Introducing Partial Differential Equations and Their Numeric Solution Prior to Transport Courses

This is the authors' abstract. We don't add key points for this paper.

The field of chemical engineering is filled with systems that exhibit multidimensional dependence. Common developments of these include position-dependent velocity distributions in viscous flow, transient thermal conduction, and diffusive mass transfer within multiphase or reacting conditions. These types of systems are introduced to undergraduate students in courses that make up the junior-level core – these are "transfer" (or transport) lecture courses: fluid mechanics (also known as momentum transfer), heat transfer, and mass transfer. Undergraduate programs in our field typically require the completion of a differential equations course by the end of the sophomore year. These courses focus on the delivery of classical techniques for the analytical solution of singular (or systems of) ordinary differential equations (ODEs), but do not provide more than a general mention of partial differential equations (PDEs). Following a 2016 curriculum revision, our department provides a required sophomore course: Computational Methods in Chemical Engineering (CMCE). This course is offered simultaneously with Material and Energy Balances (MEB). The late-semester modules which focus on time-dependent mass and energy balances in MEB coincide with the introduction of methods for the numeric solution of ODEs in the CMCE course. In CMCE, such examples are examined alongside the ODE systems which arise from chemical reaction kinetics, such as those encountered by students in a general chemistry course in the freshman year. The primary software package used in CMCE for the numeric solution of singular and systems of ODEs is MATLAB (MathWorks, Inc.), which is provided at no cost to students via a site license. The solution of ODE systems is used as a natural starting point for the recasting of PDEs as systems of ODEs within MATLAB. Following this, its built-in PDE solver (pdepe) is used to provide students with direct instruction on dimensional discretization and solution of parabolic or elliptic PDEs. Parabolic PDEs make up the plurality of those arising from transport problems (e.g., Navier-Stokes, one-dimensional transient diffusion, unsteady heat conduction), although simpler elliptic equations are also encountered (e.g., steady-state heat conduction or diffusion). In the CMCE course, students construct a customizable input file for the solution of parabolic or elliptic PDEs which allows for rectangular, cylindrical, or spherical coordinates. The program output includes a function solution for the supplied dimensions, as well as a detailed surface plot covering their discretized ranges. In this paper, the methods for introducing students to PDEs and their computer-aided solution are described with respect to learning objectives and detailed examples of student exercises. The impact of the instruction will be presented in the context of student self-evaluation and feedback from transport lecture and laboratory course instructors.