For an appreciation of what mathematical analysis implies, the reader is referred to the stipulations of SIAM (Society for Industrial and Applied Mathematics) Journal on Mathematical Analysis. Broadly, analysis may be regarded as the process of extracting logical sequences of mathematical propositions from some starting point. Such a starting point, in our context, is usually a mathematical model of a system of interest, the purpose of analysis being to determine a sequence leading to meaningful conclusions about the system. The conclusions are, of course, contingent on the starting hypotheses, whose veracity is evaluated by comparison with experiment towards an iterative process of model refinement. Most importantly, the goal of mathematical analysis is often to extract properties of the solution without necessarily having to determine it completely. The discussion that follows is in the light of comments made here.

A fundamental aspect of mathematical analysis pertaining to the solution of mathematical equations is functional analysis. The authors' interest in the area was spurred by a course in the Mathematics department at the University of Minnesota during the mid-1960s, whose contents were published later into a book by Naylor and Sell (1972). (In fact the influence wielded by the Mathematics Department at the University of Minnesota on chemical engineering even in the 1950s and 1960s was extraordinary. In particular, Hans Weinberger's text (Weinberger, 1965) and course on partial differential equations were both very popular among chemical engineering students.) An article by Ramkrishna (1979) extolled the virtues of chemical engineers learning functional analysis. The book by Ramkrishna and Amundson (1985) demonstrates methods for symmetrizing apparently nonself-adjoint operators in a variety of applications to systems in which transport and/or reaction processes take place. (It is worth noting that Davis and Thomson (2000) have recently published a book on linear algebra and linear operators that makes liberal use of Mathematica. Two other books on applied mathematics for chemical engineers, but less “operator” based in their approaches, are by Varma and Morbidelli (1997) and by Rice and Do (1994).) While several of their applications have found favor in books on transport phenomena (Deen, 1998), examples of resolving elliptical partial differential equations into self-adjoint first-order pairs have remained less noticed. The technique, originally developed by the authors for solving boundary value problems with *oblique derivative* boundary conditions (Ramkrishna and Amundson, 1979) was shown to solve analytically the Graetz problem with axial diffusion and a general class of conjugated transport problems (Papoutsakis et al., 1980; Papoutsakis and Ramkrishna, 1981). (It is interesting to note that Acrivos (1980) recovered the leading terms in the expansion of Papoutsakis et al. (1980) by singular perturbation.)

The work of Wei and Prater (1962) is an example of how spectral information of an unknown self-adjoint operator can be gleaned from dynamic data in order to determine the operator. With this interpretation, the work of Wei and Prater falls in the domain of what is known in the mathematics literature as *Inverse Scattering Theory* (see, for example, Agranovich and Marchenko (1963)). Although it is not evident to us that such work exists in the chemical engineering literature, it is of interest to cite here the work of Kravaris and Seinfeld (1985) who have solved “inverse” Sturm-Liouville problems that have a number of important applications. Here, the solution to a Sturm-Liouville problem is specified as known (usually from “noisy” measurement) and the problem is to determine the Sturm-Liouville operator by which is meant functions appearing in the Sturm-Liouville expression such as, for example, a spatially varying diffusion coefficient. Seinfeld and Kravaris (1982) present applications to flow distributions in petroleum reservoirs. Inverse problems occur in several applications in chemical engineering, and considerable scope exists for further work in this area.

#### Model simplifications

Within the scope of what we have termed “analysis” is a set of ways to negotiate a model so that its evaluation is simplified in some way. The complexity of chemical process systems is such that simplification of one form or another is frequently forced on the modeler. Besides, the level of modeling must match that of observation. There have been numerous books on modeling, some specific to chemical engineering, that make interesting reading. From the academics' perspective, we recommend Aris (1999) and Denn (1986) as excellent references on the subject. The book of Franks (1972) is of interest as the author's perspective also bears the stamp of industrial experience that should reflect the extent of influence that mathematical modeling has had on chemical engineering practice.

