Lead–acid batteries

Lead–acid batteries

CHAPTER 6 Lead–acid batteries Contents 6.1. Lead–acid battery components 6.1.1 Plates 6.1.2 Separators 6.1.3 Electrolyte 6.2. Lead–acid battery types...

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Lead–acid batteries Contents 6.1. Lead–acid battery components 6.1.1 Plates 6.1.2 Separators 6.1.3 Electrolyte 6.2. Lead–acid battery types 6.3. Electrochemistry of lead–acid batteries 6.4. Lead–acid battery applications 6.5. Governing equations 6.5.1 Conservation of mass and momentum 6.5.2 Conservation of energy 6.5.3 Conservation of charge 6.5.4 Conservation of species 6.5.5 Conservation of mass 6.6. Thermal runaway problem 6.7. Heat sources and sinks 6.7.1 Heat of reactions 6.7.2 Joule heating 6.7.3 Heat dissipation 6.8. One-dimensional model 6.8.1 Governing equations for one-dimensional model 6.8.2 Boundary conditions Potential in solid and electrolyte Chemical species Cell temperature 6.9. Physico-chemical properties 6.9.1 Electrode electrical conductivity σ 6.9.2 Electrolyte ionic conductivity k 6.9.3 Diffusion coefficients 6.9.4 Open-circuit voltage U 6.9.5 Partial molar volumes of sulfuric acid and water 6.9.6 Thermodynamic properties of different species 6.9.7 Calculation of properties in porous medium 6.9.8 Temperature dependency of parameters 6.10. Numerical simulation of lead–acid batteries 6.10.1 One-dimensional simulation without side reactions 6.10.2 One-dimensional simulation including side reactions 6.10.3 Numerical simulation of electrolyte stratification using two-dimensional modeling Simulation of Battery Systems https://doi.org/10.1016/B978-0-12-816212-5.00010-6

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Simulation of Battery Systems

6.10.4 Simulation of thermal behavior of lead–acid batteries 6.11. Summary 6.12. Problems

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Lead–acid batteries were introduced by Gaston Planté in 1859. He found out that a pair of lead–lead dioxide electrodes in sulfuric acid can produce electricity. This invention made a significant breakthrough in the history of battery technology. The positive electrode of lead–acid batteries are made of PbO2 , whereas the negative counter electrode is made of pure lead or Pb. The electrolyte of this type batteries is made of sulfuric acid. Since the invention of lead–acid batteries, the primary chemistry has remained unchanged, and only some modifications have been made to improve its performance. Using paste instead of flat plates in the construction of electrodes was one of the significant improvements. Pasted electrodes are able to provide more energy than foils or solid ones. Another major modification of the original design was usage of absorbed glass mat (AGM) separators instead of conventional separators. This invention resulted in invention of sealed lead–acid batteries, in which oxygen recombination occurs, and electrolyte concentration remains constant during its operational time. Hence the battery requires less maintenance. The most advantages of lead–acid batteries are their ability to supply high surge current and their cost. These batteries can provide a lot of power in a short time, the property that made them attractive in car industries for starting a car. In this view, lead–acid batteries have a high power-to-weight ratio, or more scientifically, they have a high specific energy. In addition to a high specific energy, lead–acid batteries are very inexpensive in comparison with other industrial technologies. Their economical cost has made them very attractive in various industries. Although a high surge current is not a matter, lead–acid batteries are a good candidate for electric storage. For example, in UPS, industrial electric cars such as lift trucks, in small cheap electric cars, and any other industry where the cost is more important than the battery weight, lead–acid batteries are the primary candidate. Another advantage of lead–acid batteries is their ability for recycling. A lead–acid battery can be recycled above 95%, and the recycled materials can be used to make new batteries. The recycling plants are very mature, and many technologies have been developed to make them environment friendly and economic. This advantage makes lead–acid batteries even more attractive for industry.

Lead–acid batteries


Figure 6.1 Conventional lead–acid batteries.

Although lead–acid batteries have many advantages, there are some drawbacks that make other technologies to be their potential competitors in the market. This type batteries have a very low energy density and specific energy. This means that they occupy a lot of space and are heavy. Hence in applications where volume and weight are essential, lead–acid batteries are not a choice of interest. For example, for car industries where the driving energy of the car is stored in batteries, lead–acid batteries are not good nominates.

6.1 Lead–acid battery components Like any other batteries, lead–acid batteries are composed of plates, separator, and electrolyte. Here we briefly discuss each component. Details of a conventional lead–acid battery is shown in Fig. 6.1.

6.1.1 Plates The plates of lead–acid batteries are usually made in three different shapes: 1. Flat plates are the most conventional type of lead–acid batteries, where the plates are pasted on a flat grid made of lead. The grid may contain different additives to improve its performance and enhance its operational life. 2. Tubular plates are another major battery type, in which the positive plates are put in some cylinders or tubes. The main advantage of this


Simulation of Battery Systems

shape is that in the tubular lead–acid batteries the active surface of the positive electrode is larger than of flat plates because the electrolyte is in greater contact with active materials; hence the battery is suitable for high current applications. However, tubular batteries have less active material comparing to flat plate types, meaning that they store less energy than flat plate types. Moreover, they are more expensive in manufacturing than flat plate batteries. 3. In another technology the battery plates are rolled and made in a spiral shape. The positive and negative plates are sandwiched in AGM separators and rolled to shape the spiral. They showed better performance and stability, but they are more expensive than flat plates. In all the configurations the positive electrodes are made of PbO2 , and the negative electrode is made of Pb. Various additives are added to the positive and negative plates so that they show better performance in different situations, for example, to have a good performance in very cold or very hot climates or to increase battery cycling life. Some additives are also added to current collectors made of pure lead. The additives differ for the positive and negative electrodes. Bismuth, calcium, silver, and many other additives are tested and practically added to improve the performance and durability of the grids.

6.1.2 Separators Separators between the positive and negative plates prevent short-circuit through physical contact, mostly through dendrites (“treeing”), but also through the shedding of the active material. Separators allow the flow of ions between the plates of an Electro-chemical cell to form a closed circuit. Wood, rubber, glass fiber mat, cellulose, and PVC or polyethylene plastic have been used to make separators. Wood was the original choice but deteriorated in the acid electrolyte. Rubber separators are stable in battery acid and provide valuable electrochemical advantages that other materials do not. An effective separator must possess several mechanical properties such as permeability, porosity, pore size distribution, specific surface area, mechanical design and strength, electrical resistance, ionic conductivity, and chemical compatibility with the electrolyte. In service the separator must have excellent resistance to acid and oxidation. The area of the separator must be a little larger than the area of the plates to prevent material shorting between the plates. The separators must remain stable over the battery operating temperature range.

Lead–acid batteries


In the absorbed glass mat (AGM) design the separators between the plates are replaced by a glass fiber mat soaked in electrolyte. There is enough electrolyte in the mat to keep it wet, and if the battery is punctured, then the electrolyte will not flow out of the mats. Principally, the purpose of replacing liquid electrolyte in a flooded battery with a semisaturated fiberglass mat is a substantial increase of the gas transport through the separator; Hydrogen or oxygen gas produced during overcharge or charge (if the charge current is excessive) is able to freely pass through the glass mat and reduce or oxidize the opposing plate, respectively. In a flooded cell the bubbles of gas float to the top of the battery and are lost to the atmosphere. In AGM configuration, however, the transport mechanism causes the produced gas to recombine and make water again. The additional benefit of the semisaturated cell is providing no substantial leakage of electrolyte upon physical puncture of the battery case that allows the battery to be completely sealed, which makes them useful in portable devices and similar roles. Additionally, the battery can be installed in any orientation, though if installed upside down, the acid may be blown out through the overpressure vent. Reducing the water loss rate, the plates are alloyed with calcium. However, gas build-up remains a problem when the battery is deeply or rapidly charged or discharged. To prevent overpressurization of the battery casing, AGM batteries include a one-way blow-off valve and are often known as “valve regulated lead–acid” (VRLA) designs. Another advantage of the AGM design is that the electrolyte becomes the mechanically strong separator material. This advantage allows the plate stack to be compressed together in the battery shell, slightly increasing the energy density compared to liquid or gel versions. AGM batteries often show a characteristic “bulging” in their shells when built in common rectangular shapes due to the expansion of the positive plates. The mat also prevents the vertical motion of the electrolyte within the battery. When a normal wet cell is stored in a discharged state, the heavier acid molecules tend to settle to the bottom of the battery, causing the electrolyte to stratify. When the battery is then used, the majority of the current flows only in this area, and the bottoms of the plates tend to wear out rapidly. This is one of the reasons a conventional car battery can be ruined by leaving it stored for a long period and then used and recharged. The mat significantly prevents this stratification, eliminating the need to periodically shake the batteries, boil them, or run an “equalization charge” through them to mix the electrolyte. Stratification also causes the upper layers of the battery to become almost completely water, which can freeze


Simulation of Battery Systems

in cold weather; therefore AGMs are significantly less susceptible to damage due to low-temperature use. Although AGM cells do not permit watering (typically, it is impossible to add water without drilling a hole in the battery), their recombination process is fundamentally limited by the usual chemical processes. Hydrogen gas will even diffuse right through the plastic case itself. Some have found that it is profitable to add water to an AGM battery, but this must be done slowly to allow the water to mix via diffusion throughout the battery. When a lead-acid battery loses water, its acid concentration increases, increasing the corrosion rate of the plates significantly. AGM cells already have a high acid content in an attempt to lower the water loss rate and increase standby voltage, and this brings about shorter life compared to a lead-antimony flooded battery. If the open-circuit voltage of AGM cells is significantly higher than 2.093 volts, or 12.56 V for a 12 V battery, then it has a higher acid content than a flooded cell; although this is normal for an AGM battery, it is not desirable for long life. AGM cells intentionally or accidentally overcharged will show a higher open-circuit voltage according to the water lost (and acid concentration increased). One Ah of overcharge will liberate 0.335 grams of water; some, but not all, of this liberated hydrogen and oxygen will recombine.

6.1.3 Electrolyte The electrolyte of lead–acid batteries are made of sulfuric acid. For different purposes, many different additives are added to the electrolyte. In lead–acid batteries, sulfuric acid or the electrolyte is an active material. In other words, the electrolyte is consumed during discharge to produce lead–sulfate and produces water. During the charging process, the reverse reaction converts lead–sulfate on both electrodes into sulfuric acid. In gelled lead–acid batteries, sulfuric acid is mixed with silica gelling agent to make a gelled lead–acid battery. The gelled electrolyte requires less maintenance, but it reduces ion mobility, which in turn reduces battery power. Therefore gelled lead–acid batteries are frequently used in energy storage devices where surge current capability is not an issue.

6.2 Lead–acid battery types From application-based viewpoint, there are two basic types, namely starting and deep cycle lead–acid batteries.

Lead–acid batteries


Starting batteries are those used in starting cars, motor cycles, boats, or any other vehicles. The main characteristic of such batteries is that in starting time, a high surge current is required. Starting batteries should not be discharged very deep, or in other words, they should work so that their state of charge is always greater than 80 percent, SoC > 80%. The reason is that their plates are thin, and if they become discharged in values higher than the limit, then lead sulfate covers all the plate area and decreases the plate conductivity. As a result, the battery cannot be charged again and looses its cyclic life. Deep cycle batteries are those used for power regulation, energy storage, and low-current applications. The plates are thicker than starting batteries; hence they can be discharged more than starting types. Consequently, they will provide more energy in a longer period. However, they are not able to provide high surge current and thus are not suitable for starting applications. From another point of view, lead–acid batteries can be divided into wet or flooded and sealed or valve-regulated types. Wet Cell (flooded) batteries are used in most automotive industries. In these batteries the electrolyte is a liquid solution of sulfuric acid where during charge and discharge is dissociated into hydrogen and oxygen. The produced gases are vented to prevent the explosion. Value Regulated (VRLA) batteries have the same configuration as flooded types, but they are sealed, so the produced gases are not able to escape from the battery. The evolved gases increase the internal pressure, which is dangerous if it exceeds some limits. To prevent the damage, some pressure valves are located at the top of each cell, and if the cell pressure reaches a designed value, the valves open and decrease the pressure. VRLA batteries require less maintenance and are commonly composed of two types: 1. AGM (Absorbed Glass Mat) In this type the electrolyte is confined in an AGM separator, so it becomes immobilized. The evolved oxygen moves from the positive electrode to the negative electrode and recombines with hydrogen ions. The oxygen cycle maintains the electrolyte concentration at its original level. The AGM cells show less internal resistance than the other types since acid migration is faster in AGM batteries. As a result, during discharge and charge, the battery can deliver higher current than any other sealed batteries.


