Mathematics and chemical engineering education

This paper is based on the report developed by the authors at the Mathematics Association of America (MAA) Curriculum Foundations Engineering Workshop held at Clemson University in May 2000. The objectives of this paper are to identify the mathematics needed by chemical engineering undergraduates, to stimulate a dialog between mathematics and chemical engineering educators on this topic, and to determine the most effective way of providing the necessary mathematics. The focus is on subject matter and not on pedagogy. The broad categories of mathematics essential to chemical engineering are pre-calculus foundations (provided by the K-12 school system or the by first-year university mathematics program), linear algebra, calculus, differential equations, and probability/statistics. Important topics within each area can be identified. The best place and time to teach this body of knowledge is open to discussion i.e.. what topics are best taught by the mathematics department, what topics should be incorporated into chemical engineering courses, what topics should be covered in the first and second years, and to what extent should the mathematics be spread out over four years? The effective use of “mathematics technology” in mathematics and chemical engineering courses is discussed. Also, various ways are presented for exposing students to chemical engineering applications in the mathematics courses. Introduction The Mathematics Association of America (MAA) has begun a major analysis of the undergraduate mathematics curriculum through the Committee on the Undergraduate Program in Mathematics (CUPM). Two subcommittees of CUPM are involved in this study: Calculus P ge 601.1 Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright  2001, American Society for Engineering Education Reform and the First Two Years (CRAFTY) and Mathematics Across Disciplines (MAD). The National Science Foundation is involved in this study through Project INTERMATH as part of the Mathematics Across the Curriculum (MATC). These organizations sponsored an interdisciplinary workshop (engineers, mathematicians, and physicists) at the United States Military Academy in November 1999 (Arney and Small, 1999). A workshop, “MAA Curriculum Foundations Engineering Workshop,” for engineers (chemical, civil, electrical, and mechanical) was sponsored and hosted by Clemson University in May 2000. Mathematicians also participated. At some of the sessions all of the disciplines met to address common problems, and then the engineering disciplines met separately to address problems common to that specific discipline. At least one mathematician participated in these disciplinespecific discussions. The participants addressed questions on (1) concepts, problem solving skills, and the desired balance between them; (2) mathematics technology; (3) mathematics education reform and education reform in the specific discipline; and (4) instructional techniques. Each of the engineering disciplines developed a report that addressed these questions. These reports and reports from workshops for the sciences and for mathematicians will be used by the MAA to produce detailed curricular recommendations for the first two years of undergraduate mathematics instruction. What follows is the report developed by the authors in the chemical engineering sessions at the Clemson University MAA Workshop. What Chemical Engineers Do Since this is a report for mathematicians, we thought an appropriate introduction would be to try to say what chemical engineers do, why we need mathematics, and how we use it. A reasonably broad definition of what we do is that chemical engineers design materials and the processes by which materials are made. Traditionally, chemical engineers have been associated with the petroleum and large-scale chemical industries, but especially in recent years, chemical engineers have been involved in pharmaceuticals, foods, polymer processing, microelectronics and biotechnology. The core subjects that underlie and unify this broad field are thermodynamics, chemical reaction processes, transport processes i.e., the spatial and temporal distribution of mass, momentum and energy, and process dynamics and control. On top of this fundamental framework, a central emphasis of chemical engineering education is model building and analysis. Good chemical engineers bring together the fundamentals to build a model of a process that will help them understand and optimize its performance. To be good at model building and analysis, students must have the mathematical background to understand and work with the core scientific areas, as well as to find solutions to the final model that they build. Here’s an example. A starting point for understanding any process is writing down the conservation laws that the process satisfies: for conserved quantities, accumulation=input – output. Depending on the level of detail of the model, this equation might be, for example, a large system of linear algebraic equations that determine the relationships between fluxes of chemical species throughout the process (a species balance), or it might be a parabolic partial P ge 601.2 Proceedings of the 2001 American Society for Engineering Education Annual Conference & Exposition Copyright  2001, American Society for Engineering Education differential equation governing the temperature of fluid in a reactor. In the thermodynamics of multiphase systems, energy is conserved but takes on a variety of forms; a good knowledge of multivariable differential calculus is essential here to keep track of everything. Math for Chemical Engineering We do not view our role here as one of prescribing the mathematics curriculum – we do not want to see taught only what students can “get by” with knowing. Nor do we want to come down on either side of the “traditional” vs. “reform” debate – it is likely that both sides are right, to an extent. The following represent our general thoughts on subject matter and emphasis: Pre-calculus Foundations By foundations, we mean basic knowledge of • families of functions (polynomial, exponential…) in terms of data, graphs, words and equations, basic trig identities, properties of logarithms, etc. • equations and inequalities • basic logic and algorithms • small linear systems of equations • coordinate systems • basic arithmetic and manipulation skills. Mastery of these areas is crucial. Probably the most important role that the mathematics education community can play is to actively critique the pedagogy of K-12 education – to help sort out which “reforms” are productive from those that are merely ed-school fads – and to encourage schools not to neglect the education of the more mathematically inclined students by focusing the curriculum too narrowly on the average performer. Another important role here is to provide programs that help K-12 math teachers understand something about the applications of the math that they teach (engineers should do more here). Linear Mathematics We feel that our students would benefit from earlier exposure to the basics of linear systems in R, particularly • the geometry of linear spaces • vector algebra (especially in 3D) • existence and uniqueness, Gaussian elimination, geometric interpretation, over determined and under determined systems and least squares problems for Ax=b • characteristic polynomial and diagonalization, Jordan form, range and null space of A, and geometry for Ax= λx. At Wisconsin, for example, there is a course on “linear algebra,” which introduces these notions and applies them to systems of ordinary differential equations (see below). Many chemical engineering students take this in lieu of the traditional differential equations class.