Time implicit model equations for the shortcut method

Time implicit model equations for the shortcut method 

The constant Ci in the Hengstebeck-Geddes equation is equivalent to the minimum number of plates, Nmin, in the Fenske equation. At this stage, the variable reflux operating mode has Gi and R, the constant reflux has and Ci, and the optimal reflux has, Ci, and R as unknowns. Summation of distillate compositions can be used to obtain Gi for variable reflux and for both constant reflux and optimal reflux operation, and the FUG equations to obtain R for variable reflux and Gi for both constant reflux and optimal reflux operations. The optimal reflux mode of operation has an additional unknown, R, which is calculated using the concept of optimizing the Hamiltonian, formulated using the different optimal control methods. 

The shortcut model is extremely efficient and reasonably accurate for nearly ideal systems and for column with neghgible holdup. For columns with severe holdup 

Literature on the optimization of the batch column is focused mostly on the solution of optimal control problems, which includes optimizing the indices of performance such as maximum distillate, minimum time, and maximum profit. However, literature on optimal design of batch distillation for performing specified operations by using the constant reflux or variable reflux policies is very limited[46]. 

As stated in Chapter 1, control refers to a closed-loop system where the desired operating point is compared to an actual operating point and a knowledge of the error is fed back to the system to drive the actual operating point towards the desired one. However, the optimal control problems we consider here do not fall under this definition of control. Because the decision variables that will result in optimal performance are time-dependent, the control problems described here are referred to as optimal control problems. Thus, use of the control function here provides an open-loop control. The dynamic nature of these decision variables makes these problems much more difficult to solve as compared to normal optimization where the decision variables are scalar. 

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