Ind. Eng. Chem. Res. 1998, 37, 4801-4807
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Shortcut Method for Multiple Task Batch Distillations Silvina D. Zamar,† Enrique Salomone,*,‡ and Oscar A. Iribarren‡ INGAR-CONICET, Avellaneda 3657, (3000) Santa Fe, Argentina
A methodology is presented for the preliminary design of separation networks of batch distillation. Analytical partition functions are derived based on the assumption that the distribution of components in an actual batch distillation can be approximated by the distribution of a batch column at total reflux. This permits computation of mass balances of multiple separation networks (allowing recycle of intermediate cuts) by solving a linear system of equations. Once the composition of the feeds to each separation task is known, the reflux ratios required to perform the tasks can be computed by using a previously developed method that relates the minimum to the actual reflux ratios and number of stages. The validity of the assumption is tested against simulation and the methodology compared with previously reported short-cut approaches. Since no new simplifications are introduced to the model, the reduction in computation effort is achieved with the same overall accuracy. Introduction Frequently, batch distillation separations involve several consecutive tasks. This is always the case when more than two products are to be obtained from the same feed, but even for the production of only two final products, it may be convenient to split this single task. This is the case when a separation requires a very high reflux ratio (very slow production rate of distillate). In such cases, operating with a smaller reflux ratio and recycling the cut out of specification results in a shorter total operating time. Further reductions of the reflux ratio beyond an optimal value increase the total operating time because with too large recycled fractions the column will spend too much time in reprocessing the slop cut. The optimization just described may yield important savings in operating time (and associated costs such as energy and labor), but it requires a large computational effort. When approached with simulations, a highly nested optimization problem results: the inner calculation consists of a simulation of the distillation run with specified reflux ratios for the production of each cut. A first loop is required to converge the mass balances. (The second run processes a mixture of the fresh feed plus the slop cuts produced in the first run, and so on until the mass balance of the recycles converge to a “steady state”.) Finally, an outer loop is required to search the set of optimal reflux ratios. The computational effort required by this approach is so large that the use of rigorous models is ruled out. Rigorous models include thermodynamic predictions, dynamics of the trays, energy balances, etc. Therefore, their use results in large computation times for each run, making the approach inappropriate as a tool to be used inside optimization loops. Starting a decade ago, the need for solving more complex configurations lead to the use of simplified models. For example, Luyben1 resorted to a process model that considers constant relative volatilities (avoid* To whom correspondence should be addressed: telephone, +54(42)53-4451;fax,+54(42)55-3439;
[email protected]. † Graduate student. ‡ Researchers at the Council for Scientific Research CONICET.
ing thermodynamic predictions and decoupling the energy balances) but still considers the dynamics of the trays. More recent approaches, as in Chiotti and Iribarren,2 disregard the dynamics of the trays (the simulation model consists of a steady-state column coupled to the dynamics of the still). Furthermore the multiple stages formulation was replaced by the Fenske-Underwood-Gilliland (FUG) approach to predict the instantaneous column performance as proposed by Diwekar and Madhavan3 and afterward developed for simulations by Sundaram and Evans.4 Finally, Salomone et al.5 proposed a method for shortcut design of batch distillations, whose accuracy is the same as that of the FUG approaches but demands at least 1 order of magnitude less computation time. It consists of the computation of minimum number of stages, Nmin, and minimum reflux ratio, Rmin, for a batch distillation task (different from the Nmin and Rmin required by continuous distillations to perform the same task). Afterward the method proposes the use of a correlation that relates Nmin and Rmin with the actual number of stages N, and reflux ratio R, which was derived for batch distillation. The information needed to use this approach is composition of the feed, relative volatilities among components, and the specification of the fractional recoveries for two components. (The fractional recovery is defined as the amount of component in the distillate after the separation, divided by the amount in the feed before the separation.) This approach suffices to do preliminary design in the case of single-pass separations. However, in the case of multiple separations one would need first to solve the mass balances in order to get the compositions of the feed to each separation. Hence, a short-cut method to solve the mass balances becomes necessary to extend the methodology. The method for mass balance estimation ought to be expedited for the whole methodology maintaining its expedite nature. In the following sections we first derive partition functions that predict the distribution of nonkey components once the recoveries of the key have been set and compare the distribution predicted with the ones obtained by simulation.
