Time Domain Model of a Lead-Acid Cell

Transcription

1 Time Domain Model of a Lead-Acid Cell D J Harrington * *BMT Defence Services Ltd, Bath, UK *Corresponding author. DHarrington@bmtdsl.co.uk / DH530@bath.ac.uk Synopsis The lead-acid battery was the first secondary (rechargeable) battery ever to be developed. Invented by French Physicist Gaston Planté in 1859, its development over the years has led to numerous applications from smaller automobile starting, lighting and ignition (SLI) batteries up to larger traction batteries for marine applications. It is important to users of a cell to know how it behaves under different load conditions, temperatures or a range of other variables. Most lead-acid cell performance data is empirical and it could be argued that since empirical data comes from performance measurements of real batteries it is more reliable than a mathematical model, in which simplifying assumptions have been made in order for it to be tractable. However, users often have to interpolate between set curves from known data and subsequently they estimate performance which reduces the accuracy. To improve engineering design, this paper seeks to develop a reliable, theoretical discharge model of a flooded 8800 Ah lead-acid cell in the time domain in order to observe how the cell behaves throughout a discharge cycle, theoretically as opposed to experimentally. Keywords: Energy storage; Electrochemistry; Mathematical modelling. 1. Introduction In development for over 150 years, lead-acid cells have become ubiquitous in the modern age representing approximately 70% of the world s secondary-battery market [1]. Despite recent developments in lithium ion/lithium polymer cells, advancements are still being made to lead-acid cells due to their numerous benefits. In an article on modern submarine technology, Dr Randolph Teppner states that Lithium ion cells provide increased energy density, better voltage stability and increased life-cycle. However the high energy density of the chemistry in a lithium ion battery results in a need for a rugged cell design with a sophisticated control system to prevent dangerous operating conditions [2]. Table 1: Advantages and disadvantage of lead-acid cell Advantages Low-cost cell (relative to Lithium based cells). Robust: tolerant to overcharging and can be left on trickle or float charge for extended periods. Good operating temperature (-40 C to 60 C). Good single cell voltage (>2V). Available in a variety of designs and specifications. Disadvantages Relatively low cycle life ( cycles). Low energy density (30-40 Wh/kg). Incorrect operation can cause irreversible damage (e.g. sulfation of battery electrode). Voltage varies significantly with charge (~ V). Electrolyte is acidic and liquid. Author s Biography Daniel Harrington is a 2 nd Year Integrated Mechanical & Electrical Engineering student, studying at the University of Bath. He is currently undertaking his year in industry at BMT Defence Services in Bath where he has carried out R&D projects as well as other surface ship related projects within the naval engineering department. 1

2 Table 1 shows the many advantages of the lead-acid cell and why they are so widely used in modern applications. Despite some disadvantages, the reliable and versatile nature of the lead-acid cell combined with low cost make it a favoured choice in many applications particularly the maritime defence industry. Lee states that all Royal Navy submarines past and present use or have used flooded lead-acid cells, and that is still the only technology qualified for the purpose [3]. This is due to the reliability and vast operating ranges of the cell which are both essential factors for use at sea, a potentially hazardous environment where cell failure could have life-threatening consequences. It is not only important that the cell does not fail but also that the crew can predict reliably how the cell is going to operate in certain conditions. In this paper, a mathematical model of a lead-acid battery is presented. The model has been developed from first principles and uses simplifying assumptions to demonstrate the different electrochemical processes at work inside the cell at any one time. Example results will be presented and compared to empirical data obtained from a proprietary battery data sheet. The paper concludes with a discussion of the results highlighting any infidelities in the model, as well as the assumptions made. 2. Fundamentals of a lead-acid cell 2.1 Structure The simple structure of a lead-acid cell comprises two electrodes immersed in an electrolyte solution of sulfuric acid (H 2 SO 4 ) the anode is made of lead (Pb) and the cathode of lead dioxide (PbO 2 ). Figure 1 shows a cutaway diagram of a lead-acid cell. Cell pillars and connectors Vent plugs Porous Lead Anode (-) PbO 2 Cathode (+) Plastic container Electrolyte: Sulfuric Acid solution Figure 1- A cutaway diagram of a lead-acid cell [4]. The porous lead anode contains many cavities to maximise the surface area of lead in contact with the electrolyte, thus maximising the Electromotive Force (EMF) produced at the terminals. While there are numerous variations of the lead-acid cell, such as valve regulated cells or cells that have lead grids coated with a paste of lead oxides, this paper will consider a flooded lead-acid cell (electrodes in an excess of electrolyte). 2.2 Chemistry Electrochemical cells produce an EMF across their terminals due to oxidation and reduction reactions occurring at the electrodes. The lead-acid cell combines Pb and PbO 2 electrodes in sulfuric acid to produce an EMF. The cell utilises the chemical reactions of lead oxidation (loss of electrons) and lead dioxide reduction (gain of electrons) to cause a flow of charge and hence produce an EMF across the terminals. 2

