{"id":946,"date":"2020-03-17T15:56:24","date_gmt":"2020-03-17T19:56:24","guid":{"rendered":"http:\/\/www.joshuagallaway.com\/?p=946"},"modified":"2020-03-27T21:46:56","modified_gmt":"2020-03-28T01:46:56","slug":"modeling-porous-electrodes-part-3-newmans-band-method","status":"publish","type":"post","link":"http:\/\/www.joshuagallaway.com\/?p=946","title":{"rendered":"Modeling porous electrodes: Part 3 (Newman’s BAND method)"},"content":{"rendered":"\n

I previously made a post about the basic formulation of porous electrode theory<\/a> (part 1), and a post about how the choice of kinetic expression<\/a> determines what the result will look like (part 2). The purpose of this post is to explain how to solve these basic problems using the BAND<\/a> formulation found in John Newman’s textbook Electrochemical Systems<\/em> (Newman and Thomas-Alyea in the Third Edition<\/a>). I also provide Python code so you can get the solution yourself and modify it for your own teaching and learning needs.<\/p>\n\n\n\n

As a model problem we’ll use the boundary value problem of Tafel kinetics in a concentration-independent porous electrode in Cartesian coordinates. The reason to choose this one is because Newman and Tobias provide the analytical solution in their 1962 paper “Theoretical Analysis of Current Distribution in Porous Electrodes<\/a>.” You can also find this solution in Fuller and Harb’s Electrochemical Engineering<\/em><\/a>, in the problems at the end of Chapter 5. The reason to choose a problem with an analytical solution is so you can check your work to know if it’s right! We lay out the problem below, which is the same as that in Newman and Tobias (1962) except one sign convention. This is four equations with four unknowns (i1<\/sub>, i2<\/sub>, \u03a61<\/sub>, \u03a62<\/sub>):<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

If you read the electrochemical engineering literature, solving problems such as this with Newman’s BAND method<\/em> often comes up, and you might wonder what that is. Technically “BAND” is some Fortran code set up to solve 2-point boundary value problems in a way that is convenient for electrochemists. (My own favorite document written about BAND is “Modeling and reactor simulation<\/a>” by Douglas Bennion in the AIChE symposium serie<\/em>s 1983, Vol 79, Num 229, pp 25-36. It’s a resource meant for teachers and is somewhat hard to find so I’ve uploaded it here.) The formulation Newman uses to set up the problem casts each of the equations in this way:<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Here c is a variable. The coefficients a, b, and d are those for the second derivative, first derivative, and the variable itself. And g is the constant term. When you linearize equation 2 above and then cast all of the four equations this way, you get something that looks like the tables below. I’ve collected some constants together as \u00df and P to keep it from looking too messy:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Once you have this, you can (a) build a block tridiagonal matrix, (b) choose initial guesses for the answers, and (c) iterate to a solution by repeatedly solving the problem until your initial guesses and final answers match. Explaining how this works is too long for a blog post, so follow these links for:<\/p>\n\n\n\n