Achievable points

Reductive Tail Gas Treatments. It was largely as a result of the effort to achieve better than 99% recovery that the reductive tail gas desulfurization processes (46) were developed in the 1970 s. The two main methods are the Beavon Sulfur Removal (BSR) (47) and the Shell Claus Off-Gas Treatment (SCOT) (48) processes. Both of these processes are now widely used as tail gas desulfurization units on sulfur recovery plants and can readily achieve point source emission levels below 250 ppm and below 100 ppm if necessary to meet regulatory standards. 

The sum of all points scored eventually represents a laboratory s performance in a single proficiency test. The maximum achievable points, however, depends on the total number of spiking chemicals used. As this number varies from test to test, the points scored cannot be used to compare performances between laboratories participating in different proficiency tests. This comparison is achieved with the final letter scoring system, and is shown in Table 9. 

A Known Achievable Point Consider if it were known (because an oracle tells you) that a toluene concentration of 0.09 mol/L is achievable without knowing what particular piece of equipment or method is needed to achieve it. Would it be possible to utilize this piece of information to deduce what equipment or methods would be needed Simply knowing that 0.09 mol/L is an achievable concentration may incentivize investigation to determine an appropriate reactor that achieves it. 

There is no need to include Xj into the set as this concentration does not expand on the set of available formulations. By comparison, both X2 and X4 expand on the set of achievable points. The inclusion of X4 allows for the synthesis of drink 7. Recall that Xj is not achievable using only bj, b2, and bj, and thus an additional drink can be gained if X4 is chosen over X2. It is therefore best to choose X4 as this maximizes the set of achievable concentrations. 

This exercise has highlighted important points about concentration and mixing. Mixing allows us to achieve new points and expand from an initially small fixed set. This idea is central to AR theory—mixing allows one to achieve many solution states from a smaller set of achievable points. Let us extend on the ideas presented so far and formalize them into concrete terminology and definitions. 

Mixing can be used to fill in regions. Interior points of a set of achievable points may be achieved via mixing of the appropriate points. Hence, the interior of the convex hull of a set of achievable points is also achievable via mixing. 

In Chapter 2, batch data for the BTX reaction were plotted in a manner that allowed us to infer useful geometric properties related to concentration and mixing. We now understand that mixing allows for two physically separate yet achievable points to be connected by a straight line in concentration space. If more than two unique concentrations are available, it is possible to produce a filled convex region in space. In doing so, concentrations that were previously unachievable can be made achievable via mixing of the appropriate points. 

Convex hulls are closely related to mixing. The convex hull assists in differentiating between unique points (those that cannot be formed from combinations of other points) and redundant ones (those that can be formed from combinations of other points). From this perspective, knowledge of the convex hull for a set of achievable points is all that is required for mixing problems. This result is important for the following two reasons ... 

There is no need to search for all possible achievable points when creating mixtures. Rather, it is sufficient to only identify the unique points that belong to the convex hull—all other achievable points may be obtained from the extreme points of the convex hull. 

Region A is the remaining set of achievable points when region is removed from the entire shaded region FGO. The union of regions A and B constitute the entire set of points. 

Let us select a point in the region and generate a batch profile from this point. Point 82 in Figure 3.2 is representative of such a point in region A. (As will be shown, selecting points for further improvement is somewhat arbitrary. ) To achieve point 82, the following three steps are required ... 

We can find the reaction time required to achieve point by plotting c versus reaction time and locating the specific time where the value of c is equal to that given by a. The line segment FGa is therefore representative of the solution trajectory obtained by running the BTX reaction from the feed until the concentration at point a is achieved. 

Figure 3.3 (a) Example of a typical mixing line used to achieve point Xj. Point Xj is located in the concave section of the profile (region B,). 

In order to achieve point X2, point y2 is required. Similarly, to achieve point j2, a starting mixture of point Xj (from experiment 2) is required. Concentration Xj, in turn, is formed from a mixture of the feed F and point yj (obtained from experiment 1). 

Figure 4.22 (a) Reactor structure required to achieve point A,... 

Point C lies on a mixing line that is connected between Cf and point B. Point B lies on the CSTR locus from the feed. Hence, to produce point C, we must first achieve point B and then mix the CSTR effluent with fi esh feed via a bypass, as shown in Figure 4.22(b). 

