Pinch point loci

It is important to recognize that every point on the ret tate column profile has two parameters associated with it, namely 

This is different from the distillation profiles generated earlier in this book, where the CMO assumption was employed those profiles had only x associated with every point. The reflux was held constant, and did not change down the length of the CS. In this scenario, however, both compositions and flows change down the length of the MGS. Hence, both x and change along a membrane column profile. 

A property of the pinch point loci, which can be mathematically proven, is that one branch will always pass through More details on the nature and 

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For a fixed there exists a unique set of pinch points for a specific as illustrated in Figure 3.5. As 7 changes, so do the locations (and sometimes the number) of the pinch points for a fixed X - Pinch point loci can therefore be defined as the loci of all pinch points obtained by varying the reflux ratio, for a set X [7]. Pinch loci help describe the path that the TTs follow as is varied because the nodes for every CPM have to lie on the respective pinch curve. In this section, we will show how the behavior of the pinch point loci affects the topology of CPMs. 

This special case in Equation 3.28 is valid for any system, regardless of the phase equilibrium properties of the system. The significance of = — 1 is that this represents the point where the vapor flowrate in the CS tends to zero, and can be seen as the switching point from a counter-current mode of operation to a cocurrent operation mode. The only pinch point at these conditions is thus at Xa and profiles will resemble the mixing only process shown in Figure 3.12b. More importantly, though, this shows that any pinch point loci will always move through Xa, irrespective of phase equilibrium. 

Figure 3.20a shows the pinch point loci for Xas in the MET, also shown in detail in Figure 3.19. In Section 3.5.2, it was discussed that CSs terminated by a reboUer or condenser which produce product streams will have the composition of the product equal to Xa for the CS. In other words, for these CSs, the design is limited to Xas in the MET, and the associated pinch point loci (Figure 3.20a). Thus, the behavior of the pinch points in simple columns is constrained to only these flow patterns and their associated topological behavior.

FIGURE 3.20 A geometric description of possible flow types associated with entries of X, with their associated pinch point loci for a constant volatility system with a = [5,1,2] for the seven different regions (labeled) depicted in Figure 2.17a. 

Figure 3.21 shows a very different behavior in the pinch point loci from those given for the constant relative volatility system. However, the same properties still exist at infinite reflux the nodes are at their RCM positions, and at R = 1 the only... 

FKiURK 3.21 Pinch point loci for the acetone/benzene/chloroform system at P=l atm (NRTL)with Xa = [0.1 0.2]. 

FKjURE 3.22 Pinch point loci produced by tangent conditions between and arbitrarily chosen and the residue curves. The RCM for the system is given, with the tangent lines di layed as dashed lines. The pinch point loci are drawn by connecting all tangent points for the (a) benzene/p xylene/toluene system, and (b) acetone/benzene/chloroform system. 

This method is simple and easy to use, and all that is required is the RCM for the system in question. However, the accuracy of the method depends on the number of residue curves used. Of course, the same method can be used directly to locate pinch point loci outside the MET, as is done in the example in Figure 3.22b for the acetone/benzene/chloroform system at 1 atm. Thus, by understanding the graphical nature of the DPE, one is able to gather a plethora of information, and many insights can be gained from this. 

Felbab, N., D. IBldebrandt, and D. Glasser, A new method of locating all pinch points in nonideal distillation systems, and its application to pinch point loci and distillation boundaries. Computers and Chemical Engineering, 2011, 35(6) 1072 1087. 

Moreover, for a fixed flux model and Xa (or x ), each Ta has associated with it a set of stationary or pinch point locations (refer to Section 3.6.). The pinch points are defined by equating the MDPE to zero, that is, when the composition is no longer changing down the length of the MGS. As / a changes, so do the positions of the pinch points. It is therefore possible to generate pinch point loci, as shown in Figure 9.6, to visualize the movement of each node. 

Analysis and visuahzation of the pinch point loci associated with each Xa gives insight into the trajectory followed by a profile. This is evident firom the above discussions. 

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