Slide 1

1 Ref: Seider et al, Product and process design principles, 3 rd ed., Wiley, 2010. Slide 2 Attainable Region Attainable region (AR) defines the achievable compositions that may be obtained from a network of chemical reactors. The attainable region in composition space was introduced by Horn (1964), and extended by Glasser and co-workers (1987- 1990). A systematic method for the construction of the attainable region using CSTRs and PFRs, with or without mixing and bypass (as presented by Hildebrant and Biegler, 1995), is demonstrated for van de Vusse kinetics as follows: 2 Slide 3 Attainable Region The van de Vusse kinetics is: At a particular temperature: Step 1: Begin by construction a trajectory for a PFR from the feed point, continuing to the complete conversion of A or chemical equilibrium. For this case we have: 3 Slide 4 Attainable Region 4 Slide 5 5 Slide 6 Step 3: The attainable region is expanded by linear arcs, representing mixing between the PFR effluent and the feed. Note that a linear arc connecting two points on a composition trajectory is expressed by the equation:,where c 1 and c 2 are vectors for two streams in the composition space, c * is the composition of the mixed stream, and is the fraction of the stream with composition c 1 in the mixed stream. The linear arcs are then tested to ensure that no rate vectors positioned on them point out of the AR. If there are such vectors, proceed to the next step, or not return to step 2. In this example, a linear arc, ADB, is added, extending the AR to ADBC. Since rate vectors computed along this arc are found to point out of the extended AR, proceed to the next step. 6 Slide 7 Attainable Region 7 Slide 8 Step 4: Since there are vectors pointing out of the convex hull, it is possible that a CSTR trajectory enlarges the attainable region. After placing the CSTR trajectory that extends the AR the most, additional linear arcs that represent the mixing of streams are placed to ensure that the AR remains convex. The CSTR trajectory is computed by solving the CSTR form of the kinetic equations as a function of the residence time, : For this example, the CSTR trajectory that extends the AR most is that computed from the feed point (curve AEF), which passes through point B. 8 Slide 9 Attainable Region 9 Slide 10 Since the union of the previous AR and the CSTR trajectory is not convex, a linear arc, AGO, is augmented. This arc represents a CSTR with a bypass stream. 10 Slide 11 Attainable Region Step 5: A PFR trajectory is drawn from the position where the mixing line meets the CSTR trajectory. If this PFR trajectory is convex, it extends the previous AR to form an expanded candidate AR. Then return to step 2. Otherwise, repeat the procedure from step 3. As shown in the next Figure, the PFR trajectory, OHI, leads to a convex attainable region. The boundaries of the region are: (a) the linear arc, AGO, which represents a CSTR with bypass stream; (b) the point O, which represents a CSTR; and (C) the arc OHI, which represents a CSTR followed by a PFR in series. It is noted that the maximum composition of B is obtained at point H, using a CSTR followed by a PFR. 11 Slide 12 Attainable Region 12 Slide 13 Example 7.4 Maleic anhydride,C 4 H 2 O 3, is manufactured by the oxidation of benzene with excess air over vanadium pentoxide catalyst: Since air is supplied in excess, the reaction kinetics are approxi- mated using first-order rate laws: A is benzene, P is maleic anhydride, and B and C are byproducts (CO 2 and H 2 O). The r i s have the units of m 3 /(kg catalyst.s). 13 Slide 14 Example 7.4 Given that the available feed stream contains benzene at a concentration of 10 mol/m 3, with a volumetric flow rate, v 0, of 0.0025 m 3 /s (the feed is largely air), propose a network of isothermal reactors to maximize the yield of maleic anhydride. Solution: First, an appropriate reaction temperature is selected. Following Heuristic 7 in chapter 6, the next Figure Shows the effect of temperature on the three rate coefficients, and indicates that in the range 366< T