Group methods are approximate calculation procedures for counter-current cascades that relate compositions of entering and exiting streams to the number of equilibrium stages. These methods were originally tailored for iterative or hand calculations and do not account for detailed changes in temperature and composition on individual stages. As a result, they have fewer variables and constraints as compared to rigorous models. Although rigorous models are capable of representing process behavior more accurately, they are more complex in terms of problem size and are often much more difficult to solve. In such cases, group methods may be used particularly if their performance closely matches that of rigorous models.

Group methods like Kremser and Edmister have been used to model absorbers, strippers, simple distillation columns etc. (Seader and Henley, 1981). The model equations for these methods often change depending upon whether the cascade acts as an absorber or stripper or is coupled with equipments like condenser (enricher) or reboiler (exhauster). Distillation columns with multiple feeds result in many counter-current cascades and it is not clear a priori whether a cascade of trays in a distillation column acts as an absorber or a stripper. Moreover, these models have not been systematically used in optimization studies of complex flowsheets where use of rigorous model either makes the problem intractable or computationally too difficult to solve.

In this paper, we propose a general cascade model that represents an aggregate model for a collection of trays in a section of distillation column. Our proposed model can be considered as an extension or improvement of the Edmister method. Some approximations in Edmister's method are replaced by more realistic constraints based on physical insight. Just like the rigorous tray-by-tray model, our model does not require knowledge about key components or about whether it behaves as an absorber or a stripper. Being an aggregate model, it can be connected to rigorous or aggregate models of other process models like feed stage of distillation column, condenser, reboiler or another such cascade. An interesting property of the proposed model is that if the number of stages in the cascade is set to zero, then the model forces the outlet vapor and liquid flow conditions to be exactly same as the corresponding inlet conditions i.e. it behaves as if the cascade does not exist. This is particularly helpful in topology optimization where sometimes minimization of capital cost may require removal of all the trays in a section (i.e. make the cascade disappear).

The capability of the proposed cascade model is demonstrated using both simulation and optimization case studies. Test problems involve distillation of binary or ternary systems where models for feed stages, condenser and reboiler are combined with cascade model to form the aggregate model and with model for all trays in a section to form the rigorous model. In the case of simulation case studies, the outlet conditions at the top and bottom are analyzed for a given set of input parameters (e.g. number of trays, column pressure, reflux ratio etc.). In the case of optimization case studies, the objective is to determine the optimal feed locations and/or optimal number of trays that minimize capital and/or energy costs while meeting the specified purity requirements. Such optimization problems have already been addressed using rigorous MINLP models by Viswanathan and Grossmann (1990, 1993a, 1993b). Their strategy involves use of binary variables which are associated with location of feed, reflux and boilup. On the other hand, our formulation using the cascade model requires that only the number of trays in each cascade (treated as integer variable) be optimized.

Results for the simulation case studies show that the outlet conditions predicted by the aggregate model are in close agreement with that of the rigorous model. For the optimization case studies, a rounding heuristic can be used i.e. the number of stages in the cascades can be relaxed as continuous variables and their optimal value can then be rounded to the nearest integer. In most cases, the integer solution obtained by this rounding heuristic was found to be the same as the integer solution of the MINLP model. Thus, use of integer variables can be eliminated for the aggregate model and its solution provides a good approximation to that of rigorous model (often missing by only one or two trays).

References

Henley, E. J. and Seader, J. D. (1981) Equilibrium-Stage Separation Operations in Chemical Engineering, Wiley, New York.

Viswanathan, J. and Grossmann, I. E. (1990) A combined penalty function and outer-approximation method for MINLP Optimization. Computers and Chemical Engineering, 14(7), 769-782.

Viswanathan, J. and Grossmann, I. E. (1993a) An alternate MINLP model for finding the number of trays required for a specified separation objective. Computers and Chemical Engineering, 17(9), 949-955.

Viswanathan, J. and Grossmann, I. E. (1993b) Optimal feed locations and number of trays for distillation columns with multiple feeds. Industrial and Engineering Chemistry Research, 32(11), 2942-2949.