Some observations on uniqueness and multiplicity of steady states in non-adiabatic chemically reacting systems

An investigation of uniqueness and multiplicity of the steady states in two non-adiabatic chemically reacting systems, the stirred tank and the tubular reactors, is presented. Exact uniqueness and multiplicity criteria are presented, and some conclusions derived therefrom, for the stirred tank reactor. For the tubular reactor, some numerical results are reported and multiplicities of the type 1-3-1, 1-3-1-3-1 and 1-3-5-3-1 noted. Criteria obtained earlier and those obtained from a lumping approximation of the distributed system are evaluated. In addition, two asymptotic solutions are obtained for the non-adiabatic tubular reactor.     

Determination of multiple steady states in oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model

A family of an enzymatically catalyzed reaction network is studied, which involves the oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model in an isothermal continuous flow stirred tank reactor (CFSTR). This system consists of 11 coupled non-linear equations and is determined to have the capacity to exhibit computational multiple steady states. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability, hysteresis and bifurcation are discussed. Moreover, the capacity of steady state multiplicity is extended to its family of reaction networks.     

Experimental and theoretical investigation of parametric sensitivity and dynamics of a continuous stirred tank reactor for acid catalyzed hydrolysis of acetic anhydride

The continuous stirred tank reactor is a dynamic system exhibiting nonlinear behavior such as multiplicity and oscillations and, in certain range of operating conditions, may exhibit a parametric sensitivity where small changes in one or more of the input parameters lead to large changes in the output variable. In the present work, hydrolysis of acetic anhydride reaction system was used to demonstrate the existence of parametric sensitivity with respect to the input parameter, the cooling water flow rate. The applications of parametric sensitivity analysis were used for detection of parametric sensitivity in a continuous stirred tank reactor using catalyses hydrolysis of acetic anhydride reaction system. Also, theoretical investigation revealed that the effect of wall capacitance has definite influence on the dynamics of continuous stirred tank reactor. The continuous stirred tank reactor showed parametric sensitivity both in the regions of uniqueness and multiplicity, and a mathematical model was developed for the reactor. The numerically simulated results are in satisfactory agreement with the experimental data.     

Analysis of the CSTR with two-phase flow under phase equilibrium conditions

A mathematical model of the flow gas-liquid continuous stirred tank reactor (CSTR) is presented under the assumption of thermodynamic equilibrium of the two-phase flow. Phase equilibrium was calculated using the Redlich-Kwong-Soave (RKS) equation of state. An efficient algorithm based on homotopy method was used to obtain a numerical solution of the system of nonlinear algebraic equations of the model. By an example of the liquid-phase hydrogenation of benzene, steady states were analyzed in isothermal and adiabatic regimes and at partial removal of heat released as a result of exothermic reaction. It was shown that under certain conditions, a multiplicity of stationary solutions is possible due to nonlinearity of the kinetic reaction, on the one hand, and deviation of the reaction mixture properties from the ideal state, on the other hand.     

Experimental study of the behaviour of an isothermal continuous stirred tank reactor in the course of an autocatalytic reaction

The dynamic behaviour of an isothermal continuous stirred tank reactor and of a two-element series of continuous stirred tank reactors in the course of an autocatalytic reaction of bistrichlormethyl-trisulphide with aniline, in methanol at 20°C, was studied. It was shown that the regime of the continuous stirred tank reactor and of the series is unstable ia a certain region of the conversion of the initial reaction component (bis-trichlormethyl-trisulphide) and that in a certain interval of the reaction mixture feed rate the reactor can operate at a single value of the feed rate in two (the series in three) stable states. The values of the bis-trichlormethyl-trisulphide conversion in steady states, as well as the dependence of the conversion on the period of stabilization of the reactor regime, determined experimentally in a continuous stirred tank reactor and in the series are in good agreement with the values computed from the kinetic data, determined in a discontinuous stirred tank reactor.     

