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SEPARATIONS Optimal Control Problem in Batch Distillation Using Thermodynamic Efficiency Jose´ C. Zavala*,† and Cristina Coronado‡ Facultad de Quı´mica, Dependencia Acade´ mica de Ingenierı´a y Tecnologı´a, UniVersidad Auto´ noma del Carmen (UNACAR), Calle 56 # 4 por AVenida Concordia, Ciudad del Carmen, Campeche, C.P. 24180, Me´ xico, and Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´ gico de Celaya (ITC), A. Garcı´a Cubas S/N, Celaya, Guanajuato, C.P. 38010, Me´ xico
In this work we have optimized a batch distillation column as an optimal control policy problem. We used the optimal constant reflux policy for the process, and the objective function is to maximize the thermodynamic efficiency for a fixed time given a product concentration. The simulation model of the column considers its complete dynamics, and it has been formulated as a nonlinear programming problem for the solution of the thermodynamic efficiency. We use a SQP based dynamic optimization technique in this work. The procedure is shown for the separation of a binary mixture Benzene + Toluene. The achieved accumulated thermodynamic efficiency was close to 80%. Also we have compared with the optimal control problem of the maximum distillate for the same separation above, and we can observe that in the problem we proposed, less reflux is required, and even when thermodynamic efficiency in the problems is quite close, the energetic requirement in the problem we propose herein is smaller. 1. Introduction The batch distillation process is widely used in the chemical industry for the separation of small quantities of liquid mixtures giving as a result high added value products. Due to the dynamic nature of the process, the energy consumption can be high, and that means that any effort directed toward the improvement of the process can have a meaningful impact in the economy of the same. In fact, the optimization of a batch distillation column operation has been considered in the literature as an optimal control policy problem. In this sense, the improvements of the process have been approached considering the optimal reflux policy using different objective functions such as maximum distillate, minimum production time, minimum quantity of energy, and maximum profit and have been mentioned in works by Converse and Gross,1 Coward,2 Robinson,3 Mayur and Jackson,4 Kerkhof and Vissers,5 Murty et al.,6 Hansen and Jorgensen,7 Diwekar et al.,8 Mujtaba and Macchietto,9 Logsdon et al.,10 Farhat et al.,11 Diwekar,12 Logsdon and Biegler,13 Furlonge et al.,14 Mukherjee et al.,15 Zavala-Lorı´a16 and Zavala et al.,17 among others. To solve these control problems, have been used different mathematical models such as: Pontriagin’s Maximum Principle, dynamic programming, genetic algorithms, and nonlinear programming. There is a particular variable that can be considered interesting in a batch distillation column; that is the energy consumption. This is the reason why it is important to determine how efficient the heat exchange is during specific conditions of operation and to modify them to find those that make it more efficient. In this work, the thermodynamic efficiency is used to measure the effectiveness of the process; as well, a new kind of problem * To whom correspondence should be addressed. Tel.: (+52-938) 38 1 10 18, ext. 2103. Fax: (+52-938) 38 2 65 14. E-mail: jzavala@ pampano.unacar.mx. † UNACAR. ‡ ITC.
related to the optimal control of the batch distillation process is suggested that maximizes the thermodynamic efficiency of the separation process by using the constant reflux policy. The approach of the control problem is obtained thanks to a mathematical model that involves the available energy to carry out the separation process, which is based on a combination of the first and the second laws of thermodynamics. To solve such problem, we used nonlinear programming (NLP), and the variable of control was customized. The NLP method that was used consists on a sequential quadratic programming (SQP). Kim and Diwekar18 and Zavala-Lorı´a16 approached the analysis of the thermodynamic efficiency in a batch distillation column; the advantage of this approach is that besides the energy conversion, the influence and restrictions of the environment are taken into account. The expression of thermodynamic efficiency appeals to the concept of exergy because it is considered that it is a basic tool to know the impact caused by the use of sources of energy on the environment, it is an effective method for the design and analysis of energetic systems, and it is a key component to obtain sustainable environments. According to this, a useful concept is the exergy efficiency, defined as follows:19
ηt )
Wmin,sep Wtotal,sep
(1)
where the minimum work of separation (Wmin,sep) is the minimum amount of work required to obtain a specific purity of the product, and the total work required to achieve the separation (Wtotal,sep) is the total exergy supplied that can be obtained by adding the minimum work of separation and the total loss of work or total loss of exergy (availability).