10.1021/ie9800795 CCC: $15.00 © 1998 American Chemical Society Published on Web 10/28/1998
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Next, we devote a section to the resolution of mass balances and compare the results of using the partition functions with published data from other authors. First we compare with a case presented by Mujtaba and Macchietto6 using SRK vapor-liquid equilibrium and to the same example, but resorting to the sharp splits assumption by Mujtaba and Macchietto.7 Another comparison is done with the results obtained with the FUG approaches by refs 3 and 4 for a case first solved with the commercial simulator BATCHFRAC by Boston et al.8 Finally, in the conclusions we discuss the merits and shortcomings of the proposed methodology. Partition Functions Geddes9 first realized that the distribution of components in actual continuous distillation columns could be approximated by straight lines in plots ln[(xt,i/xb,i)/(xt,k/ xb,k)] vs ln(Ri,k). Further work by other authors, reviewed by King10 showed that this distribution, which can be rigorously predicted for total reflux, was a good approximation to the actual distributions in a range of reflux ratios around 1.2 the minimum. Following, we derive an equation to predict the distribution of components that would be produced by a batch distillation column operating at total reflux. Afterward, we test this prediction against the distribution of actual batch distillations obtained by simulation. The mass balance for each component in a batch rectification column with no hold-up in the stages is
dni/dt ) -Dxi,t
(1)
where ni is the number of moles of component i in the still, D is the flow rate of distillate, and xi,t is the mole fraction at the top, i.e., the instantaneous composition of the distillate. Choosing any component as reference and dividing eq 1 by the equation that corresponds to the reference give
dni xi,t ) dnr xr,t
(2)
Assuming that the column is operating at total reflux, the distribution of components at each instant of the distillation is given by Fenske
xi,t xi,b ) Ri,rN xr,t xr,b
(3)
Equation 3 relates the compositions of any component i with the composition of component r through their relative volatilities Rir and the actual number of stages of this ideal total reflux column. Replacing eq 3 into eq 2
xi,b dni ) Ri,rN dnr xr,b
column) and ni ) bi (amount of i in the still after the separation) yields:
()
bi br ) fi fr
Ri,rN
(6)
To put this expression in terms of recovery fractions defined by
ni ) di/fi
(7)
we resort to the global mass balance for each component
fi ) di + bi
(8)
Finally from eqs 6, 7, and 8 the following expression is obtained
(1 - ηi) ) (1 - ηr)Ri,r
N
(9)
which is the partition functions sought for batch distillation. Note that N in eq 9 is the minimum number of stages required to obtain this separation, because any batch column operating at finite reflux will require more stages to achieve the same separation. If the recoveries of two key components are set and replaced in eq 9, the value of the constant Nmin can be computed
[
ln Nmin )
ln(1 - ηlk)
]
ln(1 - ηhk) ln Rlk,hk
(10)
Then, eq 9 can be applied setting N ) Nmin to predict the distribution of the remaining nonkey components. Comparison with Simulations
(4)
where subscript b refers to the bottom of the column (composition at the still). Multiplying and dividing the right-hand side of eq 4 by the hold up of the still and rearranging
dni dnr ) Ri,rN ni nr
Figure 1. Distribution of nonkey components in batch distillation.
(5)
Integrating eq 5 between the initial ni ) fi (feed to the
The observation in ref 9 referred to data from existing continuous columns operated in the range 1.1-1.3 times the minimum reflux required to achieve a given separation of the key components. In the same way, eq 9 is proposed here as an approximation to the distribution of components in batch distillation and was tested against simulations. The results are presented in Figure 1. The model used for simulation estimates the distribution of the components by solving a multistage model assuming constant volatilities and is described in detail in ref 5.
