NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract Session 3513 The story of polyethylene garbage bags Bruce A. Finlayson University of Washington Overview. The story of polyethylene garbage bags has been used as a structure around which to organize a course on numerical methods applied to chemical engineering problems. Starting with natural gas, a series of problems is posed that must be solved if one is to eventually end up with a garbage bag. The course begins with a theme, written about some aspect of polyethylene: raw materials, production methods, uses, business trends, recycling, etc. These themes are then put on the course Web-site so that they can be read by all the class. It is also necessary to use many differ- ent computer tools (e-mail, anonymous ftp servers, local area networks) to accommodate the diverse needs of the students, some of whom have their own computers and different software. Most problems are solved using Microsoft EXCEL™ or Mathworks MATLAB™. After the standard problems are solved, the students prepare multimedia displays illustrating the math- ematical problems a chemical engineer runs into in order to convert natural gas into garbage bags. Two displays are done: one for a technical audience (a prospective sophomore chemical engineer) and one for a general audience (middle school or high school students). Even the patent literature is used in lectures to display the role of innovation. Safety aspects in polyethyl- ene plants can be addressed by using films prepared by SACHE (Safety and Chemical Engineer- ing Education). Course Objectives. The goal of the course is to learn to apply mathematical and numerical methods to chemical engineering problems. This goal involves learning to use existing software like EXCEL and MATLAB, learning some rudimentary programming in MATLAB, and espe- cially learning to check your results numerically, since there are often no known analytical solutions for comparison. The instructor’s task is to achieve those goals within a structure imposed by a case study. In this course the structure was provided by a series of problems arising when modeling processes to obtain natural gas, separate the ethane, react it to ethylene, polymerize it to polyethylene, and extrude it to make polyethylene garbage bags. This provides the story line of the course. The sequence of problems is: sets of algebraic equations, ordinary differential equations as initial value problems, ordinary differential equations as boundary value

NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract Session 3513 Problem-centered Course in Numerical Methods Bruce A. Finlayson University of Washington Introduction. The following educational elements were all included in one course: oral and written - communication, design, generation of multimedia lessons, use of sophisticated computer software, group learning, and learning by objective. An undergraduate elective course on numerical methods and modeling has been reorganized to focus on the problem of reducing automobile pollution. The course has essentially been turned upside down. In the past, each problem was posed, and the need for its solution explained. Then the general equations were reduced to the point that a viable model was achieved. Several such problems were treated in each quarter. In the revised course, one societal problem is posed, namely automobile pollution, and a catalytic converter is chosen as one solution. The first models are very simple, but are within the reach of the students at the start of the course. Then these models are used to show the need for including additional phenomena, which necessitates more complicated numerical methods. In this way the students practiced ‘just in time learning’. Their motivation was increased, and their success at solving difficult problems was higher. Overview. The course began with an assignment of an essay on the scope of the automobile pollution problem, thereby giving them experience in written communication. One side benefit for the professor is the large number of unusual sources of information the students find. This was the first time in this professor’s experience that material on a CD-ROM encyclopedia was used as a reference in an assignment. The course went through the same topics as in the past: methods to solve algebraic equations, initial value problems, boundary value problems, and partial differential equations. However, each type of equation arose in some manner in the catalytic converter for an automobile. The programs MATLAB and EXCEL were used to solve the various numerical models. The models were improved week by week, and after seven weeks the models were much more sophisticated and powerful than had been achieved in prior courses. Then a design problem was posed. Solution of the design problem involved using the model, working in groups, and presenting the design in written and oral format. At one point in the course the class considered doing experiments to test various catalysts, but this was abandoned as too time consuming. Mathematical Models. In order to make the course manageable the problem was simplified somewhat. The class treated only carbon monoxide (CO), since the details to treat propylene, methane and other components just add complicated bookkeeping that doesn’t improve understanding. The CO kinetics were from Oh and Cavendishl. A variety of models were used, as listed here. The approach was to assign homework problems for each model that invoked some numerical procedure. The models are organized so that by the end of the course all . ..,, ?@Ai~ 1996 ASEE Annual Conference Proceedings ‘..+,yyy’,:

10.1 IntroductionThe variational principles presented in the previous two chapters show that there are problems in fluid mechanics and heat transfer for which variational principles exist. In special cases the variational integral is a minimum or maximum. Often the variational integral is only stationary, and in the general case no variational principle exists at all, except after inclusion of the adjoint problem. The goal of describing processes in terms of maximum and minimum principles is an alluring one which has occupied scientists for centuries. The first such principles were Fermat's principle of least time in optics and Maupertuis' principle of least action in mechanics (see Yourgrau and Mandelstam, 1968). In 1744 the mathematician Euler expressed the hope:As the construction of the universe is the most perfect possible, being the handiwork of an all-wise Maker, nothing can be met with in the world in which some maximal or minimal property is not displayed. There is, consequently, no doubt but that all the effects of the world can be derived by the method of maxima and minima from their final causes as well as from their efficient ones.

Many of the problems in Chapters 1–4 are solved only in a first approximation. A first approximation can be useful in that many qualitative features of the solution are revealed, but more precise answers are often desired. These can be calculated using MWR. In higher approximations the choice of method—or weighting function in MWR—is not crucial, since all methods give similar results. Another criterion can then be introduced, namely, that of convenience in doing the computations. Even the choice of trial functions is not as crucial in higher approximations, provided they satisfy certain criteria, such as a completeness and linear independence. The criterion of convenience can be introduced in the choice of trial functions, too, so that the ease of computation becomes the primary goal. Improved results can be obtained merely at the expense of additional computation without reformulation or additional intervention by the analyst. This goal is achieved below by the orthogonal collocation method, which thereby competes directly with finite difference calculations. Comparisons of computation time in representative applications below indicate that the collocation method is faster than finite difference methods, and this makes MWR of real importance as a computational tool.

We have postponed the question of convergence and error bounds because those subjects require special notation and mathematical background. In the previous chapters we considered only numerical convergence of the successive approximations. This is sufficient for many purposes, but more detailed mathematical justification is often available. We present a summary of results to acquaint the reader with the scope of information available. The theorems are stated in as much detail as space permits, and the mathematically inclined reader is referred to the original references for complete details.

The spin-up of ferrofluids (rotational motion of a magnetic fluid induced by a rotating magnetic field) is examined computationally. Key questions are the impact of the spin viscosity, a mildly non-uniform magnetic field, and the importance of the Langevin magnetization equation. Comsol Multiphysics is used with boundary layer meshes to capture the effect when the spin viscosity is small. The effect of the spin viscosity is examined as it affects the critical magnetic field for non-rotational flow to occur and the magnitude of the rotational velocity and torque. Comparisons are made for the effect of magnetic field when using the Langevin magnetization equation and a linear equation. The equations for flow, magnetic field, and spin velocity are solved in two dimensions as a representation of a long cylinder with the magnetic field oriented perpendicular to the axis of the cylinder and rotated about that axis. Solutions are obtained for spin viscosities as low as 5.8 × 10−14 kg m s−1.

This appendix contains sections titled: General Features Programming Options: Input/Output, Loops, Conditional Statements, Timing, and Matrices Finding and Fixing Errors Eigenvalues of a Matrix Evaluate an Integral Spline Interpolation Solve Algebraic Equations Integrate Ordinary Differential Equations that are Initial Value Problems Plotting Other Applications Import/Export Data Programming Graphical User Interfaces MATLAB Help Applications of MATLAB