다제품 회분증류의 최적화

함 태원, 김 영한

동아대학교 화학공학과

Optimization of Multi-Product Batch Distillation

Tae Won Ham and Young Han Kim

Dept. of Chem. Eng., Dong-A University

INTRODUCTION

Recent interest on a batch distillation in such industries as fine chemical and pharmaceutical companies is growing fast. In these industries, the separation of multiple components from a product mixture is one of major difficulties in the production processes. The batch distillation is popular in the separation processes because of its easy operation and maintenance.

Most of the optimal design techniques of batch distillation columns have been using the quasi-steady state model or the short-cut continuous distillation column model. Apparently, the batch distillation is conducted in a dynamic state from the beginning to the end of the operation. Using the quasi-steady state model or the short-cut continuous model may lead to not accurate enough solution in the design of the batch distillation process. For instance, a study [1] utilizing the short-cut model indicates that the short-cut method gives only the initial trial policy and the final optimal reflux policy is found from a rigorous dynamic model. However, many studies of the short-cut method and its extensions have shown satisfactory design results. The short-cut method using the short-cut design equations for a continuous distillation column, such as the Underwood equation, the Gilliland correlation and the Hengstebeck-Geddes' equation, gives a comparable top product composition to the rigorous plate to plate calculations. A modification of the short-cut method for a nonideal system was conducted by Diwekar and Madhavan [2]. Also, it is extended to an azeotropic batch distillation system and its simulation outcome for the compositions of a top product and a still is similar to the result of a rigorous simulation. Since the composition calculation using the short-cut method is simpler and faster than the plate-to-plate model, the computational load in an optimal design can be tremendously reduced. A different way of the short-cut method was investigated by Al-Tuwaim and Luyben [3]. In the study, many case studies using a rigorous model are conducted and summarized to find easily an optimum column design for a given set of system having a specific relative volatility, a product specification, an energy cost and a material of construction.

In this study, an optimal design procedure of a multi-product batch distillation is proposed. The procedure utilizes a dynamic process model instead of the short-cut model or the quasi-steady state model implemented in the previous studies. The dynamic process model is employed in the search of optimal operation schedule of the column and the model includes liquid holdup in a still, a reflux drum and trays as well. The optimization objective of the column operation is composed of costs of product, feed and steam and total operation time, and the successive quadratic programming is applied to obtain the optimum operation schedule. A ternary system of cyclo-hexane, n-heptane and toluene is used as an example. The solution for the optimum production is compared with that of the quasi-steady state model. In addition, the optimal number of trays of the batch distillation column is found at the maximum net profit computed by deducing an investment cost from the operational profit.

DYNAMIC PROCESS MODEL

Material balances of a batch distillation column are formulated as below using the following assumptions:

1. Vapor holdup is neglected since its amount is far less than liquid holdup.

2. Ideal stage and constant relative volatility are assumed.

3. Vapor boil-up rate is constant.

4. Total condenser is employed.

5. Holdup change in trays is negligible.

6. The variation of  vapor flow rate from top tray to bottom is linear.

Material balance for component i at the nth tray from bottom is written as

[eqn: { M}_{n } { {dx }_{i,n } } over { dt} = { L}_{n+1 } { x}_{i,n+1 } + { V}_{n-1 } { y}_{i,n-1 }- { L}_{n } { x}_{i,n }- { V}_{n } { y}_{i,n }]       (1)

where Mn is liquid holdup in the tray which is obtained using the Francis weir equation and a liquid flow rate, Ln. The liquid flow rate is found from the total material balance around the tray with the assumptions of no vapor holdup and the negligent change of liquid holdup.

DESIGN OBJECTIVE AND OPTIMIZATION

The design objective of a batch distillation system is to obtain the optimum number of trays for a given production system and to find the optimum operation schedule of the system to maximize the operational profit which includes product sale earning, feed and steam costs. By computing the profit for a unit time, not only the maximization of on-specification product but the minimization of operation time is achieved from the design.

EXAMPLE

Products are separated into on-specification products and off-specification side products and collected in different reservoirs. The switch among the containers is carried out according to an operation time schedule and product composition. The on-specification product is collected while its composition satisfies composition requirement. The product is withdrawn until the average product composition reaches the specification. If the composition is off from the specification, the top product is gathered as an off-specification product for a given time. The next on-specification product is obtained in the same manner as the first on-specification product and the procedure is continued until the last on-specification product is yielded in a still.