For the purposes of our discussion, it is convenient to regard the behavior of a system as the evolution of its state along spatial and/or temporal coordinates. The simplifications to which we referred may either apply to the description of its state or the organization of its evolutionary coordinates. Both types have been of interest to chemical engineers and practiced vigorously in the past. An example of the first type is in the use of *continuous* mixtures when the number of components in the system is very large. Similarly the numerous stages in a large multistage process, may be “imbedded” into a continuously evolving spatial coordinate. The implication here is that the continuous formulation can condense a large number of discrete model equations into a small number of distributed model equations, which can sometimes be solved more easily than the original set. This idea has emerged from the work of Zeman and Amundson (1963) and exploited in a series of succeeding articles for polymerization reactions. More recently, McCoy and Madras (2001) have used continuous formulations for polymerization systems, using the method of population balances. Continuous formulations have also been used in modeling distillation processes. There are other examples of continuous distribution of components in the literature (Aris, 1989).

An alternative mode of model simplification consists in “lumping” of chemical species, usually of homologous sets thus forsaking the distinction between elements in the lump. The lumping of monomolecular systems can be discussed with some degree of exactness and has been done by Wei and Kuo (1969). Lumping is tacitly assumed in bioreactor kinetics even in the case of nonlinear reactions. However, in dealing with chemical reaction systems, Astarita and coworkers have addressed the problem of lumping nonlinear kinetic systems in several articles of which we cite two (Astarita and Ocone, 1988; Aris and Astarita, 1989).

We shall briefly deal with model simplifications connected with temporal and spatial (evolutionary) coordinates. Simplifications of temporal behavior based on pseudo-steady-state approximations are a common occurrence in modeling, although concerns have occasionally been expressed on conditions for their validity (Heineken et al., 1967). There have been other simplifications too based on disparate time scales of system variables (Palsson and Lightfoot, 1984).

Model simplifications associated with spatial coordinates have been based on some sort of spatial averaging. We have already referred to the use of local volume-averaging in flow through porous media and in dealing with multiphase flows. (It is interesting to note that Whitaker (1969) derives Darcy's law by volume-averaging the Navier-Stokes equations, thus providing a genesis of the permeability tensor specific to this route.)

Other kinds of local averaging have also been of interest to chemical engineers. For example, transport of chemical species in conduits of constant cross-section in fully developed flows are of diverse interest to applications. In the foregoing situation, the two-dimensional (2-D) axisymmetric transport of a solute along the radial and axial (flow) directions was shown, in a noted publication, by Taylor (1953) to be closely approximated by 1-D transport along the axial direction using a solute concentration variable suitably averaged along the cross-section. This “dispersion” theory provides an *effective dispersion* coefficient for the axial direction, which depends on the dimension of the tube cross-section, the solute's molecular diffusivity in the medium and a coefficient determined by the fully developed velocity profile. The theory attracted considerable attention among researchers in chemical engineering because of its applicability to chemical reaction engineering and to a large class of dispersion processes in engineering and biological systems. (The name of Danckwerts is inextricably connected with the boundary conditions associated with axial dispersion models. Peculiarly, the boundary conditions were found to date back to an early publication by Langmuir (1908). Not withstanding this fact, these boundary conditions, continued to be named after Danckwerts, are among the most discussed and most used in the analysis of axially dispersed systems (Danckwerts, 1953). This famous article of Danckwerts introduced the concept of *residence time*, a statistical concept, that significantly contributed to predicting the performance of imperfectly mixed systems. While it was undoubtedly true that Danckwerts possessed an adeptness in mathematics far above that which was characteristic of his times, he had increasingly felt in subsequent years an overabundance of mathematics in chemical engineering (Danckwerts, 1982). Aris, however, makes the percipient observation in his Danckwerts memorial lecture in 1990 that Danckwerts' position on mathematics was that of an “intelligent mistrust” rather than of an “ignorant aversion,” an observation with which we must wholly concur.) Aris (1980) provides an interesting perspective of such model simplifications along with a healthy coverage of the literature in this area. Of particular interest are the numerous contributions of Gill and coworkers of which we cite Gill and Sankarasubramanian (1970), those of DeGance and Johns (1978a, b), and of Brenner (1980) the last of which is in periodic porous media. Simplifications, which yield “lower” dimensional models, abound in fluid mechanics; the slender body approximation (Leal, 1992) is one such example.

