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MIT10_626S11_lec35 - IV Transport Phenomena Lecture 35...

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i i i i Dc F IV. Transport Phenomena Lecture 35: Porous Electrodes (I. Supercapacitors) MIT Student (and MZB) 1. Effective Equations for Thin Double Layers For supercapacitor electrodes, convection is usually negligible, and we drop out convection terms here. Let’ s focus on effective equations governing the transports and electrostatics in electrolyte. Figure 1. Flat Electrode Surface Species conservation equations, Nernst-Plank flux constitutive equations, and Poisson equation make up Poisson-Nernst-Plank (PNP) set of equations (bold fonts indicate that the variables are in vector quantity): 0 i i c t F (1) (2) (3) 2 i i i z ec    Electrostatic constrain and flux constrains on the surface specify boundary conditions. Electrostatic constrain can be interpreted differently, given different specified variables. When there is a specified amount of surface charge, we can have Gaussian law satisfying the electrostatic constrain:
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ˆ ( ) s q   n ˆ i i R n F Lecture 35 10.626 Electrochemical Energy Systems (2011) Bazant (4) On the other hand, when given a specified surface potential, we can use the following approximation, Equation (5), instead of Equation (4), satisfying the electrostatic constrain. This boundary condition could be more simplified assuming negligible capacity in Stern layer. ˆ (x = 0) (x = 0) (x = 0) e s e  n (5) We can use either Equation (4) or (5) to satisfy the electrostatic constrain, depending on specified variables at the surface. In addition, flux constrains of species specify the rest of necessary boundary conditions: (6) We now apply the above set of equations (PNP) as well as boundary conditions to the porous electrode with double layer thickness far thinner than the pore length scale. Figure 2. Thin Double Layer in a Pore When the pore length scale is far larger than the length scale of double layer, , we have separation of length scales. The mathematical structure of thin double layer problem is well understood from the perspective of singular perturbation analysis , in which each of two regions requires a different approximation. Two different approximations are constrained by matched asymptotic expansions . We first assign notations for different variables. From now on throughout this lecture, we use the following notations. i in Double Layer : Concentration of Species ̂ i in Bulk Electrolyte : Concentration of Species
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lim c c ˆ i lim i x x ˆ  0 S i i i S i D     F Lecture 35 10.626 Electrochemical Energy Systems (2011) Bazant i in Double Layer : Chemical Potential of Species ̂ i in Bulk Electrolyte : Chemical Potential of Species Figure 3. Variables in Two Different Regions In quasi-neutral bulk electrolyte, we can use the quasi-neutral approximation, and conclude with zero divergence of current density.
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