My last post encouraged the use of case studies to survey the performance of a distillation model using the most reliable specs, boilup and reflux ratios. You may ask: why not use the Fenske-Underwood-Gilliland (FUG) methods to determine the number of stages and reflux ratio for a given separation of light key and heavy key compounds? Wouldn't this allow the direct specification of the recovery or concentration and reflux ratio for the program. The example in this post shows that this method can still fail.
The separation problem
In the previous post, a stream of ethane, propane, isobutene, n-butane and n-hexane were separated between the C4's. In this post I separate the light ends from the C4+ fraction. I've added 1% (molar) nitrogen to the stream in order to demonstrate partial condensers with liquid product in a later post. The feed stream composition is shown below.
I assumed that the bottom product was to contain 100 ppm propane (light key) as the primary specification for quality of the C4+. For the IC4 (heavy key) concentration in the overhead product, I set a value of 0.001.
The FUG procedure says that 31 stages are needed for the separation. The Underwood procedure resulted in a minimum reflux ratio of 1.24 when using a feed at the bubble point. An actual ratio of 1.3 x RRmin = 1.61 was assumed.
DCOL results using component specs
When I use RR=1.61 and a spec of 100 ppm propane in the bottom, DCOL fails to converge.
Let's look at the propane bottom concentration results from a case study.
The rows of the table correspond to x, the reflux ratio, and the columns correspond to y, the bottom/feed ratio (BF). The region where our specs fall is shown by the red box. The propane spec of 10(-4) is in a region where the values change four orders of magnitude between BF cases. This steep, almost step change could be the problem. However, specifying a concentration of say, 0.02, in the region to the right of the step change (i.e., higher BF) also causes DCOL to fail.
I tried every other component specification (concentrations and recoveries in overhead and bottom) and the only one found to converge was a spec of bottom concentration of propane <=10(-5). Many of the specs that failed involved no steep areas. Instead, the case study indicated most response surfaces were flat. The cases for C3 bottom concentration below 10(-5) are also in a flat area, but for some reason they converge.
DCOL with RR and BF
The problems encountered were most likely due to the optimization routine having to deal with both steep regions and flat regions. If the fundamental variables, RR and BF are given, optimization is not required and DCOL converges even in regions along the step change in propane bottom concentration. However, it would probably be risky with an actual column to expect stable control of the BF to achieve the requested 100 ppm propane spec. Instead, the case study shows that even better separation can be achieved in a more stable operating region at a lower BF value.
Combine FUG with the case method
The best approach is to use the FUG procedure to establish the number of stages and a minimum and center values for the RR range. The BF range can be estimated from the feed composition and the assumption of perfect separation between the light and heavy keys. Then, conduct the case study to see the performance landscape.
Contour plots may not show the steep areas. The plot algorithm assumes a fixed increment between contours that can step across a step change without indication. In this example, the steep region in the propane concentration was not evident from the contour plot. Thus, the table of results was examined. A 3D surface plot of the results would have also shown the step change but I usually use contour plots because they display numerical values better.
I should emphasize that the FUG procedure did not fail. It produced design conditions that would achieve the desired separation. The example demonstrated another reason that DCOL can fail when using component specs instead of RR and BF. I don't regard this result as a negative for DCOL because I think the optimization problem would be difficult for any routine.