*Note: This article is part of a series on a specific research project: Part 1, Part 2
*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.
3.0 Proof of the Equivalence of Methods 1 and 2
3.1.0 Geometric Basis for Equivalence
3.1.1 The Rationale of the Method for Comparison
Given that method 1 and method 2 involve plotting the exact same enriching and stripping operation lines, the enriching and stripping operation lines are by definition not parallel for any real physical system that does not involve an azeotrope**, and the combined streams q-line from method 2 also passes through the intersection of said enriching and stripping operation lines, there exists at the point of intersection of these three lines a set of three different means of determining the coordinates of the point based on the intersection of a pair of lines. Two of these lines, the enriching and stripping operation lines, are present in both methods and are defined by Eq. 1-10 and Eq. 1-45 in method 1. The third line, the combined streams q-line, is defined by Eq. 2-5 in method 2. As a matter of geometric consistency, evaluation of the ratio of x-ordinates or y-ordinates for a pair of such intersections should result in unity. In other words, when taking the ratio of a single ordinate between a pair of definitions for the intersection of the three lines, all variables should cancel exactly.
3.1.2 The Points to be Examined
The pair of definitions for the intersection of the three lines will be taken as the intersection of the enriching and stripping operation lines and the intersection of the enriching operation line and the combined streams q-line. The latter selection was made due to the relative simplicity of the definition of the enriching operation line over that of the stripping operation line.
3.2.0 Proof of Equivalence