3 The electrode equations are as follows. At the lead electrode (negative plate): (1) At the lead dioxide electrode (positive plate): (2) Equation 1 shows that the lead reacts with aqueous sulfate ions, produced by the sulfuric acid solution, to produce lead sulfate ( and two electrons. Equation 2 shows that the two electrons produced in the first reaction reduce the lead in PbO2 allowing lead sulfate and water to form. Equation 2 also shows H2SO4 being converted to water through reaction. This means that the amount of acid in the electrolyte reduces with cell discharge; an important relationship used for the model. The result of each of these chemical reactions is that each electrode has potential measured in relation to a standard hydrogen electrode. The lead electrode has a negative potential and the lead dioxide has a positive potential and this consequently causes a potential difference between them, giving the cell an EMF which can be calculated by Equation 3 below [5]. (3) The variation of the EMF forms a large part of the model and will be discussed later in the paper. 3. Modelling approach The first key aspect of designing the model was establishing known cell relationships for which equations could be derived and subsequently implemented in Matlab. Firstly, it was known that the Relative Density (RD) of the electrolyte reduced with the State of Charge (SOC) of the cell, from approximately 1.28 at full charge to approximately 1.08 at complete discharge [6]. This change in RD represents a change in concentration of the electrolyte as sulfuric acid has a much higher density than that of water (pure sulfuric acid has an RD of around 1.84 and water has an RD of 1); at full charge there is a higher concentration of acid giving it a higher RD, but as the acid reacts through discharge the solution becomes more aqueous causing the RD to approach 1. The voltage of a lead-acid cell reduces with SOC (from around 2.1V to 1.6V) with a non-linear relationship showing that the voltage is proportional to the concentration of the electrolyte solution in which the electrodes are placed. The equation that relates these variables is known as the Nernst equation, formulated by German physical chemist Walther Nernst [7]. The specific Nernst equation for the lead-acid cell can be seen below (for derivation see [8]). Where: Ecell = EMF of cell E0 = Standard cell potential R = Molar gas constant T = Temperature in Kelvin n = Number of moles of electrons involved in reactions F = Faraday s constant ah2so4 = activity of H2SO4 3 )

4 In chemical thermodynamics, activity is the measure of the effective concentration of a species in the mixture under non-ideal conditions. It is used to determine the chemical potential for a real solution rather than an ideal one. Equation 5 shows how to calculate the activity of the electrolyte solution [8]. Where γm is the mean activity coefficient and C is the molal concentration in mol/kg. Molal concentration is defined by the number of moles of solute (acid) per kilogram of solvent (water). The mean activity coefficient is generally temperature and concentration dependent and is determined experimentally. Once this relationship was recognised, the equations that governed the change in the concentration of the acid with time were derived. To simplify this task a basic model of the cell was established which can be seen in Figure 2. Semi-permeable boundary Bulk Reservoir of Acid Active acid Lead Electrode Holes in Porous lead electrode Figure 2 Simplified model of Lead-Acid Cell Figure 2 shows the simplified model which has been broken down into two separate volumes separated by a semi permeable membrane (porous boundary of electrode): the bulk volume of acid which is the large reservoir of acid not in contact with the electrode and the active volume which is the acid in contact with the electrode either on its surface or in the holes. The model only contains the lead anode as this is where the acid reacts and its porous nature allows for a diffusion model to be set up. The aim of this simplified model was to establish how the mass of acid in the active volume and bulk volume changed with time; this meant deriving equations that would give the mass of acid in each volume as a function of time. The derivation from first principles of these equations can be seen in Appendix A. For the purpose of this paper subscript 1 denotes the bulk volume, subscript 2 denotes the active volume and subscript 0 denotes an initial condition. 4

