IV. Transport Phenomena. Lecture 19: Transient Diffusion. 1 Response to a Current Step. MIT Student. 1.1 Sand s Time

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1 IV. Transpor Phenomena Lecure 19: Transien iffusion MIT Suden In his lecure we show how o use simple scaling argumens o approximae he soluion o ransien diffusion problems, which arise in elecrochemical energy sysems. In each case, we will also briefly menion he relaed resuls from exac soluions o he diffusion equaion o show how well he scaling analysis works. For furher reading, see Bard & Faulkner, Elecrochemical Mehods. 1 Response o a Curren Sep 1.1 Sand s Time Suppose we have a cell and we urn on he curren a ime = 0, I() = Iθ(), where θ() is he Heaviside sep funcion, and I is above he limiing curren, we observe he concenraion profiles indicaed in Fig. 1. This curren can be susained for an amoun known as Sand s ime, which occurs when c(x = 0) 0and V 0. This is solved in 2009 previous noes and in Bard & Faulkner, Elecrochemical Mehods. Here we give a simple scaling analysis for : F Since x and he A F x c, and hus c = c 0 a. I F c 0,F = nea ( ) c 2 0 nea I 1

2 The exac soluion o he diffusion equaion is = π ( c 0 nea ) 2. 4 I Scaling argumens are very powerful! 2

3 = 0 c(x,0)= c 0 = I > I lim a I lim in seady sae c --> 0 L x!x ) c 0!c Figure 1: Top: We display how he concenraion as a funcion of space progresses in ime afer we urn on a consan curren I >I lim a =0. is he poin in ime beyond which he curren I canno be susained anymore. since he concenraion gradien canno be mainained anymore once c(x = 0) =0. Boom: This is a magnified version of he op figure. The shaded region is he oal amoun of maerial ransporedl and grow as F.The area of he shaded region, A, is proporiona o c x. 3

4 1.2 Chronopoeniomery (Volage vs ime a consan curren) Our scaling analysis implies c(0) 1 for I >I lim.boh he Nerns c 0 equilibrium volage and he acivaion ( overpoenial ) end o have logarihmic dependence V ln c(0) ln 1 as c(0) 0, for I >I lim. c 0 V(c 0 )!V~V(c 0 sand )) I < I lim V -> seady sae 0 I > I lim Figure 2: V () for values of suddenly applied ( I<I lim, I = ) I lim and I >I lim respecively. For small imes, V V (c 0 ) 1. For large imes, I<I lim leads o a seady sae volage; I = I lim leads o a volage hiing 0; and I >I lim diverges logarihmically a Sand s ime. 1.3 Galvanosaic Inermien Tiraion Technique (GITT) Baeries can be esed by small slow curren pulses, and he relaxaion is fied o Sand s soluion of he diffusion equaion o infer he diffusiviy (c 0 ) vs sae of charge c 0.In Fig. 3, we show he inpu curren pulses and he volage response. The dashed curves indicae he open-circui volage and he volage response for some fixed I >0respecively. The wo curves 4

5 form an envelope ha bounds he acual funcion V (I). The zigzag curves have a or 1 dependence as he V jumps back and forh beween he OCV dashed curve o he I >0dashed curve. I V Q = I I > 0 OCV (I = 0) Figure 3: GITT: The dashed curves indicae he open-circui volage and he volage response for some fixed I >0respecively. 2 Response o a Volage Sep 2.1 Corell Equaion For linear response, his is like a sudden concenraion sepa he elecrode surface. We have he following boundary condiions: 5

6 = 0 c 0 c 1!x ) Figure 4: If we fix he concenraion a he boundary o be c 1,we observe how he curren responds in ime. x c(x, 0) = c 0 c(0,) = c 1 c 2 c = x 2 By scaling analysis, we have I c c 0 c 1 F = nea = x (0,) Thus, I() nea c The exac soluion is given by: ( ) c(x, ) c 1 x = erf c 0 c 1 2 c F = (0,) x 2 1 = c π 2 6

7 where erf (z) = 2 z π 0 e x2 dx. Thus, we obain he Corell Equaion: I() = nea c π Again, his resul is he same as he one obained from scaling analysis excep for a facor of π. 2.2 Poeniosaic Inermien Tiraion Technique (PITT) We can characerize a baery by slow small volage seps o infer (c 0 ). V I Figure 5: The inpu volage seps and curren response, which goes as are shown for PITT. (c 0 )can be inferred from such measuremens., 7

8 MIT OpenCourseWare hp://ocw.mi.edu Elecrochemical Energy Sysems Spring 2014 For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.