More recently, Balakotaiah and coworkers have initiated what appears to be a novel approach to model reduction (Chakraborty and Balakotaiah, 2002) by employing the Liapunov-Schmidt reduction technique (see, for example, Golubitsky and Schaefer (1984)) to perform spatial averaging. They report that the procedure leads to two coupled differential equations containing two different averages of the concentration and show a number of interesting applications.

While the subject matter of this subsection is not particularly homogeneous and can be vastly diverse, its coverage has been motivated by its being a legitimate component of analysis. We now consider another aspect of analysis that has had a profound effect on fluid mechanics, transport phenomena, and chemical reaction engineering, viz., the method of asymptotic expansions. This method, which subsumes both regular and singular perturbations, produces analytical expressions as approximate solutions, generally by linear methodology, to nonlinear problems for asymptotic limits of suitably chosen parameters. The monumental text by Leal (1992) provides an excellent account of asymptotic analysis in transport processes.

#### Analysis of nonlinear systems

Nonlinear problems occur more commonly than linear problems in chemical engineering. In chemical reaction engineering, the nonlinearity is inherited from kinetics, the temperature dependence of rate constants, thermodynamic relationships, the dependence of transport coefficients on thermodynamic state, and so on. In fluid mechanics, nonlinearity is contained in the inertial terms, interfacial boundary conditions, and, frequently, in constitutive relationships for Non-Newtonian fluids.

We focus on evolutionary equations along the time axis that are concerned with the dynamics of the state of a system. If the state of the system is in a finite dimensional space and the rate of change of that state is governed only by the local state, the model equations are a set of ordinary differential equations. This is the case with many models in chemical processes where spatial uniformity is promoted by some mechanical device. If, however, the system state is described by functions defined on some domain of other coordinates, spatial or otherwise, then the model equations are partial differential in nature. Integral and integro-differential equations are also encountered in chemical engineering in dealing with processes where a *local* change of a system variable can be effected by what is associated with a larger domain of the state space. Such is the situation in transport processes involving radiation, and in dealing with particulate systems via the method of population balances.

Of particular interest in the behavior of dynamical systems is the existence of equilibrium (or steady-state) solutions, free of temporal dependence. They are frequently the desired operating states of a process and are obtained from the model by equating the time derivatives to zero. When the dynamics is governed by ordinary differential equations, the steady states are characterized by algebraic equations. With dynamics described by partial differential equations, the steady states would be described by ordinary or partial differential equations. It is then no surprise that the nonlinear analysis of dynamic models with ordinary differential equations took precedence in the literature. Naturally, the field of chemical reaction engineering, which features the proverbial continuous, stirred tank reactor, is the first to enter the discussion.

