Ind. Eng. Chem. Res. 1997, 36, 2287-2295
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Efficient Optimization of Batch Distillation with Chemical Reaction Using Polynomial Curve Fitting Techniques Iqbal M. Mujtaba* Department of Chemical Engineering, University of Bradford, West Yorkshire BD7 1DP, U.K.
Sandro Macchietto Centre for Process Systems Engineering, Imperial College, London SW7 2BY, U.K.
A computationally efficient framework is presented for dynamic optimization of batch distillation where chemical reaction and separation take place simultaneously. An objective to maximize the conversion of the limiting reactant dynamic optimization problem (maximum conversion problem) is formulated for a representative system, and parametric solutions of the problem are obtained. Polynomial curve fitting techniques are then applied to the results of the dynamic optimization problem. These polynomials are used to formulate a nonlinear algebraic maximum profit problem which can be solved extremely efficiently using a nonlinear optimization solver. This provides an efficient framework which can be used for on-line optimization of batch distillation within scheduling programs for batch processes. The method can also be easily extended to nonreactive batch distillation and to nonconventional batch distillation columns. Introduction The use of batch distillation with or without chemical reaction is common practice in the chemical industries (Logsdon at al., 1990; Mujtaba and Macchietto, 1991; Cuille and Reklaitis, 1986; Wilson, 1987; Albet et al., 1991). Traditionally, as in many chemical industries, reaction and separation take place separately in a batch reactor followed by a batch distillation column (Charalambides et al., 1994). Therefore, the distillation of desired species cannot influence the conversion of the reactants in the reactor. However, conventional batch distillation with chemical reaction (reaction and separation taking place in the same vessel and hence referred to as reactive batch distillation) is particularly suitable when one of the reaction products has a lower boiling point than other products and reactants. The higher volatility of this product results in a decrease in its concentration in the liquid phase, therefore increasing the liquid temperature and hence reaction rate, in the case of an irreversible reaction. With reversible reactions, elimination of products by distillation favors the forward reaction. In both cases higher conversion of the reactants is expected than by reaction alone. Therefore, in both cases, a higher amount of distillate (proportional to the increase in conversion of the reactant) with desired purity is expected than that obtained by distillation alone (as in traditional approach). An extensive literature survey shows that very little attention has been given to modeling and simulation of reactive batch distillation, let alone optimization of such process. The published literature deals with the mathematical modeling and numerical integration of the resulting dynamic equations systems, with some presenting computer simulation vs experimental results. The design and/or optimal operational aspects of reactive batch distillation have been discussed by few authors. Cuille and Reklaitis (1986) considered the simulation of reactive batch distillation, with reaction occurring on * To whom correspondence may be addressed. E-mail:
[email protected]. S0888-5885(96)00573-8 CCC: $14.00
the plates, in the condenser, and in the reboiler. The model was posed as a system of differential and algebraic equations (DAEs) and a stiff solution method was employed for integration. They considered the esterification of 1-propanol with acetic acid, but the example was not suitable for use in batch distillation. Since 1-propanol (one of the reactants) is the more volatile component in the system, the removal of species by distillation causes the removal of reactant from the column thus decreasing conversion. However, the main purpose of the study was to present numerical solution techniques. Wilson (1987) discussed the optimal design of batch distillation processes using a simplified column model involving chemical reaction and using repeated simulation. For a commercially used complex parallel reaction scheme and using a simple economic model, he showed the benefit of integrating reaction and distillation. He generated a number of plots of process efficiency (in terms of product cost contribution per unit product) for a range of alternative process and design variable choices and suggested an optimal design and operation of reactive batch distillation. The paper however does not give a general procedure. Albet et al. (1991) presented a method for obtaining operational policies for both reactive and nonreactive batch distillation systems using repeated simulation techniques. Recently Sorensen and Skogestad (1994) developed control strategies for reactive batch distillation again by repeated simulation using simple model in SPEEDUP package. In this work optimal operation problem of reactive batch distillation is presented as a proper dynamic optimization problem incorporating a detailed dynamic model. The problem formulation and solution exploit the existing methods developed for nonreactive batch distillation by the authors (Mujtaba and Macchietto, 1991, 1993). However, the solution of dynamic optimization problems with detailed dynamic model by sophisticated numerial techniques can be computationally very expensive (Mujtaba and Macchietto, 1993, 1996) and therefore is not suitable for on-line optimization where a quick and reliable solution is necessary. While the © 1997 American Chemical Society
2288 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
(c) A series of product accumulator tanks connected to the product streams to collect the main and or the intermediate distillate fractions. Operation of such a column involves carrying out the fractionation until a desired amount has been distilled off. The overhead composition varies during the operation and usually a number of cuts are made. Some of the cuts are desired products while others are intermediate fractions that can be recycled to subsequent batches to obtain further separation. A residual bottom fraction may or may not be recovered as product.