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All the simulated cases consist of five components and have in common the relative volatilities and recoveries of the key components: R ) 5, η ) 0.95 for the light key and R ) 1.7, η ) 0.05 for the heavy key. Therefore, the points corresponding to the key components of all simulations coincided. In addition, the amount of the key components in the feed was kept high xf ) 0.3-0.5 to ensure that they behave as key. Then, additional nonkey components were added with varying volatilities R ) 0.25-12 and feed compositions xf ) 0.05-0.3. For each simulation run, the Rmin, required to separate the key components, was computed following the method presented in ref 5. These authors also found that optimal designs in the case of batch distillations are approximated by N ) 1.7Nmin and R ) 1.2Rmin. Thus, the simulations were performed with N ) (1.21.9)Nmin adjusting R to get the recoveries specified for the key components which led to R ) (1.02-1.7)Rmin. The coordinates of Figure 1 were chosen so that the proposed partition function is the straight line connecting the light and the heavy key components to ease comparison with the scattered points corresponding to simulations. It can be seen that the straight line overestimates the fractional recoveries for both the components heavier than the heavy key and the lighter than the light key, while it slightly underestimates the recoveries of components in between. It should be noted that the discrepancy between simulated and predicted recoveries in Figure 1 corresponds to the ratio of logarithms, while the absolute value of the differences behaves inversely. At both left and right ends, the absolute value of the differences is quite small while between the keys they are maximal. At the left end when both the predicted and the simulated values approach zero, large figures for the ratio of logarithms correspond to very small recovery values, which are essentially zero. For example, for R ) 0.05 the predicted recovery is 8.6 × 10-8 while one of the simulated values is 5 × 10-11. The same effect occurs at the right end, when both predicted and simulated values approach 1. On the other hand, for components whose volatilities lay between the key, the discrepancy is in the first place after the decimal point, for example, for R ) 2.9, the predicted recovery is 0.3190 and one of the simulated values is 0.4445. Computation of Mass Balances Using recovery fractions as variables, the mass balances at each separation task k can be written:
di,k ) ηi,k fi,k bi,k ) (1 - ηi,k)fi,k
(11)
so the mass balances for the network consist of as many equations (11) as separation tasks are contained in the network, plus the equations that connect tasks, for example:
bi,k ) fi,k+1 fi,k ) bi,k-1 + di,k+1
(12)
The first is a simple connection: the bottom collected after performing task k is going to be the feed for task
Figure 2. A network of separation tasks.
k + 1. The second represents a recycle: the distillate of task k + 1 is going to be added to the bottom of task k - 1 to conform the feed of task k. The last represents equations that connect the distillation network with the environment, i.e., the definition of feeds and products of the network. For example
Fi ) fi,1;
Pli ) di,k;
Phi ) bi,k+1
(13)
for a network with just one feed that enters to separation 1, a light product which is the distillate produced after separation k and a heavy product which is the bottom produced after separation k + 1. The separation network used here to exemplify the mass balance system of equations is illustrated in Figure 2. Shortcut Methodology for Multiple Task Distillations By use of the partition functions and the mass balances based on recoveries above, a shortcut methodology for solving multiple separation problems can be summarized as follows. Any proposed network of separation tasks has two degrees of freedom per each separation, this was well established in the literature, e.g., Mujtaba and Macchietto.11 We propose to take the recoveries of two key components per separation as the independent variables and get a value for them by optimizing a proper objective function. For a set of values of independent ηi,k generated by the optimization algorithm, eq 10 is used to get the value of Nmin for each task, and eq 9 is used to get the remaining ηi,k, which in turn makes the mass balances a linear system. Solving the mass balances makes available all the information needed by the methodology in ref 5 to get the reflux ratios Rk needed to achieve these separations and this provides all the elements needed to evaluate the optimization objective function. The mass balance constraints (product specifications) are taken care of at the optimization algorithm. Comparison with the Sharp Split Approach Next we compare our approach with the work by Mujtaba and Macchietto who used more rigorous models with a strategy similar to ours concerning the covering of degrees of freedom for the optimization problem. These authors already discussed in ref 7 the issue that each separation has two degrees of freedom and take into account that each mass balance constraint reduces them by one. Thus, it is easy for us to reproduce their examples because they explicitly inform the values for optimized and constrained variables (recoveries and/or compositions).
