Harmonic analysis in ideal ion-exchange membrane systems

The harmonic response of ion-exchange membrane systems is studied by using the network simulation method. The ionic transport processes of a binary electrolyte through a system constituted by a cation-exchange membrane and two diffusion boundary layers adjacent to the membrane, are described from the Nernst–Planck flux equations, the electrical neutrality condition in the diffusion boundary layers and the Donnan equilibrium relations at the membrane/solution interfaces. The system is perturbed with a sine current of a single frequency superimposed on a direct current, and the decomposition in a Fourier series of the resulting membrane potential is obtained. The higher-order harmonics, the waveforms and the first harmonic impedance for different values of the dc current, are analyzed and discussed. The non-linearity of the current–voltage relationship of ideal ion-exchange membrane systems in relation with the impedance measurements, is investigated. Special attention is paid to the highest values of the dc current because the diffusion impedance slightly differs from the usual Warburg impedance. The numerical results obtained for the dc resistance of the first harmonic impedance of the system are compared with those obtained from a Taylor series of the well known analytical expression of the steady-state voltage–current characteristic.