Faster Lead-Acid Battery Simulations from Porous-Electrode Theory: Part I. Physical Model

Valentin Sulzer; S. Jon Chapman; Colin P. Please; David A. Howey; Charles W. Monroe

doi:10.1149/2.0301910jes

Lead-acid batteries are the most widely used electrochemical storage technology, with applications including car batteries and off-grid energy supply. Models can improve battery management—for example, by minimizing overcharge to extend cycle life.

The most rigorous mechanistic approach to battery-cell modeling begins with a detailed microscopic description, wherein the electrolyte and electrodes occupy discrete spatial domains; volume averaging is then performed to produce a macroscopic model.^1–3 The details are beyond the present scope, but such a homogenization underpins the model presented below.

Our development of a detailed macrohomogeneous model of a typical lead-acid battery augments standard approaches by explicitly considering the balance of water, the variation of acid density with molarity, and the distribution of pressure. As well as ensuring thermodynamic consistency and retaining model closure when systems span multiple spatial dimensions,⁴ this reveals novel convective phenomena that may occur at high discharge currents.

Nondimensionalization of the model helps to assess the relative importances of different multiphysical phenomena within it. This facilitates numerical implementation because it greatly improves the conditioning of the system, and hence improves the speed of computations, by making most dependent variables close to unity. This paper focuses on the development of a detailed nonisobaric physical model of battery discharge. A novel nondimensionalization is presented, which helps to identify the key dimensionless parameters that control the battery's transient response. Finally, a numerical procedure is developed to solve the nonlinear system of partial differential equations comprising the model. The results are fit against experimental discharge data and used to show how discharge capacity depends on C-rate. The nonlinear model shows that the battery response does not satisfy Peukert's law. Instead, the capacity follows one power law at low C-rates, where average acid concentration controls the response, and a different power law at at high C-rates, where overpotentials control the response.

Below it is shown that coupled pressure effects can be neglected, on the basis that certain dimensionless parameters are small for the materials used in the battery considered here. However, excluded-volume effects remain in the model; the local variation of water concentration in the electrolyte contributes significantly to interfacial reaction kinetics and also impacts diffusion overpotentials. In cases where dimensionless parameters in a model are small or large, asymptotic analysis can be employed, producing simplified, more computationally efficient models that achieve the same level of physical accuracy. The nonlinear model presented here serves as the baseline for testing a hierarchy of computationally efficient reduced-order models, which will be developed by perturbation expansion in part II.⁵

Model

Unscaled governing system

Battery configuration and chemistry

Typically, a valve regulated lead-acid battery comprises six 2 V cells wired in series. Figure 1 depicts one such cell, which consists of five lead (Pb) electrodes and four lead dioxide (PbO₂) electrodes, sandwiched alternatingly around a porous, electrically insulating separator to produce eight electrode pairs, wired in parallel at the top edge of the electrode pile. The pile has height H, depth W, and cross-sectional area A_cs = HW. The negative (Pb) and positive (PbO₂) electrodes have half-widths l_n and l_p respectively, and the separator has width l_sep, giving each internal electrode pair total width L = l_n + l_sep + l_p. The layers repeat periodically, allowing the whole pile to be modeled via analysis of one electrode pair. Some thermal effects have been documented experimentally^6,7 and considered theoretically;^8,9 we assume an isothermal system for simplicity.

The electrodes are porous, permeated by an aqueous sulfuric acid (H₂SO₄) electrolyte that also permeates the separator. The negative and positive half-reactions are

respectively. In both reactions, solid lead sulfate (PbSO₄) forms during discharge as bisulfate anions (HSO⁻₄) leave the liquid.

Without competitive chemistry, Reaction 1 would simply reverse during recharge, as indicated. However, some authors^8,10–12 have observed that gas evolution can become important when recharging. To avoid the need for considering such side reactions, the present analysis is limited to discharges.

Liquid thermodynamics

In water (H₂O), H₂SO₄ dissociates mainly to hydrogen cations (H⁺) and HSO⁻₄; in keeping with prior models,^10,13–15 speciation to sulfate is neglected. Thus the pore-filling liquid comprises H₂O, HSO⁻₄, and H⁺, with partial molar volumes , and , molar masses M_w, M₋, and M₊, and equivalent charges 0, − 1, and + 1, respectively. Unscaled variables are denoted with a hat: for example, the species molarities are , , and .