Figure 1. Batch distillation column.
operation of a batch distillation plant being optimized it is always desirable to operate at conditions which maximize the profit. However, the profit is usually a function of product prices, e.g., raw material cost, utility costs, etc. Mujtaba and Macchietto (1993) showed that changes in these costs can significantly change the operating condition of the plant, which consequently means that an expensive dynamic optimization problem has to be solved each time there is a price change to obtain optimal operating conditions. A further complication in the model by including chemical reaction would make the solution even more expensive. Therefore, there is a potential to develop techniques for solving dynamic optimization problems cheaply for reactive or nonreactive batch distillation while taking into consideration of a detailed dynamic model. In this work we first outline briefly the features of a batch distillation column. The detailed mathematical models for the column with chemical reaction are then presented in terms of DAEs. The dynamic optimization problem is presented as nonlinear programming problems (NLP) to obtain optimal operation policies. An efficient optimal control algorithm (Morison, 1984) is used to formulate the dynamic optimization problem. Model equations are solved by efficient, stiff solution methods based on Gear (1971) while successive quadratic programming (SQP) (Chen, 1988) techniques are used to solve the optimization problem. Finally, using a suitable example in this work it is shown that by judicious use of repeated solutions of a dynamic optimization problem a-priori, an algebraic representation of the optimal solutions can be obtained, and very efficient calculation of optimal batch time, optimal reflux ratio, etc., can be performed. This can provide an on-line optimization framework for optimizing operation of batch distillation with or without chemical reaction. It is to be noted here that although formation of azeotropes are quite common in reactive distillation, we have not considered such a situation in this work, for convenience.
Model for Batch Distillation with Chemical Reaction With reference to the column configuration given in Figure 1 the model equations are presented in this section. They include mass and energy balances, column holdup, rigorous phase equilibria, chemical reaction on the plates, in the reboiler, and in the condenser but not in the product accumulator. The model is fairly detailed and assumes negligible vapor holdup, adiabatic plates, constant molar holdup on plates, and, in the condenser, perfect mixing on trays, fast energy dynamics, constant operating pressure, and total condensation with no subcooling. Some of the modeling assumptions, for example, constant molar holdup, constant pressure, fast energy dynamics, etc., can be relaxed, if needed, by adding the relevant equations (Cuille and Reklaitis, 1986; Bosely and Edgar, 1994). All thermodynamic, physical property, and reaction quantities are calculated using separate subroutines for each (eqs A.6-A.10), with all derivatives calculated analytically. A. Internal Plates: j ) 2, N - 1
total mass balance 0 ) Lj-1 + Vj+1 - Lj - Vj + ∆njHj
(1)
component mass balance
Hj
dxji ) Lj-1xj-1,i + Vj+1yj+1,i - Ljxji - Vjyji + rjiHj dt (2)
energy balance 0 ) Lj-1hj-1L + Vj+1hj+1V - LjhjL - VjhjV
(3)
equilibrium Features of Batch Distillation Column The essential features of a batch distillation column (Figure 1) are as follows: (a) A bottom receiver/reboiler which is charged with feed to be processed and which provides the heat transfer surface. (b) A rectifying column (either a tray or packed column) superimposed on the reboiler, coupled with either a total condenser or a partial condenser system.
