems 共Prentice Hall, Englewood Cliffs,
NJ, 1973兲.
17
H. Gu, T. V. Nguyen, and R. E. White, “A mathematical model of a
1
W. M. Saslow
222
lead-acid cell,” J. Electrochem. Soc. 134, 2953–2960 共1987兲.
D. M. Bernardi, H. Gu, and A. Y. Schoene, “Two-dimensional mathematical model of a lead-acid cell,” J. Electrochem. Soc. 140, 2250–2258
共1993兲.
19
H. Bode, Lead-Acid Batteries 共Wiley, New York, 1977兲.
18
20
V. Srinivasan, G. Q. Wang, and C. Y. Wang, “Mathematical modeling of
current-interrupt and pulse operation of valve-regulated lead acid cells,”
J. Electrochem. Soc. 150, A316–A325 共2003兲.
21
David Linden, Handbook of Batteries 共McGraw-Hill, New York, 1995兲,
2nd ed. See, for example, Figs. 3.6, 24.17 and 24.27.
Wheatstone’s Concertina. Sir Charles Wheatstone 共1802–1875兲 started his research career with work on musical
acoustics. In the 1820s he went into business with his brother William as a publisher of music; later the business
described itself as “Inventors and Patentees of the concertina & manufacturers of harmoniums, music sellers &
concertina makers.” In 1829 he patented the concertina. This instrument differs from the more familiar accordion by
the use of studs instead of keys to blow air from the bellows over the reeds. This example is in the collection of the
Smithsonian Institution, catalogue number 323,481. 共Photograph and Notes by Thomas B. Greenslade, Jr., Kenyon
College兲
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W. M. Saslow
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