On any scale much larger than the Debye length, local electroneutrality holds in the liquid phase. Both ion concentrations then relate to the H₂SO₄ molarity through

In an isothermal state, thermodynamic consistency of the liquid volume requires that depends on alone if the bulk modulus of the liquid is very large,¹⁶ through

where is the partial molar volume of the acid. Consequently the total liquid molarity also depends only on . Acid molarity further determines the mass density of the liquid, because

in which M_e = M₊ + M₋ stands for the acid's molar mass. Experiments show that varies almost linearly with ,¹⁷ so the partial molar volumes can be assumed constant.

The mechanical state of the liquid is described by the external pressure , while its mixing free energy is parameterized by a thermodynamic Darken factor .

Balances

The position within each electrode pair traverses three subdomains: the negative electrode (), separator (), and positive electrode (). The model structure is identical in each subdomain. Parameters describing the electrolyte are similar everywhere, while those describing pore geometry generally differ among subdomains; quantities that parameterise reactions differ between the negative- and positive-electrode subdomains, and vanish in the separator.

Liquid volume fraction and reactive solid area per volume characterize the homogenized geometry within the electrode pair. In electrode subdomains, deposition (removal) of solid PbSO₄ on pore surfaces is accompanied by removal (deposition) of solid Pb or PbO₂, so these geometric parameters can generally vary locally. Solid PbSO₄ does not tend to deposit as a compact thin film, so mechanistic lead-acid battery models usually let vary with state of charge.^10,12–15 Since the functionality of this variation is disputed, we instead let the area be constant, following the approach of Liu et al., who showed that the assumption suffices to model Li/O₂-battery discharge.¹⁸ From this perspective describes an immobile Gibbs dividing surface that partitions the layers of Pb or PbO₂, which contribute to the battery's charge state, from the current-collecting Pb layer beneath them, which does not. Thus the time change in pore volume relates to the molar volumes of the solid species in Scheme 1 through

in which is the partial molar volume of species k and is the rate at which k is generated by interfacial reactions per unit of pore area.

After homogenization, the local mass balance of species in the pore-filling liquid phase implies that

where represents the molar flux of k. With electroneutrality condition 2 and equation of state 3, all three balance Equations 6 combine to show liquid-volume continuity,

in which signifies the volume-averaged liquid velocity.

Under Faraday's law, ion balances 6 also combine to demonstrate charge continuity in the liquid,

Letting F be Faraday's constant, the liquid-phase current density is and the current density associated with interfacial charge exchange is . (Positive flows into pores.) Note that any current leaving the liquid at a given location enters the solid there. Thus

where is the solid-phase current density.

In a general isothermal setting, liquid convection is determined by a momentum balance, such as Cauchy's equation.^4,16 Such a balance governs the distribution of momentum density , which is naturally expressed in terms of the mass-averaged velocity . The kinematic relation

specifies how must relate to .

Flux constitutive laws

Two Onsager–Stefan–Maxwell flux laws govern transport in the liquid phase. The law for the thermodynamic force on water¹⁶ can be inverted¹⁹ to give

where R is the gas constant, T is the absolute temperature, t^w₊ is the cation transference number relative to the water velocity, and is the effective Fickian diffusivity of acid in water; the pressure-diffusion factor depends on through

(To put Equation 11 in the form given, one must have that and .²⁰) The law for the thermodynamic force on cations can be linearly transformed to produce a current/voltage relation:

Here is the effective ionic conductivity; is the potential measured by a reversible hydrogen electrode at .²¹ Effective properties appear in Equations 11 and 13 because pore connectivity affects apparent transport rates. Letting and represent the diffusivity and conductivity of the pure liquid, we let

following Bruggeman's tortuosity correlation.

The solid phase is electronically but not ionically conductive. Thus the current density there relates to the solid-phase potential, , through Ohm's law,

where σ^eff is the effective electronic conductivity,

Here σ^eff is constant because , the volume fraction of nonreacting Pb bounded within electron-exchange surface , does not vary. Note that does not count the volume fraction occupied by solid reactants, so .

A constitutive law for liquid stress is needed to close the model.⁴ Below, terms involving momentum will be shown to be negligible in the first approximation. To analyze the scale of these terms, it will suffice to assume that current density induces a flow with low Reynolds number, in which case the homogenization of Cauchy's equation produces Darcy's law,

in which is the liquid viscosity; pore geometry determines the Kozeny–Carman permeability .

