Marek Berezowski

Chaotic dynamics in homogeneous tubular reactors with recycle

This note presents a bifurcation analysis of the homogeneous tubular reactor with recycle and shows that the presence of recycle also may cause more complex behaviour, including chaotic behaviour.

Effect of delay time on the generation of chaos in continuous systems. One-dimensional model. Two-dimensional model – tubular chemical reactor with recycle

The present study deals with a theoretical analysis of the effect of delay time of energy transport upon the generation of complex dynamics in continuous physical system. The importance of this time for the presence of quasi - periodicity and chaos in a reactor is demonstrated. The considerations are preceded by the analysis of one - dimensional mathematical model.

Approximate characteristics of a moving temperature front in a fixed-bed catalytic reactor

Abstract The study deals with a nonstationary process of mass and heat transfer accompanied by a chemical reaction occurring in a catalytic reactor. Based on the assumptions of the ‘ideal thermal front’ in the reactor, approximate solutions are obtained for the equations that describe the process. Thus, relations are derived which define the principal properties of the thermal front, namely its propagation velocity in the bed and the maximum temperature of the front. The above relations express these properties in terms of dimensionless numbers that characterise the chemical reaction taking place in the reactor and the operating parameters of the vessel. Good agreement is found between the front properties as calculated using the approximate formulae and those yielded by the integration of the complete model equations, i.e. the exact values. A method is proposed for determining approximate temperature profiles in the bed, which is by far simpler and less time-consuming than the integration of a complete set of partial differential equations. Both the formulae derived and the method proposed for calculating the temperature profiles along the bed may be useful in the design of reactors with the periodic reversal of the feed mixture. They enable the effect to be analysed of the various operating parameters upon the propagation velocity of the thermal front and its maximum temperature, without resorting to tedious and time-consuming trial and error methods that require repeated integration of the model equations.

Spatio-temporal chaos in tubular chemical reactors with the recycle of mass

Abstract In this study the results are presented concerning the dynamics of homogeneous tubular chemical reactors with the recycle of mass. A detailed analysis shows that two types of dynamic bifurcation exist, namely, the flip bifurcation (FB) and the Hopf bifurcation (HB). It is demonstrated that each of these two types leads, for given values of the model parameters, to chaotic oscillations. Moreover, the Hopf bifurcation can also generate quasi-periodic solutions. The results are illustrated using temporal trajectories, bifurcation diagrams and Poincare sections.

Fractal solutions of recirculation tubular chemical reactors

Abstract Three kinds of fractal solutions of model of recirculation non-adiabatic tubular chemical reactors are presented. The first kind concerns the structure of Feigenbaum’s diagram on the limit of chaos. The second kind and the third one concern the effect of initial conditions on the dynamic solutions of models. In the course of computations two types of recirculation were considered, viz. the recirculation of mass (return of a part of products’ stream) and recirculation of heat (heat exchange in the external heat exchanger).Three kinds of fractal solutions of model of chemic al reactors are presented. The first kind concerns the structure of Feigenbaum_s diagram on the limit of chaos. The second kind and the third one concern the effect of initial con ditions on the dynamic solutions of models. In the course of computations two types of recircul ation were considered, viz. the recirculation of mass (return of a part of products stream) and recirculation of heat (heat exchange in the external heat exchanger).

Chaotic and non-chaotic mixed oscillations in a logistic system with delay and heat-integrated tubular chemical reactor

The paper deals with the theoretical analysis of a logistic system composed of at least two elements with distributed parameters. It has been s hown that such a system may generate specific oscillations in spite of the fact that the solutions of the mathematical method are characterized by no dynamic bifurcations. It has al so been shown that the time series of the state variables of such a system may behave in a se mi-chaotic way. This means that they have then predictable and unpredictable fragments. The a nalysis has been illustrated by two examples, viz. of a simple logistic model and of a re ctor with feedback.Abstract The paper deals with the theoretical analysis of a logistic system composed of at least two elements with distributed parameters. It has been shown that such a system may generate specific oscillations in spite of the fact that the solutions of the mathematical method are characterized by no dynamic bifurcations. It has also been shown that the time series of the state variables of such a system may behave in a semi-chaotic way. This means that they have then predictable and unpredictable fragments. The analysis has been illustrated by two examples, viz. of a simple logistic model and of a tubular chemical reactor with thermal feedback.

Dynamics of heat-integrated pseudohomogeneous tubular reactors with axial dispersion

The study concerns a theoretical analysis of pseudohomogeneous autothermal tubular reactors with axial dispersion of mass and heat. Based on analysis and simulations it is demonstrated that such a system can generate complex oscillatory profiles of temperature and concentration-periodic or chaotic. These profiles, especially those of aperiodic character, can seriously impair the performance of the system. The effect of three parameters on the reactor dynamics is studied, namely, the cooling medium temperature, the Lewis number and the Peclet number. We consider only relatively small values of the Lewis number in this paper.

A generalized analytical method for determining multiplicity features in chemical reactors with recycle

Abstract An analytical method is developed and illustrated for determining the regions of multiple steady states of a tubular adiabatic reactor with the recycle of the product. The procedure is general (it allows for a single reaction described by any kinetic expression) and does not require an iterative matching of the boundary conditions resulting from the recycle. The method enables the calculation of the so-called hysteresis varieties and the catastrophic sets determining the occurence of multiple steady states.

Generation of chaotic oscillations in a system with flow reversal

The paper is devoted to examining the effect of a reverse flow on a cascade of two nonadiabatic continuously stirred tank reactors (CSTR) connected with stream flow. A suitable numerical method was used to achieve data reduction, particularly in the region of transition between periodic oscillations and chaos. Computation results gave period doubling leading to chaos. Sensitivity to initial conditions has also been investigated. It has been shown that there is a dependence between oscillations of the cascade without reverse flow and dynamic phenomena, which can occur with flow reversal.

Dynamics of Heat Integrated Homogeneous Tubular Reactors

Abstract A detailed bifurcation analysis of heat integrated homogeneous tubular reactors is presented. The analysis is based on a one-dimensional discrete map obtained from the PDE model by means of the method of characteristics. It is shown that, under operating conditions in which the heat integration provides a positive feedback mechanism, the reactor may destabilize through a saddle-node bifurcation, resulting in multiple steady states. Under conditions in which the heat integration provides a negative feedback mechanism, the reactor may destabilize through a flip bifurcation, resulting in periodic solutions. It is shown that these periodic solutions subsequently may undergo a cascade of period doubling bifurcations into chaos. The period doubling cascade is found to follow the proposed universal Feigenbaum scenario. Higher periodic solutions and chaos is found to exist in a relatively large area of parameter space.