The Difference Point Xa

Xa is a vector of compositions which, like real compositions, is subject to the constraint that its individual components sum to unity [1]. This can be easily shown by considering the definition of Xa for individual components 1,2, and 3, shown in Equation 3.13. 

Since components and are compositions subject to the constraint that they have to sum to unity, we can rewrite the definition of Xas in terms of components 1 and 2  

Therefore, in a ternary system, for example, specifying only the two entries of the difference point enables to infer the remaining third component. This can be extended to systems of nc components nc 1) difference point compositions would have to be specified, with the remaining species being inferred by unity summation. 

Although Xa displays the summation properties of real compositions, it should in fact be interpreted as a net composition. Closer examination of its definition reveals that it is a compositional difference between vapor and liquid streams. Hence, it is referred to as the difference point for any particular CS. In other words, an individual element of the vector dictates the net composition of that specific component in the CS. 

Similarly, for any stripping section terminated by a reboder, its difference point is equal to the bottoms product composition 

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Notice that the DPEs describing the scenarios in Cases 1 and 2 (Equations 9.18 and 9.20) are mathematically similar to each other. Each comprise of the addition of two vectors, namely the separation vector, S = x —y, and the mixing vector, M = Xa — X. The only difference between the two cases is that the difference point (Xa) is equal to the composition of the product being removed in Case 1. To avoid repetition, these two cases will be discussed simultaneously, due to the similar nature of their equations. 

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