A Two‐Dimensional Mathematical Model of a Porous Lead Dioxide Electrode in a Lead‐Acid Cell

A two-dimensional mathematical model is presented for a lead dioxide electrode in a lead-acid cell. It is used to simulate the t ime dependent behavior of the electrode during discharge. The model contains six dependent variables: the concentration of the acid electrolyte, the porosity, the electrical potentials of the solid and solution phases, and the two directional components of the current density in the electrolyte. The effect of the electrode grid was included by varying the conductivity of the solid. Parameters such as electrode conductivity, electrode dimensions, and temperature are investigated to understand their effects on electrode discharge performance. The combination of low cost, versatility, and excellent rechargeability of the lead-acid battery makes it the single most widely used battery system worldwide. Its applications vary from small sealed cells for consumer use to the large load-leveling systems for electric utilities. To complement the traditional trial-and-error approach, which is expensive and time consuming, mathematical models (1-10) have been developed to provide a better understanding of the cause-and-effect relationships and the phenomena involved. However, all earlier models, being onedimensional, cannot account for the effects of nonuniformity in the vertical direction along the height of the electrode (see Fig. 1) on the electrode discharge performance. Uneven current density distribution has been known to lead to inefficient use of the porous electrodes and, subsequently, would lower the performance of each cell in a battery. A number of factors could lead to nonuniform current distribution. In a typical lead-acid cell of a car battery, the distance between current collectors is approximately 3 ram, while the height of the electrodes is close to 100 mm. This aspect ratio elicits doubts whether the current distribution is uniform in the y-direction due to electrical resistance effects, since the tab connections are placed near the center at the top of both electrodes and serve as the source and sink of current. Also, during high discharge rates and after deep discharge cycles, the changing conductivity of the solid electrode material could result in significant nonuniformity in the y-direction. Also, the internal resistance of the electrodes increases with the number of cycles (11). After numerous deep cycles, the reforming of lead dioxide during charge does not reproduce a porous matrix with the exact same structure or electrical properties (11). Therefore, it is the objective of this work to develop a two-dimensional model of a porous lead dioxide electrode that can be used as a tool by battery designers to study electrode performance under conditions where nonuniform current density distribution occurs along the height of the electrode. Figure 1 shows the six elementary parts of the lead-acid cell. A porous positive (PbO2) electrode and a porous negative (Pb) electrode, both supported by lead current-collector grids, are separated by a reservoir of electrolyte and a porous separator. The current collectors are situated at the center of each electrode. These elements are repeated alternatively to form a monopolar stack of cells. The thin electrode plates are interleaved such that each positive plate is located between two negative plates. The porous structure of both electrodes is flooded with a binary aqueous acid electrolyte. Since the bisulfate ion, H S O ( , is a rather weak acid, the electrolytic solution consists essentially of three species: H +, H S O ( , and H20 (the solvent). The electrochemical half reaction at the positive electrode (PbO2) is discharge PbO2r + H S O ( + 3H § + 2e~ PbSO4~s~ + 2H20 [i] *Electrochemical Society Student Member. **Electrochemical Society Active Member. and the one at the negative electrode (Pb) is discharge Pb<~) + H S O ( ~PbSO4<~ + H § + 2e[II] A reservoir of acid electrolyte adj acent to the positive electrode prevents the premature depletion of the acid within the PbO2 electrode due to reaction [I]. The performance of the positive electrode can be predicted by using a mathematical model for porous electrodes. The development of the general equations describing the behavior of porous electrodes can be considered to date back to the work of Newman and Tobias (12). Their model and Euler's model (13) deal primarily with current distribution and demonstrate that the polarization equation and the mass transport of the reacting species play a major role in electrode performance. Dunning et al. (14) use numerical techniques to include the diffusional transport of the active species within the porous electrodes using a macroscopic approach and apply a Butler-Volmertype polarization equation. Simonsson (1) and Micka and Rou~ar (2-5) concentrate their efforts on the porous electrodes of the lead-acid battery. They use a macro-homogeneous approach to disregard the actual geometric detail of the pores and describe the porous electrodes as a superposition of two continuous phases, liquid and solid. Simonsson predicts that a reaction layer moves inward into the lead dioxide electrode due to gradual insulation of the surface by covering lead sulfate crystals. Micka and Roufiar modeled a positive PbO2 electrode (2, 3) and a negative Pb electrode (4) separately, and afterwards combined the equations to model a complete cell (5). They show that the theoretical discharge capacity of the cell is limited by the positive (PbO2) electrode at normal temperatures and discharge rates. A model by Gidaspow and Baker (6) is used to describe the transformation of one solid phase into another in a porous electrode. Their model predicts cell failure by tab connectors ,~ /~z c urre at c~ t~ ~"--..~ __