Wednesday, November 7, 2007 - 1:20 PM

Uniqueness And Stability Of The Steady State In Flash Reactors

Yuan Xu, Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 and B. Erik Ydstie, Chemical Engineering, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213.

The uniqueness and stability of the steady state is of fundamental importance to the understanding of the operation of flash reactors. It has attracted the wide interest in the separation field. Doherty and Perkins (1982) showed that the differential equations governing the dynamic behavior of the distillation of a non-ideal multi-component mixture on a single theoretical CMO plate (flash) posses a globally asymptotically stable steady state. Lucia (1986) proved the uniqueness of the steady state in the isothermal isobaric flash reactors with the aid of Gibbs-Duhem relation and Caruchy interlace theorem. This analysis framework was extended to prove the uniqueness of the steady state for the binary homogeneous distillation (Sridhar and Lucia 1989) and the multi-component distillation (Sridhar and Lucia 1990).

In this work, we build a new uniqueness and stability proof on the geometric insight. The computationally challenging problem of finding the global minimum of the system Gibbs energy is equivalent to the geometrically intuitive problem of constructing the convex hull for all the Gibbs energy functions of candidate phases. Analytically, a convex hull is defined with Caratheodary theorem (Rockafellar 1970). The infimum operator in the theorem can be dropped under mild assumptions. The two-phase region on the convex hull is proved to a ruled surface. Its projection onto the component space is also endowed with the special structure, which allows the use of an extended level rule for the phase distribution relation. The uniqueness problem is transformed into the determination of intersection points of two trajectories. The stability proof is constructed with help of Lyapunov theorem. A ternary T-P flash is used as an example to demonstrate the idea of geometric analysis employed in this work. Future work includes the extension of the analysis in this work to reactive flash reactors and multi-staged systems.

Reference: Doherty, M. F., and J. D. Perkins (1982), On the dynamics of distillation processes IV, Chem. Eng. Sci., 37, 381-392.

Lucia A. (1986), Uniqueness of solution to single-stage isobaric flash processes involving homogeneous mixtures, AIChE Journal, 32, 1761- 1770.

Rockafellar T. (1970), Convex analysis, Princeton University Press, Princeton, NJ.

Sridhar, L. and A. Lucia (1989), Analysis and algorithms for multi-stage separation processes. Ind. Eng. Chem. Res., 28, 793-803.

Sridhar, L., and A. Lucia (1990). Analysis of multi-component, multistage separation process: fixed temperature and fixed pressure profiles, Ind. Eng. Chem. Res., 29, 1668-1675.