(500e) Single Shooting Method for Semicontinuous Distillation Design

Ternary separation using the process of semicontinuous distillation was proposed to be carried out in a single distillation column and a process vessel, which is called the middle vessel¹. During the process of separation, inlet and outlet material streams of the middle vessel are either operational or non-operational based on its state. With the help of these state-dependent discrete actions, periodic feed charges and the intermediate-boiling product discharges are carried out in the middle vessel, while the low and high volatile components are removed continuously from the column. This periodic process is thus modelled mathematically as a limit-cycle oscillating hybrid system² for design purposes. The semicontinuous distillation process design involves estimating equipment sizes, controller tuning parameters, and column design parameters which are all time-invariant parameters of the oscillating hybrid system. These system parameters are chosen to operate the process in a hydraulically feasible and an economically optimal limit cycle.

The state-of-the-art design methodology uses existing commercial process simulators, such as AspenONE simulation environment to estimate the limit cycle. In this design methodology, the semicontinuous distillation process is simulated through the process of numerical integration of the hybrid system, starting from an arbitrary initial state that does not lie on the limit cycle. This initial state is determined using the continuous separation of the same ternary mixture in a side stream column. Constrained black-box optimization is then used to find a better design. Ignoring the integer design parameters during the optimization step is typical because of the complexity of the problem. However, these design parameters are estimated using continuous design heuristics during the initial state calculation stage. Nevertheless, this design method takes a lot of CPU time (~ 2-3 CPU days).

The estimation of the limit cycle using numerical integration in the current design methodology is a time-consuming process (takes ~1-5 min to converge to a limit cycle). In the present study, we attempt to reduce this time, by estimating the initial point that lies on the limit cycle and the cycle time (period of the limit cycle) by imposing periodic boundary conditions and a phase locking condition² on the hybrid model. A phase locking condition is essential to find an isolated point on the limit cycle. Single shooting method is used to solve this periodic boundary value problem³. A sensitivity analysis on the system is also carried out to ascertain the smoothness of the state functions in the parametric space of interest in order to use a gradient-based solver for dynamic optimization in the future.

References

[1] Phimister, J. R., Seider, W. D. (2000). Semicontinuous, middle‐vessel distillation of ternary mixtures. AIChE journal, 46(8), 1508-1520.

[2] Khan, K.A., Saxena, V.P., Barton, P.I., 2011. Sensitivity analysis of limit-cycle oscillating hybrid systems. SIAM Journal on Scientific Computing 33(4), 1475-504.

[3] Parker, T. S., Chua, L. (2012). Practical numerical algorithms for chaotic systems. Springer Science & Business Media.