(243c) Passivity of Multi-Phase Reaction Systems at Chemical Equilibrium

In this work, we consider multi-phase systems and introduce the definition of a concave entropy envelop for such systems. We then show the construction of a thermodynamic-based storage function and develop passivity-based control design of a generalized multi-component flash system. A modeling example of a water gas shift reactor is given to illustrate our approach.

The equilibrium states of chemical reactions shift in response to external changes and the direction can be determined by calculation involving changes in temperature, pressure, or system constraints. Therefore, in the case of reaction systems operating near chemical equilibrium, stability analysis and resulting control design is of critical importance. We focus on a generalized multi-component flash model that operates in a similar fashion as continuous stirred tank reactors (CSTR). The stability study of such reactors are well documented in literature. Physicochemical systems are inherently nonlinear mostly due to reaction kinetics. Linearization methods are commonly used to conduct stability analysis [1] but the result is local to a fixed operating point of the original system. In the context of nonlinear systems, passivity-based approaches are also considered [2]. As an extension to the Lyapunov stability criterion, passivity based control requires a storage function that defines the so-called “energy” of the system and usually takes a quadratic form.

A thermodynamic based storage function can be constructed on the basis of irreversible thermodynamics. Dissipation of energy is a consequence of irreversible energy phenomena and the entropy tends to reach a maximum at equilibrium due to irreversibility. Chemical processes are irreversible by nature and the inherent irreversibility is quantified by the loss of available work. The dissipated availability is related to the entropy by dA = T dS. The formulation of the entropy function based on the Gibbs equation is reviewed and the concavity of the entropy function leads to the construction of a thermodynamic availability function [3][4] that satisfies the requirements of the storage function for a passive system. As for a multi-phase system, the entropy may not be concave globally and a concave envelop of the entropy function [5] is introduced.

Passivity is an input-output property of a dynamical system, and provides guidance for choosing proper variables to apply output feedback control. A modeling example of a water gas shift reactor is included to illustrate the control design. We formulated the model in a reduced invariant space, i.e. deriving mass balances of chemical elements. Under the assumption of chemical equilibrium, the full component space can be mapped from the reduced space by minimization of the Gibbs free energy, which is formulated as an optimization problem with constraints being conservation of elements. In the example, we show that the reactor can be stabilized by feedback controls of the pressure and temperature.

The plant and the controller can be considered as two subsystems that compose a process system. The interconnection property of passivity provides that the overall system inherits the passivity property if the two subsystems are both passive. As for future works, we plan to investigate the passivity of model predictive control (MPC) framework.

References

1. Uppal, A., W. H. Ray, and A. B. Poore. "On the dynamic behavior of continuous stirred tank reactors." Chemical Engineering Science4 (1974): 967-985.
2. Ramirez, Hector, Daniel Sbarbaro, and Romeo Ortega. "On the control of non-linear processes: An IDA–PBC approach." Journal of Process Control3 (2009): 405-414.
3. Alonso, Antonio A., and B. Erik Ydstie. "Process systems, passivity and the second law of thermodynamics." Computers & chemical engineering20 (1996): S1119-S1124.
4. Alonso, Antonio A., and B. Erik Ydstie. "Stabilization of distributed systems using irreversible thermodynamics." Automatica11 (2001): 1739-1755.
5. Gibbs, Josiah Willard. "A Method of Geometrical Representation of the Thermodynamic Properties by Means of Surfaces." Transactions of Connecticut Academy of Arts and Sciences(1873): 382-404.