An isothermal porous-electrode model of a discharging lead-acid battery is presented, which includes an extension of concentrated-solution theory that accounts for excluded-volume effects, local pressure variation, and a detailed microscopic water balance. The approach accounts for three typically neglected physical phenomena: convection, pressure diffusion, and variation of liquid volume with state of charge. Rescaling of the governing equations uncovers a set of fundamental dimensionless parameters that control the battery's response. For the discharge situations considered here, total volume change and nonuniform pressure effects prove to be negligible because variations occur in just one spatial dimension. A numerical solution of a simplified model is developed and exploited to predict transient cell voltages and internal concentration profiles in response to a range of Crates. The dependence of discharge capacity on C-rate deviates substantially from Peukert's simple power law: charge capacity is concentration-limited at low C-rates, and voltage-limited at high C-rates. The model is fit to experimental data, showing good agreement. (C) 2019 The Electrochemical Society.