5 (6) (7) Where: M = Mass of Acid V = Volume σ = Permeability coefficient (value between 0-1, where 0 is completely permeable and 1 is a solid membrane) t = Time As the concentration of acid is directly proportional to the mass, the equations made it possible to effectively model the variation in concentration throughout the discharge cycle and subsequently model the variation in EMF. Another aspect of the cell that had to be modelled was the heat transfer throughout the cell. This was due to the fact that the internal resistance of the cell changes with temperature, as does the electrical conductivity of the electrolyte. This in turn affects the EMF due to the T term in the Nernst equation, so modelling the heat transfer had improved accuracy over using a fixed temperature. There is an increase in internal temperature of the lead electrolyte due to I2r (lower case r denotes internal resistance) resistive heating; this heat is then transferred to the active acid solution and this heat is in turn transferred to the bulk solution. The heat transfer was modelled using the same equations derived from the concentration model as the variables map directly. The variable mapping can be seen in Table 2. Table 2: Variable comparison of concentration model and thermal model Concentration Model Variable Symbol Heat transfer Equivalent Variable Symbol Volume V Heat Capacity C Mass of acid M Heat energy Q Flow of acid dm/dt Heat Flow dq/dt Driving Head (concentration difference) H Temperature difference ΔT Permeability σ Thermal conductivity Z Once these relationships were derived, it was then possible to implement the equations in Matlab in order to test the model and obtain visual outputs of the results. To represent the code, Figure 3 shows a flow diagram which highlights the key processes and calculations carried out in the code in order to give the outputs. 5

6 Figure 3 High level flow chart highlighting the key stages in coding of lead-acid cell model. To obtain a solution, a numerical solver was used to compute the analytic solution to the initial value problem. As a positive current is being drawn from the cell, the total charge reduces by the current multiplied by the timestep, which in turn causes a proportional drop in the mass of acid in the active volume. This is equivalent to the acid being transformed into water through chemical reaction in a real cell. The first order differential equations then provide an analytic solution to the acid equalisation process within the time-step to provide new masses of acid in the active and bulk volumes. This gives new concentrations and therefore provides an analytical solution to the Nernst equation in each time-step. The method accounts for reaction and equalisation of the acid in the same time-step as two consecutive processes and uses what is essentially a simple Euler method combined with first order differential equations to solve the problem. By using a short time-step of 1s and an analytical solution from the differential equations, the inaccuracy of the method is reduced to an acceptable value consistent with the accuracy of the original data. Furthermore less computing power is required than for a higher order method such as the Runge-Kutta 4 th Order method. 6

7 4. Results This section of this paper presents the variation of different parameters throughout discharge cycles of the cell. The model was run under different conditions to see the variation in performance. The first test run on the model was to see how the cell voltage varied for time with different discharge rates. This was achieved by running the model for a number of different load currents. The plots from the model can be seen in Figure 4a, whilst Figure 4b shows empirical results from an 8800 Ah cell with the same load currents (Figure 4a shows some empirical data points overlaid for easy comparison). Due to the sensitive nature of the original data, the current values have been removed. x &!! #! # x &!! #! # 1x &!! #! # x &!! #! # x &!! #! # #. x &!! #! #.!" Figure 4a and 4b Comparison of model discharge curves to real data for a range of load currents, showing variation in voltage with respect to time 7

8 Another output from the model that can be observed is the change in concentration over one cycle; this was the main factor in changing EMF so was an important variable to consider. To show the variation in concentration, the model was run for a 32000s simulation: 8000s at a high load current, 8000s with no current, 8000s with a medium load current and another 8000s of no current. Figure 5 shows the load current profile for the simulation. The resulting variation of concentration of acid in the lead (active) volume, bulk volume and the level concentration with time can be seen in Figure 6. Figures 7 subsequently show the variation in cell EMF over the period modelled. Figure 5 Variation of load current with respect to time Figure 6 Variation in concentration of acid in the active and bulk volumes with respect to time (the Level Concentration is the total volume of acid in the cell is divided by the total volume) 8

9 Figure 7 -Variation of cell EMF with respect to time Another variation occurring in the cell is the heat transfer. The lead electrodes heat up due to the internal resistance of the cell and this in turn is transferred to the acid via conduction. Figure 8 shows the variation in temperature of the solid lead, lead (active) solution and bulk solution over a typical discharge cycle from fully charged to fully discharged with a load current of 525 A. Figure 8 Variation of temperature with respect to time 9