Simulation of Battery Systems

2. Gelled Cell In gelled cells the electrolyte is converted into a gel by addition of Silica gel and becomes a solid mass. Consequently, the electrolyte does not move, and this prevents its spillage. These batteries are charged with lower voltage to prevent evolving excess gas that eventually will damage the cell. Gel batteries are best used in very deep cycle application and may last a bit longer in hot weather.

6.3 Electrochemistry of lead–acid batteries In a lead acid batteries the following reaction takes place: discharge

−−  PbO2(s) + HSO−4 + 3 H+ + 2 e−  −− −− −− − − PbSO4(s) + 2 H2 O



at the positive electrode and discharge

+ − −  Pb(s) + HSO−4 −  −− −− −− − − PbSO4(s) + H + 2 e



at the negative electrode. Then the whole battery reaction is written as discharge

−−  Pb + PbO2 + 2 H2 SO4  −− −− −− − − 2 PbSO4 + 2 H2 O.



It is remarkable that both electrodes convert to lead sulfate during discharge and recover again when they are charged. Also, in the lead–acid batteries, electrolyte acts as an active material. In many battery technologies, this is not the case. In other words, the electrolyte is just a medium for ionic transportation and does not contribute to electrochemical reactions. In the case of lead–acid batteries the energy content depends on the volume and concentration of the electrolyte and on the mass of positive and negative electrode active materials. When a lead–acid cell is discharged, as discussed before, both electrodes convert to lead sulfate and recover when it is charged. In an ideal case, when the battery is fully charged, all lead sulfates are converted to original materials. However, it is not a real case in practice. Actually, when a battery cell is charged, some lead sulfate remains intact and does not recover and forms a passive layer on the electrode surface. This fact repeats in succeeding cycles, and in each cycle the amount of passive lead sulfate increases. The passivation process continues until the cell becomes unusable.

Lead–acid batteries


The passivation process takes place due to many reasons. It should be noted that there are two common lead sulfate crystals α and β . It is known that α crystals are needle-shaped and resistive; hence they do not contribute in the charge and discharge process. In contrast, β crystals have soft and round edges and contribute to charge and discharge. The more α crystals are generated, the more passivated the electrode becomes. Although the main reason for the generation of α crystals is not yet clearly known, it is observed in practice that by increasing the discharge current density the rate of its generation becomes faster. Also, deep discharge has a positive effect on α crystal generation. Besides the main reactions, some side reactions occur in the lead–acid batteries. For the positive electrode, the following side reactions accompany the main reaction: discharge

+ − −  2 H2 O −  −− −− −− − − O2 + 4 H + 4 e ,




+ − −  H2 −  −− −− −− − − 2H + 2e ,



and the following side reactions accompany the main reaction of the negative electrode: discharge

−−  O2 + 4 H+ + 4 e−  −− −− −− − − 2 H2 O,




−  2 H+ + 2 e− −  −− −− −− − − H2 .



Side reactions are important when SoC > 0.6 and can be ignored otherwise. Especially, they play an essential role in overcharge when SoC  1 and the cell is still being charged. The importance of side reactions in battery dynamics will be discussed in more detail in Section 6.10. As a result of side reactions, hydrogen and oxygen release. The mixture is an explosive material and should be vented from the battery. In the flooded lead–acid types, small flame arrester equipped orifices are located at the top of each cell to vent the mixture. Flame arresters are for further safety because the hydrogen/oxygen mixture is highly explosive. Fig. 6.2A shows the schematic gas evolution in a flooded lead–acid battery, where evolved oxygen bubbles are not able to move through the separator and reach the negative electrode. In contrast to flooded lead–acid batteries, in AGM or gelled lead–acid types the electrolyte does not let the evolved gases to rise to the headspace,


Simulation of Battery Systems

Figure 6.2 Hydrogen and oxygen evolution in lead–acid batteries. (A) Flooded lead– acid batteries. (B) Sealed lead–acid batteries.

but the evolved oxygen moves toward the negative electrode, where recombination takes place to produce water according to Eq. (6.6). The oxygen recombination cycle compensates water dissociation, and the electrolyte composition remains intact. Fig. 6.2B shows the moving direction of evolved gases in a sealed lead–acid battery.

6.4 Lead–acid battery applications Without a doubt, the automobile industry is one of the largest markets of lead–acid batteries since each car has a lead–acid battery for its starting,

Lead–acid batteries


lighting, and ignition; due to the applications, the battery is called SLI battery. SLI batteries are made of thin plates that can produce very high surge current about 500 A or more. Flooded, gelled, and spiral wound batteries are made as SLI batteries. Valve-regulated lead–acid (VRLA) and tubular batteries are commonly used to supply the necessary power for large factories, telephone and computer centers, and for off-grid energy storage. Especially with the growth of renewable energy sources such as solar cells or wind turbines, VRLA batteries are getting more and more attention. Traction batteries are used in industrial cars such as lift–trucks, submarines, emergency devices, and so on. Also, to power up small electric cars, traction lead–acid batteries are the choice because of their low cost and proper recycling. Many other applications use lead–acid batteries mostly because of their inexpensive cost, especially where the weight and volume is not a matter. For instance, in many playing devices such as small cars, toys, and so on, lead–acid batteries are good choices.

6.5 Governing equations Lead-acid batteries types and applications were studied in the previous chapter. Although the technology is relatively mature and the batteries are produced in huge numbers, they are still not satisfactorily optimized. Traditionally, the optimization process is based on experimental tests that are costly and time-consuming. To complement the conventional trial-anderror method of improving the performance and cycle life of the lead-acid battery, mathematical models have been developed to predict discharge and charge behavior and the effect of cycling. Many efforts have been made to improve mathematical methods so that they could be applied on lead–acid batteries with minimum error. The first comprehensive study of porous electrodes was developed by Newman and Tiedemann [46]. They applied the theory to simulate the discharge behavior of a lead–acid cell. To correct the uniform acid concentration assumption considered in that study, Sunu [47] extended the model to the case of nonuniform acid concentration in the electrolyte reservoir. The first model that was able to simulate both discharge and charge process of a lead–acid cell was introduced by Gu et al. [48]. The model was extended so that it was able to predict the cell behavior during the rest and charge cycles in addition to the discharge cycle. Moreover, the dynamic behavior


Simulation of Battery Systems

of acid concentration, the porosity of the electrodes, the state of charge (SoC) of the cell, and the dependency of cell performance on electrode thicknesses and operating temperature were also investigated. They solved the obtained system of equations using a finite volume scheme. Later, Esfahanian and Torabi [49] applied the Keller–Box method to the coupled one-dimensional electrochemical transport equations and showed that the method is very efficient in solving such a system of equations. Although it was approved that electrolyte movement makes acid stratification in the lead–acid batteries, none of the previous modelings was able to simulate electrolyte movement. To overcome the weakness, for the first time, Alavyoon et al. [50] developed a two-dimensional mathematical model for mass transfer and electrolyte motion during charge process. Later, Gu et al. [37] introduced a model with an integrated formulation for battery dynamics. In this approach the whole battery was considered as a model volume, and the transport equations including electrochemical, mass, and fluid flow were derived for the whole cell volume. Also, the model was capable of predicting the transient behavior of the battery during discharge, rest, and charge. Esfahanian et al. [44] introduced an improved and efficient mathematical model for the simulation of flooded lead-acid batteries based on computational fluid dynamics and equivalent circuit model (ECM). This model inherited not only the accuracy of the CFD model, but also the physical understanding of ECM, which made it quite suitable for real-time simulations. Torabi and Esfahanian [31,32] studied thermal-runaway in batteries theoretically and, as a result, presented a general set of governing equations by which the thermal behavior of batteries could be obtained. Since the proposed system of equations was general, it can be used for any battery systems. This literature review shows in short the efforts that have been done on simulation of lead–acid batteries. In all the literature the system of governing equations was applied to a cell model, and the system was solved using different numerical schemes. It is necessary to note that the numerical scheme should be chosen such that the numerical stability is provided because the obtained system of equations is nonlinear with exponential source terms resulting in a stiff system of algebraic equations. Also, it should be noted that the selection of a numerical scheme should not affect the results, meaning that if we choose two different methods, then we have to obtain the same results.

Lead–acid batteries


The governing equations of lead–acid batteries result in an ill-posed system of equation as discussed in Chapter 5. In addition to ill-posed behavior of the governing equations discussed before, simulation of the system of nonlinear governing equations requires some attention. The problem of illness of the systems is tackled by the compatibility equation as discussed before. In addition to the compatibility equation, it is remarkable that the potential field has no unique value and is a relative quantity. Hence a reference point should be selected as the reference state in all the simulations. In a battery, usually, the potential of the current collector or lug of the negative electrode is taken as the reference state. By this selection the potential of the positive electrode is the voltage of the cell. However, we can assume that the reference point of potential is the current collector of the positive electrode. This assumption leads us to negative values for the potential of the negative electrode. In any case the selection of reference point should not make any difference in results. In the related literature, both assumptions are used without any problem. Another difficulty arises from the fact that a battery cell is a multiregion domain, where special care should be taken to resolve proper numerical values at regions’ boundaries. Therefore, for an accurate numerical simulation with reasonable numerical cost, a nonuniform mesh should be generated with specific attention on the uniformity of mesh size at the boundaries. An inappropriate mesh results in inappropriate numerical values. The governing equation of lead–acid batteries can be obtained from the general governing equations of battery systems discussed in Chapter 3. A typical lead–acid cell is shown schematically in Fig. 6.3; it consists of the following regions: a lead–grid collector at x = 0, which is at the center of the positive electrode; a positive PbO2 electrode, an electrolyte reservoir for providing more electrolyte for the positive electrode; a porous separator; a negative Pb electrode; and finally a lead–grid collector at x = l, which is at the center of the negative electrode. The reservoir region in practice is made by constructing some ribs on one side of the separator. It is worth noting that in some designs the separator has ribs on both sides, meaning that the separator provides reservoir for both electrodes. Thus the model should contain electrolyte reservoir on both sides. The positive and negative electrodes consist of porous solid matrices whose pores are flooded by a binary sulfuric acid H2 SO4 and of a gaseous phase made up of O2 and H2 . During charge and discharge, the following electrochemical reactions occur:


Simulation of Battery Systems

Figure 6.3 A Typical lead–acid cell model.

• The positive electrode (PbO2 /PbSO4 ): discharge

−  PbO2(s) + HSO−4 + 3 H+ + 2 e− −  −− −− −− − − PbSO4(s) + 2 H2 O,




+ − −−  2 H2 O  −− −− −− − − O2 + 4 H + 4 e ,




+ − −−  H2  −− −− −− − − 2H + 2e .



• The negative electrode (Pb/PbSO4 ): discharge

+ − −−  Pb(s) + HSO−4  −− −− −− − − PbSO4(s) + H + 2 e ,




−  O2 + 4 H+ + 4 e− −  −− −− −− − − 2 H2 O,




−  2 H+ + 2 e− −  −− −− −− − − H2 .



Eqs. (6.8) and (6.11) are the main reactions of a lead–acid battery, and the other equations are side reactions. Side reactions are not important in the flooded lead–acid batteries unless they are in overcharge mode. However, in valve–regulated, sealed, and gelled lead–acid batteries, side reactions play a significant role in oxygen recombination. Therefore in the flooded lead–acid batteries, only main reactions exist, and all the side reactions are neglected. In studying side reactions, it is customary to neglect hydrogen reactions (i.e., Eqs. (6.10) and (6.13)) because of their poor kinetics.