Interfacial constitutive laws

Electrons are the only solid-phase charge carrier in Scheme 1, making interfacial electronic current a proxy for the half-reaction rates. No interfacial reactions occur in the separator domain, so uniformly there and the reactive area does not need to be defined there.

In an electrode subdomain where a single half-reaction involving electrons occurs, the rate of change of the reaction extent is and hence every is given by

where s_k is the signed stoichiometric coefficient of species k in the half-reaction. (For a reduction half-reaction, s_k is positive for a product and negative for a reactant.)

Relationship 18 permits simpler expressions to be used in place of general material balances 6. First, Equations 5, 6, 7 and 18 combine to give

This liquid-phase volume balance introduces the volume of reaction , related to half-reaction stoichiometry by

Second, one can combine Equations 2, 5, 6, 8, 11, 18, and 19 to show that the acid is governed by a form of the convective diffusion equation:

The generation term here includes a single additional parameter,

Three aspects of acid-balance Equation 21 are new. First, the convection term, and second, the volume of reaction , appear because state Equation 3 imposes constraints on balances 6. Finally, a pressure-diffusion term appears because the flux laws are based on thermodynamic forces.

To complete the model, a chemical-kinetic constitutive law must be introduced to govern the local current density across pore surface . Generally such laws involve the voltage difference between the liquid and solid, the equilibrium potential of the half-reaction, and the chemical activities of the reactants. We assume the half-reactions in Scheme 1 are elementary, following Butler–Volmer kinetics. Butler–Volmer laws naturally include a symmetry factor, which we take to be one half,¹⁴ yielding

where is the concentration-dependent exchange-current density,

and is the surface overpotential

Here stands for the half-reaction's open-circuit potential (OCP) relative to a particular reference electrode. (The reference must be the same for all half-reactions.) Following Newmann and Tiedemann,¹² we use the formulas of Bode²² for the open-circuit potentials. Terms involving interfacial capacitance help to smooth out numerics, but have a negligible effect on model predictions because the capacitive time-scale is very short.²³

Equations 5, 8 to 10, and 13 to 25 comprise a three-dimensional model with closure at every interior point within an electrode pair.

Boundary conditions

Symmetry and insulating boundary conditions demand that no species in the liquid phase flows through the centers, sides, and bottom of the electrode pair, so

Kinematic relation 10 shows that at these boundaries too. Flux laws 11, 13, and 17 further require the surface-normal gradients , and to vanish at these boundaries.

Above the electrodes, there is a region of free electrolyte, with height . At the top surface of this region, we impose a known external pressure , and the absence of flux relative to the surface, which moves with velocity :

Note the final condition in Equation 27 determines the a priori unknown height .

Hereafter, subscripts n and p denote property values in the negative- and positive-electrode subdomains. We choose the negative electrode to be the ground state, and define the cell voltage to be the potential at the positive electrode tab:

One can either control the voltage, or consider a current-controlled discharge where the voltage is determined by

where is the current drawn from the battery, which is positive for a discharge; the factor of 8 appears because the cell comprises eight electrode pairs in parallel.

This paper focuses on experiments under 'galvanic control', following condition 29, which allow to be any function of time. Since six cells are connected in series, the voltage in the external circuit is .

Relationships between subdomains

The liquid phase permeates all three subdomains. Therefore scalar variables , , and , as well as the normal components of all the flux vectors, are continuous across electrode/separator boundaries.

There is no solid-phase current across the boundary of the separator subdomain, nor is there any pore-surface charge exchange within it, so vanishes uniformly there. Further, since the separator subdomain electronically isolates the positive and negative electrodes, is not continuous across it.

Integrating the interfacial current distributions in Equation 9 and applying the divergence theorem, boundary conditions 29 and the fact that vanishes at the electrode/separator interfaces lead to integral constraints,

In short, these say that the total current leaving the negative electrode domain must enter the positive electrode domain. Equation 30 will help to assess the scales of pressure and velocity in the Dimensionless model section.

Initial conditions

The initial electrolyte concentration and electrode porosities are spatially uniform, but depend on the state of charge, q, which we define as

Let c^max, ε^max_n and ε^max_p be the values of electrolyte concentration, negative electrode porosity, and positive electrode porosity, respectively, at full state of charge. In a lead-acid battery, the state of charge is closely linked to the concentration of the electrolyte. Hence q is chosen to be unity when the concentration of the electrolyte is at its maximum value, c^max, and zero when the concentration of the electrolyte is zero, so that the initial conditions are

Parameters q^max and are chosen to make Equations 31 and 32 consistent with Equations 5 and 21 (see Appendix A for details):

Finally, because interfacial capacitance effects are included, initial conditions are needed for the potentials; we choose spatially homogeneous values, letting

so that everywhere at .