Wtotal,sep ) Wmin,sep + LW
10.1021/ie0710972 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/15/2008
(2)
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( ) ( )
T0 T0 d(BBB) ) 1QB - 1 Q - DtBD - LW dt TB TC C
(9)
where the terms on the right side of this equation represent the difference between the exergy of flows into and out of the system, considering the loss of exergy as well. The term on the left side represents the change of exergy in the reboiler. The function of availability or exergy is defined as B ) h - T0s. It is considered that the state of reference is liquid at 25 °C and 1 atm. The change of exergy at the bottom can be calculated with eq 3 of the mathematical model for the process; the final equation shall be:
d(BB) d(BB) d(BBB) d(B) )B + BB )B - DtBB dt dt dt dt
Figure 1. Volume of control for the discontinuous process. Table 1. Mathematical Model of a Batch Distillation Column
dB V ) -Dt ) dt Rt + 1
(3)
dx(i) B V (i) L (i) x - y(i) [x - x(i) ) B + B] dt B B V 1
( ){
()
}
(4)
Total exergy is calculated from its physical and chemical components. The physical exergy comprises the physical processes that involve thermal interactions and its surroundings, whereas chemical exergy takes into account the transference of mass and heat and its surroundings; the main contribution of this kind of energy is a result of the effects of the mixture. According to Kim and Diwekar,18 physical exergy could be eliminated in the derivation of exergy because it is relatively less than chemical exergy and can be considered as constant for all chemical species. So, the change of exergy in the reboiler can be calculated as follows:
i ) 1,...,n
( ){
dx(i) j
V ) dt Hj
(i) yj-1
-
y(i) j
}
L (i) + [x - x(i) j ] V j+1
()
BB ) BBchem ) RT0 (5)
( )
(i) (i) y(i) j ) K j xj n
∑y i)1
i ) 1,...,n
i ) 1,...,n
j ) 1,...,N
(6)
d(BB) (7)
)
∑x
)
d(BBchem)
dt
n
(i) j
(i) j
)1
i ) 1,...,n
∑i x(i)B ln x(i)B
(11)
For an ideal mixture with constant relative volatilities, the derivation of the last equation is:
i ) 1,...,n ; j ) 1,...,N
dx(i) D V ) [y(i) - x(i) D] dt HD n
(10)
(8)
i)1
j ) 1,...,N
During the obtaining of the mathematical model that corresponds to the conventional batch distillation column, the following considerations where taken into account: feeding a liquid mixture of “n” components at saturation temperature, feeding through the dome, negligible vapor holdup in each one of the stages of separation, constant vapor flow, constant liquid holdup (trays and reflux), theoretical trays, constant operation pressure, adiabatic column with “N” stages of equilibrium, and total condenser. The mathematical model shown in Table 1 is in accordance with the work of Zavala et al.20 These equations are arranged according to the sections of the column in Figure 1. 2.1. Availability and Thermodynamic Efficiency. By analyzing the availability of the energy of the system shown in Figure 1, we can obtain an expression corresponding to the thermodynamic efficiency. To apply the first and second laws of thermodynamics, it is considered that during the process an exchange of energy with the environment can take place;1 however, no tree work is done. As a result, the balance of exergy or availability in the system is:
) RT0
∑i x(i)B ln x(i)B ] (12)
dt
The term on the right side of eq 12 can be represented in terms of eq 4 that forms the model of the column:
d[ 2. Mathematical Model
dt
d[
∑i x(i)B ln x(i)B ]
)
dt )
∑i
{
[1 + ln
( )∑ { V
B
x(i) B]
}
dx(i) B dt
(i) x(i) B - yB +
i
() L
V
}
(i) [x(i) 1 - xB ] ×
[1 + ln x(i) B ] (13) By substituting the last equations at the right side of eq 10, the result is as follows:
d(BBB) dt
) VRT0
∑i
{
(i) x(i) B - yB +
() L
}
(i) [x(i) 1 - xB ] × V (i) [1 + ln x(i) x(i) B ] - Dt{RT0 B ln xB } (14)
∑i
In the same way, the exergy of the current of production that shall be used in the eq 9 is calculated with:
BD ) RT0
∑i x(i)D ln x(i)D
(15)
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The transference of exergy associated to the transmission of energy as heat during the process can be calculated with the help of the balances of energy in the reboiler and the condenser. The same assumptions given by Kim and Diwekar18 are taken into account; such assumptions state that ∆Hvap is the same for each component and is not related to the temperature.
