University of California, Davis
Department of Chemical Engineering and
Materials Science

ECH 198: Numerical Methods for Chemical Engineers
(CRN# 42116)

Spring Quarter 2008

TR (6:10-8:00 PM)
TBA

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Instructors: Brian G. Higgins
Office : Bainer 3012
Office Hours: TBA
Room 1116 Academic Surge
F 11-12 Noon
Bainer 3012
Phone: 752-8780
e-mail: bghiggins@ucdavis.edu

Course Overview

The following Concepts & Skills will be emphasized in ECH198
    GOALS
  1. Expose students to the finite difference and collocation method for solving PDEs
  2. Learn how to write efficient programs (in Mathematica) for solving systems of algebraic equation to which the PDEs reduce.
  3. Learn how to debug complex programs
  4. Learn how to assess the accuracy of your numerical solution.
  5. Learn how to connect the physics with the numerical solutions
  6. Iteration and Recursion
  7. Solution of algebraic and differential equations
  8. Visualization of data - graphics programming
    TOPICS
    The underlying theme for the class will be the numerical solution of partial differential equations using finite difference/collocation methods for solving problems in chemical engineering. Student will also be exposed to Mathematica's advanced programming tools/functions.
    The first two weeks we will examine several programming functions for solving boundary value problems in fluid mechanics, and methods for simulating adsorption of particles on a surface. Then we will consider linear PDEs of the type studied in ECH 140 (the material covered in Chapter 6 from Haberman's textbook)
    In the first part of the course we will consider the heat conduction equation. In this way we can readily ties the numerical results to the physics of the problem.
  1. Random Sequential Adsorption We will illustrate how Mathematica can be programmed to describe the adsorption of particles onto a surface.
  2. Shooting Method for Solving BVP We will review the shooting method and show how the method can be implemented in Mathematica. As an example wee will study the fluid flow over a reotating disk.
  3. Finite Differences
    4. The finite difference method for solving PDEs is based on the premise that you can represent a partial derivative in terms of a finite difference approximation. We will show how various finite difference approximations can be derived from Taylor series expansions, and how one goes about selecting the appropriate difference formula for a particular application.
  4. Parabolic PDES
    5. The first set of PDEs we will study will be related to the transient heat conduction equation thatw as studied in ECH140 and ECH142. We will show how the PDE can be discretized by finite diffrence methods that are defined on a specified finite difference grid to yield a system of linear algebraic equations. Issues of stability, and efficient matrix methods will be discussed. We will also examine how one can program the difference equations in Mathematica and show what functions built into Mathematica can be used to display and interrogate the finite difference solutions . We will examine both explicit methods(method of lines) as well as implicit methods (e.g. Crank-Nicolson methods)
  5. Elliptic PDES
    6. . The next set of PDEs we will study will be elliptic equations in 2- dimensions. We will discretize the the PDE on a specified finite difference grid defined over the rectangle 0 ≤x≤Lx, 0≤y≤Ly, We will show how the PDE can be represented by a system of linear algebraic equations. Issues of stability, and efficient matrix methods will be discussed. We will discuss various finite difference methods to approximate the PDE in a specified rectangle. These are the direct method and ADI method. The importance of efficient methods for handling matrices and their solution becomes crucial when we solve PDEs in more than one spatial dimension
  6. Advection PDES
    7. The next set of PDEs we will study are called the advection equations (sometimes called the convective-diffusion equation). Such equations describes not only how heat is distributed by diffusional processes but also how thermal heat is convected by the fluid velocity U. The relative importance of diffusion to convection terms determines the numerical method used to solve the PDE. When convection dominates the solution acn take on the form of a shock wave.
  7. Application 1: Nonlinear Reaction Diffusion PDEs
    8. Reaction diffusion equations are generally nonlinear PDES- they may be elliptic or parabolic in time. Again we make use of finite differences but now the equations are nonlinear. Thus we solve the nonlinear algebraic equations using a Newton's method. It is now also possible to have multiple steady states, some of which may be unstable. We will show how this equation and other variants can be solved using the method of lines, in conjunction with NDSolve and FindRoot. Issues of bifurcation and tracking of multiple solutions in a parameter space will be reviewed.
  8. Application 2: Spin coating
    9. We solve the time dependent Navier Stokes equations that describe the similarity solution for spin coating using a Crank-Nicolson method on a variable grid.
  9. Application 3: Flow in a 2-D Lid driven Cavity We will solve the Navier-Stokes equations in a rectangular cavity in which mass transfer and chemical reactions acn take place The governing equations are highly non-linear equation that can be solved by a relaxation (iteration) method.

There is no text for the course. Course notes will be provided by Professor Higgins- see Class Notes link above.

Course Grade

The overall letter grade for the course will be determined as follows: Take-home Exam 40% , and homework 60%. Grades will not be assigned on the basis of a curve. The approximate, absolute scale is given by
A: 70 - 100%   B: 55 - 69%
C: 40 - 54%   D: 25 - 39%
F: 0 - 24%