Multiplicities and instabilities in chemically reacting systems — a review

Ignition–extinction analysis of catalytic reactor models

A detailed analysis of the ignition–extinction and hysteresis behavior of the two widely used catalytic reactor models (packed-bed and monolith) for the case of a single exothermic reaction is presented. First, limiting models are used to determine the minimum adiabatic temperature rise and/or catalyst activity needed to observe hysteresis behavior. Next, explicit expressions are provided for estimating the feed temperature or space time at ignition (light-off) and extinction (blow-out) as a function of the adiabatic temperature rise (or inlet concentration of limiting reactant), effective thermal conductivity, time and length scales (reactor, tube/channel diameter, effective diffusion length and pore size), catalyst activity (or dilution) and heat loss. It is shown that various limiting reactor models such as the thin-bed, long-bed, lumped thermal, adiabatic and strongly cooled cases that are defined in terms of various inter- and intraphase heat and mass dispersion time scales can be used to derive scaling relations that are useful in predicting the ignition/extinction loci for both laboratory scale (with heat exchange) and large scale (near adiabatic) reactors.

Application of Cholette’s model in non-ideal mixing CSTR: a simulation study on dynamic behavior

Non-ideal mixing phenomena are widely found in industrial chemical reactors. In this work we derived the bifurcation formulas for a non-adiabatic CSTR with an irreversible exothermic first order reaction with the non-ideal mixing effect. This is investigated via dynamic behavior simulations based on Chollete’s model. The results show that the non-ideal mixing parameter n (the fraction of the feed entering the perfect mixing zone) determines the variation between six classified regions and dominates the dynamic behavior patterns in the steady-state response diagram. On the other hand, the phase portraits of examples verify the formulas derived in this work. We note that the non-ideal mixing effect has significant importance in CSTR design and control steps. For example, in the safe operating region for an ideal mixing CSTR, non-linear dynamics are obtained by the system under non-ideal mixing conditions (n ≠ 1). The present study has significance and help for chemical reactor design and CSTR control.

Biodiesel Production by Reactive Flash: A Numerical Simulation

Reactive flash (RF) in biodiesel production has been studied in order to investigate steady-state multiplicities, singularities, and effect of biodiesel quality when the RF system approaches to bubble point. The RF was modeled by an index-2 system of differential algebraic equations, the vapor split (ϕ) was computed by modified Rachford-Rice equation and modified Raoult’s law computed bubble point, and the continuation analysis was tracked on MATCONT. Results of this study show the existence of turning points, leading to a unique bubble point manifold,(xBiodiesel,T)=(0.46,478.41 K), which is a globally stable flashing operation. Also, the results of the simulation in MATLAB® of the dynamic behavior of the RF show that conversion of triglycerides reaches 97% for a residence time of 5.8 minutes and a methanol to triglyceride molar flow ratio of 5 : 1.

Injectivity, multiple zeros and multistationarity in reaction networks

Polynomial dynamical systems are widely used to model and study real phenomena. In biochemistry, they are the preferred choice for modelling the concentration of chemical species in reaction networks with mass-action kinetics. These systems are typically parametrized by many (unknown) parameters. A goal is to understand how properties of the dynamical systems depend on the parameters. Qualitative properties relating to the behaviour of a dynamical system are locally inferred from the system at steady state. Here, we focus on steady states that are the positive solutions to a parametrized system of generalized polynomial equations. In recent years, methods from computational algebra have been developed to understand these solutions, but our knowledge is limited: for example, we cannot efficiently decide how many positive solutions the system has as a function of the parameters. Even deciding whether there is one or more solutions is non-trivial. We present a new method, based on so-called injectivity , to preclude or assert that multiple positive solutions exist. The results apply to generalized polynomials and variables can be restricted to the linear, parameter-independent first integrals of the dynamical system. The method has been tested in a wide range of systems.