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.