## Optimal Feed Locations

Finding the optimal feed locations can be formulated as an optimization problem in which the vapor rate is minimized by varying the feed tray locations.

minimize VS

Because the total tray number NT is finite, we can find the optima by exhausting all NT possibilities. It is reasonable to restrict the search space to the reactive zone so that the possible choices are further reduced to NRx. Here, we take a brute force approach by fixing NF,A first while varying NF,B until a minimum VS is found. Next, NF,A is changed, and the procedure repeats itself until a global minimum is located. Figure 18.4 shows the variation of the vapor rate for various values of the two feed locations. The results show that we should move the feed location of heavy reactant B down to NF B = 15 (from 19) and move the feed tray of light reactant A up to NF,A = 11 (from 9). This corresponds to a 10.9% energy

Figure 18.4 Feed tray locations and corresponding energy consumption (compared to base case) from optimization results.

savings compared to the conventional feed arrangement (see Table 18.2). Furthermore, simulation results from this and many other examples reveal that the feed location of the heavy reactant should not be placed lower then the feed tray of the light reactant. This reduces the search space to (Ngx + 1)Nrx/2.

In addition to the percentage of energy savings, comparisons are also made in terms of the profiles of temperature, compositions, fraction of total conversion, and forward and reverse specific reaction rates. Figure 18.5 shows that the optimal feed arrangement (Fig. 18.5b) has a much sharper temperature profile in the reactive zone than the conventional feed locations (Fig. 18.5a). The tray temperature almost reaches 390 K in the optimal case, but barely reaches 380 K in the conventional case. Further, the profiles of the tray conversion and rate constant also take a qualitative shape similar to that of the temperature.

The composition profiles in Figure 18.5 explain what is happening. First, as a result of moving Nf,b downward and Nfa upward, we have nonmonotonic reactant distributions for the optimal case as opposed to the monotonic reactant distribution for the conventional one. This is advantageous for the forward reaction. Second, we obtain an almost monotonic product distribution for the optimal case, especially for heavy product D. The xD (tray composition of product D) almost reaches 60% at the bottom of the reactive section, which has a profound effect on the temperature profile. In contrast, the mole fraction of D gives a nonmonotonic profile for the conventional case, and xD takes a downturn toward the bottom of the reactive tray as a result of dilution from the excess light reactant A that is introduced in the bottom of the reactive zone. The results presented in Figure 18.5 reveal the complicated interaction between temperature and composition in the reactive zone,

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