(279d) A Note On Stability and Uniqueness of Solutions to Multi Component Flash Under Equilibrium
AIChE Annual Meeting
2009
2009 Annual Meeting
Engineering Sciences and Fundamentals
Thermodynamic Properties and Phase Behavior V
Tuesday, November 10, 2009 - 1:30pm to 1:50pm
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by many authors but not yet proved for a general system [6, 4, 1, 7]. In this work, we
present a proof of the global stability and uniqueness of the solution to a multi-component
flash system with two phases and arbitrary number of independent reactions. The only
assumption made is that the system is under thermodynamic equilibrium with respect to
vapor-liquid equilibria and chemical reactions.
Our method is based on the concept of convergent systems [5] and Lyapunov's second
theorem on stability. The basic argument is that if all the solutions of a system "forget" their
initial conditions and converge to each other then the system converges to a unique solution
only. We use the second law of thermodynamics to define a Lyapunov function W and prove
that the variation or distance between two nearby trajectories of the process goes to zero
with time. W is closely related to the Availability function which, as defined by Keenan [3],
is the maximum amount of work done by a system in equilibrating against another system.
W represents the maximum work that can be extracted if two states of the process converge
to each other. It can be interpreted as a metric in the thermodynamic space.
W is equivalent to the second order variation of entropy. In this regard it is similar to
the Lyapunov function used by Glansdorff and Prigogine [2]. They analyzed local stability
around an equilibrium point while we are able to prove global stability and uniqueness.
Furthermore, our analysis addresses the semi-definite nature of the Lyapunov function due
to the degeneracy in the entropy surface. This allows us to study stability with respect to
convective fluxes of systems with multiple phases.
We use the entropy balance to derive the balance for the Lyapunov function and show
that its rate of change with respect to time is negative and less than or equal to a factor of
its own value. This implies exponential convergence and hence proves Lyapunov stability.
The Lyapunov function is only positive semi-definite in the extensive variable space therefore
we need to apply inventory control for the convergence of extensive variables. Using Gibbs
phase rule, it is shown that for the system to converge to a steady state we need to control as
many independent extensive variables as there are phases present. Finally we present some
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The aim of this paper is to introduce a novel method to analyze stability of chemical
processes. The method is not limited to the multi-component flash and can be applied to a
more general class of processes like heterogeneous systems with multiple phases. It can be
easily used to analyze the stability of process networks like a distillation column which is
of much practical significance. We propose the extension to kinetic systems operating away
from chemical equilibrium.
ness and stability of the steady state in homogeneous continuous distillations. Chemical
Engineering Science, 37(3):381 -- 392, 1982.
[2] P. Glansdorff and I. Prigogine. Thermodynamic Theory of Structure, Stability and Fluc-
tuations. Wiley-Interscience, New York, NY, 1971.
[3] J. H. Keenan. Thermodynamics. John Wiley & Sons, Inc., 1941.
[4] A. Lucia. Uniqueness of solutions to single-stage isobaric flash processes involving homo-
geneous mixtures. AIChE Journal, 32(11):1761 -- 1770, 1986.
[5] A. Pavlov, A. Pogromsky, N. van de Wouw, and H. Nijmeijer. Convergent dynamics, a
tribute to boris pavlovich demidovich. Systems & Controls Letters, 52:257--261, 2004.
[6] H. H. A. Rosenbrock. Theorem of dynamic conversion for distillation. Trans. Inst. Chem.
Eng., 38:279--287, 1960.
[7] P. Rouchon and Y. Creff. Geometry of the flash dynamics. Chemical Engineering Science,
48(18):3141 -- 3147, 1993.
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