Local stability of tubular reactors

Abstract

Liapunov analysis techniques employing a general quadratic functional are used to derive stability conditions for tubular reactor systems. The adiabatic tubular reactor without axial dispersion is shown to be locally stable, which excludes the possibility of multiple steady states, and the reactor with axial dispersion is proven locally stable if a condition involving only system parameters and steady state values is satisfied. Peclet numbers for heat and mass transfer are not specified equal for the latter proof.Results of simulation studies are used to confirm the validity of the derived stability condition, and it is shown that the parametric region of multiplicity is quite well defined. For the nonlinear equations, single steady state cases appear to possess nonuniform stability.This paper investigates the stability of tubular reactors using both theoretical and computational techniques. Liapunov's direct method employing a functional based on a general quadratic norm is used as the theoretical method. The theoretical analysis yields through straightforward algebraic manipulation stability results without the need for a solution of the partial differential equations describing the system. Dynamic simulations were computationally carried out by finite differencing the describing equations and using the quasilinearization technique.In the literature much attention has recently been devoted to stability studies of distributed parameter systems. Murphy and Crandall (19) applied Liapunov stability theory to the catalyst particle problem using a theorem somewhat similar to that to be presented here. Their analysis yielded stability conditions depending on system parameters and steady state values. Han and Meyer ( 1 5 ) used bounded-input-bounded-output concepts to analyze the stability of a nondispersive, catalytic bed reactor. They obtained a stability condition based on discretized nonlinear equations which was different from previous results based on linearized equations. Other work has been done on nondispersive models, and an example is the recent article by Agnew and Narsimhan (1). that is, positive away from a given steady state and zero only at the steady state is specified, and stability is ensured if it can be shown that the time derivative of this function is negative definite. This time derivative involves time derivatives of the state variables, and these are conveniently available in the system differential equations.Liapunov's first method, according to Bertram and Kalman ( 7 ) , "deals with stability questions via an explicit representation of the solutions of a differential equation," whereas Aris ( 4 ) is more specific in stating, "His so-called first method shows how the process of linearization about the steady state may be justified and exploited."The difficulty in the application of the second method is usually found in the search for the Liapunov function. An inverse formulation involving first the specification of a negative definite Liapunov function derivative has been widely used for line...

Local stability of tubular reactors

“…and The idea of Clough and Ramirez (1972) was to choose the constants Ki in a manner such that the most liberal stability condition is reached. We will show here that the best choice is K1 = rlr K2 = r2 where rl and r2 are the Peclet numbers for heat and mass transfer, respectively.…”

“…This whole approach is, in fact, shown to yield more conservative stability criteria than those of Varma and Amundson (1972). Conclusions of Liou et al (1972b), since they were based on their earlier work, are therefore also incorrect for rl z r2.In the notation of Clough and Ramirez (1972), the negative definiteness of the symmetric matrix -C corre- …”

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Some difficulties, which existed due to the nonsymmetric nature of matrix C, in a recent paper of the above title by Clough and Ramirez (1972) have been aptly noted by Liou et al (1972a) who attacked these and presented suitably modified stability criteria. Unfortunately, however, there are still some problems in the case of the tubular reactor with axial dispersion (point 3 of Liou et al, 1972a) and the stability criterion reached for this case, inequality (6) of Liou et al (1972a), is incomplete.

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“…The above analysis implies that if the choice of functions P i ( x ) (16) are frequently made (compare Berger and Lapidus, 1968;Varma and Amundson, 1972) to make the spatial differential operators in Equations (B6) and (B7) of Clough and Ramirez (1972) self-adjoint. In the transformed variables, the Liapunov functional in Equation (15) takes the form which is the simplest Liapunov functional that comes to mind.…”

“…In the transformed variables, the Liapunov functional in Equation (15) takes the form which is the simplest Liapunov functional that comes to mind. We can then summarize that if the approach of Clough and Ramirez (1972) is to give better stability conditions than the ones that can be obtained by the most obvious choice of a Liapunov functional, one must look for functions P i ( x ) other than exp (-K i x ) . Varma and Amundson ( 1972) have recently considered several problems concerning the nonadiabatic tubular reactors, including those concerning uniqueness and asymptotic stability of the steady states.…”