Modal analysis and optimization of isothermal autocatalytic reactions

Abstract A comprehensive and systematic analysis of optimization of autocatalytic reactions is presented in this article considering a very general objective function. The objective functions of common interest, such as maximum yield, maximum selectivity, maximum productivity and minimal reaction time, are some of the special cases of this general objective function. The following reaction scheme is considered in this work: A+ n B ⇌ ( n +1)B, A ⇌ B, B→C. The problem of maximization of the general objective function is posed as that of the optimal control of a variable-volume reactor with time-variant feed and effluent flow rates as the control variables. The systematic ‘modal’ analysis enables identification of feasible modes of reactor operation and conditions under which each feasible mode is admissible. Of all the possible modes, only a few are proved to be feasible and only certain transitions are permissible from each feasible mode. The policy for programmed manipulation of the feed and effluent flow rates is obtained employing the optimal control theory. Candidate optimal policies are identified as individual feasible modes or certain sequences of these. A singular fed-batch operation in which the feed rate is decided exclusively by the state of the reactor is part of some of the candidate optimal policies. Conditions for admissibility and feasibility of singular control and the feeding policy during singular control interval are obtained analytically for arbitrary kinetics of these reactions. The concentration trajectories in a fed-batch operation converge to or diverge from one or more limit points. Operation at each limit point is characterized by invariance of concentrations of all species, the feed rate being an exponential function of time. A linearized stability analysis, supported by numerical illustrations, reveals that some or all of the limit points may be accessible during fed-batch operation. The optimal operation strategy is linked to the disposition of batch reactor trajectories in the space of concentration variables. The rich behavioral patterns of autocatalytic reactions are revealed by specific numerical illustrations.

[1]  David T. Lynch.  Chaotic behavior of reaction systems : consecutive quadratic/cubic autocatalysis via intermediates , 1993 .

[2]  Juan Hong,et al.  Optimization of autocatalytic reactions , 1993 .

[3]  A. Sapre.  Diffusional enhancement of autocatalytic reactions in catalyst particles , 1989 .

[4]  R. Aris,et al.  Modelling Cubic Autocatalysis by Successive Bimolecular Steps , 1988 .

[5]  S. Scott.  Reversible autocatalytic reactions in an isothermal CSTR: Multiplicity, stability and relaxation times , 1983 .

[6]  H. Maurer.  On Optimal Control Problems with Bounded State Variables and Control Appearing Linearly , 1975, Optimization Techniques.

[7]  M. Morita,et al.  A New Permanganate–Nitrite–Formic Acid–Methanol Oscillator , 1988 .

[8]  F. M. Kirillova,et al.  High Order Necessary Conditions for Optimality , 1972 .

[9]  M. Grzesik,et al.  Enhancement of heterogeneous autocatalytic reactions by intraparticle diffusion , 1993 .

[10]  Alison S. Tomlin,et al.  Period doubling and other complex bifurcations in non-isothermal chemical systems , 1990, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[11]  Yield optimization for multiple reactions , 1988 .

[12]  D. T. Lynch,et al.  Chaotic behavior of reaction systems : parallel cubic autocatalators , 1992 .

[13]  M. Grzesik,et al.  Enhancement of heterogeneous autocatalytic reactions by external diffusion , 1993 .

[14]  Arthur E. Bryson,et al.  Applied Optimal Control , 1969 .

[15]  D. T. Lynch,et al.  Chaotic behavior of reaction systems: mixed cubic and quadratic autocatalysis , 1992 .

[16]  K. F. Lin.  Multiplicity, stability and dynamics for isothermal autocatalytic reactions in CSTR , 1981 .

[17]  Satish J. Parulekar,et al.  Analytical optimization of some single-cycle and repeated fed-batch fermentations , 1992 .

[18]  A. Sapre,et al.  Analysis of autocatalytic reactions in isothermal catalyst particles , 1990 .

[19]  K. Showalter,et al.  Critical slowing down in the bistable iodate-arsenic(III) reaction , 1983 .

[20]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C , 1984 .

[21]  Stephen K. Scott,et al.  Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability , 1983 .

[22]  B. Brooks,et al.  Autocatalytic reactions in semi-batch reactors with no catalyst in the feed stream , 1988 .

[23]  S. Scott.  Isolas, mushrooms and oscillations in isothermal, autocatalytic reaction-diffusion equations , 1987 .

[24]  Irving R. Epstein,et al.  Sustained and damped pH oscillation in the periodate-thiosulfate reaction in a continuous-flow stirred tank reactor , 1989 .

[25]  H. Lintz,et al.  26 The study of mixing in a continuous stirred tank reactor using an autocatalytic reaction , 1980 .

[26]  S. Scott,et al.  Oscillatory chemical reactions in closed vessels , 1986, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[27]  Irving R. Epstein,et al.  A General Model for pH Oscillators , 1991 .

[28]  J. Ross,et al.  Multiple ranges of flow rate with bistability and limit cycles for Schlögl’s mechanism in a CSTR , 1983 .

[29]  K. Showalter,et al.  Period Doubling and Chaos in a Three-Variable Autocatalator , 1990 .

[30]  B. Wojciechowski,et al.  The mechanism of paraffin reactions on HY zeolite , 1989 .

[31]  B. Brooks.  Semi-batch reactors and their use for autocatalytic reactions , 1988 .

[32]  O. Hougen,et al.  Kinetics of the reduction of nickel oxide by hydrogen , 1962 .

[33]  Ralph G. Pearson,et al.  Kinetics and mechanism , 1961 .