Lead–acid batteries


According to the main electrochemical reactions of lead–acid batteries, sulfuric acid is produced and consumed in the positive and the negative electrodes during charge and discharge. However, the rate of production and consumption in a positive electrode is much faster than the negative electrode due to its higher stoichiometric coefficient. Because of this fact, it is customary to build some ribs on the separator to provide more electrolyte on the positive side. The rib space is called “Reservoir” as shown in Fig. 6.3. In some cases, ribs are designed in either side of the separator; hence in both sides, we have to consider one reservoir. In the presence of side reactions, a lead–acid cell is a three-phase system consisting of the solid matrix, the liquid electrolyte, and a gas phase. During charge and overcharge, oxygen is generated at the PbO2 /electrolyte interface and evolves into the gas phase after exceeding its solubility limit in the electrolyte. The generated oxygen moves to the negative electrode via the liquid and gas phases. At the negative electrode the oxygen gas reduces at the electrode interface. This process forms an internal oxygen cycle in VRLA cells. Governing equations of lead–acid batteries are deduced from general governing equations explained in Chapter 3. In this chapter, physical characteristics of lead–acid batteries are applied to the general governing equations, and a specific set of equations is obtained suitable only for lead–acid batteries. The obtained system of equations is defined in vector form that can be used in one-, two-, or three-dimensional modelings. In the next section, we derive a simplified system of equations for the one-dimensional case, since one-dimensional modeling is very common and is used in many cases.

6.5.1 Conservation of mass and momentum Electrolyte movement occurs in a lead–acid cell mostly due to the interaction of gravity and concentration gradient in a form of natural convection. Since electrolyte is confined in porous media, usually, the electrolyte movement can be neglected. However, in the rib space or reservoir region the electrolyte movement cannot be neglected; in fact, it has a very strong role in producing stratification. Electrolyte movement is modeled using the Navier–Stokes equations in the form of porous media as is shown in the following equation:   μ ∂ρv + v · ∇(ρv) = −∇ p + ∇ · (μ∇v) + g 1 + β(c H − c◦H ) + (εv). (6.14) ∂t K


Simulation of Battery Systems

Eq. (6.14) is the famous Navier–Stokes equation with two additional terms on its right-hand side. These terms are added to the original equations to include the effect of porosity. The third term adds the effect of buoyancy force induced due to the concentration difference from a reference state c◦ . The term clearly shows that in places where electrolyte concentration is greater than c◦ , it moves toward gravity, and where it is less than c◦ , it moves upward. The last term in Eq. (6.14) indicates the exerted drag force due to the porous media on electrolyte. This term, also known as the Darcy term, is proportional to the velocity v and porosity ε. The parameter μ is the viscosity of the electrolyte, and K indicates the permeability coefficient and is calculated from the Kozeny–Carman relation K=

ε 3 d2 , 180(1 − ε)2


where d is the size of solid particles that are making the porous medium. Example 6.1. Calculate the permeability of a porous electrode with ε  0.5 and particle size d  0.1 µm. Answer. From the Kozney–Carman relation or Eq. (6.15) we have K=

0.53 × (10−7 )2 ε 3 d2 =  3 × 10−17 . 180(1 − ε)2 180(1 − 0.5)2

The value indicates that the porosity of a porous electrode is such that it exerts a lot of force on electrolyte, and we can assume that the electrolyte is nearly stagnant inside the electrodes. The example shows that stratification only happens in separator rib regions or inside the separators with larger porosity and particle size. In VRLA batteries the whole electrolyte is either confined in porous media (electrodes and separator) or immobilized using a gelled electrolyte. In either case, either the electrolyte is stagnant, or its velocity is very low. In this paper the movement of the electrolyte is neglected. Hence the momentum and continuity equations are not considered. This means that the convective mass transfer is neglected because of very small velocity field of the electrolyte.


Lead–acid batteries

6.5.2 Conservation of energy Neglecting the electrolyte movement, the general heat balance over a representative element results in ρ Cp

∂T = ∇ · λ∇ T + q, ∂t


where ρ and Cp are the density and specific heat capacity, respectively, T is the temperature, t is time, λ is the thermal diffusion coefficient, and q represents the total heat source. Eq. (6.16) is defined in spatial form by which the temperature distribution inside a battery can be obtained. To do this, the heat source should also be expressed in distributed form. This term will be discussed in more detail in the following section.

6.5.3 Conservation of charge Conservation of charge in solid and liquid phases is represented according to the following relations: • Conservation of charge in solid: ∇ · (σ eff ∇φs ) = Amain Jmain + AO2 JO2 + AH2 JH2 .


• Conservation of charge in liquid: H ∇ · (keff ∇φe ) + ∇ · (keff D ∇(ln c )) = −(Amain Jmain + AO2 JO2 + AH2 JH2 ),

(6.18) where Jmain , JO2 , and JH2 are exchange current densities from the solid matrix toward electrolyte associated with main, oxygen, and hydrogen reactions, respectively. These exchange current densities can be obtained from the well-known Butler–Volmer equation and are represented for different reactions as follows [38]:  main Jmain = i◦, ref

 O2 JO2 = i◦, ref

cH H cref cH

H cref

γ main   main    αa F main αcmain F main η η exp − exp − ,


γ O2   O2  αa F O2 η exp −



ceO2 2 ceO,ref



⎤   O 2 α F O2 ⎦ exp − c η ,




Simulation of Battery Systems

 H2 JH2 = −i◦, ref

cH H cref

γ H2

 exp −

αcH2 F






In these equations, ηmain = φs − φe − U main , ηO2 = φs − φe − U O2 , and = φs − φe − U H2 represent the overpotentials that act as the driven force for driving each reaction. It should be noted that the coefficients and open-circuit potentials U are different at positive and negative electrodes. Eq. (6.21) is simplified regarding the fact that the anodic reaction related to recombination of hydrogen ions is extremely poor and with good accuracy can be neglected.


6.5.4 Conservation of species Other conservation equations are obtained using the mass balance for hydrogen ions in the electrolyte phase, the dissolved oxygen gas in the electrolyte phase, and the oxygen gas in the gas phase. These equations are well known as the conservation of species equations: H H ∂(εe c H ) = ∇ · Deff ∇ c + a1 Amain Jmain + a2 (AO2 JO2 + AH2 JH2 ), ∂t   ∂(εe ceO2 ) O2 2 = ∇ · DeO,eff ∇ ceO2 + a3 Amain Jmain − Jeg , ∂t   ∂(εg cgO2 ) O2 2 = ∇ · DgO,eff ∇ cgO2 + Jeg . ∂t

(6.22) (6.23) (6.24)

In these equations, a1 , a2 , and a3 are the coefficients related to each species at each electrode, and JegO2 represents the evaporation rate of oxygen at the electrolyte/gas interface. It should be noted that in the separator region, Eqs. (6.22), (6.23), and (6.24) can be used knowing that Jmain , JO2 , and JH2 are zero. However, since JegO2 is not related to the electrochemical reactions and just represents the thermophysical equilibrium at the electrolyte/gas interface, it is not zero in the separator region and acts as a source term in Eqs. (6.23) and (6.24). A good correlation for modeling the interfacial evaporation rate of dissolved oxygen from liquid phase to the gas phase can be presented as JegO2 = k(ceO2 − H  cgO2 ).


In this equation, k is the interfacial mass transfer coefficient related to dissolved oxygen referred to electrolyte side, which can be obtained using the concept of diffusion length proposed by Wang et al. [29], and H  is


Lead–acid batteries

the Henry constant [38]. In the equilibrium condition, H  cgO2 , which represents the interfacial concentration of dissolved oxygen, is equal to ceO2 , which is the average concentration of oxygen in the gas phase. Every reduction or enhancement of oxygen in either liquid or gas phase causes a net mass transfer at the interface.

6.5.5 Conservation of mass To account the porosity change of the electrodes, conservation of mass for solid phase can be used. This balance results in ∂εs = a4 Amain Jmain ∂t


∂εe = a5 Amain Jmain + a6 AO2 JO2 + a7 AH2 JH2 ∂t


for solid electrodes and

for electrolyte. Again in Eqs. (6.26) and (6.27) the coefficients a4 to a7 are constant values but different at different electrodes [38]. The porosity of gas the phase εg is a function of solid and liquid porosity satisfying the relation εe + εs + εg = 1


The effective or specific area of each reaction is approximated using simple correlations: • During discharge for main reaction in both electrodes, Amain = Amain,max × SoC ξ .


• During charge for main reactions in both electrodes,

Amain = Amain,max × (1 − SoC ξ ).


• For secondary reactions during both charge and discharge processes,

Aj = Aj,max × SoC ξ ,


where the index j refers to either O2 or H2 . In Eqs. (6.29) to (6.31), SoC is the local state of charge of a microscopic volume inside the electrodes, and its value is limited to one when the whole active material in that volume is


Simulation of Battery Systems

Figure 6.4 Typical thermal behavior of a lead–acid battery [54]. (A) Thermal Rise. (B) Thermal Runaway.

fully charged. The rate of SoC at each point can be explained as follows: ∂ SoC Amain Jmain =± . ∂t Qmax


In this equation, Qmax is the maximum theoretical capacity of electrodes, and the positive and negative signs correspond to PbO2 and Pb electrodes, respectively.

6.6 Thermal runaway problem Thermal behavior of lead–acid batteries is of great interest. When a battery is charged (usually under a float charge at constant voltage), its temperature rises due to the internal chemical and electrochemical reactions and Joule heating. When the generated heat is balanced by heat dissipation to its surrounding, the temperature rise stops at a moderate temperature as shown in Fig. 6.4A. This is the normal behavior of a VRLA battery, and this fact is called temperature rise (TR). However, at some floating voltages the internal heat generation exceeds the heat dissipation, and as a consequence, the temperature of the battery increases dramatically and out of control. Consequently, the temperature may exceed 60◦ C as illustrated in Fig. 6.4B [51–53]. At this stage and under some critical conditions (e.g., electrolyte saturation) the battery could go into a unstable state, which triggers the spontaneous rise in cell temperature and current. As a result, the battery undergoes a nonstationary and self-accelerating state at which the temperature of the battery rises out of control. This phenomenon is known as thermal-runaway or TRA. TRA is usually considered to be the result of positive feedback of current (chemical and electrochemical reactions) and temperature when a cell

Lead–acid batteries


Figure 6.5 Different mechanisms of oxygen reduction in negative electrode [54]. (A) Pure electrochemical reaction. (B) Electrochemical and chemical reactions.

is under float charge at a constant potential. The initial float current flowing through the cell causes the cell temperature to increase, which in turn causes an increase in current, which further increases the temperature until both current and temperature reach high values. This phenomenon continues until the self-accelerating conditions are interrupted. Then the cell temperature and current decrease, and the battery reaches a new stationary state. To understand the cause of TR or TRA in a valve-regulated lead–acid batteries, it would be essential to study the involved physical phenomena. During the charge process [54], at the states of charge (SoC) over 0.6, water dissociated at the positive electrode through a successive rather complex mechanism on which no common mechanism exists. However, we can consider the overall reaction with its initial and end products: H2 O −−→ 2 H+ + 2 e− + 12 O2 ,

H = +285.8 kJ mol−1 . (6.33)

The generated oxygen moves toward the negative electrode and reduces by a set of chemical and electrochemical reactions for which two different accepted mechanisms exist as illustrated in Fig. 6.5. The first mechanism shown in Fig. 6.5A states that the oxygen reduces through the following pure electrochemical reaction: 1 2 O2

+ 2 H+ + 2 e− −−→ H2 O.


The second accepted mechanism shown in Fig. 6.5B states that at the negative electrode, oxygen reduces by the following chemical and elec-


Simulation of Battery Systems

Figure 6.6 Self accelerating heat generation mechanism [54].

trochemical mechanism: Pb + 12 O2 −−→ PbO, PbO + H2 SO4 −−→ PbSO4 + H2 O,

H = −219.08 kJ mol−1 , −1

H = −172.71 kJ mol .