Q ) V∆Hvap
( ) ( )
(
T0 T0 1 1 1Q - 1Q ) V∆HvapT0 TB B TC C TC TB
(16)
)
1-
T0 T0 Q - 1Q ) TB B TC C VRT0 ln
(17)
[
]
x(1) D (R1,2 - 1) + 1
x(1) B (R1,2 - 1) + 1
[ [ {
LW ) VRT0 ln
x(1) B (R1,2 - 1) + 1 (i) x(i) DtRT0 B ln xB i
∑
VRT0
∑i
]
(18)
(i) x(i) B - yB +
( )[ L
+
∑i x(i)D ln x(i)D
(i) x(i) 1 - xB
V
}
]
-
][
]
1 + ln x(i) (19) B
And the minimum amount of work required is calculated as follows:
Wmin ) DtRT0 VRT0
∑i
[∑
{
(i) x(i) D ln xD -
i
x(i) B
-
y(i) B
+
∑i x(i)B ln x(i)B
( )[ L
V
x(i) 1
-
x(i) B
] ]}[
+
Dt
V ηt )
∑i
{
[∑
ln
x(i) D
-
( )[
i
L
(i) x(i) B - yB +
V
[
∑i
x(i) B
1 + ln
x(i) B
ln
(i) x(i) 1 - xB
]
(20)
]}[
]
+
1 + ln x(i) B
]
x(1) B (R1,2 - 1) + 1
[∑ ∑{ [
(i) x(i) D ln xD -
i
]
∑i x(i)B ln x(i)B
}[
(21)
]
+
(i) (i) (i) Rt x(i) 1 + ln x(i) 1 - yB + xB - yB B
ηt )
]
x(1) D (R1,2 - 1) + 1
V ln
i
(Rt + 1)ln
[
(23)
tf
2.2. Optimal Control Problem. The optimal control problem is posed considering Rt g 0.0, in a period of time 0.0 e t e tf, to obtain an average concentration of product that has been previously established within the valid limits, 0.0 e x e 1.0 hence, the problem is: Subject to:
∫0t η(t)dt
MaxRt,N,tf ηaverage )
tf
∫0t Dtx(i)D dt / ) ) xD ∫0t Dtdt f
(i) xD,average
f
(24)
The restrictions include all the equations that form the mathematical model corresponding to the distillation column (eqs 3-8). To solve the control problem, we used NLP through a SQP with the discretization of the state variables by using the technique of direct orthogonal collocation. The variable of control was parametrized through the optimal constant reflux. In this work, the results are compared with the solution of maximum distillate problem using the same restrictions and the following objective function: Subject to:
MaxRt,N,tf D )
∫0t Dtdt ) ∫0t Rt V+ 1dt f
f
∫0t Dtx(i)D dt / ) ) xD ∫0t Dtdt f
x(i) B
As a consequence, the thermodynamic efficiency or eq 1 is defined by:
x(i) D
ηaverage )
f
Therefore, the term of lost exergy LW can be obtained from eq 9
x(1) D (R1,2 - 1) + 1
∫0t η(t)dt f
By using the equation of Clausius-Clapeyron to calculate ∆Hvap, the last equation can be expressed as follows:
( ) ( )
Equation 22 can represent the thermodynamic efficiency of the dynamic behavior of the batch distillation. As can be seen, its value depends not only on the reflux ratio, Rt, but also on the amounts and compositions of streams resulting from the process whose values determined the energy consumption. To optimize the process, an average efficiency is defined and evaluated based on the total time of distillation:
]
x(1) D (R1,2 - 1) + 1
x(1) B (R1,2 - 1) + 1
] (22)
(i) xD,average
f
(25)
3. Results and Discussion To show the solution to the control problem corresponding to the maximum thermodynamic efficiency, we have used a binary mixture of benzene/toluene (BT). The conditions and parameters of operation for distillation are given in Table 2. The (average) relative volatility calculated from DECHEMA22 data is 2.4721. The distillation column was carried to the steady-state considering:20 feed through the dome at temperature of saturation, negligible vapor holdup, constant flow, liquid holdup, constant pressure at the column, total condenser, and adiabatic column. The operation time (tf ) 1.0 h) does not consider the start up, namely, tf is the time after steady state. The thermodynamic efficiency obtained along this work is a complex function that directly involves: relative volatilities, compositions of the dome and bottoms as well as the reflux ratio. The vapor flow, speed of production, and remaining amount in the reboiler are indirectly involved. As was mentioned before, the reflux ratio, Rt, has a direct effect on the thermodynamic efficiency of the process since when Rt increases the thermodynamic efficiency decreases.16,18 From eq 19 we can
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2791
Figure 4. Total efficiency profile. Figure 2. Reflux ratio optimal profile.
Figure 5. Total heat profile. Figure 3. Profile of concentration of distilled regarding the lightest component.
observe that an increment in Rt rises the availability of energy required for the separation due to an augmentation in the losses of exergy, which diminishes the thermodynamic efficiency of the process. Figures 2 and 3 show the profiles of optimal constant reflux and distillate concentration of the lightest component, respectively, for the maximum thermodynamic efficiency problem compared to the maximum distillate problem. In Figure 2 it is possible to observe that the average value of the profile corresponding to the reflux ratio for the maximum thermodynamic efficiency problem (3.812) is less than the average value of the maximum distillate problem (4.196) due to the fact that the first problem requires lower quantities of reflux to maintain a high rate of thermodynamic efficiency during the process, whereas the second problem aims to get the highest levels of distilled. Figure 3 shown the impact of the reflux ratio in the concentration of product; in other words, when a higher reflux ratio is used, the normal decrease of concentration can change through time to the point where the behavior of such concentration can be reversed and increased or at least be softened. A comparison between the total thermodynamic efficiency of the process of both problems is shown in Figure 4. In this case, the maximum distillate problem shows an efficiency 0.5% lower than the problem of maximum thermodynamic efficiency; however, as shown in Figure 5, the energetic requirement is 9.5% lower; then a small variation in the processed thermodynamic efficiency decreases the energy requirements, which have an influence in the process economy. We can conclude that when solving the problem of optimal control, a higher need of energy during the process does not
Table 2. Input Data for the Problem of Optimal Control variable
amount
variable
amount
V N Hj HD F
120 mol/h 10 Tray 1 mol 1 mol 100 mol
R1,2 zBenzene zToluene xD,aVerage tf
2.4721 0.55 0.45 0.999 1.0 h
mean a decrease of thermodynamic efficiency. From Figures 4 and 7 we can observe that the amount of energy required for the process is lower in the problem of maximum thermodynamic efficiency even though the total efficiency of the process is practically the same for both problems. It is due to another variable that affects the concept, the losses of work or lost energy LW (see Figure 6). In other words, the thermodynamic efficiency of the process reduces the losses of work, which provokes that the supplied energy can be used in a better way during the separation of the mixture.