10 5. Discussion Figure 4a & 4b form the most important evidence for the validity of the model designed. It is observed that the model gave intuitive results when compared to empirical data, with the same characteristic shape of curve. This shows that that the basis for the variation in EMF caused by the changing concentration of the electrolyte is valid for the model. Overall the accuracy of the model is good with all plots having less than 7% inaccuracy when compared to the empirical data. The least accurate plot was the 3.5x current plot which had a mean difference of 6.3% and the most accurate plot was the 0.3x plot which had a mean difference of 0.86%. As the model contains simplifications of the highly complex electrochemical and physical processes in the cell, this level of accuracy supports the validity of the model. Figure 6 shows that when a load current is applied there is an initial steep drop in the concentration of acid in the active volume. This is because the active volume is the location of the chemical reaction where acid reacts to produce an EMF. After an initial drop the rate of change reduces as acid from the bulk volume diffuses into the active volume to replenish the lost acid. After some time the rate of loss of acid from the active volume is equal to the rate of diffusion of acid which causes the two lines to have the same gradient. The Level Concentration is the concentration obtained if the total volume of acid in the cell is divided by the total volume. As there is a constant current being applied the level concentration has a constant negative gradient. Once the load current is removed, Figure 6 shows an equalisation process at 8000s. As there is no chemical reaction occurring once there is no current, the concentration of the acid in the active volume does not reduce. However, the cell does not remain at a steady state as there still remains a difference in concentration between the bulk and active volumes. This causes an equalisation process to occur as the concentrations in the two volumes tend towards a new steady state value where the concentrations are equal. The change in the active volume is much more noticeable due to it having a significantly smaller volume. At 16000s a new lower current is applied. This causes a softer initial drop to the active volume concentration as less acid is being removed per unit time; this is also the reason for the smaller rate of change of concentration. Figure 6 also shows a smaller separation between the parallel bulk and active concentration lines this is due to less time passing before the rate of loss of acid from the active volume equals the rate of diffusion. The same equalisation can be seen at 24000s however as the concentration head is smaller, less time is needed for equalisation. Figure 7 shows the variation in EMF for the same discharge cycle. There is an initial drop in EMF when the current is applied due to an initial drop in concentration of the active acid. However, the concentration of the active acid does not drop low enough in the cycle for the significant non-linearity of the EMF curve to be observed, meaning that the gradient remains relatively constant. Once the current is switched off and the equalisation occurs the EMF can be seen to increase slightly; this is due to the concentration of acid in the active volume increasing slightly when the equalisation process occurs. As the EMF is directly related to the concentration of active acid, there is a subsequent increase in EMF. The same process can be seen when the lower current is applied however this time the rate of change of EMF is reduced as there is less acid reacting per unit time due to the lower current. The second equalisation causes a much smaller increase in EMF as there is a smaller increase of the active volume concentration to the new steady state value. Figure 8 shows the variation in temperature of the constituent parts of the cell over a discharge cycle. As the lead electrode is the part of the cell that is being heated directly by I 2 r heating, an initial increase in temperature can be seen as heat energy enters the mass directly. As the heat is dissipated into the active acid volume the rate of change reduces. As the electrolyte has a much higher heat capacity compared to the solid lead the rate of heat transfer from the solid lead to the active solution is slow, therefore it takes some time before a change in temperature can be detected. As the bulk volume is large compared to the active volume, an increase in the heat energy of the active volume has almost no effect on the overall temperature of the bulk volume, hence the temperature of the bulk volume remains almost constant. 10