(6.35) (6.36)

The negative sign in enthalpy change indicates that these are exothermic reactions and increase the battery temperature. It is worth noting that when the evolved oxygen flows from the positive to the negative electrode through the separator micropores, it pushes the electrolyte out causing the separator dry-out. During the TRA, the separator dry–out enhances the oxygen flow toward the negative electrode. This means that more reaction occurs at the negative electrode, and hence more heat is released. Moreover, separator dry-out means increasing the internal resistance, which in turn means increasing the Joule heat. The first step on studying TRA is determining the origin of heat sources in a battery cell. A review of the literature reveals that there is no unanimous idea about the heat generation inside the battery. Some [51–53, 55,54,56] believe that heat generation inside the battery is due to the oxygen reduction reactions at the negative electrode (according to reactions (6.34), (6.35), and (6.36)). Besides that, Joule heating increases the battery temperature. The increase in battery temperature accelerates the rate of electrode reactions, which in turn increases the battery temperature more and more. This phenomenon constitutes a self-accelerating mechanism seen in Fig. 6.6. This viewpoint is accepted by most researchers, and let us call it the first paradigm. Unlike the first paradigm, the second paradigm [57,58] states that the closed oxygen cycle cannot produce any heat. Since it is a closed system and according to Hess’ law, the net generated heat during a closed cycle is

Lead–acid batteries


Figure 6.7 Born–Haber cycle illustrating the essential parts of the closed oxygen cycle [57].

zero regardless of the intermediate steps. For example, drawing the Born– Haber diagram for the second mechanism of oxygen reduction, as shown in Fig. 6.7, we can conclude that the net generated heat is zero. Accepting this statement, the only mechanism remained in a battery is the Joule heating. Therefore this paradigm states that the only mechanism that can contribute to the temperature rise is the Joule heating. To validate the second paradigm, Catherino [57] has made some tests on TRA and showed that in some cases the first paradigm cannot explain the cause of TRA, but the second paradigm is capable of explaining TRA in all the situations. This is the conflict point between the first and second paradigms. The second paradigm states that the enthalpy of the reaction cannot produce any heat, because there is a cycle in the battery and water is dissociated at the positive electrode and recombined at the negative electrode. Therefore, the only mechanism which is responsible for temperature rise is Joule heating. Torabi and Esfahanian [32] studied the mentioned conflict and showed that by dividing the generated heat into reversible and irreversible parts a new concept called the generalized Joule heating can be defined that is the sum of classical Joule heating and the irreversible part of electrochemical reaction. Then they showed that generalized Joule heating is the only mechanism responsible for thermal rise in lead–acid batteries. Details of the study are given in Section 6.10.

6.7 Heat sources and sinks The heat source q in Eq. (6.16) is the sum of heat generation and dissipation. The generation is due to the electrochemical reactions and Joule heating.

6.7.1 Heat of reactions The heat generation inside the battery is due to the chemical and electrochemical reactions. As it was discussed before, each reaction (Eqs. (6.8)


Simulation of Battery Systems

to (6.13)) produces a specific amount of heat, which can be divided into reversible and irreversible parts: qrev = Jmain T

dU main dU O2 dU H2 + JO2 T + JH2 T dT dT dT


and qirrev = −Jmain U main − JO2 U O2 − JH2 U H2 ,


where J and U are the current density and local cell voltage, respectively. In calculating the irreversible heat, it should be noted that the following thermodynamic relationship between the open circuit voltage U and the entropy change s for each reaction can be used: s = nF

dU dT


On the other hand, Lampinen and Fomino [34] showed that to calculate the entropy change of a half-cell reaction, we can use an absolute scale for noncharged species together with a semiabsolute scale for charged species considering the entropy of an electron. They used this semiscale to calculate the entropy of an electron: 1 s[e− (Pt)] = sa [H2 (g)] = 65.29 J mol−1 K−1 , 2


where s and sa are semiabsolute and absolute entropies, respectively. For example, consider the half main reaction at the positive electrode (for a charge process): charge

PbSO4 (s) + 2 H2 O −−−→ PbO2 (s) + HSO4− + 3 H+ + 2 e− .


The entropy change is + − s =sa [PbO2(s) ] + s[HSO− 4 ] + 3s[H ] + 2s[e ]

− sa [PbSO4(s) ] − 2sa [H2 O].


In Table 6.2 the entropy of all the necessary charged and noncharged species are given.

Lead–acid batteries


6.7.2 Joule heating Besides the heat generation due to the chemical and electrochemical reactions, Joule heating is also a source of heat generation in the lead–acid batteries. The resistance of the battery can be obtained in the same manner as discussed in Section 4.2.2. From the concept of Joule heating defined in that section we can calculate the Joule heating part of heat generation as qJoule = |φ · ik |.


The net heat generation is the sum of reversible, irreversible, and the Joule heats: qgen = qrev + qirrev + qJoule .


6.7.3 Heat dissipation In addition to heat generation in an electrochemical cell, heat dissipation plays a vital role in its thermal behavior. There are different heat dissipation mechanisms in a battery cell: 1. Heat dissipation to surrounding by convection. 2. Radiation to ambient. 3. Dissipated heat by exhausting gas. The convected heat to the surrounding from the battery outer walls is compensated by heat conduction through battery walls as well as electrical connections and supporting mechanical structure. Therefore the resistance model developed in Section 4.2.2 can be used to account for all these mechanisms. The total heat dissipation can be calculated as qdiss. = qconv. + qrad. + qexhaust .


In the case of lead–acid batteries the radiation can be neglected since the cell temperature is not very high. Moreover, it is assumed that the present battery model is sealed and there is no exhausting gas. Hence we also omit the dissipated heat by exhausting gas. Consequently, only the convective heat transfer to the ambient is simulated. When the battery case temperature exceeds the ambient temperature, the convection starts to dissipate the thermal energy. The amount of this dissipation can be obtained from the formula qconv. = hA(Tcase − Tambient ),



Simulation of Battery Systems

Figure 6.8 Illustration of thermal resistances in the modeled battery.

where h is the convection coefficient, A is the area of the case, and Tcase and Tambient are the case and ambient temperatures, respectively. This equation shows that the convective heat dissipation is proportional to the temperature difference between the battery case and the ambient. The temperature of the case surface usually differs from the bulk temperature and can be found by solving the energy equation inside the battery case. Moreover, the Navier–Stokes equations should also be solved inside the headspace, since this space is filled with gaseous material. However, in most situations the captured air at the headspace is almost stagnant, and the thickness of the battery case is small. Therefore we can use the one-dimensional thermal convective and conductive resistance models illustrated by Fig. 4.2 for headspace and battery cases, respectively. The values of these resistances illustrated in Fig. 6.8 can be obtained from the equations Rconv. =

1 , h∞ d

Rcond. = , λ

(6.47) (6.48)

where h∞ is the convection heat coefficient of the gas, d is the thickness of the case, and λ is the conduction heat coefficient of the battery case. As can be seen from Fig. 6.8, the middle cells have less dissipation, which results in higher temperatures. Therefore the assumption of a bulk temperature for the whole battery is not valid in all cases. Moreover, it should be noted that heat can also be transferred from one cell to its neigh-


Lead–acid batteries

bor cell. To calculate the amount of this heat transfer, we can use the same resistance methodology described before. The amount of thermal resistance can also be calculated from Eq. (6.48).

6.8 One-dimensional model The electrodes of a lead–acid battery usually have a very high aspect ration. This means that the width of the plates are of order of 1 mm, whereas their other dimensions are about 10 cm. In other words, the aspect ratios of the plates are of order of 100. As a consequence, the assumption of onedimensional model usually fits well, and we may apply a one-dimensional model for simulations. The literature survey shows that in most cases, this assumption has proven to be accurate enough for many purposes including optimization loops and design. The assumption of one-dimensional model greatly simplifies the governing equation. For instance, in a one-dimensional model, the fluid flow is dropped out of the equations. This fact means that the natural convection and stratification phenomena cannot be captured by a one-dimensional model.

6.8.1 Governing equations for one-dimensional model Applying the previous assumption results in the following system of governing equations: 1. Conservation of charge in solid electrodes:    ∂ ∂φs (Aj) = 0. σ eff − ∂x ∂x All



2. Conservation of charge in electrolyte:      ∂ ∂ eff ∂φe eff ∂ ln c k kD (Aj) = 0. + + ∂x ∂x ∂x ∂x All



3. Conservation of chemical species for electrolyte concentration H+ :  H ∂(εe c H ) AjH 1 − t+◦  ∂ H ∂c = Deff + (Aj). + a2 ∂t ∂x ∂x 2F F Side reactions



Simulation of Battery Systems

4. Conservation of chemical species for dissolved oxygen in concentraO tion ce 2 :  O2  ∂ ∂(εe ceO2 ) 1 O2 ∂ ce O2 = De,eff (Aj)O2 − Jeg . + ∂t ∂x ∂x 4F

(6.52) O

5. Conservation of chemical species for dissolved oxygen in gas phase cg 2 : ∂(εg cgO2 ) ∂t

 O2 ∂ O2 ∂ cg O2 + Jeg = Dg,eff . ∂x ∂x


6. Energy balance: ∂ρ Cp T ∂φs ∂φe 1 = − is − ie + ∂t ∂x ∂x Vc



A η−T

∂U ∂T



hAc (T − T∞ ) . Vc


These equations are the main fundamental governing equations that depict a one-dimensional model of a battery cell. The solution of the system by numerical techniques will simulate the battery subjected to the fact that proper boundary conditions are provided.

6.8.2 Boundary conditions The solution of any system of differential equations strongly depends on initial and boundary conditions. Here we discuss the appropriate boundary conditions for a one–dimensional model. It should be noted that Eqs. (6.49) and (6.50) are steady state and require only boundary conditions; however, Eqs. (6.51) to (6.54) are transient, and initial conditions should be defined in addition to boundary conditions.

Potential in solid and electrolyte Although Eqs. (6.49) and (6.50) are steady state and do not require any initial conditions, a proper initial guess is strongly recommended for potential distribution both in solid and electrolyte phases. The reason is that with a badly defined initial guess, source terms in all the equations may become very large, which in turn results in unstable numerical calculations. Therefore we need to have good initial guesses for these values.

Lead–acid batteries


There are two main methods for defining proper initial guesses for solid and electrolyte potentials φs and φe : a) neglecting other equations and solving only Eqs. (6.49) and (6.50) with a specific constant value for each species or b) simulating the whole system of equations with uniform concentration for all species with a very small time step (e.g., 10−8 s). These methods will give a proper initial guess for potential fields that will be used for simulation. Note that these are referred to as initial guesses instead of initial values since a steady-state differential equation does not require initial values. For boundary conditions, we need to define proper values at the center of the positive electrode, x = 0, and at the center of the negative electrode, x = l. For φl , the proper values for both sides are ∂φl = 0. ∂x


This condition indicates that the current in the electrolyte phase is zero at both boundaries due to the symmetry of the domain. In other words, at the boundaries, the whole current is in the solid phase, and no current is in the electrolyte phase. The boundary conditions defined by Eq. (6.55) cause some illness in the system of governing equations. Basically, from the mathematical point of view, an elliptic differential (ordinary or partial) equation with Newman boundary condition on all its boundaries is an ill-posed equation. Eqs. (6.49) and (6.50) are mathematically elliptic and hence defining all the boundaries of Newman type (i.e., Eq. (6.55)) results in an ill-posed equation and will have infinitely many solutions, where only one of which is the correct answer. To find the correct answer, we need to impose another constraining equation together with the original equations. This constraining equation is called the compatibility equation and will be discussed in more detail in this section. Compatibility equation is a mathematical description of electroneutrality, which means that no charge is produced or consumed in electrochemical reactions. For solid potential, we can define different boundary values depending on operational conditions. For a voltage defined operation, we have: φs = V (t), φs = 0,

x = 0, x = l.

(6.56) (6.57)

Eq. (6.57) indicates that the zero level of potential is the potential of the current collector of the negative electrode. On the other hand, Eq. (6.56)


Simulation of Battery Systems

shows that the potential of the current collector of the positive electrode can be changed with time. This boundary condition enables us to define any defined voltage discharging or charging operation. If the operation condition is defined with a defined current function, we have to impose the following boundary conditions: ∓σ eff

∂φs = I (t). ∂x


The left-hand side of Eq. (6.58) is the current in the solid phase that enters the current collector, and the right-hand side is the applied operation value. The positive and negative signs also indicate the current at the negative and positive current collectors, respectively. It is also clear that in predefined current density the equation of potential in solid phase again becomes ill-posed. As discussed before, we need to impose the compatibility equation to overcome the ill-posed condition.