Dimensionless model

Nondimensionalization of the model presented above helps to determine the dominant physical phenomena in the system. If is sufficiently small and the electrode conductivity is sufficiently high, one can assume uniformity in the plane normal to , reducing the problem to one spatial dimension. In this case, the current density is uniform in the current collectors at and , and so boundary condition 29 becomes

which introduces the variable to stand for the total current density at the cell level. Let , , , and represent the -components of vectors , , , and . A first set of dimensionless variables is formed by the natural rescalings

where .

Reduce the number of parameters by identifying the scale of interfacial current density, , and discharge time-scale, . Hence nondimensionalize interfacial current density, exchange-current density, and time as

After nondimensionalization, and become functions. To nondimensionalize the potentials, note that the dominant exponents in the hyperbolic-sine terms of Equation 23 should be , since the terms multiplying the sinh functions are all . Equation A1 can be exploited to define dimensionless open-circuit potentials

Then, to ensure that the exponents in Equation 23 are and noting that at , introduce the dimensionless potentials

Quantities (and hence ) and are appropriately scaled with their values at c = c^max:

where and . The ionic conductivity and Darken thermodynamic factor rescale as

where the quantity is the excluded-volume number defined by Liu and Monroe.⁴

In Equation 24 for the exchange-current density, the reference concentrations are taken to be and . The dimensionless widths of the negative electrode, separator and positive electrode become ℓ_n = l_n/L, ℓ_sep = l_sep/L, and ℓ_p = l_p/L, respectively.

In velocity Equations 10, 17 and 19, and are scaled with their values at c^max, and , with d², where d is the characteristic pore size at full charge. The reaction velocity scale is , and the Darcy pressure scale μ^maxv^rxnL/d², so that

where p^ref is a reference pressure (e.g. atmospheric). This scaling transforms the equations governing v^□, v and p to

in which the dimensionless parameters β, ω_c and ω_i are defined in Table I. Further, the effect of pressure gradients in Equations 13 and 21 is smaller than the effect of concentration gradients (ignoring the functions of concentration χ and ψ) by a factor of

Since β takes different values in the two electrodes, Equation 43a does not admit a one-dimensional solution where v^□ vanishes at both electrode centers (see Appendix B). That is, the y- and/or z-component of v^□ must be non-zero, and there must be a flow in the directions normal to the x-axis. In order to systematically account for this inherently three-dimensional nature of velocity in a one-dimensional model, one must conduct a detailed analysis of the three-dimensional model. Our preliminary analysis of the higher-dimensional problem suggests that the distributions of the normal components of v^□ depend strongly on geometric properties of the battery, such as relative porosities in different domains and aspect ratio of the cell sandwich. Such analysis is beyond the scope of this paper; the limit of finite β will be addressed in a future study.

Table I. Dimensionless parameters, relative to the C-rate, . is the diffusional C-rate.

			Value
Parameter	Definition	Interpretation	n	sep	p

ι_s				-
β^surf		Change in porosity associated with local half-reaction going to completion	0.084	-	− 0.064
β		Change in acid volume fraction associated with local half-reaction going to completion	0.033	-	0.040
γ_dl			2.1 × 10^{− 5}	-	1.7 × 10^{− 4}
ω_c		Diffusive kinematic relationship coefficient	0.70
ω_i		Migrative kinematic relationship coefficient	0.41
π_os

At present, we note that the dimensionless parameters β and π_os are very small, and therefore assume a limit where both of these parameters are zero (and hence ), in which case v^□ vanishes everywhere, and v and p decouple from the other variables. The resulting one-dimensional full model can then be solved to find the voltage without needing the velocity and pressure.

