Adaptive collocation on finite elements models for the optimization of multistage distillation units

The size of the model for multistage distillation processes can be reduced by orthogonal collocation on finite elements (OCFE) techniques. The accuracy of the OCFE models can be improved by adaptively placing the breakpoints between the elements so that the approximation error is equally distributed among the elements within each column section. The location of the breakpoints depends on the features of the composition and temperature profiles in the column. Two different approaches are used for the estimation of the approximation error resulting from the OCFE solution. The first approach is based on the equidistribution of the residuals of the material and energy balances around envelopes that include predefined regions in the column. The additional constraints, generated by the adaptive grid procedure, are embedded into the economic optimization problem. Both the optimal operating conditions and the optimal breakpoint sequence are determined simultaneously. The second method uses the derivatives of the approximate solution and determines the element lengths by equidistributing the estimated error in an iterative procedure. The residual-based approach is more efficient than the derivative method in determining an element partition that results in a feasible optimal solution close to the optimal solution obtained by a tray-by-tray model. The adaptive placement of the breakpoints allows a more compact OCFE model for which an optimal solution exists. Multiple locally optimal element partitions may be obtained using the residual-based approach.