(421c) Nonsmooth Simulation of Dry and Vaporless Tray Distillation Columns

Rigorous simulation of tray distillation columns requires the solution of the system of MESH (Mass balance, Equilibrium, Summation and Energy balance) equations, which assume liquid-vapor equilibrium exists at the conditions of each stage. However, as one or more input parameters such as the reflux ratio are varied, vapor and liquid streams can disappear within the column, which in turn allows for the remaining vapor/liquid phases to become superheated/subcooled. In that case, the “E” and “S” equations can no longer be satisfied simultaneously, leading to failure of any MESH-based simulation tool such as Aspen Plus’ RadFrac model. This is especially concerning for flowsheet simulation with recycle streams and for optimization, since model convergence is required under multiple input conditions.

Alternative formulations can be proposed to create a single model that is valid both in the 2-phase and 1-phase regimes, with the correct equations being enforced automatically without prior knowledge of the effective regime for a given set of inputs. However, these strategies introduce nonsmooth function behavior and require special mathematical tools for model simulation.

To the best of our knowledge, dry/vaporless distillation simulation has only been directly addressed in the literature in [1], [2]. In [1], slack variables are introduced and the original equation-solving task is transformed into an optimization problem, solved with a penalty-based iterative linear programming strategy. In [2], the KKT conditions for this optimization problem, with the corresponding complementarity constraints, are solved iteratively using smoothing approximations. However, all of these approaches introduce artificial variables, increase problem complexity and require the solution of a sequence of equation systems or of an optimization problem. Additionally, simulation results in the 1-phase regime were presented only for a couple of isolated input conditions.

Instead, we propose the reformulation of the MESH model by introducing explicitly nonsmooth equations, which allows for simulation in the 2-phase and 1-phase regimes by direct equation-solving and doesn’t introduce any extra variables. As similarly done in [3] for the simpler problem of single-stage flash calculations, automatic lexicographic directional differentiation [4] can be employed to obtain exact generalized derivative elements for the nonsmooth MESH equation system, which can in turn be solved with a suitable method such as the semismooth Newton or the linear programming Newton methods.

In this work, we will present simulation results for dry/vaporless tray distillation columns using novel nonsmooth modeling approaches. The results include detailed input parameter sensitivity analysis and a new type of bifurcation behavior which hasn’t been reported in the distillation literature so far. Additionally, we will present nonsmooth strategies to avoid infeasible input specification automatically and create a distillation simulation model that is robust to failure associated with the disappearance of liquid and/or vapor phases.

References

[1] 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.

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

[3] H. A. J. Watson, M. Vikse, T. Gundersen, and P. I. Barton. Reliable flash calculations: Part 1. Nonsmooth inside-out algorithms. Industrial & Engineering Chemistry Research, 56(4):960-973, 2017.

[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.