(112b) Stability Analysis of Bifurcations of Nonsmooth Differential-Algebraic Models for Dry Distillation Columns

Chemical engineering systems with multiple physical regimes are inherently non-differentiable, or nonsmooth, with each regime being modeled by different equation systems. An important example is the phase equilibrium problem, i.e., determining the number of coexisting phases in a given process stream or within an equipment. For each equilibrium stage in a distillation column, or even for a single-stage flash vessel, three regimes are possible: 2-phase vapor-liquid equilibrium, 1-phase superheated vapor, and 1-phase subcooled liquid. However, the distillation models used in standard process simulators, such as Aspen Plus’ RadFrac block, assume that all stages are operating in the 2-phase regime and thus fail to converge whenever the column becomes dry or vaporless, that is, when one or more stages fall into the 1-phase regimes. Although possible for a single-stage flash, trying to guess the regime of each stage in a distillation column is of combinatorial nature and thus impractical. In order to simulate dry and vaporless distillation columns, we need a modeling strategy that can automatically “switch” between equation systems without a priori knowledge of the regime.

In previous work [1], we modified the traditional MESH equations to create a system of nonsmooth algebraic equations that can model dry and vaporless distillation columns at steady state. Unlike other modeling strategies in the literature [2,3], our explicitly nonsmooth approach yields a single, compact system of model equations without adding extra variables, parameters or constraints. Furthermore, given recent developments in nonsmooth analysis [4], generalized derivative elements can be systematically computed for these models and so direct equation-solving methods can be used for nonsmooth process simulation. Thanks to the purely algebraic nature of our nonsmooth MESH model, we were able to employ detailed parametric analysis with special continuation methods to reveal a novel nonsmooth bifurcation behavior in dry and vaporless columns. At a critical parameter value, these systems exhibit a continuum of infinitely many steady states forming a bounded segment, which can be described by a nonsmooth 1-dimensional manifold.

In this work we use a nonsmooth system of differential-algebraic equations (DAEs) to describe the dynamical behavior and characterize the stability of this new type of bifurcation in distillation columns. Compared to the nonsmooth algebraic MESH equations, the more complex DAE model introduces molar and enthalpy phase holdups as extra variables. Even though routinely eliminated from steady-state simulation, we show that phase holdups become crucial to fully determine the steady state of dry distillation columns, corresponding to the equilibrium solutions of the DAEs. In this study we employ analogous continuation procedures to show that these equilibria exhibit the same bifurcation behavior, and we analyze the stability of the corresponding nonsmooth segment of equilibria in contrast to the standard notion of stability of an isolated equilibrium point. In particular, we consider the mathematical conditions that give rise to this bifurcation behavior and discuss how local information in terms of the generalized sensitivities of the nonsmooth DAEs can be used to predict global stability behavior of the equilibrium set.

References

[1] S. M. Cavalcanti and P. I. Barton. Nonsmooth Simulation of Dry and Vaporless Tray Distillation Columns. Oral Presentation, 2018 AIChE Annual Meeting, Pittsburg, PA.

[2] L. G. Bullard and L. T. Biegler. Iterated linear programming strategies for non-smooth simulation: a penalty based method for vapor-liquid equilibrium applications. Computers & Chemical Engineering, 7(1):95-109, 1993.

[3] V. Gopal and L. T. Biegler. Smoothing Methods for Complementarity Problems in Process Engineering. AIChE Journal, 45(7):1535-1547, 1999.

[4] K. A. Khan and P. I. Barton. A vector forward mode of automatic differentiation for generalized derivative evaluation. Optimization Methods and Software, 30(6):1185-1212, 2015.