11 This model contains many assumptions and simplifications of the physical and electrical behaviour of a real cell. This was necessary due to the extremely complex electrochemical processes occurring within the cell. The model aims to effectively represent the processes in the cell and give the best possible outputs whilst adhering to the computing power available. Two key assumptions made were the values of the electrode permeability coefficient and the ratio of volumes of the bulk to the active acid. These values were required to accurately model the cell; however it was not possible without experimentation to obtain values for these variables. Values were therefore estimated, and subsequently iterated, to achieve the most accurate outputs. It was estimated that the active volume was 8% of the bulk volume (5% was the initial rough estimate and the value was then iterated) and the permeability coefficient of the boundary was set at Whilst these are estimates, the real value of each is cell dependant and varies with design. Moreover these values gave accurate outputs from the model with an error of less than 7%; therefore they can be taken as adequate for the purposes of this paper. Another assumption was the manner of heat transfer in the cell. This was modelled as firstly heat energy going directly into the lead electrode then in a two stage conduction from the lead electrode to the active volume, and then from the active volume to the bulk volume. In a real cell there would also be convection and mixing of the two volumes which would cause heat transfer but these processes are highly complex and therefore would be hard to model accurately. Despite the inaccuracies, the changing temperature of the active acid did not have a large effect on the EMF produced as the term in the Nernst equation had only a 9.7% variation over the temperature range in the model. Finally an important relationship in the model was that between molal concentration and γ m. The relationship was defined in the model from experimental data [5] using interpolation. This data is recorded at a constant temperature of K and the model uses a changing electrolyte temperature, therefore this will cause some errors. Nevertheless, the temperature change was not significant and as γ m cannot be recalculated at each time step the assumption was necessary to enable modelling of the system. 6. Conclusion In conclusion, the model gave accurate and intuitive results when compared to empirical data. This validates the model as it used simplified processes of the action within the cell yet still gave the same characteristic outputs. Nevertheless the overall simplification of the electrochemical processes meant that results did not match up perfectly, particularly at high and low load currents. The concentration profile shows that the first order differential equations derived are correct as they follow the right shape for the load conditions of the cell (i.e. equalisation at 0 current, constant gradients with a load current). The subsequent EMF curve supports the proportional relationship between concentration of the electrolyte and the cell EMF. The temperature profile showed that the equations determining the transfer of heat gave intuitive outputs despite the simplifications; however this part of the model had little effect on the results and therefore could have been neglected as the temperature change of a lead-acid cell remains within a relatively small range. With some improvement, this model could be used as an on-board tool to predict battery performance. By inputting a known load cycle into the model, the user could observe predicted model performance and use this to adjust the actual loading in order to meet a set of desired behaviour. Acknowledgements The guidance, advice and support of Prof Chris Hodge OBE FREng are acknowledged with thanks. The kind permission and resources granted to the author by BMT are acknowledged with thanks. All findings, ideas, opinions and errors herein are those of the author and are not necessarily those of BMT Defence Services Limited. 11

12 References 1. Jung, J.J, Lead-Acid Battery Technologies: Fundamentals, Materials and Applications. 1st ed.: CRC Press. 2. Teppner, R, Modern Submarine Technology. Marine Engineers Review (MER), Issue July/August 2013, Page Lee, J.M, Faulkner, D & Rowlinson, P, Increasing the battery voltage on a nuclear submarine. Proceedings of International Naval Engineering Conference (INEC). Portsmouth, 11-13th May Yoder, J.A, DOE Primer on Lead Acid Storage Batteries. 1st ed. Page 25: U.S. Department of Energy. [DOE-HDBK ] 5. Pavlov, D.P, Lead-acid Batteries Science and Technology: a handbook of lead-acid battery technology and its influence on the product. 1st ed. Pages 30-33: Amsterdam: Elsevier Science Ltd Staples, B.R, Activity and Osmotic Coefficients of Aqueous Sulfuric Acid at K. J. Phys. Chem. Ref. Data, Vol. 10, No. 3, Pages Available at: [Accessed 18 September 2016]. 7. Orna, Mary Virginia; Stock, John (1989). Electrochemistry, past and present. Columbus, Ohio: American Chemical Society. 8. Berera G.P, Materials Laboratory Module. Lead-Acid Storage Cell, [Online]. 1,4. Available at: [Accessed 19 September 2016]. APPENDIX A Figure 9 - Schematic Representation of Cell Diffusion The total volume of the cell available for acid The bulk volume of the reservoir of acid The volume of the acid contained in the porous lead (active region) The concentration of the acid if fully equalised between the two volumes The concentration of the bulk volume acid The concentration of the active acid The total mass of the acid The mass of the acid in the bulk volume acid The mass of the acid in the active volume 12

13 The initial mass at time zero of the acid in the bulk volume acid The initial mass at time zero of the acid in the active acid H The concentration head (concentration difference) The permeability of the membrane (given by a factor between 0-1, where 0 is no boundary and 1 is solid wall) Relationships Volume of cell is constant no loss of acid Mass is constant (conservation of mass) Concentration is mass per unit volume The rate of change of mass is given by permeability factor multiplied by the concentration head (difference in concentration). (Note: a positive flow of acid from the bulk volume to the porous lead reduces the mass in the bulk volume hence the negative sign of the RHS.) Differential Equation Substitute in to remove M2 term Rearrange to get M1 terms on the left hand side Integrate both sides 13

14 Combine LHS integral to a single fraction Exponentiate both sides to remove log Rearrange & simplify to isolate M1 (8) Substitute in (8) to get equation for M2 Gives: Which is, as expected, the total mass and constant. (9) Adding the expressions for Rearrange & simplify Note: 14