Chemical species When the battery is on rest and the electrochemical cell is not charging or discharging, the concentration of any chemical species becomes uniform across the cell because of mass diffusion. Consequently, the proper initial condition for chemical species (i.e., Eqs. (6.51) to (6.54)) can be defined as follows: c H = c◦H , O2

c O2 = c◦ , H2

c H2 = c◦ ,

(6.59) (6.60) (6.61)

where the zero subscript indicates a uniform value. Symmetric boundary conditions properly define physical boundary conditions for these equations: ∂ cH = 0, ∂x ∂ c O2 = 0, ∂x ∂ c H2 = 0. ∂x

(6.62) (6.63) (6.64)

Lead–acid batteries


Cell temperature Since the energy equation is transient, we need an initial condition and proper boundary conditions. For a battery staying in a uniform temperature, the proper initial condition is T = T◦ .


Symmetry boundary conditions are proper physical choices because the domain of solution has symmetry on both sides. In other words, ∂T = 0. ∂x


Note that since transient equations Eqs. (6.51) to (6.54) are parabolic rather than elliptic, defining Newman boundary condition on all the boundaries will cause no illness. Therefore, Eqs. (6.62)–(6.64) and (6.66) do not impose illness to the system, and the system can be solved properly.

6.9 Physico-chemical properties Physico-chemical properties of lead–acid batteries are temperature dependent. As stated before, the dependency of the parameters are described by Arrhenius’ equations or Eq. (6.85). Besides temperature, some properties are functions of electrolyte concentration as well. In this section, properties of lead–acid batteries are summarized.

6.9.1 Electrode electrical conductivity σ The electrical conductivity of electrodes is defined in units of S cm−1 . The electrical conductivity of electrodes varies during charge and discharge due to the conversion of lead and lead-dioxide into lead-sulfate and also porosity change. Typical values for pure materials of the electrode are: σ = 500 σ = 4.8 × 10


for PbO2 ,


for Pb.


For porous electrodes, these value should be corrected according to porosity. The relation is given by Brugmann [29]: σ eff = σ ε 1.5 .



Simulation of Battery Systems

The exponent 1.5 is normally used in literature, but in general, it is a case of study. In all the governing equations proposed in this chapter the effective conductivity is used instead of pure conductivity of materials.

6.9.2 Electrolyte ionic conductivity k The ionic conductivity of sulfuric acid is a function of concentration and temperature. Newman and Tiedeman [59] expressed the value of ionic conductivity as an experimental relation in units of cm2 s−1 : 

3916.95 7.2186 × 105 − k = c exp 1.1104 + T T2    4 9.9406 × 10 2 c − 16097.781 c . + 199.475 − T


6.9.3 Diffusion coefficients The diffusion coefficient of sulfuric acid is a function of acid concentration, where its value at T = 25◦ C is [60] DH = (1.75 + 260 c ) × 10−5 .


To include the effect of temperature on the diffusion coefficient, Newman and Tiedeman [59] gave the following relation: DH = (1.75 + 260 c ) × 10−5 exp

2174.0 2174.0 − . 298.15 T


It is quite evident that Eq. (6.72) gives a relation for sulfuric acid binary diffusion in a nonporous medium. In a porous medium the value should be corrected according to the Brugmann relation, that is, Eq. (6.84). The diffusion coefficient of dissolved oxygen in electrolyte is [61] O

De 2 = 0.8 × 10−5 .


The activation energy for this property is equal to EO2 = 14000 J mol−1 .


Therefore, the diffusion coefficient for dissolved oxygen is O

De 2 = 0.8 × 10−5 exp

14000 1 1 − R 298.15 T



Lead–acid batteries


6.9.4 Open-circuit voltage U Open-circuit voltage of a lead–acid battery is not unique since in general potential does not have an absolute value. Therefore we have to define a reference state for potential measurement. Normally, the potential of the negative electrode is chosen as the reference state, and any other voltage is measured with this scale. For lead–acid batteries, Bode [62] gave the following relation for calculation of open-circuit voltage at T = 25◦ C: UPbO2 = 1.9228 + 0.147519 log m + 0.063552 log2 m + 0.073772 log3 m + 0.033612 log4 m,


where m is the molality of the electrolyte at T = 25◦ C. The molality and concentration of sulfuric acid is also given by Bode in the same reference: m = 1.00322 × 103 c + 3.55 × 104 c 2 + 2.17 × 106 c 3 + 2.06 × 108 c 4 . (6.77) By selecting the negative electrode as the reference point, the opencircuit potentials of side reactions are: UH2 = 0.356,


UO2 = 1.649.


These values can be modified by the Arrhenius relation (6.85) to include the effect of temperature. The activation energy for these values in kJ mol−1 are: EactH2 = 54,


EactO2 = 70.


6.9.5 Partial molar volumes of sulfuric acid and water Partial molar volumes Ve and V◦ of sulfuric acid and water, respectively, are given by Bode [62]. A part of these values that are functions of acid concentration are given in Table 6.1. To have a better understanding of these values, the data are plotted in Figs. 6.9 and 6.10. For numerical simulation, it is better to give a relation between partial molar volume values, instead of tabulated data. Using a curve fitting, we have


Simulation of Battery Systems

Table 6.1 Partial molar volumes of sulfuric acid and water as functions of acid molality. Partial molars Density Molality Ve V◦ ρ m cm3 mol−1 cm3 mol−1 kg l−1 l3 kg−1 35.10 18.07 0.9970 − 39.44 18.03 1.0300 0.526 39.49 18.01 1.0640 1.133 17.94 1.1365 2.549 40.85 43.45 17.76 1.2150 4.370 17.57 1.2991 6.797 45.29 45.90 17.47 1.3911 10.196

Figure 6.9 Partial molar volume of sulfuric acid.

Figure 6.10 Partial molar volume of water.

⎧ 3 −1 ⎪ ⎨ (0.02538165 − 5.4367785e − 5 m ) Ve = 34.877248 + 2.962056 m ⎪ ⎩ −0.262232 m2 + 0.007625 m3

m <= 2.549, (6.82) m > 2.549.

Lead–acid batteries

Table 6.2 Thermodynamic properties of different species. species G◦ cP◦ S◦ − 1 − 1 − 1 − 1 (kJ mol K ) (kJ mol−1 ) (kJ mol K )


H ◦ (kJ mol−1 )

H2 H+ O2 H2 O(g) H2 O(l)

6.889 0 7.016 8.025 17.995

31.208 0 49.003 45.107 16.710

0 0 0 −54.634 −56.687

0 0 0 −57.796 −68.315

OH − SO41− HSO4− H2 SO4 Pb

−35.5 −70

−2.57 4.4 28.45 37.501 15.49

−37.594 −177.83 −180.55 −164.936

−54.970 −217.32 −212.08 −194.548



2.5 15.9 35.509 22.1 18.26

−5.83 −45.16 −194.36 −51.94 −52.34

0.4 −52.34 −219.82 −63.52 −66.12

Pb 2+ PbO PbSO4 α − PbO2 β − PbO2

33.20 6.32 10.95 24.667 −


⎧ 2 ⎪ ⎨ 18.042475 − 0.019430 m − 0.0081488 m V◦ = 18.210753 − 0.105352 m ⎪ ⎩ −0.0015226 m2 + 0.000464 m3

m <= 2.549, m > 2.549. (6.83)

The tabulated data and the curve fittings are plotted in Figs. 6.9 and 6.10. The figures show that the curve fittings have good accuracy for calculations.

6.9.6 Thermodynamic properties of different species In many calculations, thermodynamic properties of different species are required. The necessary values for different species that are involved in lead–acid battery reactions can be found in different thermodynamics handbooks. Table 6.2 summarizes sum of the values.

6.9.7 Calculation of properties in porous medium In all the above equations, the physicochemical properties like the diffusion coefficient and ionic conductivity of electrolyte should be modified because of the porosity of the media.


Simulation of Battery Systems

Figure 6.11 Geometry of a lead–acid battery.

In the case of porous media, all the properties are modified such that the effect of porosity comes into account. For any property such as  , the effective value and its pure value are related to each other according to the Brugmann equation  eff = εjξ .


In this relation, j indicates phase j, and ξ is an exponent that shows the tortuosity of the porous medium. In battery systems, ξ = 1.5 is usually chosen for all the simulations.

6.9.8 Temperature dependency of parameters The temperature dependency of physicochemical properties is obtained using the Arrhenius equation [30] 

Eact,φ  = ref exp R

1 1 − Tref T



In this equation, ref is the physicochemical property at the reference temperature (including the effect of tortuosity), and Eact,φ is the activation energy of the evolution process of . The activation energy is a measure of the sensitivity of the parameter to temperature. The higher the value, the more the parameter is sensitive to temperature.

6.10 Numerical simulation of lead–acid batteries Just like other battery systems, a lead–acid cell is shown in Fig. 6.11, where the domain of solution consists of a current collector at the center x = 0

Lead–acid batteries


of the positive electrode, which is a porous medium made of lead dioxide PbO2 , an electrolyte region near the positive electrode, a porous separator, the negative electrode that is made of pure, and finally the negative current collector at the center x = l of the negative electrode. The whole system is filled with sulfuric acid, which reacts with electrodes to produce lead sulfate. The electrolyte region attached to the positive electrode is an extra space to provide more electrolyte for the positive electrode since the stoichiometric of the reaction of the positive electrode requires three times more electrolyte than the negative electrode. This extra space is created by attaching some ribs on the separators. In some separators the rib is located at both sides of the separator, and hence the electrolyte space should be also considered at the negative electrode side. Although the governing equations of lead–acid batteries are expressed in vector form and are valid in three dimensions, one- and two-dimensional simulations are more common in practice. This is because one- and twodimensional assumptions are accurate enough for many purposes. In some cases, one-dimensional simulation is even more than enough because many distributed parameters are not required for those applications. Consequently, depending on the application, we can choose the desired dimension and start solving the equations with proper physical assumptions. For example, for simulation of acid stratification, a two-dimensional model is appropriate, whereas for monitoring purposes, in many applications or for optimization and design purposes, a one-dimensional model is sufficient. In this chapter the simulation of governing equations is discussed in one and two dimensions separately, and the main issues and assumptions are discussed in more detail. Since three-dimensional simulation is rarely required, it is not considered in this chapter, but the same procedure discussed here can be applied for three-dimensional simulations.

6.10.1 One-dimensional simulation without side reactions In some batteries, except in overcharge situations, side reactions are not involved in discharge or even in charge. Hence in these situations the battery can be simulated without side reactions, that is, by dropping Eqs. (6.52) and (6.53) from the system of equations. The resulting system of nonlinear equations can be solved using different numerical techniques including the finite volume method (FVM), the finite difference method (FDM), the finite element method (FEM), or any other known methods. Each method has advantages and drawbacks. Although FVM is a popular method amongst battery simulation teams, Esfahanian


Simulation of Battery Systems

and Torabi [49] showed that the Keller–Box method is a good choice for these types of numerical computations. To show the ability of the method, here the system of equations is solved using the Keller–Box method. In the next section, the FVM method is used for simulation of battery behavior when side reactions are present. The main advantages of Keller–Box method can be summarized as follows: 1. In the Keller–Box method, both function and its derivative are presented in the final system of equations. Hence in battery systems, where both function values and their derivatives are present at all boundaries, the Keller–Box method is a good choice. 2. The Keller–Box method provides second-order accuracy both in time and space, whereas the normally used numerical methods provide the first-order accuracy in time. 3. The resulting coefficient matrix in the Keller–Box method is always tridiagonal, which means that the Thomas algorithm can be used for its solution. This advantage gives a fast algorithm for numerical integration. 4. Mixed boundary conditions such as the simultaneous implementation of potential and current at boundaries are possible. 5. Since in the Keller–Box method, each point depends on its previous numerical point, a nonuniform grid can be easily applied to the numerical domain without any difficulty. Other characteristics of the Keller–Box method can be found in CFD books such as [63,64] and is vastly used in boundary layer theories [65]. The method and its implementation to system of nonlinear partial differential equations (PDEs) are discussed in Appendix F. By neglecting side reactions in a constant temperature situation, the system of governing equations is limited to Eqs. (6.49)–(6.51). By solving the system we are looking for the values of the solid potential φs , the potential distribution in electrolyte phase φe , and the electrolyte concentration distribution c H . In Keller–Box method, we have to convert the second–order PDEs to a system of first-order PDEs. To do that, we introduce the following assumptions: ∂φs = φsx , ∂x ∂φe = φex , ∂x