An Approach for Achieving Unstable Convergence for Non‐Isothermal CSTRs

The nonlinearity introduced by the temperature variation in the non‐isothermal continuous stirred tank reactor (CSTR) exhibits concentration multiplicity for certain parameter ranges. The dynamics of reactor temperature under these conditions would be very difficult to observe and maintain. In this work, a novel algorithm is proposed to stabilize the system by designing a cascade of CSTRs that are capable of achieving this inaccessible steady state. Optimization of reactors parameters is performed in an iterative manner to achieve this solution. For a first‐order reaction rate, this method was successful in achieving the inaccessible steady state temperature of 312.5 K using three CSTRs cascades. The suggested algorithm is presented both graphically as well as using computational optimization techniques. The transient simulation studies using the above three CSTRs showed that the unstable steady state is achieved. The newly designed cascade meets the design criteria and achieves the locally unstable steady state temperature to a high degree of accuracy.     

The regions of multiple stable steady states of a biological reactor with wall growth,utilising inhibitory substrates

The stirred tank biological reactor with wall growth and an inhibitory substrate is analysed. The necessary and sufficient conditions for the existence and stability of a multiplicity of steady states are given. Diagrams are presented showing the effect of wall growth, reactor feed concentration and a kinetic inhibition parameter on conversion and on the existence of multiple steady states.     

Multiple steady states,viscosity, and high conversion in continuous free-radical polymerization

Using existing literature data on the rate of a bulk polymerization of styrene in a batch reaction carried to high conversion, it is mathematically demonstrated that there is a clear possibility of the existence of multiple steady states induced by viscosity effects in isothermal continuous stirred tank reactors. In solutions of high viscosity, the rate of free-radical polymerization increases with conversion, reaching a peak at very high viscosity, then falling off rapidly. Given this sort of behavior, it is demonstrated mathematically that steady-state mass balance solutions are possible at three levels of conversion. The lower and higher steady states are stable while the middle steady-state condition is shown to be necessarily unstable. This multiplicity of steady states with its particular problems of stability is analogous to the much studied phenomena of temperature stability. It is closely related to the problems of concentration stability characteristic of autocatalytic and heterogeneous catalytic reactions. This multiple steady-state problem is qualitatively discussed in relation to reactor stability, control, and optimization.     

Parametric Analysis of the Models of a Stirred Tank Reactor and a Tube Reactor

A parametric analysis is performed for a basic model of the combustion theory with dimensional parameters. The model describes the dynamics of a firstorder exothermic reaction in a wellstirred tank reactor. Parametric dependences of steady states on dimensional parameters, curves of multiplicity and neutrality of steady states, and parametric and phase portraits of the system are constructed. Regions of multiple steady states and selfexcited oscillations and a region of technological safety are distinguished. Combustion regimes in a continuous stirred tank reactor are compared to those in a tube reactor.     

Modeling of the Nitration of Amyl in a Continuous Stirred Tank Reactor and a Tube Reactor

Parametric analysis of mathematical models of amyl nitration in a continuous stirred tank reactor and a tube reactor was erformed. Curves of steady states versus control parameters, multiplicity and neutrality curves of the steady states, and parametric and phase portraits of the system were plotted. The critical values of the input concentration and the heat-transfer coefficient leading to disturbance of the steady-state nitration of amyl were obtained for the tube reactor.     

精解全混流反应器热衡算

依热力学理论 ,建立了科学衡算反应器的热力学方程。就国内外一些论著中的疑点、误点进行了辨析、论证 ,得出了理想流动反应器之一的全混流反应器 (continuousstirredtankreactor)新型的热量衡算方程精确解 :(△Hr) T0 FA0 XA+FtCpt(T -T0 ) =U(Tm-T)A ,说明了其应用 ,为反应器严谨的热计算提供了可靠的方法。本文论述谨以全混流反应器为代表 ,但其方法不失通用性 ,可望在反应器热量衡算中广泛应用     

The determination of multiple steady-states for a family of catalytic reactions in an isothermal CFSTR

The Subnetwork Corollary provides some sufficient conditions for which a complex reaction network could be determined to have capacity of admitting multiple positive steady-states, if it contains a subnetwork that also admit multiple positive steady-states. A family of reaction networks, involving catalytic reactions in an isothermal continuous flow stirred tank reactor (CFSTR), is determined to have the capacity to exhibit steady-state multiplicity by implementation of the Deficiency One Algorithm and the Subnetwork Corollary. A set of rate constants and two corresponding positive steady-states are constructed for several cases.     

The determination of possibility of multiple steady states for complex catalytic reactions in an isothermal CSTR

A class of complex reaction networks with deficiency from two to six, involving isothermal catalytic reactions in a continuous flow stirred tank reactor (CSTR), is determined to have the capacity to admit multiple positive steady states by implementation of the combination of the Deficiency One Algorithm and the Subnetwork Corollary. A set of rate constants and two corresponding positive steady states, one stable and the other one unstable, are constructed for four examples.     

