Applied Mathematics and Transport
A comprehensive account of mathematics in chemical engineering over the past several decades has appeared in this journal.1The mathematical culture that prevailed in the latter half of the twentieth century spurred the use of analytical tools in chemical engineering applications. While Amundson and Aris were the chief architects of this movement, there were others such as Horn, Acrivos, Brenner, Churchill, and Stewart, to mention only a few, whose publications reflected a significantly higher level of mathematics. It is interesting to note that scholarship pervaded in the early sixties even among industrial circles such as the Socony Mobil Oil (now a part of ExxonMobil) hosting Truesdell’s lectures in continuum mechanics.2 Numerous in-depth mathematical applications emerged from industry such as the determination of rate constants of catalytic reaction systems using spectral information from suitably designed experiments,3 the analysis of lumping reaction systems,4 and so on. Such applications thrived from an edifice that espoused analysis as well as computation.
Fluid mechanics thrived through a major pioneering effort by Acrivos and his group. In particular, Acrivos established the use of matched asymptotic expansions for problems in transport with several analytical results.5 Aris’ Vectors, Tensors and the Equations of Fluid Mechanics provided a rare perspective of the subject more akin to a physicist’s view.6 Scriven’s study of fluid interfaces and Marangoni instability and subsequent involvement with interfacial driven flow set a tone in fluid mechanics distinctly higher than the prevailing level.7 Transport Phenomena by Bird, Stewart, and Lightfoot most effectively changed the teaching of Transport.8 The growth of this development continued with the publication of their (long delayed) second edition. The books of Leal9 and Deen10deserve special mention in preserving the graduate version of transport phenomena. It is puzzling, however, that undergraduate courses in transport phenomena have suffered some loss of level. There are notable exceptions to this observation but consensus may be said to exist on this impression. The unique expertise of chemical engineers on mass transfer is at risk with dwindling attention to this subject. Often a single semester course in graduate Transport is unable to include mass transfer, in spite of its high importance to chemical and biological systems encountered in research and in industrial practice.
Mathematical software such as Matlab and Comsol have enormously eased the application of mathematical models in engineering. The capabilities of symbolic software such as Maple and Mathematica are prodigious in that computation can be postponed to the very final stage when parameter values are to be inserted for numerical evaluation of a mathematical model. Matlab is routinely used in undergraduate courses. Sometimes their premature use has been at the expense of time that could be better spent on issues of formulation and analysis that harbor the essence of innovation. Notwithstanding the strongly positive role of mathematical software in education, most undergraduate instructors sense a drop in the comfort level of chemical engineering students in mathematics. Under these circumstances, elimination of a first-level core course in Applied Mathematics in ChE graduate programs would appear to be patently unwise. Schools strong in the systems area are, however, an exception in this regard since this field of research has made notable strides over the years.
Generally, computational work can be said to be thriving but often without recognition that it can sometimes be coaxed to gain from a timely input of analytical reasoning. For example, instances can be cited in which parameter estimation can be notably improved by local analysis of nonlinear dynamic behavior with bifurcation methods. It is unclear whether the age-old practice of inaugurating the introductory chemical engineering course with dimensional analysis and the Buckingham Pi Theorem is still in vogue. Clearly, parameter estimation of nonlinear models can be aided by non-dimensionalization, as the number of dimensionless parameters is less than the original set; furthermore, a better rationale is frequently available for their initial estimates for iterative computation.
The argument in favor of analytical reasoning is not made to diminish in any way the importance of computational methods but rather to point to its effectiveness in organizing computation. A culture of analysis has the facility to cultivate creative ideas in the profession, for in its absence, there would not be the likes of Gavalas,11who introduced topological methods in reactor analysis, Balakotiah and Luss,12 for their work on singularity theory, Feinberg,13 who made intriguing contributions to chemical reaction network theory, and Jackson,14 who published many interesting papers on particle flows as well as in chemical reaction engineering. Of course, there are many more who deserve mention, but the goal here is to argue for more analysis than provide an exhaustive list of creative analysts.