Satish J. Parulekar

Analysis of axially dispersed systems with general boundary conditions—I: Formulation

Abstract Transient analysis in the past of finite axially dispersed systems with or without a rate process has been subject to the use of the Danckwerts boundar conditions. We present here a formulation which suggests a general set of boundary conditions from which the Danckwerts conditions arise as a special case. The resulting boundary value problems neatly fit the mold of self-adjoint operator theory. An integral transform approach is outlined for those not familiar with operator theory. Discussion of actual solutions to different cases is deferred to two other papers which appear as Parts II and III.

Tubular reactor stability revisited without the danckwerts boundary conditions

Abstract Tubular reactor stability is reconsidered for a wide variety of situations where there is dispersion outside the reactor.A complete set of boundary con

Analysis of axially dispersed systems with general boundary conditions. II: Solution for dispersion in the appended sections

Abstract The axial dispersion problem formulated in Part I[1] of this series is solved here for “case 1” which corresponds to the situation of finite, non-zero dispersion in both semi-infinite appended sections. The solutions are obtained in terms of the spectrum of the operator consisting of a continuous spectrum and frequently a finite set of discrete eigenvalues. Transient solutions are presented for a first order reaction in the axially dispersed reactor for a variety of initial conditions demonstrating where and where not the Danckwerts boundary conditions may be valid. It is shown that “controlling” eigenvalues may be exploited from persisting tails of transient concentration measurements on tracer experiments to obtain dispersion parameters.

Analysis of axially dispersed systems with general boundary conditions—III: Solution for unmixed and well-mixed appended sections

Abstract The axial dispersion problem formulated in Part I[1] of this series is solved for cases 2–4 and 7 identified therein. The cases analyzed here conside appendages to the finite axially dispersed reactor which are either semi-infinite sections of finite dispersion or well-mixed sections of finite capaci The findings possess features akin to those found in Part II[7] for case. 1 but obviously displaying variations in detail.

Dynamics of continuous commensalistic cultures—I. Multiplicity and local stability of steady states and bifurcation analysis

The population dynamics of continuous mixed cultures with pure commensalism or commensalism plus competitive assimilation are investigated in detail. As many as seven steady states are possible in these systems when the growth processes of the two species are inhibited by the substrates they prey on (self-inhibition). The seven states comprise of a complete washout state, two partial washout states and four coexistence states. A priori information about the number and types of steady states possible for a given set of parameters is obtained by dividing the entire multi-dimensional parameter space into several regions. Up to three steady states can be locally, asymptotically stable in these systems. The conditions under which transition takes place from a stable steady state to an unstable steady state (and vice versa) (static and Hopf bifurcation points) have been identified with the application of static and Hopf bifurcation theories. The necessary and sufficient conditions for existence of periodic solutions have been derived. It is shown that bifurcation to periodic solutions is possible for only two steady states.

Optimal Control of Fed-Batch Bioreactors

Optimal feed rate profiles for fed-batch bioreactors have been characterized for the production of cell mass and metabolites using various kinetic forms. A complete nonlinear feedback control based on the state variables and a kinetic model may be realized for certain limiting cases.

Dynamics of continuous commensalistic cultures—II. Numerical results for steady-state multiplicity regions and transient behaviour in-the-large

Abstract The multiplicity and stability characteristics of commensalistic cultures grown in a continuous stirred tank biological reactor have been analysed in detail in Part I. Several numerical examples are presented in this paper to illustrate the important features of the static and dynamic behaviour in these systems. The illustrations provided here pertain to several special cases of the work reported earlier in Part I and supplement its findings adequately. Further, these examples show that the significant two-way interaction between the host and commensal populations in systems with competition leads to the occurrence of oscillatory states in some situations even when the yield coefficients are constant and the maintenance requirement is negligible.

Interfacial surfactant concentrations on an oscillating droplet: Solution of a singular boundary-initial value problem

Abstract Using singular spectral theory, a boundary-initial value problem is solved for the interfacial concentration of a surfactant on a liquid droplet oscillating in a surrounding second immiscible liquid phase. Besides being of value to the specific application the methodology of this paper is useful for a variety of boundary-initial value problems of interest to chemical engineers in which material domains have infinite or semi-infinite extents.

Transients in adiabatic tubular reactors. Axial dispersion models with well-mixed appended sections

Abstract Using boundary conditions more general than those due to Danckwerts, the authors recently showed that significant variations may be encountered in the global stability characteristics of an axially dispersed adiabatic tubular reactor. The numerical results presented in this communication demonstra that for a fixed initial state of the reactor, steady states very different from those predicted with Danckwerts conditions are reached depending on th initial state of the reactor appendages. The cases where the reactor is preceded or succeeded by well-mixed (non-reactive) sections are considered. The findings will be very useful in regulation of the dynamics of tubular reactors.