Product simplexes

Figure 3.7. (a) Product simplexes Regsi p for ternary azeotropic mixture (shaded), and (b) two-column sequence (product points - 2,13,1 for feed point xp( ) or 2,13,3 for feed point xp(2)). 

Further, we call triangles 1-2-13 and 3-2-13 product simplexes Regsmp- This notion has great significance for separation flowsheets synthesis, because for a feed point xp located inside the product simplex one can get all the components and azeotropes that are vertexes of this simplex in a sequence of (n - 1) columns. 

However, the above-described method is unfit for splits other than the direct and indirect ones, and the number of such splits grows dramatically with the increase of n. The general methods of splits set determination are based on the usage of structural matrix and of method of product simplex for distillation subregions (Petlyuk, Kievskii, Serafimov, 1979). 

Let s call the above-stated method of determination of the set of feasible splits at R = 00 and A = 00 the method of product simplex. 

Product simplex for three-component mixtures is a triangle for four-component mixtures, it is a tetrahedron for five-component mixtures, it is a pentahedron etc. Inside one distillation subregion at m > n, product simplexes cross each other (i.e., one and the same feed point can simultaneously enter several product simplexes). 

Figure 3.15. An example of product simplex Regsimp of four-component azeotropic mixture (shaded).

Figure 3.16. Regions Reg°° and subregions Regsa of distillation and product simplexes B.egsimp of some structures of three-component mixtures. 1,3.4a,..., classification according to Gurikov (1958). Dotty lines with arrows, boundaries of distillation regions thin lines, boundaries of distillation subregions dotted lines, boundaries of product simplexes.

The notion of product simplex coincides with the notion of region of batch distillation that was used in Safrit Westerberg (1997). 

A method similar to that of product simplex and considering azeotropes as pseudocomponents was proposed for synthesis of separation flowsheets in Sargent (1994). 

For n-component mixtures, the method of product simplex is based on application of structural matrix. It includes the following steps ... 

Determination of the product simplexes Regs,> p from each bonds chain, for which m> n... 

Checkup of which product simplexes feed point belongs to (xf e Regsimp)... 

To illustrate the first step, determination of the first and second bonds chain from structural matrix is shown in Figs. 3.17a, b, respectively. Let s note that the attempt to use bond 1 4 at creation of the second bonds chain does not lead to positive result (bonds chain 12->-1 4 5 25 does not include component 3). To isolate the product simplexes Regstmp from bonds chain with m> n (second step), the combinations of n stationary points from m are being examined. For example, for the chain 12->l- -3- 4 5 25(12 25), it is possible to get the following product simplexes Reg p ... 

The other combinations of six stationary points 12,1,3,4,5, and 25, five at a time, do not form product simplexes because they do not contain all the components. The checkup of belonging of the feed point to one or another product simplex (third step) should be performed for all the product simplexes. This checkup is... 

The system [Eq. (3.8)] is an expression for center of gravity of the product simplex, when gravity is applied only to its vertexes (stationary points). For all this, the center of gravity coincides with feed point xp and relative distances of vertexes of the simplex Aj from feed point are equal to the corresponding coefficients aj. 

If the feed point belongs to product simplex Reg p being examined, then solution of the system [Eq. (3.8)] relative to coefficients uj should answer inequalities 0 < fl < 1. If part of coefficients aj does not answer these inequalities, then feed point is located out of the product simplex being examined. This method of determination of whether the feed point belongs to one or to another product simplex was proposed in Petlyuk et al. (1979). 

If it is ascertained by means of such analyses that feed point belongs to several product simplexes, then the mixture at R = 00 and N = 00 can be separated into pseudocomponents at any border between two adjacent key stationary points of each product simplex if the rule connectedness is not broken. The splits with one pseudocomponent being distributed between products is also feasible. 

An example of determination if the feed composition belongs to this or that product simplex, which was formerly depicted in Fig. 3.14a, is shown in Fig. 3.19. Besides the direct and the indirect splits that were shown in Fig. 3.14a, we have three other possible splits an intermediate spht and two sphts with one distributed component. In Fig. 3.19, lines of material balance are shown for all possible sphts of the feed composition under consideration at / = oo and N=oo. 

The rule of product simplex gives us the instrument that allows not only to determine feasible splits in the first column, but also gives an opportunity to determine at once the compositions of the products in the sequence of (n - 1) columns. Compositions corresponding to the vertexes of the product simplex the feed point belongs to can be obtained as columns system products. 

If w > n, then feed point belongs to several product simplex. Therefore, several sets of products, corresponding to vertexes of each product simplex, can be got from such feeding. Let s examine a few examples. 

Let s return to the example of industrial mixture that is a by-product of wood pyrolysis. It is seen from the examination of product simplex Regsmp = 1 234 23 38 568 58 78 89 8 that in a column sequence consisting of... 

What is the difference between product simplex Reg p and distillation subregion... 

Feed point belongs to product simplex Regsimp = 12->l->3->4at Fig. 3.15. Enumerate all the boundary elements of dimensionality 0,1, and 2 of this product simplex. Which splits are feasible for this feeding ... 

For a given feed point Xf in the product simplex 1-13-2-4 for the mixture of exercise 1, find the bottom point xb for the best direct separation. 

For a mixture of acetone(l)-methanol(2)-chloroform(3)-ethanol(4), determine the location of the boundary between distillation regions. List all the distillation subregions and product simplexes for this mixture. For each product simplex, state all feasible sphts without distributed components or pseudocomponents. 

Product simplex (Reg /mp) simplex whose vertexes are stationary points of distillation subregion and their number is equal to the total number of components. 

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