Page 374 - Design of Simple and Robust Process Plants

9.5 Closed Loop Steady-state Optimization 361
The technique is applicable to process systems/units that have more degrees of
freedom available than output specifications, and the optimal solution of a suitable
criterion function is unconstrained. The work was evaluated for an integrated
Petlyuk distillation column (Petlyuk et al., 1965). The Petlyuk column, which in a
more practical form was presented by Kaibel (1987)as a divided wall column
(DWC), is a distillation column that separates three products at required purity (Fig-
ure 9.6). A DWC has a lower capital investment as well as lower energy cost, up and
to 30% compared to conventional distillation in two successive columns, (Figure 5.7
in Chapter 5), (Triantafyllou and Smith, 1992). The DWC is an excellent example of
simple and robust design, less equipment and related less capital, less energy con-
sumption and as such a lower cost of operation.
The self-optimizing control study was carried out for a DWC where the top and
bottom qualities were controlled, as well as the quality of the side stream, with
regard to one impurity. Previous studies have shown that such a column does not
necessarily have strong interactions for the control of product qualities. Optimiza-
tion of the DWC showed the optimum of such a column to be very sharp (Halvorsen
and Skogestad, 1997). During the early investigations, researchers considered that
the optimal operation point of such a column would be difficult to maintain in case
of disturbances. However, this was questioned by Halvorsen and Skogestad (1999),
who subsequently evaluated a self-optimizing control for this application. The two
manipulated variables (DOF)available for optimization (not utilized for control)
were the liquid and vapor split between both separated parts of the column (pre-
fractionator and main column). The objective of the optimization for this application
was translated as, minimization of the energy stream to the reboiler, translated in
vapor flow V. In a self-optimizing control, the key is to determine the effect of dis-
turbances on the optimal operation point, and to find a common measurable vari-
able with a near-constant value at these optimal conditions. In case such variable
can be found, a control loop can be implemented based on a simple feedback con-
troller. The disturbances (d)studied were: feed flow rate, feed composition, feed liq-
uid fraction, and product purities The choice was made to select the liquid split (u)
as the only manipulated variable, and to fix the vapor split. Eight feedback variables
were selected for evaluation. The optimal solution was found by minimizing V (u, d)
with respect to u. The optimal value of the criterion function V opt and the corre-
sponding solution u opt will be a function of d

V opt (d)= min V(u,d) = V(u opt (d),d (2)
u
The methodology was evaluated for a chosen system. The boil-up V and the liquid
split u (at a fixed vapor split)with self-optimizing control for different feedback
cases for the selected disturbances were compared to the overall optimal boil-up val-
ues. The overall conclusion was that self-optimizing control was a good method for
optimization of a Petlyuk column. The selection of the final feedback variable
depends on the most probable disturbances, but three of these were selected as the
best with errors within 1%. The three selected feed back variables were: