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SEPARATIONS Simplified Calculation for the Liquid Holdup Effect in Batch Distillation Jose´ C. Zavala,* Rosa M. Cero´ n, Ramo´ n J. Palı´, and Atl V. Co´ rdova UniVersidad Auto´ noma del Carmen, Dependencia del AÄ rea de Ingenierı´a y Tecnologı´a, Facultad de Quı´mica, Calle 56 No. 4 por AVenida Concordia, Colonia Buro´ cratas, C.P. 24180, Cd. del Carmen, Campeche, Me´ xico
One of the processes that has received major attention during past years is the batch distillation process; among other reasons this is because of its flexibility and versatility during the separation process of small amounts of mixtures for both obtaining products of great added value and recovery of compounds or solvents that if spilled into the environment would have a considerable impact on ecosystems. Efforts aimed to improve operation policies concerning the above-mentioned process have been made; this has been considered because any small change may represent important variations regarding the costs of the process, and as a result the present work aims to study the liquid holdup in the columns, as well as in the reflux tank. It is worth mentioning that previous studies had concluded that liquid holdup can affect the quality of the final product as well as the energy requirements needed to carry out the process. A number of research works have been studied to confirm the conclusions of similar studies and to establish a policy regarding the liquid holdup without using complex mathematical methods. Introduction The batch distillation process is widely used mostly for the separation process of small amounts of mixture to obtain products of high purity and added value; it is also used for the recovery of solvents and reagents as well as for the elimination of pollutants from wastewaters.1,2
Figure 2. Scheme of the concept of numeric experimentation.
Figure 1. Scheme of conventional batch distillation process.
Flexibility can be considered as one of the advantages of this process because in only one column it is possible to carry out either the separation of different mixtures or the separation of one mixture with different concentrations. It is important to * To whom correspondence should be addressed. Tel.: (+52-938) 38 1 10 18, ext. 2103. Fax: (+52-938) 38 2 65 14. E-mail:
[email protected].
mention that the main disadvantage of this process is represented by the energy requirements. Figure 1 shows the configuration of a unit for conventional batch distillation consisting of a reboiler or pot, a column of trays, a system for heat exchange, a reflux tank, and accumulative recipients for the products and wastes. With the purpose of solving the obstacles that the process may present and proposing new strategies of operation to lower the costs during the process as well as to improve the quality of the products and the experimentation work in the laboratory, researchers are taking advantage of new technology computers to carry out research works consisting of numeric experimentation starting from the solution of the mathematical model of the process to evaluate the performance of the system by virtually modifying the behavior of one or more variables. A proposal for the numerical experimentation concept is given by Zavala-Lorı´a et al.;3 Figure 2 exemplifies a modification of the above-mentioned experimentation. In the present research, numeric experimentation has the main purpose of knowing the effect of the liquid holdup on
10.1021/ie061190a CCC: $37.00 © 2007 American Chemical Society Published on Web 06/20/2007
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Figure 3. (a) Profile of light component concentration using total reflux policy (0-0.25 h interval). (b) Profile of light component concentration using total reflux policy (0.25-1.0 h interval). Table 1. Mathematical Model of a Conventional Batch Distillation Column dB V ) -D ) dt R+1 dx(i) B dt
)
(1)
{
}
L (i) V (i) (i) x - y(i) B + [x1 - xB ] ; B B V
{
dx(i) j
}
dx(i) D dt
)
V (i) [y - x(i) D ]; HD n
(i) (i) y(i) j ) Kj xj ; n
i ) 1, ..., n
i ) 1, ..., n;
j ) 1, ..., N
j ) 1, ..., N
(3)
(4) (5)
n
∑y ) ∑x (i) j
i)1
i ) 1, ..., n;
(i) j ;
i ) 1, ..., n;
j ) 1, ..., N
component
φV
component
φV
cyclohexane toluene
0.9708 0.9644
methanol ethanol
0.9845 0.9784
(2)
i ) 1, ..., n
L (i) V (i) (i) y - y(i) ) j + [xj+1 - xj ] ; dt Hj j-1 V
Table 2. Fugacity Coefficients of Components Used for this Research
and vapor are important factors that affect the quality and quantity of the obtained product.4 On one hand, this effect has an impact on the quantity of the intermediate products (wastes) affecting the obtaining of nonspecified products; on the other hand, when there is little holdup compared to the remaining quantity in the reboiler and the receiver an effect of rejection can occur, and such a phenomenon takes place when short and semi-rigorous methods are used.
(6)
i)1
both the plates and the reflux tank in a conventional batch distillation column, regardless of the kind of reflux (constant, variable, or optimal) used during the process; liquid holdup
Methodology To obtain the mathematical model of the column of conventional batch distillation, the following considerations have been taken into account: feed through the dome at temperature of
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Figure 4. Profile of the light component during the production stage.
Figure 5. Energy requirements in the reboiler when using a total reflux policy.
Table 3. Data Used for the Separation of the CT Mixture variable V N HP HD CD CW
amount 120 10 2-25 2-25 0.991 0.206
mol/h % % $/mol $/mol
variable xD,average (cyclohexane) F Zcyclohexane Ztoluene CF
amount 0.998 100 0.55 0.45 0.103
mol
equation where parameters:
A,
Kj(i) )
B,
and
C
[
represent
Bi 1 exp Ai P Tj + C i
]
Antoine’s
(7)
$/mol
saturation, negligible vapor holdup, constant flow, liquid holdup as percentage of the initial feed (trays and reflux tank), theoretical trays, constant pressure at the column, total condenser, and adiabatic column. Table 1 shows the equations of the mathematical model of the batch distillation process. These equations are displayed following the sections of the column shown in Table 1. A less complicated way to obtain the value of the constant of “liquid-vapor equilibrium” K(i) j is by using Antoine’s
If eq 7 is solved with the bubble point method, the iterative Newton-Raphson method shall be used too. To do so, the first derivative of eq 7 should be obtained; this derivative is shown as follows. The fact that the pressure is constant and equal to the unit shall be considered.
BiKj(i) ∂Kj(i) ) ∂Tj (Tj + Ci)2
(8)
The strategy followed for the solution of the mathematical model of the column has also considered a mixture whose behavior
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Figure 6. Energy requirements (reboiler) during the production stage.
Figure 7. Energy requirements and total production obtained when using different percentages of liquid holdup.
can be described as ideal. Table 2 shows the volatility coefficients of each one of the components used during this research. Previous studies regarding the behavior of the liquid holdup during a conventional batch distillation process5-11 have stated that the vapor holdup can be considered as negligible because it only represents a very small quantity compared to the liquid holdup; it also has been established that the liquid holdup has a direct effect over the purity of the products that are obtained. Diwekar10 states that the methods used to solve equations of the mathematical model can show erroneous results when using small holdup quantities. To know the impact of the liquid holdup on the conventional batch distillation column an interval of percentages has been considered regarding the initial supply that goes from 2% to 25%. Results and Discussion The first case study shows the separation of a binary mixture of cyclohexane/toluene (CT) which has been previously considered by Domenech and Enjalbert,12 Logsdon and Biegler,13
Zavala-Lorı´a,4 and Zavala-Lorı´a et al.3 Table 3 shows both the conditions and the operation parameters that were used considering a constant reflux ratio of 5. Obtaining a product with an average concentration of 99.8% of cyclohexane has been taken into account. In this case, it has been considered that the bottom product is not waste, and therefore has a commercial value greater than that of the feed mixture. The results of the simulation process for the analysis of the influence of the liquid holdup in the conventional batch distillation column are shown in Figures 3-7. According to the results obtained, when a low quantity of liquid holdup is used, the process stabilizes in a shorter period of time (Figure 3), which has a direct impact on the energy consumption (Figure 5); when a higher quantity of liquid holdup is used throughout the column, the composition slowly changes, and this effect is known as “flying effect” or “flywheel effect”.8 Similarly, when using a lower liquid holdup the product is obtained in a shorter period of time (Figure 4), and as a consequence, the energy expenses drop; however, it is worth mentioning that the amount of product with the desired quality that is obtained is smaller (Figure 7). Because the results do
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Figure 8. Profit profile for CT mix.
Figure 9. Profit profile for ME mix.
to 2324 J/kg (474.1 kJ/(kg‚mol)), and its current price (December 2006) per liter is $0.52 USD; the total expenses of fuel can be calculated as follows:
Table 4. Data Used for Separation of the ME Mixture variable V N HP HD CD CW
amount 100 15 2-30 2-30 0.51 0.01
mol/h % % $/mol $/mol
variable xD,average (methanol) F Zmethanol Zethanol CF
amount 0.96 100 0.35 0.65 0.056
mol $/mol
QBtotalCQB QE
)
Q (0.8450.52 × 2324)
Btotal
)
2.65 × 10-4QBtotal (10)
not allow knowing a priori the right percentage of liquid holdup during the process, different criteria shall be used to obtain the right percentage that will make it possible to obtain better profits. Next, there is an equation that shows the relationship between the economic profit during the process and the percentage of liquid holdup to be used:
P ) DCD + WCW - FCF - QBtotalCQB - BCC
Qtotal )
(9)
In eq 9, the supplying costs and the expenses caused by the consumption of energy as well as the profits obtained after selling the product have been considered. If we take into account that the density of the fuel (diesel) is 0.845 kg/L, the average molecular weight is 204 kg/(kg‚mol), its heating is equivalent
where QBtotal includes the startup and production heat. QE is the equivalent heat in J/L. Qtotal is total heat cost in $. The batch capital cost (BCC) is given as a modification of annual capital cost from Mujtaba and Macchietto.14
BCC ) (C1V 0.5N 0.8 + C2V 0.65)toperation
(11)
where C1 ) 5.41 × 10-3, C2 ) 1.22 × 10-2, and V is in mol/h. Equation 11 has a small impact on the capital cost and can be neglected, but in this work it was considered; moreover, the time is a holdup function. In eq 10 the heat is holdup function too. Along with eq 10, eq 9 can be expressed as follows:
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P ) DCD + WCW - FCF - 2.65 × 10-4QBtotal - BCC
(12)
The behavior of eq 12 is shown in Figure 8. The profile given by the profit clearly shows that there is a percentage of liquid holdup equal to 3.5%. The following case study consists of the separation of a binary mixture of methanol/ethanol (ME). Table 4 shows the conditions and operation parameters considering a constant reflux ratio equal to 5. During the process, it is possible to obtain a product with an average concentration of 96% of methanol. The obtained results in this case are similar to those obtained in the first one. It is worth mentioning that, when the quantities of liquid holdup increase, the time of startup as well as the production time shall increase too, and this has an impact over the energy requirements; see Table 5. Table 5. Results for the ME Mixture %H
TO, h (operation)
D, mol (total)
QB, J × 10-6 (total)
2 5 10 15 20 25 30
1.90 2.45 3.27 4.01 4.72 5.43 6.15
18.30 21.12 24.31 26.30 27.46 28.06 28.18
4.78 6.48 8.73 10.50 11.95 13.15 14.15
With the profile obtained after using eq 12, it is possible to show that the percentage of liquid holdup that allows the maximum profit is 20% (see Figure 9). Similarly to the first case, the profit shows a maximum value; therefore, it represents a very simple way to obtain the best value of the liquid holdup to be used during the batch distillation process. Concluding Remarks The liquid holdup in a batch distillation column is considered to be important because it has a direct impact on both the quality and the quantity of the final product which certainly affect the energy consumption during the process. This research presents a proposal that has been developed to find the adequate liquid holdup for the process of separation of