A ternary system of cyclo-hexane, n-heptane and toluene is adopted from Farhat et al. [4] and its result is compared with the optimization outcome of this study. The relative volatilities are obtained by fitting to the equilibrium data calculated from the UNIQUAC model. The product prices are based on the current market price. For the steam cost, it is assumed that process maintenance is directly related to steam consumption, and so higher than actual cost is applied to include the maintenance cost.

RESULTS AND DISCUSSION

The outcome of the optimal design of a batch distillation column using a dynamic process model is compared with that of the design employing the quasi-steady state model in two different reflux ratio patterns: a constant reflux ratio and combined exponential/constant  reflux ratios. The numerical comparison between two studies indicates that the design with the dynamic model produces 5 % more product in total with 3.7 % more operation time than the quasi-steady state model.

One apparent difference between two studies is that the first product of this study is more than that of the quasi-steady state model while the last product is less. It is because of the liquid holdup in trays and a reflux drum in the beginning of an operation. Unlike the quasi-steady state model or the short-cut model, the liquid holdup is counted in this study. In addition, the variation of overhead product composition indicates the improvement of the accuracy of the composition calculation. It is apparent when a sudden change of reflux ratio is applied. In Farhat et al. [4] using the quasi-steady state model, the composition varies discretely at a sudden reflux ratio change, while this study shows a continuous composition variation. Though the significance of liquid holdup is much less than those of previous studies, the improvement of the computation of product composition due to the inclusion of liquid holdup is largely responsible for the improvement of this study.

A similar comparison is conducted for the combined exponential/constant reflux ratios. In this operation, seven parameters are used for two exponential profiles of reflux ratio, two constant reflux ratios and one operation time for the first side product. The comparison of total amount of products shows that the dynamic model gives 4.2 % more product with 0.3 % less operation time than the study of the quasi-steady state model. The combined exponential/constant reflux ratio operation gives more difference than the constant reflux ratio operation in the comparison with Farhat et al. [4]. Notice that a batch distillation is a dynamic process and is operated more dynamically with the combined reflux ratio operation. It is evident that the dynamic model can give more accurate design of a batch distillation process. Though the manipulation of reflux ratio in an exponential variation is more difficult than the constant reflux ratio operation, the production improvement is more significant. The increase of the concentration of cyclo-hexane during the production period of the first side product is due to the high reflux ratio. Some high concentration product can be produced with the high reflux ratio, but its production amount is very small because of the high reflux ratio and the added operation time reduces the optimization objective. It is found that a 23-tray column gives the highest net profit for the example system of this study. The operation schedule and product amounts are computed for the case.

In practice, the off-specification products from the first operation are recycled for the next batch operation. In order to examine the variation of optimal design when the recycled operation is conducted, the off-specification products are returned to a still for the following operation. In this case, the amount of fresh feed is the first charge, the still capacity, minus the recycled amount of the previous operation due to the limitation of the fixed still capacity. Since column trays and a reflux drum contain liquid holdup from the previous operation, the composition profile of the liquid holdup in the trays and reflux drum is assumed to be same to the end composition profile of the previous operation. In this case, the equilibrium operation time is replaced with the preparation time for the slop recycle which is assumed to be same to the equilibrium time for the balanced comparison of optimization results. While the optimization objective is not varied much, the amount of off-specification products is drastically reduced in the recycle operation and no significant difference is observed as the number of cycle increases.

CONCLUSION

An optimal design and operation of a batch distillation column for a multi-product system are investigated with a dynamic process model. The difference between the dynamic model and the quasi-steady state model is examined by comparing the results of the optimization of an operational profit for an example system. Also, the optimum number of trays of the system is obtained.

It is found that the design with the dynamic model gives more product and the design improvement is more apparent for the operation with combined exponential/constant reflux ratios which involves more dynamic operation than the constant reflux ratio operation. It means that the dynamic model gives more accurate design for the operation of a batch distillation column. The optimization procedure does not require long computation time in spite that a relatively rigorous process model is utilized. In addition, an optimum number of trays for the example system is found to be 23 by optimizing a net profit computed from an operational profit and an investment cost, and the optimal operation schedule for the case is yielded. The recycle of side products increases the amount of on-specification products with longer operation time to result in a similar optimization objective to a fresh feed operation.

ACKNOWLEDGMENT

Financial support from the Dong-A University and the Korea Science and Engineering Foundation through the Automation Research Center at POSTECH is gratefully acknowledged.

LITERATURE CITED

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