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

A mathematical model of the sodium/iron chloride battery containing a molten A1C13-NaCI electrolyte is presented. A cylindrical cell consisting of a positive iron electrode, an electrolyte reservoir, a separator, and a negative sodium electrode is considered. The analysis uses concentrated-solution theory within the framework of a macroscopic porous electrode model. The effects of the state of discharge, the cell temperature, the precipitation and dissolution rates of NaC1, and the current density on the current-potential relation during the discharge and charge cycles are discussed. The major influences on battery performance are changes in porosity and component volume fractions during cycling.

Section: Introductionmentioning

confidence: 99%

Mathematical Modeling of the Sodium/Iron Chloride Battery

Sudoh

Newman

1990

“…They take into account the finite rate of the charge-transfer reactions, and ohmic as well as diffusional resistances. To cite a few, Gu et al 1 7 These models can predict the discharge period of a battery once a cutoff voltage is specified. However, these models have focused on predicting the performance of the battery at normally encountered ambient temperatures.…”

mentioning

confidence: 99%

Simplified Mathematical Model for Effects of Freezing on the Low-Temperature Performance of the Lead-Acid Battery

Gandhi¹,

Shukla

Martha

et al. 2009

Discharge periods of lead-acid batteries are significantly reduced at subzero centigrade temperatures. The reduction is more than what can be expected due to decreased rates of various processes caused by a lowering of temperature and occurs despite the fact that active materials are available for discharge. It is proposed that the major cause for this is the freezing of the electrolyte. The concentration of acid decreases during battery discharge with a consequent increase in the freezing temperature. A battery freezes when the discharge temperature falls below the freezing temperature. A mathematical model is developed for conditions where charge-transfer reaction is the rate-limiting step, and Tafel kinetics are applicable. It is argued that freezing begins from the midplanes of electrodes and proceeds toward the reservoir in-between. Ionic conduction stops when one of the electrodes freezes fully and the time taken to reach that point, namely the discharge period, is calculated. The predictions of the model compare well to observations made at low current density ͑C/5͒ and at −20 and −40°C. At higher current densities, however, diffusional resistances become important and a more complicated moving boundary problem needs to be solved to predict the discharge periods. The lead-acid battery was invented as early as 1859 by Raymond Gaston Planté. Today, it happens to be the most widely used storage battery for a range of applications, from powering bicycle headlights to electric vehicles. It offers several advantages: the highest cell voltage among aqueous electrolyte batteries, ability to operate over a wide range of temperatures, an acceptable energy efficiency of over 80%, an acceptable level of charge retention, and nearly 100% recyclability of spent batteries. Predicting the performance of leadacid batteries is therefore of importance, and several models have been developed for this purpose. They take into account the finite rate of the charge-transfer reactions, and ohmic as well as diffusional resistances. To cite a few, Gu et al. 1 7 These models can predict the discharge period of a battery once a cutoff voltage is specified. However, these models have focused on predicting the performance of the battery at normally encountered ambient temperatures. References ͑see, for example, Chapter 23 in Ref. 8 and the book by Bode 9 ͒ do point to severely curtailed discharge periods when the battery is operated at temperatures well below the freezing point of water ͑i.e., 0°C͒. The same is demonstrated by data on discharge at constant current density obtained in our laboratories and shown in Table I: The discharge period is reduced from 5 h at 25°C to 2.75 h at −40°C. As evident from the longer discharge period at 25°C, the discharge period at low temperatures is reduced even though the battery still has a considerable amount of unreacted active materials. The objective of the present paper is to present a simplified model to explain the drastic reduction in discharge periods at low temperatures. A major reason f...

“…Charging must stop when sulfate is not available. To account for this, Gu et al 19 equated the 'effective' area available for charge transfer reaction to the actual area available for charge transfer multiplied by fraction of active material used: a = a o (1 − r ) 1.5 r . Others proposed the multiplication factor, called morphological factor, to be some function of fraction utilized: Gu et al 20 use 1 − (1 − r ) 0.55 while Srinivasan et al 21 use r 0.6 .…”

Section: Mathematical Modelmentioning

confidence: 99%

Modeling of Effect of Double-Layer Capacitance and Failure of Lead-Acid Batteries in HRPSoC Application

Gandhi¹

2017

Lead-acid batteries fail faster in partial state-of-charge start-stop technology than in SLI application. Accumulation of lead sulfate on negative electrode's surface has been identified as the cause. It is also known that life can be enhanced by increasing capacitance of negative electrode. A bench-marking test cycle is used to explain these observations through a one-dimensional model. It is shown that, at the large discharge current densities used, faradaic reactions in the electrodes are spatially inhomogeneous, and charging is unable to reverse its effects. Consequently, lead sulfate deposit is larger on electrode's surface than at its center. Model uses a rate expression for charging modified to include diffusion of Pb 2+ , and predicts that sulfate continues to accumulate with cycling. A portion of electrode becomes inactive when volume fraction of sulfate reaches a critical value there. Battery fails when inactive area becomes large. It is shown that double-layer capacitance suppresses the non-uniformity in the faradaic reaction and alters the pattern of accumulation of sulfate. Negative electrode does not benefit from this since its capacitance is low. Sulfate accumulates in positive electrode also, but does not reach critical levels since positive electrode's capacitance is large. A large fraction of the demand for lead-acid batteries from automotive industries is for SLI application. But the demand is expected to decline as the auto industry moves toward Li batteries. However, as specifications become more stringent both on fuel economy and on emissions of green house gases, lead-acid battery has found a new application in stop-start technology of electric vehicles. Here, battery should be capable of rapid enough discharge to permit starting of the vehicle along with maintenance of other on-board electrical loads. Further, to facilitate charging by regenerative braking, the battery has to be in partial state of charge, which can be as low as 50%. This has come to be known as high rate partial-state-of-charge or for short, HRPSoC, duty. In the start-stop operation, the frequency of charging-discharging cycles is large. Equally importantly, while the discharge rate during cranking is about the same as in conventional SLI application, charging rate can be as large as thirty times more. The challenges faced in meeting this application have been well summarized by Moseley and Rand 1 and Moseley et al. 2 The main problem faced is the faster degradation of battery than encountered in the conventional SLI application. The diminished life has been attributed to accumulation of lead sulfate on the surface of the negative electrode. Conclusive evidence for this has been reported by Lam et al.3 based on tear-down analysis of failed batteries. Interestingly, it has been found that increasing the capacitance of the negative electrode enhances its life, and development of ultrabattery 4 is a well known demonstration of this. While a complete description of this device is not available in the open literature, it is s...