Point F is located on a mixing line joining points C and E. We already know that point C is achieved by a CSTR with bypass from answer (b). Point E lies on a PFR tfajectory, although this trajectory is not the same as in answer (a). A CSTR operating up to point D followed by a PFR in series is needed to achieve point E. Point E is therefore a mixture between points E and C, and thus E is obtained by... 

Similar to that of point 1, point 2 may also lie on a mixing line. If a PFR were to be initiated with a feed concentration from point 2, the resulting trajectory would travel, at least initially, out of the region. This trajectory would hence serve to further expand the set of achievable points. 

The rays associated with points 1 and 2 touch the candidate region at points x and y, and are thus achievable by a CSTR. (Point x is the feed concentration to a CSTR that achieves point 2, and point y is the feed concentration to a CSTR that achieves point 1.) Note that the ray associated with point 3 does not intersect the candidate region, and as a result it is not directly achievable by a CSTR. 

Observe that the CSTR locus is contained entirely within the region generated by the PFR, and hence a CSTR from the feed does not serve to expand the region of achievable points further. From Chapter 4, it is known that PFR trajectories cannot cross. Hence, running PFRs from the CSTR locus will not expand the region—any PFR trajectory extended from points on the CSTR locus in Figure 5.3 are contained... 

Now that the optimal reactor stmcture is known for aj > a2, it would be useful to compute the volumes of the reactors necessary to achieve point P. Given the information in Table 5.4, calculate the following ... 

Theorem 6.1 (exposed points on the AR boundary are either PFR trajectories or feed points) Suppose that we have a specified feed set F in R" and a convex set of achievable points given by C, also contained in R". The rate function r(C) associated with this region is assumed to be continuously differentiable and also defined on R". Furthermore, the set of concentrations in C is assumed to comply with the complement principle. If it is found that all rate vectors on the boundary ofC do not point outward, then any protrusion in C that is separate (disjoint) from F is the union of PFR trajectory segments. The solution trajectories then satisfy the PFR equation dC/dr =r(C). 

The collection of achievable points obtained from both the PFR trajectory and CSTR locus can be combined and the convex hull for the entire set is then determined, which is shown in Figure 7.2(a). It is evident that the candidate region is enlarged by the inclusion of the CSTR locus. 

Let us consider the role of critical reactors in the formation of the AR boundary. The aim of this example is to provide a complete set of all achievable points, and so we shall continue investigating further expansion of the region. In order to do this, critical reactors must be introduced, which will require us to use ideas and theory developed previously in Chapter 6. 

In this scenario, the AR intersects the plane at Cg = 0.4 mol/L at a number of points. However, we seek to find the unique point on the plane that maximizes Cg. This occurs on the AR boundary at point G in Figure 7.7(b). In this instance, the desired concentration of component D is larger than that obtained at the critical point B. The reactor structure needed to achieve point G is therefore different when Cg = 0.3 mol/L described previously as structure 1 (given by Figure 7.6) is now required. This is true even though scenarios (a) and (b) share similar objective functions—0.4 mol/L in component D in scenario (b) compared to 0.3 mol/L in scenario (a). A critical DSR fed at the feed point operated to point F, followed by a PFR to point G is now required. The reduced path AFG shown in Figure 7.7(b) is needed in this instance. Again, if it is required to only achieve point G, then structure 2 need not be considered. 

The DSR trajectory from point C travels in a direction opposite to the CSTR locus and terminates at the first DSR equilibrium point from Section DSR from the feed point (point B). The path taken from the feed point to the second DSR equilibrium points is then A to C (in a CSTR) and then C to B (in a critical DSR). This DSR trajectory further expands the set of achievable points. 

In Figure 7.21(b), the PFR trajectory, and the shape of candidate region, is shown for an additional PFR. The point at which the base trajectory touches the current boundary is now given by point G. In order to achieve point G, two PFRs in series are required with interstage bypass of feed to the exit of each reactor. The PFR trajectory GHD achieves a lower overall residence time than before. Again, the inclusion of an additional PFR increases the complexity of the structure however, a lower system residence time is obtained. 

The two following batch structures are thus required to achieve all achievable points ... 

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