Thermal modeling of supercapacitors

From a physical point of view, heat transfer has its origins in temperature differences. Thus, a transfer of energy in the form of heat occurs any time that a temperature gradient exists within a system, or when two systems at different temperatures come into contact. There are three modes of heat transfer for a supercapacitor heat conduction, heat convection and radiation. Inside the supercapacitor, conduction is the dominant mode of heat transfer therefore, to begin with, we can discount the other two modes of heat transfer. However, it is helpful to take account of convective heat transfer between the ambient air and the outer surface of the supercapacitor. 

For optimal design of the supercapacitor and its cooling system, when dealing with an assembly of many cells, it is important to know the change in temperature over time and space. For this pnrpose, we need to solve the heat balance equation. In the case of a cylindrical supeicapacitor, we use the cylindrical coordinates. This equation is given by the following expression  

The temperature following the angle is assiuned to be constant, and in view of the difficulty of solving the equation in the transitory regime, we consider ordy the operation in the permanent regime. This means there is zero variation in the temperature over time and P is constant. In these conditions, the heat equation is given by  

To begin with, we can make the following hypothesis the snpeicapacitor is made of a single material with thermal properties (heat condnctivity, calorific capacitance, etc.) which are equivalent with account taken of the respective thicknesses of the materials. The solution to the system can be found numerically in the permanent regime. In these conditions, the calculation of the thermal resistance and thermal capacitance of supercapacitor is the same as that for a tube whose internal and external radii are respectively r, and r +i, and whose length is L. The expression of these two values is given by [GUA 11]  

PjCp represent the volumetric density and calorific capacitance, and a is the radius of the interface i. 

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