Having decoupled flow velocity and pressure from the rest of the model, the following dimensionless system for c, j, ε, i, Φ, i_s and Φ_s results:

with boundary conditions

and initial conditions

Integral condition 30 nondimensionalizes to

Composition dependences of the properties , χ, , j₀, , and are established through the functions listed in Table AII. The four dominant dimensionless parameters, , ι_s, β^surf and γ_dl, are defined in Table I, which also states physical interpretations and typical values. In particular, the diffusional C-rate can be written as

where 8A_csc^maxFL is the volumetric capacity of the battery (in Ah), Q is the capacity of the battery (in Ah), and is the C-rate of operation. Alternatively, one can identify to be the ratio between the applied current scale, , and the scale of the liquid-phase limiting current, i_L = c^maxD^maxF/L.

Table AI. Dimensional parameters from the literature. Parameters with several values indicate different values in negative electrode (n), separator (s) and positive electrode (p). (^*) Reported by .¹⁴ P.M.V. = Partial Molar Volume.

		Value
Parameter	Name	n	sep	p	Units	Refs.
l	Subdomain width	0.9 × 10^{− 3}	1.5 × 10^{− 3}	1.25 × 10^{− 3}	m	36
	Electrode porosity	0.53	0.92	0.57	-	23
H	Battery height	11.4 × 10^{− 2}			m	Measured
W	Battery depth	6.5 × 10^{− 2}			m	Measured
s₊	Stoichiometery of cations	− 1	-	− 3	-	1
s₋	Stoichiometery of anions	1	-	− 1	-	1
s_w	Stoichiometery of Water	0	-	2	-	1
	Number of electrons	2	-	2	-	1
	P.M.V. of water	1.75 × 10^{− 5}			m³ mol^{− 1}	10
	P.M.V. of cations	1.35 × 10^{− 5}			m³ mol^{− 1}	35
	P.M.V. of anions	3.15 × 10^{− 5}			m³ mol^{− 1}	35
	P.M.V. of lead	1.83 × 10^{− 5}			m³ mol^{− 1}	37
	P.M.V. of lead dioxide	2.55 × 10^{− 5}			m³ mol^{− 1}	37
	P.M.V. of lead sulfate	4.82 × 10^{− 5}			m³ mol^{− 1}	37
M_w	Molar mass of water	1.8 × 10^{− 2}			kg mol^{− 1}	37
M₊	Molar mass of cations	0.1 × 10^{− 2}			kg mol^{− 1}	37
M₋	Molar mass of anions	9.7 × 10^{− 2}			kg mol^{− 1}	37
F	Faraday constant	96485			C mol^{− 1}	-
R	Ideal gas constant	8.314			J mol^{− 1} K^{− 1}	-
T	Room temperature	298.15			K	-
t^w₊	Transference number	0.72			-	14,17
σ	Solid conductivity	4.8 × 10⁶	-	8 × 10⁴	S m^{− 1}	15
j^ref	Reference exchange-current	8 × 10^{− 2}	-	6 × 10^{− 3}	A m^{− 2}	38 (*)
c^max	Initial electrolyte concentration	5.6 × 10³			mol m^{− 3}	36
	Surface area per volume	2.6 × 10⁶	-	2.05 × 10⁷	m^{− 1}	38 (*)
d	Pore size at full charge	10^{− 7}	-	10^{− 7}	m	15
C_dl	Double-layer capacity	0.2	-	0.2	F m^{− 2}	23
q⁰	Full initial SOC	1			-	-
Q	Manufacturer-specified capacity	17			Ah	-

Table AII. Dimensional functions of concentration, (measured in mol/m³) and (dimensionless). (^†) Empirical formulae are given by Gu et al.,¹⁵ citing Tiedemann and Newman,³⁸ and agree with data from Chapman and Newman.¹⁷ (^*) Our fit to data of given reference(s). (^‡) Molality m is defined as .

Name	Formula	Value at	Units	Refs.
Electrolyte diffusivity		3.02 × 10^{− 9}	m² s^{− 1}	(†)
Thermodynamic factor		2.8	-	17,39 (*)
Electrolyte conductivity		77	S m^{− 1}	(†)
Electrolyte density		1.32 × 10³	kg m^{− 3}	40
Permeability		-	m²	15
Electrolyte viscosity		2.5 × 10^{− 3}	Pa s	17 (*)
Water concentration		4.27 × 10⁴	mol m^{− 3}	3
Exchange-current density			A m^{− 2}	18
Open-circuit potential (neg)	− 0.295 − 0.074log m − 0.030log ²m − 0.031log ³m − 0.012log ⁴m (‡)	− 0.41	V	22
Open-circuit potential (pos)	1.628 + 0.074log m + 0.033log ²m + 0.043log ³m + 0.022log ⁴m (‡)	1.76	V	22

In the Results section, q⁰ will be taken to be unity (the battery starts from a fully charged state) unless explicitly stated.

