Michael F. Doherty

On the dynamics of distillation processes—I: The simple distillation of multicomponent non-reacting, homogeneous liquid mixtures

Abstract The mathematical theory of multicomponent simple distillation processes is presented. Through the analysis it is possible to deduce the characteristics of this inherently dynamical process. It is shown that every azeotropic point and pure component vertex corresponds to a singular point and that both elementary and non-elementary singular points may arise. It is further shown that the temperature and pressure surfaces are naturally occurring Liapounov functions for this system. The latter part of the paper is concerned with design methods. Rayleighs design equation is shown not to extend to multicomponent mixtures and an alternative procedure is presented.

Vapor-liquid phase equilibrium in systems with multiple chemical reactions

Abstract We introduce a new set of composition variables for treating phase equilibria in multicomponent, multi-reaction systems with or without inert components present. These variables provide a way of reducing the dimensionality of the problem and simplifying the analysis. We find that reactive azeotropes occur at points of equal transformed composition in each phase, but not equal mole fraction. Therefore, phase diagrams for complex reacting mixtures are greatly simplified when represented in terms of these new variables, and it is especially easy to identify the presence or absence of azeotropes in the (reacting) mixture. Three examples are given in detail.

The simple distillation of homogeneous reactive mixtures

Abstract The equations describing the simple distillation of homogeneous reactive mixtures are derived, and residue curve maps are computed for ideal and non-ideal systems. These maps show that, by allowing the components of a liquid mixture to react, we can either create or eliminate distillation boundaries. It is also shown that not all non-reactive azeotropes appear as products of the distillation process. Knowledge of these features is fundamental for the design and synthesis of sequences of reactive distillation columns.

Design and minimum-reflux calculations for single-feed multicomponent reactive distillation columns

Abstract A new set of transformed composition variables is introduced to simplify the design equations for single-feed, multicomponent reactive distillation columns. Based onthese equations, a general method of calculating minimum reflux ratios for reactive distillation columns is presented. The new composition variables are also used to derive simple relationships between the dependent design variables which are not evident when the design equations are written in terms of mole fractions.

On the dynamics of distillation processes—III: The topological structure of ternary residue curve maps

Abstract The differential equations describing the simple distillation of azeotropic ternary mixtures place a physically meaningful structure (tangent vector field) on ternary phase diagrams. By recognizing that such structures are subject to the Poincare-Hopf index theorem it has been possible to obtain a topological relationship between the azeotropes and pure components occurring in a ternary mixture. This relationship gives useful information about the distillation behavior of ternary mixtures and also predicts situations in which ternary azeotropes cannot occur.

Homogeneous nucleation of methane hydrates: unrealistic under realistic conditions.

Methane hydrates are ice-like inclusion compounds with importance to the oil and natural gas industry, global climate change, and gas transportation and storage. The molecular mechanism by which these compounds form under conditions relevant to industry and nature remains mysterious. To understand the mechanism of methane hydrate nucleation from supersaturated aqueous solutions, we performed simulations at controlled and realistic supersaturation. We found that critical nuclei are extremely large and that homogeneous nucleation rates are extremely low. Our findings suggest that nucleation of methane hydrates under these realistic conditions cannot occur by a homogeneous mechanism.

The influence of equilibrium chemical reactions on vapor—liquid phase diagrams

Phase diagrams for simultaneous chemical reaction and phase equilibrium are presented for ideal and non-ideal systems. It is shown that reactive-azeotropes can occur even for ideal mixtures. The conditions for formation of reactive-azeotropes in constant-volatility systems are derived. These conditions show that for such systems reactive-azeotropes can occur only when the volatilities of the reactants are either all higher or all lower than the volatilities of the products.

Geometric behavior and minimum flows for nonideal multicomponent distillation

A simple algebraic method for calculating the minimum flows for nonideal multicomponent distillation columns has been developed. The method, which is based on an analysis of the geometry of the composition profiles in the state space, is quite general and applies to ideal, nonideal and azeotropic mixtures. Using the set of composition transforms developed by Barbosa and Doherty (1988, Chem. Engng Sci.43, 1523–1537), the procedure can be extended to include multicomponent reactive distillation columns. For systems with constant relative volatilities, our method and Underwoods method are shown to be identical. The technique provides an accurate and efficient procedure for determining the minimum flows without the necessity of lengthy iterative schemes, as is the current practice. The method is demonstrated using several different multicomponent systems, including the mixture acetaldehyde—methanol—ethanol—water, which exhibits a pair of binary azeotropes, a ternary azeotrope, and a simple distillation boundary in the three-dimensional composition state space.

Chaos in deterministic systems: Strange attractors, turbulence, and applications in chemical engineering

Abstract It is now well established that seemingly innocuous dynamical systems, dissipative or not, can produce complicated phase trajectories and, eventually, chaos. There is now an enormous amount of mathematics and physics literature on this subject, little of which has permeated into the chemical engineering community at large. This article is divided into two parts: the first one considers dissipative systems, the second Hamiltonian systems. One of the major achievements of the research in these areas has been the recognition that both difference and differential models share a group of universal features that go quite beyond the formal details of the models. For example, one of the fingerprints of many dissipative systems is the Feigenbaum cascade of period-doubling. The limit of the cascade is a chaotic situation resembling turbulence. Feigenbaums analysis provides“universal” scaling laws and statistics which characterize“turbulent motions” arising via the period-doubling route. A sound mathematical framework characterizes the behavior of Hamiltonian systems. Chaos arises due to transversal intersections of stable and unstable manifolds belonging to hyperbolic points and prevented from becoming widespread by surviving invariant tori whose behavior is controlled by the Kolmogorov-Arnold-Moser theorem. In this article, emphasis is placed on differential models since they arise naturally in chemical engineering applications. An interesting example, in the context of fluid mechanics, is provided by mixing problems. The particle trajectory of every fluid particle in every fluid mechanical problem is represented by a dynamical system of three ODEs, autonomous if the flow field is steady, nonautonomous if it is not. In two dimensions, the problem can be framed in terms of Hamiltonian mechanics. An analysis of the particle trajectories (flows) provides a rational way to create chaotic (or good) mixing. Other applications can be drawn from fluid mechanics, transport processes, and reaction engineering. It is important to recognize that these new techniques provide additional ways of interpreting experimental data and they suggest new experiments for discriminating between possible mechanisms and routes to complex or turbulent behavior.

Computing Azeotropes in Multicomponent Mixtures

Abstract The problem of computing the temperatures and compositions of all azeotropes predicted by thermodynamic models for nonideal, multicomponent mixtures can be formulated as a multi-dimensional root-finding problem. This problem is complicated by the presence of multiple solutions, constraints on the compositions and the complexity of realistic vapor—liquid equilibrium descriptions. We describe a homotopy method which, together with an arc length continuation, gives an efficient and robust scheme for finding solutions. The homotopy begins with a hypothetical ideal mixture described by Raoults Law for which all of the solutions to the problem are known trivially, since they are simply the pure components. There are as many solution branches for the homotopy as there are pure components, and one or more of the branches shows a bifurcation when azeotropes are present in the mixture. Solutions for the azeotropes are found from the limiting behavior of the homotopy and we show that azeotropes containing c components can be found from a series of c -1 bifurcations in the solution branches of lower dimensions. There is no restriction on the dimension of the problems other than the availability of an accurate thermodynamic model; examples containing up to five components are described.