Kapitza pendulum effect in a continuously stirred tank reactor
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TICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Kapitza Pendulum Effect in a Continuously Stirred Tank Reactor N. I. Vaganova and E. N. Rumanov* Institute of Structural Macrokinetics and Materials Science, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia * email: [email protected] Received September 2, 2011
Abstract—The equations of a continuously stirred tank reactor in a wide range of the parameters have three stationary solutions describing the hot, cold, and intermediate unstable states. Similarity with the equations for onedimensional motion indicates the possibility of the stabilization of instability by small highfrequency perturbation (Kapitza pendulum effect). This stabilization has been obtained in numerical simulation. DOI: 10.1134/S1063776112070199
1. INTRODUCTION
2. FORMULATION OF THE PROBLEM
First investigations of the continuously stirred tank reactor [1, 2] showed that the space of the parameters is divided into two parts; each “point” in one of them corresponds to one stationary state, whereas each point in the other part corresponds to three such states. Depending on the initial conditions, the reactor passes either to the hot regime, in which almost com plete transformation of the introduced mixture occurs, or to the cold regime, in which a substance flows through the system almost without transforma tion. Formally, there is an intermediate state, but it is unstable. Under certain conditions, relaxation oscilla tions are developed [3–5]: after a burst, a pause occurs during which products are replaced by a fresh mixture and a new burst then follows.
We represent the reactor equations in the form con venient for the subsequent numerical simulation:
Zeldovich [6] stated that it is desirable to find a method for the stabilization of the unstable regime in order for transformation to continue to valuable inter mediate products and compared a reactor and a star in the main sequence (e.g., the Sun). The balance of a nuclear fusion source and radiant removal of the energy also provides three stationary states, but the middle state is stable because the specific heat of the star is negative (see [7]). We consider the stabilization of the unstable regime using a mathematical analogy between the reactor equations and onedimensional motion in the field of the potential force and friction force. The potential is maximal in the unstable state. A highfrequency perturbation creates a minimum of the effective potential energy at the place of the maxi mum, which ensures stabilization (Kapitza pendulum effect [8]).
dη = Φ ( η, θ ) – η , dt D
(1)
dθ + θ. = ZΦ – Z dt S
(2)
Here, η is the concentration of the reaction product 2 (transformation depth), 0 < η < 1; θ = (E/ T b )(T – Tb), where the activation energy E and temperature T are expressed in the same units, Tb = Ta + Q/c, Ta is the ambient temperature coinciding with the inflow tem perature, Q is the reaction heat, and c is the specific heat. The reaction rate in the simplest case is given by the ex
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