In dealing with steady states of a continuous, stirred tank reactor, the issue of primary interest is that of multiplicity of solutions of the algebraic equations. (While there had been discussions about periodic phenomena in chemical reaction equations and some other unusual happenings, there had been little discussion about whether a given system could exist in more than one steady state and whether there was a rational procedure to predict such a thing. This, along with stability, apparently had never been considered. Although it seems almost miraculous that it wasn't the clue to being able to discuss, this was the earlier study of nonlinear mechanics as developed and used by Poincare, Liapunov, Minorsky, and some others. In principle, this only involved the idea of considering what happens to a steady-state system when perturbed only infinitesimally from that state. This idea was exploited in some chemical systems with a continuous stirred-tank reactor.) Looking at the various kinds of steady states, van Heerden (1953) produced the familiar sigmoidal shaped diagram, which suggests the existence of multiple steady states for even a very simple first-order exothermic chemical reaction. What had remained was to apply the idea of perturbing the steady state by a small amount and following the transient path or rather to carry out the computation from a variety of initial states and determine where the steady states were. This was accomplished by Bilous and Amundson (1955) with computations on an analogue computer. (A second article by Bilous and Amundson (1956) investigated the empty tubular reactor for parametric sensitivity and stability using the Laplace transform, recycle, and perturbation methods.) It was no surprise that the profiles led to the steady states predicted from van Heerden's sigmoidal curve and that there was, in general, for one simple reaction, one stable steady state which corresponds to a single state in van Heerden's diagram. (More generally, for more complex systems, transient profiles would produce an odd number of states and, in a rare case, no stable steady states, but rather a limit cycle implying perpetual oscillation about one of the steady states. With a more detailed and precise analysis by considering the algebra of the linearized system and plotting in the phase plane the full equation, there can be a great array of transients showing there is convergence to stable steady states and transient avoidance of unstable steady states. This analysis was carried out in many very complicated cases with multiple reactions, multiphase systems, polymerization systems, and more. In one interesting case with a polymerization system, where there might be many possible states in the same system, it was found that the difficulty of control was the attempt to operate about an unstable state.) The three part article by Aris and Amundson (1958a) considers control of the stirred tank reactor and the manipulation of steady states and their stability through variation of the tuning constant for proportional control. It may be regarded as the first article on bifurcation analysis with respect to the tuning constant as the parameter although the results were not presented in the usual bifurcation plot in bifurcation theory.

Fredrickson and Tsuchiya (1977) provide some of the early developments in the field of bioreaction engineering not only with respect to the nonlinear issues under discussion, but also from several other fundamental viewpoints.

There was considerable interest in predicting via analysis the conditions under which a system has a unique steady state or multiple steady states. A mathematical monograph by Gavalas (1968) provides an outstanding initiation into nonlinear analysis of chemically reacting systems, in which the nature of steady states and their stability is presented in considerable depth. This monograph represents a milestone in the use of higher mathematics by chemical engineers. We note two other significant and early contributors to the issue of uniqueness and stability of steady states, Luss and Varma, the former with special focus on catalyst particles (Luss, 1977) and the latter more concerned with reactors (Varma and Amundson, 1972). A most comprehensive treatment of reaction and diffusion in porous catalysts is available in Aris (1975). A detailed nonlinear study of diffusion and reaction in carbon burning is made in Amundson and Mon (1980). A review by Varma and Aris (1977) provides a comprehensive account of the nonlinear behavior of stirred and tubular reactors. Following contributions in the area of linearized stability analysis, there were also early attempts to apply Liapunov's “direct” method to determine the region of stability via the construction of Liapunov functions. Warden et al. (1964) construct Liapunov functions for control of simple reactions, second-order reversible reactions, and for polymerization systems in a stirred-tank reactor. Limit cycles are studied in detail and extensive numerical work is presented. We also call attention to an extensive review article on periodic phenomena in chemical reaction systems by Bailey (1977).

The literature in the application of nonlinear analysis to chemical reaction engineering has undergone prolific growth ever since the publication of Bilous and Amundson (1955). We shall, therefore, cite only certain articles that we hold to be landmark contributions in this area, indeed regretfully mindful of likely omissions in the process.

Although nonlinear analysis with regard to multiple equilibria and complex transient behavior had taken root among chemical engineers, the systematic application of bifurcation theory did not occur until after the notable publications of Uppal et al. (1974, 1976). Ray (1977) provides a detailed perspective of bifurcation theory in chemically reacting systems. The bifurcation analysis of stirred-tank reactors was followed by that of tubular reactors by Jensen and Ray (1982). Ray and coworkers are noted for their outstanding contributions to the nonlinear analysis of polymerization reaction systems together with an impressive experimental demonstration of many of the phenomena (Ray and Villa, 2000). The application of bifurcation theory to biological reactors first appeared in the publication of Agrawal et al. (1982).