Numerical Solution

The system of Equations 45 was solved numerically. Code used to solve the model and generate the results below is available publicly on GitHub.²⁴

To facilitate numerical solution, the model was first manipulated into a form suitable for application of the Finite Volume Method. Letting ϕ = Φ_s − Φ and noting that i_s = i_cell − i, one can replace Equations 45d to 45f with

We also eliminate ∂Φ/∂x from Equation 47 to find i as a functional of c and ϕ:

The result is a closed system of partial differential equations for c, ε and ϕ:

where j = ∂i/∂x is obtained by differentiating Equation 48a in each electrode and vanishes in the separator, with boundary conditions

and initial conditions derived from Equation 45j.

Equation system 48 is solved by discretising the spatial domain using Finite Volumes, choosing the spatial discretisation to be uniform within each subdomain and as uniform as possible across subdomains (100 points in the negative electrode, 170 in the separator, and 140 in the positive electrode were deemed sufficient). The results are robust to the total number of points chosen; 410 were chosen here to achieve grid independence in one minute.

Having discretized in space, the resulting system of transient ordinary differential equations is solved using scipy.integrate in Python.²⁵ Finally, Φ is calculated as

where the final term comes from the fact that Φ = −ϕ at x = 0. The voltage drop across the electrode pair is computed with

Crucially, the partial differential equation system 48 is much easier to solve than the differential-algebraic system 45.

Results

Figure 2 shows the (dimensional) voltage against capacity used for a range of C-rates. As is to be expected, the total capacity available decreases as the C-rate increases. This dependence is further elucidated by exploring the total capacity of the battery for constant-current discharges across a higher range of C-rates, summarized on Figure 3. By Peukert's Law, one would expect the graph to be linear on this log-log plot. However, in this case deviations from linearity occur because there are two distinct capacity-limiting mechanisms. At low C-rates (below 1C), the battery capacity is concentration-limited, with full discharge occurring when the electrolyte concentration reaches zero. At high C-rates, the battery is voltage-limited, because the cutoff voltage of 10.5V is achieved before the bulk concentration falls to zero.

**Figure 3.** Log-log plot of battery capacity against C-rate, using the parameters from literature (Table AI). Dashed lines indicate the expected behavior from Peukert's law.

The model also allows internal variables to be explored, particularly the local water concentration. Electrolyte and water concentrations are depicted in Figure 4. At a very low C-rate of 0.1C (Figure 4a), both concentrations remain almost uniform throughout the discharge; electrolyte concentration decreases, and water concentration increases proportionally. At higher C-rates of 0.5C (Figure 4b) and 2C (Figure 4c), the concentrations become spatially inhomogeneous, which leads to concentration overpotentials that limit the accessible capacity.

**Figure 4.** Electrolyte and water concentrations at various States of Charge (SOCs) for a constant-current discharge using the parameters from literature (Table AI), for a range of C-rates. Opacity increases with decreasing SOC.

A Derivative-Free Gauss-Newton method²⁶ was used to fit the model to data from a series of constant-current discharges of a 17 Ah BBOXX Solar Home battery at intervals of 0.5 A from 3 A to 0.5 A (Figure 5). Each constant-current discharge was followed by a two-hour rest period at open circuit (zero current). The model yields a good fit to the data, with a root-mean-square error of 43 mV, but this approach is slow since it requires solving the full system of partial differential equations at each iteration. A faster approach will be developed in part II.⁵

**Figure 5.** Comparing data (dots) with results from full numerical model (lines) for a range of currents, with parameters fitted using DFO-GN. The root-mean-square-error of the fit is 43 mV.

Conclusions

Three novel phenomena were included in a porous-electrode model for lead-acid batteries. First, the mass-averaged and volume-averaged velocities of the electrolyte were both considered, the former associated with momentum transport and the latter with kinematics. Due to density variation in the liquid, and volume changes associated with the electrode reactions, neither of these velocities remains solenoidal after homogenisation. Second, an extra convective term, associated with the volume-averaged velocity, drives cation transport. Third, a pressure-diffusion term drives cation transport, and appears also in the MacInnes equation (modified Ohm's law) describing the liquid. Although these terms are small in magnitude for lead-acid batteries discharged at low-to-moderate rates, they could be important for other chemistries where large volume changes occur during charge/discharge, such as lithium-ion batteries with silicon anodes,^27–32 or to understand other scenarios such as the impedance signature of electromechanical/transport coupling.³³ Nondimensionalization of this model led to the identification of key parameter groupings, a methodology that could extend easily to other models and chemistries.