(6.86) (6.87)

Lead–acid batteries

∂ cH = cx . ∂x



Substituting Eqs. (6.86) to (6.88) into Eqs. (6.49) to (6.51), we get:  ∂ eff σ φsx − (Aj) = 0, ∂x All


   ∂ eff ∂ ln c ∂ k φex + keff (Aj) = 0, + D ∂x ∂x ∂x All


∂ H ∂(εe c H ) AjH 1 − t+◦  = Deff cx + a2 + (Aj). ∂t ∂x 2F F Side





Eqs. (6.86)–(6.91) constitute a system of first–order nonlinear PDEs with six unknowns φs , φsx , φe , φex , c H , and cx . The nonlinear source terms in these equations should be linearized so that the resulting numerical matrix becomes stable. The details can be found in Appendix F. Cell-II from Appendix A is chosen for simulation. The parameters of the cell are tabulated in Table A.3 and Table A.4; the other required lead– acid characteristics are taken from Section 6.5. The results of the simulation are shown in Fig. 6.12. Fig. 6.12A shows the cell voltage versus time, where the simulation results are compared with the results of other researchers. It is clear from the figure that the present simulation has a good agreement with the results of others, and also it is obvious that the cell reaches the cut-off voltage at t = 105 s. This indicates that the chosen cell is a starter battery. The variation of electrolyte concentration in times 60 and 105 s is shown in Fig. 6.12B. As we can see, at t = 105 s, where the battery reaches the cutoff voltage, electrolyte concentration reaches zero at the positive electrode. Considering the amount of active materials at both electrodes as shown in Fig. 6.12C, it can be definitely concluded that electrolyte is the limiter of the cell voltage. Although we still have plenty amount of active material in both electrodes, the lack of acid concentration causes the cell voltage to drop. Also from Fig. 6.12B it is clear that the net electrolyte concentration in all the cell regions is not zero since there is still enough acid concentration at the negative electrode. We can conclude that if we put the cell in rest for a while, then the electrolyte becomes uniform again due to diffusion, and the cell can become discharged again. Fig. 6.12C also indicates that the active material in the positive electrode is consumed more than the negative counterpart. Also, it is evident that in


Simulation of Battery Systems

Figure 6.12 Simulation results for Cell-II. (A) Cell voltage vs time. (B) Electrolyte concentration in different time levels. (C) Variation of active material in different time levels. (D) Electrolyte potential distribution.

both electrodes, about 80% of active materials are still available in both electrodes, indicating that the state of charge of the battery is SoC  0.8. Again it emphasizes that we are dealing with a starter battery in which active materials should not be consumed more than 20–30%. From Figs. 6.12B and 6.12C we can conclude that the design of the battery is not appropriate since while the positive electrode stops delivering power, the negative electrode is still alive and has potential to provide more energy. By a better design the thickness of the negative electrode can be minimized to reduce the battery weight resulting in less cost. The variation of electrolyte potential distribution is shown in Fig. 6.12D, in which it is clearly indicated that the electrolyte potential drops in all re-

Lead–acid batteries


gions resulting in cell potential drop. It is also evident that the electrolyte potential is a little bit higher in the negative electrode in comparison with the positive counter electrode.

6.10.2 One-dimensional simulation including side reactions When side reactions are present, we need to simulate the whole system of Eqs. (6.49)–(6.54). The system can be solved using any numerical methods including the Keller–Box method, as explained in the previous section or any other methods. Choosing the numerical method is quite arbitrary; just we have to keep in mind that the numerical method should not alter the final results. In other words, regardless of the numerical method, the same results should be obtained. In this section, to show the capability of FVM, we used this method to solve the system of governing equations. The details of FVM in one dimension is explained in Appendix D; it is based on the method of Patankar [45]. To verify and validate the numerical scheme, Cell-V from Appendix A is chosen. The cell belongs to a sealed lead–acid battery, in which side reactions become important in the overcharge period. The cell is under constant current charge until side reactions become important, and after that, the charging process continues. The results of simulation are shown in Figs. 6.13 and 6.14. The simulated cell voltage during charge is compared and shown in Fig. 6.13A. It takes about 15 hours for the cell to become fully charged. The difference between the simulation results of different researchers is because they used different values for simulation due to the lack of information about the original test. The experiment carried out by Bernardi [67] shows that during the charging process, the cell voltage rises to a pick value at about t = 8 hr and falls to a specific value about 2.4 V. From then on, the cell voltage reaches a plateau, which is an indicator of a steady state. All the results of the other researchers including present simulation predict the same behavior, but the value differs a little bit from the experimental data. The difference can be translated to different simulation parameters used since the real values are not properly tabulated in the open literature. The cell voltage behavior in charge process is attributed to the side reactions. Without simulation of side reactions, the voltage decrease at t = 8 hr cannot be resolved. Fig. 6.13B illustrates the charging process with and without the simulation of side reactions. We can see that neglecting side reactions causes the cell voltage to rise exponentially when the active


Simulation of Battery Systems

Figure 6.13 Simulation results for Cell-V. (A) Cell voltage during charge. (B) Cell voltage with and without side reactions. (C) Current density of reactions at t = 0 hr. (D) Current density of reactions at t = 8 hr. (E) Current density of reactions at t = 10 hr. (F) Current density of reactions at t = 13.8 hr.

Lead–acid batteries


Figure 6.14 Simulation results for Cell-V. (A) Electrolyte potential distribution. (B) Electrolyte concentration in different time levels. (C) Concentration of oxygen in gas phase. (D) Concentration of oxygen in electrolyte phase. (E) Overpotential distribution. (F) Solid phase potential.


Simulation of Battery Systems

materials are completely charged. If there were not any side reaction in the real case, it would be the real behavior of the system. However, when the cell voltage reaches about 2.4 V, side reactions become dominant and drop the cell voltage as discussed before. The share of different reactions are plotted in Figs. 6.13C to 6.13F for different time levels. It should be noted that the applied current is the only driving source for all side reactions. It means that the integral of current density J over each electrode on all the reactions is equal to the applied current. From the figures we can see that at the beginning of charge process (t = 0 hr), side reactions can be neglected since all the applied current is dedicated to main reactions. From another perspective the cell voltage at the beginning of the charge is about 1.9 V, and as we know, water dissociation is very slow at that voltage. By increasing the cell voltage to 2.3 V, or t  8 hr, the share of the main reaction becomes less, and side reactions start showing up. The process continues until t  10 hr, where most of the current goes to side reactions rather than to the main reaction. Finally, at t = 15 hr, it is clear that the share of the main reaction is almost negligible due to the fact that almost all active materials are charged; hence all the applied current drives the side reactions. It is clear that the oxygen side reaction is almost dominant, and the kinetics of hydrogen side reaction is very poor, meaning that we can neglect the hydrogen side reaction in their simulation. In Fig. 6.14, other simulation parameters are shown. For instance, Fig. 6.14A shows the variation of electrolyte phase potential. As it can be seen, the electrolyte phase potential increases in charge process and reaches a maximum value at about φe  2.21 V, meaning that the cell is not charged anymore. Fig. 6.14B shows the variation of electrolyte concentration at different cell regions and different time levels. Again, the electrolyte concentration reaches a maximum, which is another indicator for a fully charged cell. A very good result is shown in Figs. 6.14C and 6.14D. In these graphs the concentrations of oxygen in gas and electrolyte are respectively plotted. As it is clearly seen, at the beginning of charge process, the oxygen concentration is not considerable; however, by continuing the charging process the oxygen concentration becomes higher and higher due to the fact that the share of side reactions becomes important in benefit of oxygen reaction. This fact can be further investigated if we plot the overpotentials of the main and oxygen reactions as we did in Figs. 6.14E and 6.14F. It is evident that as the charging process continues, the share of the main reaction

Lead–acid batteries


becomes more and more negligible, whereas the share of oxygen reaction becomes more and more dominant. Comparison of Figs. 6.14C and 6.14D from another viewpoint indicates that the oxygen in the gas phase has flatter distribution than that in the electrolyte phase. This is due to the fact that the diffusion coefficient of oxygen in the gas phase has a higher value than that in the electrolyte phase. In other words, oxygen in the gas phase becomes homogeneous much faster than in the electrolyte phase. The results of the figures prove this fact.

6.10.3 Numerical simulation of electrolyte stratification using two-dimensional modeling In previous sections, we studied one-dimensional modeling of lead–acid batteries. Although the one-dimensional model is very accurate and much useful information can be obtained, in some cases, at least two-dimensional modeling should be performed. Simulation of electrolyte stratification is an example of such cases. In this phenomenon, natural convection takes place inside the battery cell due to the electrolyte concentration gradient. Since during charge or discharge, electrolyte concentration takes place (as was discussed in Fig. 6.12B), a more concentrated electrolyte becomes heavier and sinks, whereas a less concentrated electrolyte rises due to gravity force making a natural convection movement. The induced natural convection causes the electrolyte to become stratified, which in turn results in nonuniform usage of electrodes. To numerically capture the phenomenon, the Navier–Stokes equations should be coupled with governing the electrochemical system of equations. In this case, at least a two-dimensional space should be modeled because electrolyte movement has no meaning in one dimension. Electrolyte movement occurs due to the following reasons: 1. In portable devices like cars, the case of the battery moves, and so does the electrolyte. 2. Released gases inside the battery cause the electrolyte to move. 3. As mentioned before, electrolyte concentration is a major source of electrolyte movement. 4. The temperature gradient in a battery may be a driving force for electrolyte movement. Independently of the mechanism responsible for electrolyte movement, the Navier–Stokes equations should be coupled with battery governing equations to simulate the electrolyte movement. In the lead–acid batteries, electrolyte moves in porous media such as electrodes and separators.


Simulation of Battery Systems

Normally, the porosity of the regions exerts a lot of force on electrolyte and renders the acid movement; however, in rib regions of separators the electrolyte has enough space for circulation and natural convection. Since stratification occurs in porous media, the Navier–Stokes equations should be written in the form such that the effect of porosity would be included in the equations. The proper form is given in Eq. (6.14). Electrolyte stratification is studied in discharge process under constant temperature condition. In this case the side reactions are dropped out of the governing system of equations, and the Navier–Stokes equations are added to the system. The simplified system of equations is as follows: ∇ · (σ eff ∇φs ) − Aj = 0, ∇ · (k ∇φl ) + ∇ eff

· (keff D ∇(ln c )) + Aj = 0,

∂(ε c ) Aj + v · ∇ c = ∇ · (Deff ∇ c ) + a2 , ∂t 2F ∂ρv μ + v · ∇(ρv) = −∇ p + v · (μ∇v) + ρ g[1 + β(c − c◦ )] + (εv), ∂t K ∂ρ + ∇ · (ρv) = 0. ∂t

(6.92) (6.93) (6.94) (6.95) (6.96)

The existence of Navier–Stokes equations and the continuity equation require special attention for numerical solution. Patankar [45] was one of the pioneers of FVM and gave a proper algorithm called SIMPLE to solve such systems. The details of the method are given in Appendix E, and more can be found in CFD textbooks such as [45,68]. To show the numerical simulation of acid stratification, we chose CellIV from Appendix A. All the necessary parameters such as geometrical dimensions and electrochemical characteristics are given in the same appendix. Alavyoon et al. [50] was the first who used the cell to investigate the effect of electrolyte stratification. They used the holographic laser interferometry method for measuring electrolyte concentration and the laser Doppler velocimetry (LDV) for measuring the flow field. The cell consists of three regions, namely a positive electrode, a free space for electrolyte, and a negative electrode. The electrodes and the free space have 2-mm thickness, and the charging current is very low, about 9.434 mA cm−3 . Since the charging current is low, the temperature of the cell does not vary too much during the test, and we can assume an isothermal model at T = 25◦ C. Alavyoon et al. [50] proposed a system of equations for simulation of electrolyte stratification, in which instead of solving the full Navier–Stokes