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Figure 3. Example by Mujtaba and Macchietto.
In that paper, they also try a sharp splits approach for the balances. The method consists of choosing adjacent key components (no components with volatility between the keys). Then it is assumed that nonkey components lighter than the light key are depleted from the bottom while nonkey components heavier than the heavy key are absent in the distillate. This sharp splits approach can be seen as a simpler or “first level” partition functions approach. To do the comparison we took the example ref 6 that publishes more complete balances obtained using SRK liquid-vapor equilibrium and apply the sharp split approach in ref 7 to this example. The example is the fractionation of a mixture of cyclohexane, n-heptane, and toluene in an open network schematized in Figure 3 (the intermediate slope cuts are produced but not recycled). The balances obtained by the different approaches are presented in Table 1. Recoveries and compositions in boldface were either specifications or optimal values reported in ref 7. Light and heavy keys are underlined Table 1. Comparison with the Sharp Splits Approach
in the recoveries table. These key component recoveries are in our approach the original 8 degrees of freedom. The three key component recoveries constrained or optimized in ref 6 were set at their reported values. The five remaining key component recoveries were fixed minimizing the sum of the squared differences between calculated (by solving the mass balances eqs 15, 16, and 17) and reported (specified or optimal) compositions. Therefore, the reported recoveries and compositions are the same for the three approaches. The recoveries and compositions under the heading “SRK Prediction” are the ones reported in ref 6. The recoveries shown under the heading Partition Functions are the result of satisfying the reported specified/optimal compositions driving the nonspecified/ optimal key component recoveries, while the nonkey component recoveries are computed through eq 9. This produced the compositions under the same heading. The recoveries and compositions under the heading Sharp Splits are the result of setting the recoveries for toluene in separations 1 and 2 equal to 0, and for cyclohexane in separations 3 and 4 equal to 1, which is the approach tried in ref 7. Then the nonspecified/ optimal recoveries are driven to match the reported compositions. In the first two separations, it can be appreciated that the partition functions are in line with the rigorous simulation. They predict similar recoveries for the nonkey toluene while the sharp splits assumption sets it to zero and thus predicts zero composition in D1 and R1. In the third separation, the three approaches coincided in predicting recovery 1 for the nonkey cyclohexane. In the case of SRK and the partition functions this was the result of predicted values over 0.99999, while the sharp splits approach worked quite well in this case setting it to 1. Consequently, after the depletion of the
Ind. Eng. Chem. Res., Vol. 37, No. 12, 1998 4805 Table 2. Comparison with FUG Approaches and BATCHFRAC
lighter component, the compositions necessarily coincide because the degrees of freedom (two per separation) coincide with the number of components. Overall, it can be said that SRK and partition functions gave very similar balances with the last predicting higher recoveries for the nonkey toluene as expected because the nonkey components are either lighter than the light key or heavier than the heavy key.