Figure 6. Profile of the losses of work during the process.
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The results obtained have a similar behavior with other ideals binaries mixtures. For example, Figures 8, 9, and 10 shown the profiles of optimal constant reflux of the lightest component of Cyclohexane + Toluene mixture, total efficiency profile, and total heat profile, respectively, for the maximum thermodynamic efficiency problem compared to the maximum distillate problem. In this case are observed similar results; moreover, the energetic requirement is 25% major for the maximum distillate problem compared to the maximum thermodynamic efficiency problem. Equation 22 does not apply for non-ideal mixtures. For non-ideal mixtures, it is necessary to use a state equation or solution model for the thermodynamic model. Figure 7. Total distillate profile.
Figure 8. Reflux ratio optimal profile.
4. Concluding Remarks The use of the concept of thermodynamic efficiency as developed in this work is quite useful for posing a function that includes not only the concept of energy per-se but also purity and amount distilled. Through the concept of thermodynamic efficiency we have proposed and solved a new kind of optimal control for the batch distillation process. The aforementioned process includes optimal constant reflux that can be considered as variable along the process, which fulfills a restriction of average purity. From the results of applying the problem of control that we developed in the present work in relationship with the binary mixture benzene/toluene, we can observe that the relationship between reflux and purity of product really affect the thermodynamic efficiency of the process. On the other hand, considering there is a direct relationship between thermodynamic efficiency and operation costs for the process, a diminished equivalent to 9.5% in the consumption energy for the maximum thermodynamic efficiency problem comparing to the maximum distillate problem resulting from the separation of the binary mixture benzene/toluene is promising even when the amount of distillate that is obtained is 3% less. Acknowledgment We thank the financial support received through the research grants: P/PROMEP UACAR-99-03 FOLIO UACAR-05; COSNET 539.01P and 539.01PR (Mex.); CONACYT (Mex.); Instituto Tecnolo´gico de Celaya (Mex.).
Figure 9. Total efficiency profile.
Figure 10. Total heat profile.
In Figure 7 is shown the behavior of the accumulation of product during the operation; 28.16 mol was obtained for the maximum distillate problem and 27.30 mol was obtained for the maximum thermodynamic efficiency problem.
Nomenclature R ) relative volatility B ) amount of moles in the reboiler (mole) B ) function of availability D ) total distillate (mole) Dt ) distillate flow rate (mol/h) ∆Hvap ) vaporization heat Hj ) molar hold-up on tray j HD ) molar hold-up on the condenser h ) enthalpy (J/mol) K(i) j ) equilibrium constant L ) liquid flow rate (mol/h) LW ) lost work (J) n ) number of components N ) number of trays in the column η ) thermodynamic efficiency Q ) heat (J) R ) ideal gas constant Rt ) reflux ratio s ) entropy (J/mol K)
Ind. Eng. Chem. Res., Vol. 47, No. 8, 2008 2793
T ) temperature t ) time (h) V ) vapor flow rate (mol/h) W ) work (J) x(i) j ) mole fraction in the liquid of component i at plate j x(i) B ) mole fraction in the liquid of component i in the reboiler x(i) D ) mole fraction in the vapor of component i in the distillate y(i) j ) mole fraction in the vapor of component i at plate j z ) mole fraction in the feed Subindex 0 ) reference 1 ) benzene 2 ) toluene C ) condenser f ) end D ) distillate B ) reboiler chem ) chemical Literature Cited (1) Converse, A. O.; Gross, G. D. Optimal Distillate-Rate Policy in Batch Distillation. Ind. Eng. Chem. Fundam. 1963, 2(3), 217-221. (2) Coward, I. The Time-Optimal Problem in Binary Batch Distillation. Chem. Eng. Sci. 1967, 22, 503-516. (3) Robinson, E. R. The Optimal Control of an Industrial Batch Distillation Column. Chem. Eng. Sci. 1970, 25, 921-928. (4) Mayur, D. N.; Jackson, R. Time Optimal Problems in Batch Distillation for Multicomponent Mixures Columns with Holdup. Chem. Eng. J. 1971, 2, 150. (5) Kerkhof, L. H. J.; Vissers, H. J. M. On the Profit of Optimum Control in Batch Distillation. Chem. Eng. Sci. 1978, 33, 961-970. (6) Murty, B. S. N.; Gangiah, K.; Husain, A. Performance of Various Methods in Computing Optimal Control Policies. Chem. Eng. J. 1980, 19, 201-208. (7) Hansen, T. T.; Jorgensen, S. B. “Optimal Control of Binary Batch Distillation in Tray or Packed Columns”; The Chemical Engineering Journal 1986, 33, 151-155. (8) Diwekar, U. M.; Malik, R. K.; Madhavan, K. P. Optimal Reflux Rate Policy Determination for Multicomponent Batch Distillation Columns. Comput. Chem. Eng. 1987, 11(6), 629-637.
