Transient Heat Diffusion in a Semi-Infinite Resistor-Capacitor Transmission Line in Response to a Voltage Step

Temperature change in electrochemical energy devices is a very important effect that takes place in the bulk ion-conducting electrolyte, electron-conducting electrodes, and at the interface in between. However, not enough theoretical studies are done on this problem. In this work, we assume a porous electrode/electrolyte system modeled as a typical one-dimensional, distributed RC transmission line (TL), where the resistors are used to represent the resistance of the bulk electrolyte, and the capacitors are used to represent the electric double-layer capacitance at the electrode/electrolyte interface. In response to a step voltage excitation, we derive first the resulting expression for the heat source term as a result of the interaction of current with the electric field. From there we solve analytically the unsteady heat conduction equation with the distributed source term using the Laplace and Fourier cosine integral transforms. The numerical simulations reveal the dependence of the electrical variables (voltage, current and power) on the electrical diffusion time, whereas, the temperature distribution shows a co-dependence on both the electrical and thermal diffusion times. These results can be used to optimize thermal management and prevent overheating in supercapacitors and batteries.