A mathematical model of heterogeneous electrochemical supercapacitors ͑HES͒ is proposed which makes it possible to develop capacitors with optimal designs and make calculations of their energy, capacity, and power parameters. Special attention is paid to the processes which occur in electrodes with double electric layer ͑DEL͒ of HES capacitors during their charge and discharge by different currents. It is demonstrated theoretically that in the efficiency of HES capacitors' operation, an important role is played by distribution of the potential along the thickness of electrodes with DEL. In order to build capacitors with improved parameters and optimal design, it is of paramount importance to take account of the type and value of conductivity of the material of the electrode with DEL and conductivity of the electrolyte in its pores. Analysis is performed to determine the efficiency of PbO 2 ͉H 2 SO 4 ͉C system capacitors' operation subject to thickness, conductivity, and specific capacitance of the electrode with DEL. It is established that energy and capacity parameters of the capacitors depend to a great extent on conductivity of their negative electrodes and electrolytes. It is demonstrated that there are different mechanisms of losses of energy and charge of the capacitors during their charge, discharge, and storage.

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The paper contains the results of the research of the mechanism of "shuttle" self-discharge determined by the manganese and titanium ions in the electrolyte of a PbO 2 ͉H 2 SO 4 ͉C heterogeneous electrochemical supercapacitor ͑HES͒. Research was made with respect to the effect of the shuttle self-discharge determined by the manganese and titanium ions on the energy, capacity parameters, and the charge and discharge voltage of HESs. Using the methods of the research of the energy, capacity parameters, self-discharge of the capacitors, and optical spectra of their electrolytes, it was established that, in the process of the shuttle, self-discharge of the capacitors with the electrolytes containing manganese and titanium, mostly MnO 4 − , Mn 2+ and Ti 3+ and TiO 2+ ions, respectively, is involved. It is shown that the shuttle self-discharge determined by the manganese ions brings about the self-discharge of an HES down to the minimal allowable voltage, and the titanium ions to the voltage of 1.62 V. It was established experimentally that the self-discharge current and the power of energy losses during the self-discharge related to the manganese ions in the electrolyte depend nonlinearly on the concentration of the manganese ions. When the concentrations of iron, titanium, and manganese are identical in the electrolyte of HES, the value of the capacitors' self-discharge decreases during the transition from the iron to the manganese. In the spectra of the optical density of HESs electrolyte containing manganese and titanium in the wave band of 200-1100 nm, absorption bands are discovered which are related to the ions of the manganese, titanium, and the level of the capacitors' state of charge. During the charge and discharge of an HES with the electrolyte which contains the titanium, Ti 2+ ions whose concentration is substantially lower than the concentration of Ti 3+ ions are also formed in the electrolyte.

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This work contains the results of research on the mechanism of the shuttle self-discharge determined by Fe2+ and Fe3+ ions in the electrolyte of a heterogeneous electrochemical supercapacitor (HES) of a PbO2∣normalH2SO4∣C system, as well as the results of the measurements of the effect of the shuttle self-discharge current on the energy, capacity, and operation parameters of HES capacitors. It is shown that the presence of the iron ions in the electrolyte of a HES capacitor brings about shuttle self-discharge, which continues until the voltage decreases to the value of 1.05V . The shuttle mechanism is related to oxidation and reduction of Fe2+ and Fe3+ ions in the positive and negative electrodes, respectively. An analytical expression is obtained for the calculations of the shuttle self-discharge current determined by different types of impurity ions in the electrolyte of HES capacitors. Analysis is made with respect to the obtained experimental and theoretical data of the self-discharge current, energy, and capacity parameters, as well as their dependence on the concentration of iron ions in the electrolyte, the separator’s thickness, and modes of the capacitors’ charge. It is established that the processes of oxidation and reduction of iron ions take place in the near-surface layers of the capacitor electrodes. It is shown that the linear increase of the separator thickness does not bring about any linear decrease of the shuttle self-discharge current determined by Fe2+ and Fe3+ ions.