(9) Mujtaba, I. M.; Macchieto, S. Optimal Control of Batch Distillation. Proceedings 12th IMACS Wold Congress; Paris, July 18-22, 1988. (10) Logsdon, J. S.; Diwekar, U. M.; Biegler L. T. On the Simultaneous Optimal Design and Operation of Batch Distillation Columns. Trans. IChemE 1990, 68(A), 434-444. (11) Farhat, S.; Czernicki, M.; Piboleau, L.; Domenech, S. Optimization of Multiple-Fraction Batch Distillation by Nonlinear Programming. AIChE J. 1990, 36(9), 1349-1360. (12) Diwekar, U. M. Unified Approach to Solving Optimal Design Control Problems in Batch Distillation. AIChE J. 1992, 38(10), 1551-1563. (13) Logsdon, J. S.; 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) Furlonge, H. I.; Pantelides, C. C.; Sorensen, E. Optimal Operation of Multivessel Batch Distillation Columns. AIChE J. 1999. 45, 781-801. (15) Mukherjee, S.; Dahule, R. K.; Tambe, S. S.; Ravetkar, D. D.; Kulkarmi, B. D. Consider Genetic Algorithms to Optimize Batch Distillation. Hydrocarbon Process. 2001, 9, 59-66. (16) Zavala-Lorı´a, J. C. Optimizacio´n del Proceso de Destilacio´n Discontinua. PhD Thesis, Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´gico de Celaya, Celaya, Guanajuato, Me´xico, 2004. (17) Zavala-Lorı´a, J. C.; Co´rdova-Quiroz, A. V.; Robles-Heredia, J. C.; Anguebes-Franseschi, F. Efecto de la derivacio´n del reflujo en Destilacio´n Discontinua. ReV. Mex. Ing. Quı´m. 2006, 5(S1), 109-113. (18) Kim, K. J.; Diwekar, U. M. Comparing Batch Column Configurations: Parametric Study Involving Multiple Objetives. AIChE J. 2000, 46(12), 2475-2488. (19) Agrawal, R.; Herron, D. M. Optimal Thermodynamic Feed Conditions for Distillation of Ideal Binary Mixtures. AIChE J. 1997, 43(11), 2984-2996. (20) Zavala Jose´, C.; Cero´n Rosa, M.; Palı´ Ramo´n, J.; Co´rdova, Atl. Simplied Calculation for the Liquid Holdup Effect in Batch Distillation. Ind. Eng. Chem. Res. 2007, 46(15), 5186-5191. (21) Seider, W. D.; Seader, J. D.; Lewin, D. R. Process Design Principles: Synthesis, Analysis, and EValuation; John Wiley & Sons, Inc.: New York, 1998. (22) Gmehling, J.; Onken, U.; Arit, W. Vapor-Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series: Frankfurt, Germany 1980.
ReceiVed for reView August 10, 2007 ReVised manuscript receiVed February 8, 2008 Accepted February 13, 2008 IE0710972
