  Supercapacitors are promising electrochemical energy storage devices due to
their prominent performance in rapid charging/discharging rates, long cycle
life, stability, etc. Experimental measurement and theoretical prediction on
charging timescale for supercapacitors often have large difference. This work
develops a matched asymptotic expansion method to derive the charging dynamics
of supercapacitors with porous electrodes, in which the supercapacitors are
described by the stack-electrode model. Coupling leading-order solutions
between every two stacks by continuity of ionic concentration and fluxes leads
to an ODE system, which is a generalized equivalent circuit model for zeta
potentials, with the potential-dependent nonlinear capacitance and resistance
determined by physical parameters of electrolytes, e.g., specific counterion
valences for asymmetric electrolytes. Linearized stability analysis on the ODE
system after projection is developed to theoretically characterize the charging
timescale. The derived asymptotic solutions are numerically verified. Further
numerical investigations on the biexponential charging timescales demonstrate
that the proposed generalized equivalent circuit model, as well as companion
linearized stability analysis, can faithfully capture the charging dynamics of
symmetric/asymmetric electrolytes in supercapacitors with porous electrodes.