Comparison with FUG Approaches These approaches use the Fenske-Underwood-Gilliland method to predict the instantaneous separation performance of the batch column. The approach was first introduced by Diwekar and Madhavan3 who integrated the differential material balance equation using Hengstebeck-Geddes equation (assuming C ) Nmin
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from Fenske) but on a finite differences basis to predict new values given old values of bottom compositions. Afterward, Sundaram and Evans4 further developed the approach for simulation. Among their improvements, they derived an explicit form for Fenske distribution of components that was later used by Salomone et al.5 inside the prediction of Nmin for batch distillations. Both FUG approaches compared their results with the ones obtained with BATCHFRAC by Boston et al.8 The case study is a fractionation of a mixture of propane, butane, pentane, and hexane in five operating steps with stopping criteria for each. The balances are presented as moles in distillate after each separation step in Table 2. The FUG approaches used the reflux ratios of BATCHFRAC and compared the separations predicted by them with the ones obtained with BATCHFRAC. Our approach requires the specification of the recoveries of two key components, so we used the recoveries obtained with BACTHFRAC and compared our prediction for the distribution of nonkey components with both BATCHFRAC and the FUG approaches. The light and heavy key components are indicated at the right-hand side of the table. Our moles in the distillate for these components coincide with the ones of BATCHFRAC because we use the same recoveries. With respect to the nonkey components, our recoveries are always somewhat larger than those of BATCHFRAC as expected. It can be said in general, that the balances in ref 3 are more in line with our approach in overestimating the presence of nonkey components in the distillate, while the ones in ref 4 tend to underestimate this presence. In view of our own testing against simulations, we interpret that the underestimation in the balances published by Sundaram and Evans4 is due to their choice of key components. They take as light and heavy keys, the lighter and heaviest components present in the column. From Figure 1, taking as key the lighter and heaviest component, the prediction (the straight line connecting the keys) underestimates the recoveries of the components with intermediate volatilities. At the bottom of Table 2, we present the reflux ratios required to achieve the specified recoveries of key components. These ratios were estimated by using the methodology presented by Salomone et al.5 after solving the mass balances through the partition functions proposed in the present paper. The reflux ratios used in BATCHFRAC are also presented. The estimated values are smaller for some separations and larger for others. The percent discrepancies are in line with the (35% reported in ref 5, and the scattering tends to compensate when computing the boilup requirement of the separation. The amount of material that needs to be evaporated to perform the separations is
boil up )
∑k dk(Rk + 1)
(14)
where dk is the amount of distillate produced at each step k and Rk is the reflux ratio used at each step. This required boil up determines the design of a column (the diameter required) or the operating time required by an existing column. For the case presented in Table 2, the boil up estimated by us was 1272.1 mol, while the simulation with BATCHFRAC required 1305.3 mol.
Conclusions We derived an analytical prediction of Nmin for batch distillations as a function of specified recoveries of two key components and propose to use it as a partition function to predict the recovery of nonkey components after a batch distillation. We propose to write the mass balances of multiple separation networks in terms of recoveries and select two key component recoveries per each separation as optimization variables for the network. In doing so, the analytical prediction of nonkey component recoveries mentioned above makes the network mass balances a linear system of equations. This short cut calculation of the mass balances permits extending the Rmin, Nmin methodology for batch distillation presented in a previous paper5 to multiple separation networks. The analytical prediction of Nmin presented here supersedes the algorithmic procedure to compute it presented in ref 5. At that time we did not realize that a rigorous analytical solution could be found. We started wandering after detecting that the procedure was insensitive to feed composition and only depended on the recoveries specified for the key components. The recoveries predicted with this methodology were compared with recoveries predicted by simulations. It was found that the prediction overestimates the recoveries of components with volatilities lighter than the light key and heavier than the heavy key, while it underestimates the recoveries of components with volatilities among the keys. This study also added insight to the behavior