Hlavacek and coworkers have a sustained record of contributions to the nonlinear analysis of chemical reactors and associated computational methods (Hlavacek and Vanrompay, 1981; Seydel and Hlavacek, 1987). From a computational point of view, Gilles and coworkers have also contributed significantly to nonlinear dynamics of chemical processes (Holl et al., 1988; Kroner et al., 1990). We also cite the book of Finlayson (1980) in this regard. Schmitz and coworkers provide a very comprehensive account of oscillatory phenomena in catalytic reactions (Schmitz et al., 1980; Sheintuch and Schmitz, 1977).

While the focus in the foregoing discussion on nonlinear analysis has been primarily on chemical reactors, there also has been an abundance of contributions in other areas of chemical engineering. An example in fluid mechanics is to be found in the work of Kevrekidis et al. (1994) who made a thorough investigation of the classical Benard convection instability. Shaqfeh (1996) provides a review of fluid instability from purely elastic sources. We also cite the work of Kumaran (1995), among several other publications of his, on the stability of flow in a tube surrounded by a flexible viscoelastic medium. Joshi et al. (2001) have most recently reviewed work on the hydrodynamic stability of multiphase reactors. Steady-state uniqueness and multiplicity have been of interest to several workers in distillation processes (Lucia, 1986; Kienle and Marquardt, 1991; Jacobs and Krishna, 1993; Kienle et al., 1995). Morari and coworkers have contributed significantly to the nonlinear analysis of both homogeneous and heterogeneous distillation processes with numerous publications of which we cite but two (Bekiaris et al., 1993, 1995).

Another landmark event in the application of nonlinear methods in chemical engineering is in the use of singularity theory developed by Golubitsky and Schaefer (1984). Balakotaiah, Luss and coworkers stand out in the global investigation of multiplicity using singularity theory with numerous publications of which we quote only the early ones (Balakotaiah and Luss, 1983, 1984). Polizopoulos and Takoudis (1986) applied singularity theory to reactions on catalytic surfaces. Guttinger and Morari (1997) have recently used singularity theory for predicting multiple steady states in distillation processes.

The issue of progression of periodic behavior of deterministic systems to chaos appeared first in the chemical engineering literature towards the latter half of the 1970s (Schmitz et al., 1977). This subject of transition to chaos and its applications have been discussed in a comprehensive review article by Doherty and Ottino (1988). Khakhar et al. (1986) provide an application of the theory to mixing. The contributions of Ottino and coworkers in the area of chaotic mixing are indeed unique and are best represented in the book by Ottino (1989).

A most interesting issue associated with nonlinear systems is their capacity to form spatial or spatio-temporal patterns. The issue of patterns first arose with the famous work of Turing (1952). Patterns may occur in a set of similar systems (forming either a discrete or continuous family) which interact directly and/or indirectly through a shared environment. Such an ensemble could display a hierarchy of symmetries in which the most symmetric would be a “homogeneous pattern” that is capable of “splitting” to more asymmetric ones under suitable circumstances. Scriven and coworkers were the first to consider pattern formation in the chemical engineering literature. The setting here was a string of cells in which chemical reactions took place and interaction between them was facilitated through transport (Gmitro and Scriven, 1966; Othmer and Scriven, 1969, 1974). Schmitz and Tsotsis (1983) have studied pattern formation in an interacting assembly of catalyst particles, while Arce and Ramkrishna (1991) show that indirect interaction between catalyst particles through the intervening fluid can itself lead to pattern formation. Hudson et al. (1993) show pattern formation occurring in the electrodissolution of iron. Also noteworthy is the work of Shinbrot et al. (1999), who provide an interesting study of chaotic mixing in granular flows. Qin et al. (1994) show an interesting motivation for studying pattern formation by introducing a control element in it. We note in passing that the use of group theory to derive bifurcation equations for studies of pattern formation may be of interest (Sattinger, 1979).

The University of Minnesota, through the Institute of Mathematics and its Applications, has actively sponsored symposia on the application of mathematics to chemically reacting systems and has had one on pattern formation (Aris et al., 1991).