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MIT10_626S11_lec16

MIT10_626S11_lec16 - IV Transport Phenomena Lecture 16...

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1 IV. Transport Phenomena Lecture 16: Concentration Polarization MIT Student We have previously discussed open circuit voltage, which can be derived from the Nernst equation, and activation overpotentials, which can be de- rived from the Butler-Volmer equation. This can accurately describe the behavior of electrochemical cells at low currents, but for suﬃciently large currents, the transport of reactants to the reaction sites begins to become limiting, and the concentration at the reaction sites will be significantly lower than the bulk concentration. As the current gets even larger, the re- action will completely run out of reactants, and the voltage required to push the reaction will diverge as the current approaches a limiting current, I lim . This behavior is commonly described by a concentration polarization. For this lecture, we will examine how the diffusion of reactants can be modeled to describe this process. Linear Diffusion and Convection In a dilute (quasi-binary) solution, the transport of solutes can be described in terms of a ﬂux density F , which in the absence of convection can be described be Fick’s Law: F = D C , where D is the diffusivity of the particles, and C is their concentration. In the presence of ﬂuid velocity u , the more general expression for Fick’s Law becomes: F = D C + uC. (1) The diffusivity D can be derived from a molecular random walk deriva- tion, which yields a result of the form Δ x D = , (2) t 1

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Lecture 16: Concentration polarization 10.626 (2011) Bazant whereΔ x is the mean free path of the solute andΔ t is the mean time between collisions of the solute. Keep in mind that this expression would need to be modified for concentrated solutions because it relies upon the fact that there are no interparticle interactions in order to obtain independent random walks. In order to apply Fick’s law, we need a final relationship between ﬂux and concentration, which we obtain from conservation of mass. There are many good derivations of the conservation of mass relation available in textbooks, [1] but for our purposes, it is adequate to observe that for a point in space, the ﬂux inwards plus any local particle generation must be equal to the ﬂux outwards, or else there will a change in local concentration. For the case where there are no volumetric sources, this relationship is described by: ∂C + ∇ · F = 0 . (3) ∂t If we assume a constant diffusivity and incompressible ﬂow, so that ∇ · ( D C ) = D 2 C and ∇· u = 0, we can combine equations 1 and 3 to obtain the linear convection-diffusion equation: ∂C + u · ∇ C = D 2 C, (4) ∂t which is the
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