A radial cell model for a tubular polymerization reactor

A cell model for the prediction of temperature and concentration gradients in a nonisothermal tubular polymerization reactor at steady state is presented. Both radial and longitudinal gradients are considered. The complete molecular weight distribution is calculated as well as the leading moments of the distribution. The model is easily reduced to predict the performance of a plug-flow tubular reactor, batch reactor, and continuous stirred tank reactor (CSTR). The specific polymerization mechanism application consists of free-radical initiation, propagation, and combination termination.     

Dynamics of a CSTR for styrene polymerization initiated by a binary initiator system

The steady state and dynamic behavior of a continuous stirred tank reactor has been analyzed for free radical solution polymerization of styrene initiated by a mixture of two initiators having different thermal stabilities. From the steady state analysis of the reactor model with a mean residence time as a bifurcation parameter, four unique regions of steady state solutions are identified in an operating parameter space for a given initiator feed composition. A variety of complex bifurcation behavior such as multiple steady states, Hopf bifurcation and limit cycles have been observed and their stability characteristics have been analyzed. The effects of feed initiator composition and the concentration of the initiator in the feed stream on the reactor dynamics are also presented.     

MULTIPLICITY OF STATES IN CONTINUOUS STIRRED COPOLYMERIZATION REACTORS: ITS EXISTENCE AND CONSEQUENCES*

The dynamic behaviour of a continuous copolymerization stirred tank reactor is analyzed. The evolution of multiple steady states due to Trommsdorf or gel effect in free radical bulk copolymerization systems is illustrated for the first time taking into consideration the changes in the cross termination rate constant [or the φ factor] at high extent of reaction. The effect of concentration multiplicity on cumulative copolymer composition is illustrated. The consequences of the results of the present analysis in copolymerisation reactor operation and design have been discussed.     

Multiplicity,stability and dynamics for isothermal autocatalytic reactions in CSTR

Multiplicity, stability and dynamics for autocatalytic reactions of the type A? → rR + ? with overall rate expression r_A = kC_A^mC_Rⁿ in a continuous stirred tank reactor (CSTR) are studied. Exact multiplicity criteria are defined in the parameter space. Stability analysis shows that no periodic oscillation is possible for the system. When multiplicity occurs, some minimum of R present initially in the reactor is required in order for the high steady state to be achieved. Loci of transient extreme for product are also investigated. Multiplicity and uniqueness criteria are further compared with reported experimental data in literature.     

A method for achieving inaccessible conversions by use of cascade CSTR's

The hysteresis of steady state could be found in a single isothermal, continuous stirred tank reactor (CSTR) for reaction rates of form R = kC/(1 + KC)². Clearly, a conversion (the metastable point) between the two stable solutions is never achieved. A numerical method for cascades of CSTR's to achieve this inaccessible conversion is presented along with a graphical procedure illustrating this method.     

OPERATION POLICIES FOR A GAS—LIQUID STIRRED TANK REACTOR

The operation of a gas-liquid stirred tank reactor is simulated, including batch operation, steady and unsteady state, continuous operation and cyclic variable volume operation. A pseudo first order reaction is considered. Analytical expressions for the enhancement factor of gas absorption rate are derived for each case, and productivity obtainable under different operation policies is evaluated. Results obtained are shown graphically for specific cases for the purpose of demonstrating the influence of some of the parameters on the system

The equations as presented can be easily extended to other kinetics, and serve as a useful guide for optimum operational policies in a gas-liquid stirred tank reactor.     

OPERATION POLICIES FOR A GAS—LIQUID STIRRED TANK REACTOR

The operation of a gas-liquid stirred tank reactor is simulated, including batch operation, steady and unsteady state, continuous operation and cyclic variable volume operation. A pseudo first order reaction is considered. Analytical expressions for the enhancement factor of gas absorption rate are derived for each case, and productivity obtainable under different operation policies is evaluated. Results obtained are shown graphically for specific cases for the purpose of demonstrating the influence of some of the parameters on the system

The equations as presented can be easily extended to other kinetics, and serve as a useful guide for optimum operational policies in a gas-liquid stirred tank reactor.