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Dynamical analysis of distributed parameter tubular reactors. (English) Zbl 0979.93077
The authors describe dynamics of tubular reactors corresponding to two models: the so-called plug model, and the model which considers axial dispersion. The kinetics of reactions depend only on concentrations of the reactants. The reactions of the form \(C_1\to bC_2\) are governed by mass balance equations \[ \partial x_1/\partial t= D_a\partial^2 x_1/\partial z^2- v\partial x_1/\partial z- r(x_1, x_2), \partial x_2/\partial t= D_a\partial^2 x_2/\partial z^2- v\partial x_2/\partial z+ br(x_1, x_2), \] with appropriate boundary conditions. \(z\in [0,L]\) is the axial variable. Everything is happening in the sum of \(L^2\) spaces. All terms on the right-hand side of these equations are chemical concentrations, and so obviously must be terms like \(\partial x_i/\partial t\). Assuming that \(r= k_0 x_1\), the authors present this partial differential equation dynamic model of the reaction. They define the operators \[ A_2= \begin{pmatrix} D_a & 0\\ 0 & D_a\end{pmatrix},\quad A_1= \begin{pmatrix} -v & 0\\ 0 & -v\end{pmatrix},\quad A_0= \begin{pmatrix} -k_0 & 0\\ k_0 & 0\end{pmatrix} \] and the operator \(A= A_1+ A_2+ A_0\). The operator \(A\) is a generator of a continuous semigroup. Bounded control and observation operators are introduced. The physical domain is shown to be reachable, thus proving controllability in this sense, while any dominant set of modes is shown to be observable, under certain conditions. Some finite-dimensional approximations are discussed and the authors note that lumped parameter approximations produce results that are similar to the distributed parameter case when only the dominant mode is considered.

MSC:
93C95 Application models in control theory
93C20 Control/observation systems governed by partial differential equations
80A30 Chemical kinetics in thermodynamics and heat transfer
93B07 Observability
93B05 Controllability
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[1] Benoist-Gueutal, P., & Courbage, M. (1993). Mathématiques pour la Physique — T3: Opérateurs Linéaires dans les Espaces de Hilbert. Paris: Eyrolles.
[2] Callier, F.M.; Winkin, J., LQ optimal control of infinite dimensional systems by spectral factorization, Automatica, 28, 4, 757-772, (1992) · Zbl 0776.49023
[3] Cho, Y.S.; Joseph, B., Reduced-order steady state and dynamic models. parts I and II, A.i.ch.e. j., 29, 2, 261-276, (1983)
[4] Cohen, D.S., Multiple solutions and periodic oscillations in nonlinear diffusion processes, SIAM journal of applied mathematics, 25, 1, 640-654, (1973) · Zbl 0279.35008
[5] Curtain, R. F., & Zwart, H. J. (1995). An introduction to infinite-dimensional linear systems theory. New York: Springer. · Zbl 0839.93001
[6] Danckwerts, P.V., Continuous flow systems: distribution of residence time, Chemical engineering science, 2, 1, 1-13, (1953)
[7] Dieudonné, J. (1960). Eléments d’Analyse, Tome 1. Paris: Gauthier-Villars.
[8] Dochain, D. (1994). Contribution to the analysis and control of distributed parameter systems with application to (bio) chemical processes and robotics. Thèse d’Agrégation de l’Enseign. Supérieur. UCL, Belgium.
[9] Dochain, D., & Winkin, J. (1995). Dynamical analysis of a class of distributed parameter fixed bed reactors. Proceedings of the 34th CDC (pp. 3225-3230).
[10] Dochain, D., Winkin, J., & Ligarius, Ph. (1997). Dynamical analysis of axial dispersion reactor models. Proceedings of the IEEE Conference on computer cybernetics and simulation SMC’97 (pp. 2420-2425).
[11] El Jai, A., & Pritchard, A. J. (1988). Sensors and controls in the analysis of distributed systems. Chichester: Ellis-Horwood. · Zbl 0637.93001
[12] Emirsajlow, Z., & Townley, S. (1997). From PDEs with boundary control to the abstract state equation with an unbounded input operator: A tutorial. Manuscript. Department of Mathematics, University of Exeter (UK), European Journal of Control, 6, nb. 1, 2000 (to appear).
[13] Fattorini, H.O., Boundary control systems, SIAM journal on control, 6, 349-385, (1968) · Zbl 0164.10902
[14] Froment, G. F., & Bischoff, K. B. (1990). Chemical reactor analysis and design. New York: Wiley.
[15] Grabowski, P.; Callier, F.M., Admissible observation operators. duality of observation and control using factorizations, Dynamics of continuous, discrete and impulsive systems, 6, 87-119, (1999) · Zbl 0932.93009
[16] Lasiecka, I.; Triggiani, R., Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, Journal of differential equations, 47, 246-272, (1983)
[17] Pazy, A. (1983). Semigroups of linear operators and applications to partial differential equations. Berlin: Springer. · Zbl 0516.47023
[18] Ray, W. H. (1981). Advanced process control, Series in chemical engineering. Boston: Butterworth.
[19] Rudin, W. (1974). Real and complex analysis. New York: MacGraw-Hill. · Zbl 0278.26001
[20] Russel, D. L. (1978). Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions. SIAM Review, 20, 639.
[21] Sagan, H. (1961). Boundary and eigenvalue problems in mathematical physics. New York: Wiley. · Zbl 0106.37303
[22] Slemrod, M., Stabilization of boundary control, Journal of differential equations, 22, 402-415, (1976) · Zbl 0304.93022
[23] Waldraff, W.; Dochain, D.; Bourrel, S.; Magnus, A., On the use of observability measures for sensor location in tubular reactor, Journal of process control, 8, 497-505, (1998)
[24] Winkin, J., Dochain, D., & Ligarius, Ph. (1999). Dynamical analysis of distributed parameter tubular reactors. Report 99/09. Department of Mathematics, University of Namur (FUNDP). · Zbl 0979.93077
[25] Young, R. M. (1980). An introduction to nonharmonic fourier series. New York: Academic Press. · Zbl 0493.42001
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