In the discussion up to this point, we have been considering the "design problem," namely, finding the optimum number of stages. A second type of optimization problem of equal importance is the "rating problem": finding the optimum operating conditions for a given column with a fixed number of stages.

There are several types of rating problems. One of the most common is finding the product purities that maximize profit. In the design problem considered in previous sections, we assume that the product purities were given. In many columns the purity of one product may be fixed by a maximum impurity specification, but the other product may have no set purity. For example, suppose that the propane product is more valuable than the isobutane and has a maximum impurity specification of 2 mol% isobutane. We know that distillate flowrate should be maximized and that as much isobutane as possible should be included in this stream, up the impurity constraint. This can be achieved by minimizing the concentration of propane that is lost in the bottoms. But reducing xB requires an increase in reboiler heat input, which increases energy cost. Therefore, there is some value of xB that maximizes profit. The optimization must take into account the value of the propane product compared to the bottoms and the cost of energy.

The steady-state simulator can be used to find this optimum operating condition. The distillate composition is held constant using a Design Spec/Vary. A value of the bottoms composition is specified in a second Design Spec/Vary, and the simulation is run to find the corresponding reboiler heat input, the distillate flowrate, and the bottoms flowrate. The profit is calculated for this value of xB by multiplying the price of each product ($/kg) by its mass flowrate (kg/ s), multiplying the price of the feed by its mass flowrate and multiplying the reboiler heat input (MW) by the cost of energy ($/MW . s). Profit ($/s) is defined as the income from the two products minus the cost of the feed minus energy cost. Then a new value of bottoms composition is specified and the calculations are repeated.

Figure 4.2 shows the results of these calculations using the following parameter values:

As the bottoms composition decreases, the reboiler heat input and the distillate flowrate increase. There is a rapid rise in reboiler heat input below 0.2 mol% propane. The maximum profit is obtained with a bottoms composition of 0.25 mol% propane.

This type of optimization is a "nonlinear programming" problem (NLP), which can be performed automatically in Aspen Plus. Click Model Analysis Tools on the Data Browser window and select Optimization. Click the New button and then OK to create an ID. The window shown in Figure 4.3 opens, which has a number of page tabs.

On the Define page the variables to be used in calculating the profit are defined. Type a variable name under the Flowsheet label. Figure 4.4 shows that several variables have been entered. The mass flowrates of feed, distillate, and bottoms are FW, DW, and BW (in Flowsheet column) in kilograms per second. Reboiler heat input is QR in watts.

Placing the cursor on one of the lines and clicking the Edit button open the windows shown in Figure 4.5, where the information about that variable is specified. For example, FW is edited in Figure 4.5a. Under the Category heading, Streams is selected. Under the Reference heading, the type is Stream-Var, the stream is F1, the variable is Mass-Flow.

Figure 4.5b shows the editing for the reboiler heat input. Since it is in the C1 block, Blocks under the Category heading is selected. Figure 4.5c shows that all variables have been defined. Clicking the Objectives & Constraints page tab opens the window shown in Figure 4.6, on which PROFIT is specified to be maximized. This variable is defined by clicking the Fortran page tab and entering the equation for profit as shown in Figure 4.7:

(where 4.7e-9 = 4.7 x 10"9). Selecting the final page tab Vary opens the window shown in Figure 4.8, in which the variable to be manipulated is defined. The distillate composition is being held constant by manipulating distillate flow using a Design Spec/Vary.

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