MIT10_626S11_lec36

MIT10_626S11_lec36 - VII. Porous Media Lecture 36:...

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VII. Porous Media Lecture 36: Electrochemical Supercapacitors 1. Transmission Line Model for Linear Response Last time, we took the ‘supercapacitor limit’ of a general porous medium theory for thin double layers based on three assumptions: 1) The electrolyte concentration remains nearly constant during charging of the electrode, (ionic conductivity in pores). 2) Faradaic reactions are negligible or can be lumped together with double-layer capacitance as an additional “pseudocapacitance”, as we will discuss in the next lecture. 3) Voltage applied is small enough ( ), that the total interfacial capacitance per area is roughly constant C D const . Under these conditions, the model reduces to two elegant linear PDEs: where is the electron potential, is ion potential, is macroscopic electron conductivity in conducting phase, is macroscopic ionic conductivity in pores, is double layer area per volume. Then we define resistance and capacitance as sketched in Fig .1. Electron resistance per length is , ion resistance per length , and double layer capacitance per length , where A is the macroscopic electrode area, and L is the electrode length, from separator to current collector. Substituting these definitions above, we arrive at a pair of linear PDES MIT Student (and MZB) which can be interpreted as an RC transmission line , as shown in Fig .2.
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Once these PDEs are solved for the potentials of ions and electrons, the charge density (per length) stored in the double layer capacitors is given by Boundary conditions for these two PDEs are as follows: We set reference pore potential at the separator , ions carry current from separator, thus . Similarly, at the back side of the electrode (current collector) , where is the half of the total voltage if symmetric electrodes are used for the whole cell. At the current collector, the current is totally carried by electrons, therefore . Initial condition: suppose electrode is in equilibrium with I=0, and charge density is constant
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MIT10_626S11_lec36 - VII. Porous Media Lecture 36:...

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