Two distinct mechanisms determine the total cell capacity: concentration limitation at low C-rates, and voltage limitation at high C-rates. In addition to concentration limitation and voltage limitation, capacity could be limited by additional physical effects, such as pore occlusion at very high C-rates.¹⁸

The model developed in this paper provides physical detail about the electrochemical processes occurring in a lead-acid battery during discharge, but is ultimately too computationally intensive to underpin an advanced battery management system. In part II,⁵ asymptotic analysis is used to derive three simplified models valid at low-to-moderate discharge rates. These can be solved much faster than the detailed model developed here, while giving more physical insight than an equivalent circuit.

Acknowledgments

This publication is based on work supported by the EPSRC center For Doctoral Training in Industrially Focused Mathematical Modelling (EP/L015803/1) in collaboration with BBOXX. JC, CP, DH and CM acknowledge funding from the Faraday Institution (EP/S003053/1).

ORCID

Valentin Sulzer 0000-0002-8687-327X

S. Jon Chapman 0000-0003-3347-6024

Colin P. Please 0000-0001-8917-8574

David A. Howey 0000-0002-0620-3955

Charles W. Monroe 0000-0002-9894-5023

Appendix A.: Parameters

The dimensional parameters are given in Table AI, and the concentration dependences of coefficients are laid out in Table AII. Formulae for open-circuit potentials were obtained empirically by Bode.²² Note that these could be written as

to reflect their relation to the Nernst equation (following Treptow³⁴), with U^⊖_Pb = −0.295 and .

Appendix. Consistent initial conditions

For consistent initial conditions, we take 'starting at x% SOC' to mean an internally equilibrated (i.e. spatially homogeneous) initial state at uniform concentration corresponding to this charge. The following determines initial conditions for and consistent with this choice.

Considering a one-dimensional model and integrating Equation 21 in across the whole electrode pair and using the no-flux conditions 26, and the integral condition 30,

Both sides of Equation A2 should be zero when q = 0; hence choose

which, with the parameter values in Table AI, gives a maximum capacity of 26.1 Ah for the battery. This compares favorably to the stated battery capacity of 17 Ah.

Now, integrating Equation 5 in across the whole negative electrode and using the integral condition 30,

Then, assuming that ε⁰_n is spatially uniform,

Similarly,

Appendix B.: Velocity

Integrating Equation 43a for the volume-averaged velocity from x = 0 to x = 1, and using integral condition 30 for the interfacial current density and Equation 45h for v^□ at x = 0, one finds that

which contradicts Equation 45h for v^□ at x = 1 since the sum of the volume changes across the whole cell is non-zero. It follows that the model can have no exact solution in a one-dimensional setting. This can be resolved by considering a multi-dimensional problem with a free electrolyte surface; this solution is beyond the present scope. For the present purposes, we take the limit in which β_n and β_p are zero (note, from Table I, that β_n and β_p are small).

Faster Lead-Acid Battery Simulations from Porous-Electrode Theory: Part I. Physical Model

Article metrics

Author affiliations

Author notes

ORCID iDs

Dates

Abstract

Model

Unscaled governing system

Battery configuration and chemistry

Liquid thermodynamics

Balances

Flux constitutive laws

Interfacial constitutive laws

Boundary conditions

Relationships between subdomains

Initial conditions

Dimensionless model

Numerical Solution

Results

Conclusions

Acknowledgments

ORCID

Appendix A.: Parameters

Appendix. Consistent initial conditions

Appendix B.: Velocity

Faster Lead-Acid Battery Simulations from Porous-Electrode Theory: Part I. Physical Model

Article metrics

Share this article

Author affiliations

Author notes

ORCID iDs

Dates

Abstract

Model

Unscaled governing system

Battery configuration and chemistry

Liquid thermodynamics

Balances

Flux constitutive laws

Interfacial constitutive laws

Boundary conditions

Relationships between subdomains

Initial conditions

Dimensionless model

Numerical Solution

Results

Conclusions

Acknowledgments

ORCID

Appendix A.: Parameters

Appendix. Consistent initial conditions

Appendix B.: Velocity