mixtures using an equation that shows the relationship between the monetary profits and the liquid holdup. The results indicate that by applying this proposal it is possible to obtain the percentages of liquid holdup to be used during the separation process without using a formal method of optimization based on mathematical calculations. What other authors have proposed regarding the fact that small quantities of liquid holdup in a batch distillation column favor short periods of time regarding the point of stabilization and the production process to obtain a product with the desired concentration and low requirements of energy has been proven; however, it is worth mentioning that small quantities of liquid holdup give as a result small quantities of product. Acknowledgment J.C.Z. would like to thank the financial support received through Research Grant F-PROMEP-36/Rev-02 (Me´xico). Notation A, B, C ) Antoine’s equation coefficients C ) cost, $/mol or $/J
D ) product, mol F ) initial feed, mol H ) liquid holdup, mol BCC ) impact of capital cost, $ K ) vapor-liquid equilibrium constant L ) liquid flow into column, mol/h N ) number of equilibrium stages n ) components P ) pressure P ) profit, $ Q ) heater flow, J/h T ) temperature, K t ) time, h V ) vapor flow, mol/h W ) bottom, mol x ) liquid-phase mole fraction y ) vapor-phase mole fraction z ) feed mole fraction Superscripts and Subscripts B ) reboiler C ) condenser E ) steady state i ) component j ) stage k ) tray N ) dome P ) production Literature Cited (1) Barolo, M.; Guarise, B. Batch Distillation of Multicomponent Systems with Constant Relative Volatilities. IchemE 1996, 863-871. (2) Klingberg, A. Modelling and Optimization of Batch Distillation. Master Thesis, Department of Automatic Control, Lund Institute of Technology, Lund, Sweden, 2000. (3) Zavala-Lorı´a, J. C.; Co´rdova-Quiro´z, A. V.; Anguebes-Franseschi, F.; Robles-Heredia, J. C. Experimentacio´n Nume´rica Aplicada a un Proceso de Destilacio´n Discontinua. Inf. Tecnol. 2006, 17 (1), 53-60. (4) Zavala-Lorı´a, J. C.; Optimizacio´n del Proceso de Destilacio´n Discontinua. Tesis Doctoral, Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´gico de Celaya, Celaya, Guanajuato, Me´xico, 2004. (5) Rose, A.; Welshans, L. M. Sharpness of Separation in Batch Fractionation. Ind. Eng. Chem. 1940, 32 (5), 668-676. (6) Rose, A.; Williams, T. J.; Prevost, C. Holdup in Batch Distillation. Ind. Eng. Chem. 1950a, 42 (9), 1876-1879. (7) Rose, A.; Johnson, R. C.; Williams, T. J. Batch Fractional Distillation. Ind. Eng. Chem. 1950b, 42 (10), 2145-2149. (8) Pigford, R. L.; Tepe, J. B.; Garrahan, C. J. Effect of Column Holdup in Batch Distillation. Ind. Eng. Chem. 1951, 43 (11), 2592-2602. (9) Rose, A.; Johnson, R. C.; Williams, T. J. Effect of Column Holdup and Reflux Ratio. Chem. Eng. Prog. 1952, 48 (11), 549-556. (10) Diwekar, U. M. Batch Distillation: Simulation, Optimal Design and Control; Taylor & Francis: Washington, DC, 1995. (11) Seader, J. D.; Henley, E. J. Separation Process Principles; John Wiley & Sons, Inc.: New York, 1998 (12) Domenech, S.; Enjalbert, M. Modele Mathematique D’une Colonne de Rectification Discontinue - I. Etablissement du Modele. Chem. Eng. Sci. 1974, 29, 1519-1528. (13) Logsdon, L. R.; Biegler, L. T. Accurate Determination of Optimal Reflux Policies for the Maximum Distillate Problem in Batch Distillation. Ind. Eng. Chem. Res. 1993, 32, 692-700. (14) 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 (1), 27-36.
ReceiVed for reView September 11, 2006 ReVised manuscript receiVed May 17, 2007 Accepted May 22, 2007 IE061190A
