*Note: This article is part of a series on a specific research project: Part 1, Part 2, Part 3, Part 4
*Note: This article assumes you have rudimentary knowledge of how a simple distillation column operates. Though much of the basics of the simple model for a distillation column will be covered below, it should not be considered as a stand alone reference.
Since there were several motivations for this derivation, the conclusions that may be drawn from it overall will be broken down into two relevant categories for discussion and summary.
There was sufficient intrigue to the question of how far the simple models typically taught in a separations course can be abstracted to describe general systems to move forward with an extended derivation. However, the original motivation was to double chek on the validity of two very different, though equvalent, methods to solving a single problem. I admit freely that when I sat down and wrote out the proof I was a bit flustered and, moreover, concerned with succintly epressing my opinion, that I had a correct solution, in a manner that left no room for dispute. Two broad catagories for discussion stand out when viewing these, and other factors, in concert: analytical conclusions and pedagogical conclusions.
There are a few pedagogical conclusions that may be drawn from the manner and degree of this mathematical exercise.
One is that the professor, by enforcing a rigidly narrow set of acceptable routes to the solution, did indeed force me to learn the course material at a significantly greater depth than was expected. This is one of the many attributes of a good teacher. He made me critically evaluate the solution methods available and gain deeper understanding of them. I was not happy with the means or the outcome with regard to the scoring, but the knowledge gained would likely have otherwise never have been pursued so explicitly, or with such enthusiasm.
Another is a concern for those who did not think to employ the same reasoning I uses when approaching this distillation problem. In searching myself for the quickest correct method for a solution during the test I happened to remember how to quickly solve the problem by using a transform via lever rule. This made finding a solution much faster. Given the lack of explicit direction in the prompt, this should have been an acceptable method. In retrospect, no such means of transforming systems to simply reduce the required degrees of freedom during analysis was taught as a general method during the separations course, which unfortunately deprives students of a very powerful tool.
I would go further, but I feel I have already wandered far enough off topic. To sum up the pedagogical conculsions:
- Both methods are well suited for an introductory separations course AND should be taught in sequence
- Students should be highly encouraged to plum the depths of fundamental solution methods
- Use of advanced solution methods on an exam should be investigated to assess extent of understanding, rather than punished with a reduced score
- Abstraction of a technique to other system topologies should accompany the general overview of solution methods covered
- Mathematical tools that may expediate a solution method should be discussed and investigated in tandem with fundamentally derived solution methods
There were several interestng analytical conclusions made during the course of the derivation, the greatest of which may be that the same derivation for the original McCabe-Thiele method may be extended to a distillation column with an arbitrary topology. Since many of these conclusions have already been explored, I will summarize them below with links to each where possible.
- The derivation for a single feed column may be simply extended to cover multiple feeds/side draws
- A simple transform allows for reduction of the number of feeds/side draws
- The transform is analytically identical to its basis
- The transform takes advantage of geometry to expediate graphical analysis
- The derivation for a column with two feeds/side draws may be extended to an arbitrary topology
- Restrictions on the use of the model for an arbitrary topology may be overcome
- Overcoming these restrictions may be computationally prohibitive
- Reductions in degrees of freedom for the generalized model are dependent on the boundary conditions imposed by individual problems
That’s all for this proof. I hope that it is of use to those interested in modeling distillation systems.