of the FUG methods depending on their choosing of key components. If two components with adjacent relative volatilities are taken as key, the nonkey component recoveries are overestimated, while if the lightest and heaviest components present in the column are taken as key, then the nonkey component recoveries are underestimated. We think that the distribution of nonkey components deserves further work. When analyzed using the representation described in this work, insightful information can be obtained and a parametric study might yield additional evidence. FUG methods3,4 and the one presented here are based on the same process model. The FUG methods use it to predict instantaneous performance at discretized times while our approach integrates it to get analytical predictions. This later approach is appropriate (more expedite) if one is interested in end effects (distribution after separations) rather than on the trajectory of the distillation. The methodology was also compared with a rigorous and a “sharp-splits” approach.6,7 The sharp splits approach could be seen as a first level or simplified partition functions, although it cannot predict recoveries for components of volatilities among the keys. The form proposed for the mass balancesswith task balances and connecting equationssis helpful in reducing the programming effort for the setup of new cases to explore. Although an ad-hoc formulation for each case would probably lead to fewer variables and equations, the extra computational load is highly compensated by having a general framework for the generation of alternatives. A particular feature of the approach is that fractional recoveries are monotonically increasing quantities (compositions are not). The recovery of any component increases from 0 to 1 with the advance of the rectifica-
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tion. As the separation capacity of the column increases (higher N, R), the trajectories for the recoveries of individual components becomes more steep and separate one from each other. For systems with constant volatilities the specification of two recoveries unambiguously defines the separation, and it will be always possible to find a design (number of stages, reflux ratio) that produces that separation. (i.e., the trajectories of the recoveries for the two components specified reach the target value simultaneously). Acknowledgment Funding for this work by CONICET through Project PMT-PICT 0232 Ingenierı´a de Procesos Batch is gratefully acknowledged. Nomenclature b ) moles in bottom after the separation (kmol) d ) moles in distillate after the separation (kmol) D ) flow rate of distillate (kmol/h) f ) feed: moles in still before separation (kmol) F ) feed to separation network (kmol) n ) moles in still during the separation N ) number of separation stages of the column Nmin ) minimum number of separation stages to perform a given separation task Ph ) heavy product exiting the network (kmol) Pl ) light product exiting the network (kmol) R ) reflux ratio Rmin ) minimum reflux ratio to perform a given separation task t ) time (h) x ) composition (molar fraction) Greek Symbols R ) relative volatility η ) recovery defined by eq 7 Subscripts b ) bottom of the column
f ) feed i ) component k ) separation task lk ) light key hk ) heavy key r ) reference component t ) top of the column
Literature Cited (1) Luyben, W. L. Multicomponent Batch Distillation. Ternary Systems with Slop Recycle. Comput. Chem. Eng. 1988, 27, 642647. (2) Chiotti, O. J.; Iribarren, O. A. Simplified Models for Binary Batch Distillations. Comput. Chem. Eng. 1991, 15, 1-5. (3) Diwekar, U. M.; Madhavan, K. P. Multicomponent Batch Distillation Column Design. Ind. Eng. Chem. Res. 1991, 30, 713721. (4) Sundaram, S.; Evans, L. B. Shortcut Procedure for Simulating Batch Distillation Operations. Ind. Eng. Chem. Res. 1993, 32, 511-518. (5) Salomone, H. E.; Chiotti, O. J.; Iribarren, O. A. Shortcut Design Procedure for Batch Distillations. Ind. Eng. Chem. Res. 1997, 36, 130-136. (6) Mujtaba, I. M.; Macchietto, S. Simultaneous Optimization of Design and Operation of Multicomponent Batch Distillation Column. Single and Multiple Separation Duties. J. Process Control 1996, 6, 27-36. (7) Mujtaba, I. M.; Macchietto, S. An Optimal Recycle Policy for Multicomponent Batch Distillation. Comput. Chem. Eng. 1992, 16, Suppl. 273-280. (8) Boston, J. F.; Britt, H. J.; Jirponghan, S.; Shah, V. B. An Advanced System for the Simulation of Batch Distillation Operations. In Found. Comput. Aided Chem. Process Des.; Mah, R. S. H., Seider, W. D., Eds., 1981, 2, 203. (9) Geddes, R. L. A General Index of Fractional Distillation Power for Hydrocarbon Mixtures. AIChE J. 1958, 4 (4), 389-392. (10) King, C. J. Separation Processes; McGraw-Hill: New York, 1980; 433-436. (11) Mujtaba, I. M.; Macchietto, S. Optimal Operation of Multicomponent Batch Distillation. Multiperiod Formulation and Solution. Comput. Chem. Eng. 1993, 17 (12), 1191-1207.
Received for review February 9, 1998 Revised manuscript received August 28, 1998 Accepted August 31, 1998 IE9800795