Lead–acid batteries


equations, they used a creeping flow and reduced the momentum equation. Moreover, they made many simplifying assumptions: 1. The kinetic rates of reactions were assumed to be constant in the direction of cell thickness. 2. Electrolyte diffusion was assumed to be constant. In reality the diffusion coefficient is a function of both concentration and electrode porosity. 3. It was also assumed that the porosity of electrodes is constant, which is not an accurate assumption. They solved the resulting system of equations using FDM and compared their results with experimental test. It is good to notice that before putting the battery cell under test, Alavyoon et al. made a preparation routine: 1. After preparation of the setup, the cell was filled with 5 M sulfuric acid, and the cell was discharged with I = 9.34 mA cm−2 until the cell reaches the cut-off voltage Vcut = 1.5 V. 2. Then the cell was filled with 2 M sulfuric acid and kept for 48 hours so that the electrolyte becomes uniform all across the cell. On the other hand, Gu et al. [37] investigated this problem once again using full Navier–Stokes equations. In this case, the model proposed by Gu was more accurate than that of Alavyoon. The only thing that was not considered in their simulation was the preparation process. They did not model the preparation process, and as we will see, the process makes changes in initial conditions. We show that the preparation process can be modeled using a one-dimensional model, and as we will see, it affects the results. Here the preparation process is simulated using a one-dimensional model, and the results are shipped to a two-dimensional model. Figs. 6.15 and 6.16 show the results of a one-dimensional simulation. Fig. 6.15A shows the cell voltage variation. It shows that the cell requires about 5.5 hr for full discharge. The shares of current density in solid and electrolyte phases are plotted in Fig. 6.15B. In the same graph the sums of both current densities are also plotted. It is quite evident that the sum of both current densities is constant and equal to I = −9.34 V, which is the result of electroneutrality. The electrolyte concentration variation is shown in Fig. 6.15C, and as it can be seen in this cell, the electrolyte concentration reaches zero almost in all regions except about 0.4 M in the negative electrode, which is not considerable. Fig. 6.15C illustrates the variation of porosity during discharge. As can be seen, the preparation process results in a nonuniform


Simulation of Battery Systems

Figure 6.15 Simulation of discharge phase of preparation process for Cell-IV. (A) Cell potential during discharge. (B) Share of current density. (C) Electrolyte concentration. (D) Porosity change. (E) Active material distribution. (F) Distribution of SoC.

porosity distribution. This result also can be seen in the active material distribution shown in Fig. 6.15E and state of charge in Fig. 6.15F.

Lead–acid batteries


Figure 6.16 Simulation of discharge phase of preparation process for Cell-IV. (A) Cell potential during discharge. (B) Share of current density. (C) Electrolyte concentration. (D) Porosity change. (E) Active material distribution. (F) Distribution of SoC.

Fig. 6.16 shows the same results for the rest process, where the cell is put in rest for 48 hours. The voltage of the cell remains constant (Fig. 6.16A),


Simulation of Battery Systems

Figure 6.17 Cell-IV model and numerical grid. (A) Cell model. (B) Numerical grid.

and as it can be seen in Fig. 6.16B, both solid and electrolyte current densities are zero. The only parameter that changes during the rest is the electrolyte concentration because the cell is filled with 2 M sulfuric acid, and from Fig. 6.16C it is clear that it takes 48 hours for the electrolyte to become uniform. From Figs. 6.16D to 6.16F we can see that the porosity, active area, and SoC do not change during the rest period. Therefore the initial values for stratification simulation should be taken from these figures. The fluid flow is simulated using a SIMPLE algorithm given in Appendix E. The simulated domain is shown in Fig. 6.17A and the numerical grid is shown in Fig. 6.17B. As can be seen, a non–uniform mesh is used for simulation. Also, note that for giving a proper visualization, the x and y axes are independently scaled. The results of the simulation are shown if Figs. 6.18 and 6.19 for time levels t = 15 and t = 30 min, respectively. Figs. 6.18A and 6.19A show the velocity vectors at electrolyte region. It is clear that the electrolyte tends to move downward near the electrodes because, during the charging process, acid is produced inside the electrodes according to the electrochemical reaction of electrodes. But it is evident that the electrolyte near the positive electrode is denser than the negative electrode because of stoichiometric coefficients of lead–acid main reactions. Figs. 6.18B and 6.19B show the natural convection that takes place inside the electrolyte region. Some vortexes are seen near the top of the cell because of electrolyte movement. The result of electrolyte movement is translated into electrolyte stratification as is plotted in Figs. 6.18C and

Lead–acid batteries


Figure 6.18 Results of simulation at t = 15 min. (A) Velocity vectors. (B) Velocity field. (C) Electrolyte contours.

Figure 6.19 Results of simulation at t = 30 min. (A) Velocity vectors. (B) Velocity field. (C) Electrolyte contours.

6.19C. The movement of electrolyte causes the denser electrolyte to sink and the lighter ones to rise. Therefore, along the vertical cross-sections of the cell, we see an acid gradient, also known as acid stratification. If we do not couple the Navier–Stokes equation with the other governing equations, then electrolyte stratification could not be captured. To show this argument, we draw the same results of Fig. 6.19 in the absence


Simulation of Battery Systems

Figure 6.20 Results of simulation at t = 30 min without fluid flow. (A) Velocity vectors. (B) Velocity field. (C) Electrolyte contours.

Figure 6.21 Comparison of electrolyte concentration between with and without electrolyte movement at section A − A.

of electrolyte movement in Fig. 6.20. As can be seen, since we do not have any velocity field (comparing Figs. 6.20A and 6.20B), the electrolyte does not show any gradient in a vertical direction. The vertical contour lines in Fig. 6.20C support this argument. Fig. 6.21 shows the electrolyte concentration gradient at midheight cross-section of the battery cell. The figure shows that including elec-

Lead–acid batteries


Figure 6.22 Comparison of vertical velocity component at section A − A.

trolyte movement and neglecting it make a significant difference in final results. Hence if in a battery, free electrolyte exists, then the simulation of electrolyte movement is crucial even though the electrolyte movement is slow and creepy. The vertical component of the velocity field plotted in Fig. 6.22 at the same height supports this argument. The velocity at most reaches about 0.1 mm s−1 , which is a slow movement. Finally, electrolyte concentration gradients in the vertical direction at the center of the electrolyte region in different time levels are plotted in Fig. 6.23. It is clear that as time passes, the gradient becomes more significant.

6.10.4 Simulation of thermal behavior of lead–acid batteries Lead–acid batteries, just like any other battery technology, faces with thermal problems. During charge or discharge, according to electrochemical reactions and Joule heating, the battery gets warm and may undergo thermal runaway, as was discussed with more details in Section 6.5. Studying thermal rise (TR) or thermal runaway (TRA) requires an insight into understanding the involved physical phenomena. This study can be done by numerical simulation if a proper system of governing equations is modeled and solved. The purpose of the present section is investigating the thermal behavior of lead–acid batteries by solving the governing equations. Governing equations of lead–acid batteries are fully described in Section 6.5. For the simulation of thermal behavior of lead–acid batteries,


Simulation of Battery Systems

Figure 6.23 Comparison of vertical velocity component at section A − A.

Eqs. (6.16)–(6.32) should be solved. Since the oxygen recombination and consequently thermal-runaway problem occur in the sealed or gelled lead– acid batteries, electrolyte movement can be neglected; hence the conservation of momentum or Navier–Stokes equations are not going to be modeled. The heat generation in the cell is obtained using Eq. (6.44) with details given in Section 6.5. It should be noted that heat dissipation should also be simulated because dissipation of energy plays an important role in thermal behavior. In this study the resistance model described in Section 6.5 is used for simulation. To numerically illustrate the present modeling, Cell-V from Appendix A is chosen and simulated. This cell was introduced by Srinivasan [39] for simulation of pulse charging. Since all the necessary data for this cell are available, it is used for the present thermal modeling as a test case. The chosen cell is a VRLA battery in which the side reactions accompany the main reactions according to Eqs. (6.8)–(6.13). The rates of reactions are determined by the Butler–Volmer equations (6.19)–(6.21). All the required data and parameters of the battery are given in Appendix A. Fig. 6.24A shows a physical illustration of the cell. As can be seen in this figure, the ambient temperature is constant (Tambient = 25◦ C), and heat is dissipated via convection. Since these electrodes are located in the middle of an electrode set inside the middle battery cell, with a good approximation,

Lead–acid batteries


Figure 6.24 A typical valve-regulated lead–acid cell model.

we can consider that the left and right electrodes are at the same temperature, and hence no heat transport exists between the electrodes. This means that the left and right boundary conditions are of adiabatic type and heat is dissipated into ambient only from the top and bottom of the battery case. The proper numerical boundary conditions are illustrated in Fig. 6.24B. To numerically solve the governing equations, they should be discretized through a computational domain. In this study, we use the semiimplicit method for pressure-linked equation (SIMPLE) introduced by Patankar [45]. A nonuniform computational grid (Fig. 6.24B) is generated to reduce the computational cost and improve the accuracy where the gradients of variables are high. A comprehensive grid study has been performed to insure that the results are grid independent. Fig. 6.25 illustrates the parameter sensitivity study results. Fig. 6.25A shows that when the time step is equal to t = 5 seconds, changing the grid size in j-direction affects the final results (in this case the positive electrode terminal temperature). However, if the time step is reduced to t = 1 second, them the results would be independent of grid size in j-direction (Fig. 6.25B). Reducing the time step does not alter the accuracy of the simulation, as can be seen in Fig. 6.25C. Finally, increasing the grid size in i-direction does not give any more accurate


Simulation of Battery Systems

Figure 6.25 Grid study procedure. (A) Sensitivity to mesh size (t = 5). (B) Sensitivity to mesh size (t = 1). (C) Sensitivity to time step. (D) Sensitivity to mesh in x-direction.

results (Fig. 6.25D). As a consequence, using a grid size of the order of Im × Jm = 15 × 35 and choosing the time step equal to t = 1 second guarantees that the result would be grid independent. This mesh size and time step are used for the rest of the simulation. The governing equations contain highly nonlinear sources, which are linearized by Newton method to obtain a stable solution. The system of equations is solved with a segregated method, and iterations are performed until the relative error is less than 10−8 . The charging voltage of a battery is very important in TRA. Culpin [51] has carried out a set of experimental tests and indicated that TRA appears at float charging about 2.63 V per cell, whereas the batteries at 2.40 V per cell showed a much lower and more uncertain trend to TRA. In this study the battery cell is simulated at three different floating voltages 2.26, 2.36, and 2.46 V. The cell is overcharged from a fully charged state (SoC = 1) to ensure that the main reactions do not contribute to heat

Lead–acid batteries


Figure 6.26 Time history of simulated results for float charging V = 2.26 V. (A) Voltage and current. (B) Cell Temperature.

generation. Therefore all the input current is spent on driving the side reactions, and the effect of side reactions can be better understood. Figs. 6.26–6.28 show the results of simulation at floating voltage of V = 2.26 V. Fig. 6.26A shows the applied voltage of the cell and the resulting current density. Since the cell is being charged from SoC = 1, all the input current goes into the generation and consumption of oxygen and hydrogen. At the beginning of charge the generation of oxygen at the positive electrode and its consumption at the negative electrode are not equal. Hence the battery requires a proper current to drive the reactions. When the charging continues, these rates balance each other, and the applied current reaches a limit, which can be seen in Fig. 6.26A. The current drop follows an exponential trend since it is governed by the Butler–Volmer equation (6.20) and is known as the natural behavior of lead–acid batteries. Fig. 6.26B shows the variation of terminal temperature versus time. This figure shows that the terminal temperature reaches an equilibrium after a while and the cell temperature does not vary after that. The temperature plateau forms because the heat generation and dissipation come to an equilibrium. Fig. 6.27 shows the variation of different parameters at the cell halfheight y = 12 H at different time levels. Figs. 6.27A and 6.27B show the current densities of the main and side reactions. It can be seen that the current of the main reaction is zero since the battery is fully charged and no current goes into the main reaction. On the other hand, the oxygen current has a moderate value, which is decreasing until it reaches a specific


Simulation of Battery Systems

Figure 6.27 Variation of different properties at the cell half-height vs. for V = 2.26 V. (A) Jmain and JO2 . (B) JH2 . (C) Concentration of oxygen in liquid phase. (D) Concentration of oxygen in gas phase.

equilibrium amount. Moreover, we can see that the current that drives the hydrogen evolution reaction is small, which shows that the hydrogen reaction has a slow kinematic rate and we can even neglect the hydrogen evolution reaction, as can be clearly seen from Fig. 6.27B. As a conclusion, in this cell, all the charging current enters the oxygen generation and consumption reactions. In Figs. 6.27C and 6.27D the oxygen concentrations in liquid and gas phases are shown, respectively. From these figures we can see that at the beginning of the charge, oxygen evolves very fast and its rate decreases as charging continues.

