Attainable region boundary

Around the same time, Glasser et al. (17) retrieved and extended the insightful methods of Horn (18) and presented graphical procedures known as the attainable region (AR) method. Their approach requires the graphical construction of the convex hull of the problem and helps to exemplify the need for a systematic and general methodology. In principle, the reactor network with maximum performance in terms of yield, selectivity, or conversion can be located on the boundary of the AR in the form of DSR and CSTR cascades with... 

Some computational methods are based on finding the boundary of the attainable region using the reactor types that characterize the attainable region extreme points. Hybrid methods involving superstructures and geometric considerations have also been proposed [15], ... 

However the region at the left of the point P is not convex. But we may continue from the point P with a PFR, as indicated in Fig. 8.27B. The computation is done by simply Integrating the differential equations of a PFR, this time input concentrations supplied by the exit of the CSTR. The new augmented region is convex. This time all three conditions are fulfilled. No other mixed reactors can be found above the boundary that could give a higher amount of B. This is the final Attainable Region. 

As it has been shown, in AR approach the construction of the boundary is essential. Feinberg and Hidebrandt (1997) demonstrated that the boundary of attainable region could be assembled only by means of combinations of PFRs, CSTRs, and DSRs (differential side-stream reactors). The calculation of these elementary reactors is simple. The construction of an AR is not complicated, if the analysis is restricted at two dimensions. Moreover, 3-D representation could be visualised relatively easy with an appropriate computer tool. In the case of systems of higher dimensions the method relies rather on numerical techniques, and is not easy to apply. 

Fig. 6.3 Selectivity to B against conversion of A for a t A/o-phase CSTR and for the boundary of the attainable region...

Reactions 1, 2, and 3 are first-order in A, B, and B, respectively, while reaction 4 is second-order in A. The rate constants at a particular temperature are ki = 0.01 s , ki = i s , 3 = 10 s , and = 100 m /kmol -s . The boundary of the attainable region, shown in Figure 6.7, is composed of arcs, each of which results from the application of a distinct reactor type, as described next. 

CSTR (point O), and a CSTR followed by a PFR (curve D). Within the region bounded by the three arcs and the horizontal base line Cg = 0), product compositions can be achieved with some combination of these reactor configurations. The appropriate reactor conhguradoo along the boundary of the attainable region depends on the desired effluent concentration of A. When 1 > > 0.38 kmol/m, a CSTR with bypass (curve C) provides the maximum... 

Step 2 When the PFR trajectory bounds a convex region, this constitutes a candidate attainable region. When the rate vectors at concentrations outside of the candidate AR do not point bad into it, the current limits are the boundary of the AR and the procedure terminates. In Figure 6.12, the PFR trajectory is not convex, so proceed to the next step. 

Step 5 A PFR trajectory is drawn from the position where the mixing line meets the CSTR trajectory. When this PFR trajectory is convex, it extends the previous AR to form an expanded candidate AR. Return to Step 2. Otherwise, repeat the procedure from Step 3. As shown in Figure 6.12, the PFR trajectory (line 7) leads to a convex attainable region. The boundaries of the region are (a) the linear arc (line 5) from A to C, which represents a CSTR with a bypass stream (b) the point C, which represents a CSTR and line 7 from C to B, which represents a CSTR followed by a PFR in series. Note that the maximum composition of CO is obtained at point D, using a CSTR and PFR in series. The maximum selectivity, defined by the ratio of CO/CH4, is also achieved at point D, where the ratio is 0.47, as compared to point C, where the ratio is only 0.30. ... 

A key difference between the complement method and the LP method is that the latter requires the solution of a linear program, whereas the former is a direct application of the CSTR attainability condition. In the LP approach, all points on the AR boundary are computed simultaneously—via the solution of a large linear program—in a single calculation step. In order for this result to be achieved, the candidate region boundary points must be expressed in terms of all other boundary points in space using a superstructure formulation, which is termed the total connectivity model. 

The solution of the gas flow and temperature fields in the nearnozzle region (as described in the previous subsection), along with process parameters, thermophysical properties, and atomizer geometry parameters, were used as inputs for this liquid metal breakup model to calculate the liquid film and sheet characteristics, primary and secondary breakup, as well as droplet dynamics and cooling. The trajectories and temperatures of droplets were calculated until the onset of secondary breakup, the onset of solidification, or the attainment of the computational domain boundary. This procedure was repeated for all droplet size classes. Finally, the droplets were numerically sieved and the droplet size distribution was determined. 

Theory would indicate, therefore, that electrodes should indeed have pores (and hence large numbers of menisci), but they should be very thin or else the catalyst should be concentrated only in a small region of high current density near the higher part of the three-phase boundary. The practical attainment of such a model was first reached in research carried out at the university level (Texas A M), and in a government research institute (Los Alamos National Laboratory). 

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