Lead–acid batteries


Figure 6.28 Contribution of reversible and irreversible heat of different reactions. (A) Heat of main reaction. (B) Heat of hydrogen reaction. (C) Heat of oxygen reaction. (D) Joule heating. (E) Total heat source inside the cell. (F) Variation of cell heat capacity.

To have a better understanding of the heat sources and sinks in a lead– acid battery, the generated heats of different reactions and heat dissipation


Simulation of Battery Systems

are plotted in Fig. 6.28. As expected, according to Fig. 6.27A, the generated heat of the main reactions is zero. The same argument is true for hydrogen reaction, as we can see from Fig. 6.28B. The main source of heat inside the battery is the heat generation of the side reaction shown in Fig. 6.28C. In this figure the reversible (Eq. (6.37)) and irreversible (Eq. (6.38)) heats of the reaction are plotted for different cell regions (i.e., positive electrode, separator, and negative electrode). The total reversible and irreversible heats are also plotted in the same figure. We can see that the net reversible heat in all the regions is zero. Moreover, from the figure we can see that only the irreversible heat is responsible for heat generation inside the battery cell. Fig. 6.28D shows the Joule heating in different cell regions. As it was stated before, the Joule heating is small since the battery is in a fully charged state, and hence its resistance is negligible. If the charging process begins at SoC < 1, the Joule heating will have a large value. The other point about the Joule heating is that it exists even in the separator region because the potential field exists inside the separator. The heat dissipation from the upper and lower walls are plotted in Fig. 6.28E. The total heat generation is also shown in this figure and compared with heat dissipation. This figure shows that at the beginning of the charging process, the total heat generation inside the battery is positive resulting in battery thermal rise. When the charging continues, the battery cell temperature increases. As a result, the temperature difference between the battery and ambient becomes larger. The larger the temperature difference, the larger the heat dissipation. After about 10 hr, the heat generation and dissipation balance each other out, and the battery temperature stays constant (see Fig. 6.26B). Finally, the variation of internal heat capacity of the cell at the middle height y = 12 H is shown in Fig. 6.28F. The heat capacity of the cell does not change significantly since the initial composition of the cell does not change. Although the oxygen generation and acid concentration variation affect this value, we can see from the figure that the variation is negligible. Figs. 6.29 and 6.30 show the same results for float charging V = 2.36 V. From Fig. 6.29B we can see that increasing the charging voltage increases the battery temperature; but at this voltage the cell temperature rise will not exceed the limit (i.e., 60◦ C), hence thermal runaway would not be expected. The higher temperature rise happens since the oxygen generation rate becomes faster and more current passes through the cell. The increase in cell current means that the irreversibility becomes larger and more heat is

Lead–acid batteries


Figure 6.29 Time history of simulated results for float charging V = 2.36 V. (A) Voltage and current. (B) Temperature profile.

Figure 6.30 Contribution of reversible and irreversible heat of different reactions. (A) Heat of oxygen reaction. (B) Total heat source inside the cell.

generated, as clearly shown in Fig. 6.30A. It is worth noting that other heat sources can be neglected, as discussed for the case of float charging 2.26 V. Fig. 6.30B shows the sum of all heat sources and dissipations inside the cell. Comparing with Fig. 6.30A, it is clear that the net reversible heat generation is zero (as in the case of float charging 2.26 V) and only irreversible part remains for heating the cell. We can also see in the figure that the generation and dissipation rates will balance each other, which results in a stable cell temperature after about 5 hr (comparing with Fig. 6.29B).


Simulation of Battery Systems

Figure 6.31 Time history of simulated results for float charging V = 2.46 V. (A) Voltage and current. (B) Temperature profile.

Figure 6.32 Contribution of reversible and irreversible heat of different reactions. (A) Heat of oxygen reaction. (B) Total heat source inside the cell.

Figs. 6.31 and 6.32 show the same results for float charging V = 2.46 V. In this voltage the heat generation becomes too strong, and according to Fig. 6.31B, the cell temperature exceeds the limit in less than 1 hr, that is, TRA occurs. From Fig. 6.32A we can see that the irreversibility becomes much larger than the previous charging. Therefore, the heat generation is much more than the previous cases. It should be restated that in all the simulations the total reversible heat is zero and only the irreversible heat is responsible for the temperature rise.

Lead–acid batteries


Figure 6.33 Temperature contours at different time levels for V = 2.46 V. (A) At time level t = 0 min. (B) At time level t = 20 min. (C) At time level t = 40 min. (D) At time level t = 60 min.

Moreover, after about 1 hr, the cell temperature exceeds the limit, and the battery enters the unstable situation. Fig. 6.32B shows that in this period the total irreversible part is not balanced by dissipation. Meanwhile, the net reversible part in the positive and negative electrode has been canceled. It worth noting that in the present study the cell temperature is not considered as a bulk temperature and the distribution of temperature inside the battery is also modeled. Fig. 6.33 shows the temperature contours inside the cell at different time levels for float charging of V = 2.46 V. Since


Simulation of Battery Systems

the chosen cell is slender and its aspect ratio (hight over width) is large, the temperature contours cannot be clearly displayed. Therefore, for better presentation, the x and y axes are scaled differently. From the figure we can see that the temperature inside the battery is not uniform, and there exists a temperature gradient inside the battery. This means that using a bulk temperature for modeling of lead–acid batteries is not quite accurate. From these results we obtain important information. For example, as mentioned before, there are different theories about TRA, all of which can be categorized into two distinct paradigms that conflict with each other. The first paradigm [55,54,56], which is most accepted by many researchers, states that the heat generation due to oxygen reduction at the negative electrode is the main source of temperature buildup, which can result in TRA. This paradigm further states that the Joule heating not only contributes to temperature buildup but also accelerates the internal oxygen reduction cycle. The second paradigm [57,58] states that according to the Hess law, the closed oxygen cycle cannot produce any heat since it is a closed cycle. This paradigm further discusses that the only heat generation mechanism in a VRLA battery is the Joule heating. From the present study and numerical simulation another paradigm can be deduced, which links both mentioned paradigms. It has been shown that heat generation of each electrode reaction can be split into reversible and irreversible parts. The reversible part can be found by considering the enthalpy of each reaction and changes sign during charge and discharge. On the other hand, for each reaction, the irreversible part defined by Eq. (6.38) should also be considered. This term can be found by considering the local current density and the open-circuit voltage of the reaction noting that it is always positive. Besides the heat generated due to the electrochemical and chemical reactions, the Joule heating is also a source of heat in a battery. The Joule heating can be found by solving Eq. (6.43). A closer examination of the Joule heating and the irreversible part of heat generation due to the electrochemical reactions shows that these parts have the same nature. To be more specific, the Joule heating can be found by multiplication of current and voltage: qJoule = IV . On the other hand, the irreversible part of heat generation of each reaction is qirr. = JU. Comparing these equations, we conclude that these parts have the same nature or qirr. is of Joule heating type in which I is replaced with the local current density J, and V with local open-circuit voltage U. From Eq. (6.38) we can obtain the irreversible part of heat generation of each reaction: qirrev = −Jmain U main − JO2 U O2 − JH2 U H2 .


Lead–acid batteries


According to these results, a new notion is proposed, called the generalized Joule heating, which is the sum of classical Joule heating and the irreversible part of the generated heat due to the chemical and electrochemical reactions: qGJH = |φ · ik | + qirrev = qJoule + qirrev .


By defining this notion we can conclude that instead of the classical Joule heating, the generalized Joule heating qGJH is the only responsible mechanism for TR or TRA in batteries or specifically VRLA batteries. In addition to the heat sources, heat sinks also play an important role in the thermal behavior of a battery. Heat sinks have also been introduced to the governing equations with a resistance model developed in Section 6.5. The results of the numerical simulation give detailed information about the internal physical phenomena. Firstly, they show that the temperature distribution inside the battery is significant and the assumption of a bulk temperature, which is commonly used in VRLA battery simulations, is not quite proper. The temperature distribution depends on the boundary conditions and can be worse in different operational conditions. However, we can obtain this temperature gradient using the proposed model. Secondly, the results show that the total reversible heat of the reactions at the positive and negative electrodes cancel each other out, and hence the net reversible heat generation is zero inside the battery. Therefore we can conclude that in contrast to the first paradigm, we should not consider the heat generation due to the reactions at only one electrode. If we add up the heat sink at the other electrode, then the total amount of heat generation would be zero. Finally, the results show that besides the Joule heating, the irreversible part of heat generation due to the chemical and electrochemical reactions contributes to temperature rise. Hence, in contrast to the second paradigm, the Joule heating is not the only responsible mechanism for TRA.

6.11 Summary In this chapter, we introduced lead–acid batteries in more detail. Different aspects of batteries are discussed. Active materials, electrochemistry, main and side reactions, and other issues related to this type of batteries were explained. Finally, applications of lead–acid batteries were introduced. The governing equation of lead–acid batteries are obtained out of the general governing equations of batteries. The properties are also presented


Simulation of Battery Systems

via tables, equations, and relations. This information was used to simulate lead–acid batteries under different operational conditions. Finally, different cells were simulated and studied using numerical simulation of lead–acid batteries governing equations derived in this chapter. The first study showed the capability of numerical methods in predicting the behavior of a lead–acid battery under discharge condition. Since it that case, side reactions were not important, a simplified one-dimensional system of equations was simulated using the Keller–Box method. After that, the problem of charging in which side reactions are important was considered. Again the problem was simulated using a one-dimensional model. The second problem was solved using the finite volume method instead of the Keller–Box method. It is quite evident that the problem could be solved also by using the Keller–Box method. Selection of the solution method is a choice of interest. The problem of electrolyte stratification was considered, in which electrolyte movement was important. Therefore the problem could not be solved in a one-dimensional space; thus a two-dimensional model was considered for simulation. Again finite volume method was chosen, and SIMPLE method introduced by Patankar was used for simulation of flow field. The last sample was chosen for studying thermal behavior of lead– acid battery in overcharge process. In this type of batteries, side reactions play an important role in estimation of battery behavior. It was shown that without simulation of side reactions, we cannot have a good estimation of cell voltage and other physical phenomena.

6.12 Problems 1. What are the main advantages of lead–acid batteries? 2. Although lead–acid batteries do not have the highest energy density and specific energy among other battery technologies, they still are the most manufactured types. Investigate and compare the total number of produced lead–acid batteries with other technologies. 3. Investigate difference between gelled and AGM batteries. 4. In the case of constant properties in one-dimensional model, simplify the governing equations (6.49)–(6.54). 5. Between different heat dissipation mechanisms, which one is not considerable and can be neglected in lead–acid batteries? Why? 6. Calculate the efficiency of lead–acid cells under standard conditions. 7. Plot the diffusion coefficient of sulfuric acid versus temperature in the range 25–55◦ C.

Lead–acid batteries


8. Plot the diffusion coefficient of dissolved oxygen in electrolyte versus temperature in the range 25–55◦ C. 9. The finite element method is very popular in numerical simulations. Try solving Cell–II using FEM. 10. In case of one-dimensional simulation including side reactions, obtain an appropriate system of equations suitable for the Keller–Box method. 11. Regarding Fig. 6.12B, if the battery is put under rest after t = 105 s until the electrolyte become uniform all over the cell: (a) Give an estimation for electrolyte concentration. (b) Using the Bode relation, give an estimation for battery voltage. 12. Studying Fig. 6.12, how can we check if the proper boundary conditions are satisfied? 13. Comparing Figs. 6.14C and 6.14E, how can you explain the difference between the shape of oxygen gradients in gas phase, which is very flat, and in liquid phase, which has a stepwise shape? 14. Explain why in Fig. 6.33 the upper part of the cell is hotter that the lower parts. 15. Estimate an average heat dissipation for Figs. 6.33A–D. Compare your results with numerical values shown in this chapter.