yji ) Kjixji
(4)
∑yji ) 1
(5)
restrictions
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2289
relations defining physical properties and chemical reactions Kji ) Kji(yj,xj,Tj,P)
(6)
hjL ) hjL(xj,Tj,P)
(7)
hjV ) hjV(yj,Tj,P)
(8)
rji ) rji(kr,xj)
(9)
∆nj )
∑rji
(10)
(20)
h1L ) h1L(xDi,T1,P)
(21)
Dynamic Optimization Problem Formulation The batch distillation model described by the system of DAEs (presented earlier) can be written as:
(22)
with consistent initial conditions
dHN/dt ) LN-1 - VN + ∆nNHN
dxNi ) LN-1(xN-1,i - xNi) - VN(yNi - xNi) + dt rNiHN - ∆nNHNxNi (12)
energy balance 0 ) LN-1(hN-1L - hNL) - VN(hNV - hNL) + QR (13) The other equations are same as eqs 4-10 where j is replaced by N. C. Condenser and Distillate Accumulator: j ) 1
accumulator total mass balance dHa/dt ) LD
(14)
component mass balance (a) accumulator dxai ) LD(xDi - xai) dt
(15)
(b) condenser holdup tank dxDi ) V2y2i + r1iHc - (V2 + ∆n1Hc)xDi (16) dt
energy balance 0 ) V2h2V - (V2 + ∆n1Hc)h1L - QC
(17)
other equations L1 ) rf(V2 + ∆n1Hc)
f(t0)0,x′0,x0,u0,v) ) 0
(11)
component mass balance
Hc
T1 ) T1(xDi,P)
f(t,x′,x,u,v) ) 0
total mass balance
Ha
(19)
The other equations are the same as eqs 9 and 10 with j replaced by 1, and rf is the internal reflux ratio, defined by eq 18.
B. Reboiler: j ) N
HN
LD ) (V2 + ∆n1Hc)(1 - rf)
(18)
where x is the set of all the variables (differential and algebraic), x′ values are the time derivatives of the differential variables, u values are the time dependent control variables, v is a set of constant parameters, and t is the time. The dynamic optimization problem is formulated and solved using the optimal control algorithm of Morison (1984). The optimization problem for the DAE system (given by eq 22) is formulated as a nonlinear programming problem with the time-varying controls parameterized into a finite set of control parameters (control levels and switching times). Other optimization variables are the time independent design parameters, initial conditions, and the final time tf. These are optimized to give a minimum (or maximum) of a general objective function F(tf,x′(tf),x(tf),u(tf),v) ) 0, subject to any constraints. Details of the algorithm are in Morison and in Mujtaba and Macchietto (1991, 1993, 1996) and are not repeated here. A robust code is used for DAE integration, with an efficient SQP method used for optimization of the finite number of parameters. In this work, we consider a maximum conversion problem for batch distillation, subject to given product purity constraints. The reflux ratio is selected as the control parameters to be optimized for a fixed batch time to so as to maximize the conversion of the limiting reactant. The optimal amount of product and condenser/ reboiler duties are thus also calculated. The optimization problem can be stated as follows: Given: the column configuration, the feed mixture (B0, xB0), condenser vapor load (V2), and a separation task (i.e., achieve the product with purity specification for a key distillate component (x*D). Determine: the optimal reflux ratio profile rf(t) for the operation. So as to maximize: an objective function defined for instance the conversion (or the amount of distillate). Subject to: equality and inequality constraints. Mathematically the optimization problem (P1) can be written as: P1 subject to and
Max rf(t) xD(tf) g x*D f(t,x′,x,u,v) ) 0
with
f(t0,x′0,x0,u0,v) ) 0 linear bounds on reflux ratio
C (inequality constraint) (model equations, equality constraints) (initial conditions, equality constraints) (inequality constraints)
2290 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 1. Input Data for Ethanol Esterification Using Conventional Batch Distillation no. of ideal separation stages (including a 10 reboiler and a total condenser) 5.0 total fresh feed, B0, kmol feed composition (acetic acid, ethanol, ethyl 0.45, 0.45, 0.0, 0.1 acetate, water) xB0, mole fraction column holdup, kmol: condenser 0.1 internal plates 0.0125 condenser vapor load, kmol/h 2.5 column pressure, bar 1.013
where C is the conversion of the limiting reactant, xD(tf) is the composition of distillate at the end of the operation (tf), rf(t) is the reflux ratio as a funtion of time (t). For the solution of problem P1 each “function evaluation” of the optimizer requires a full integration of the DAE system. “Gradients” of the objective function and constraints with respect to optimization variable(s) are evaluated in an efficient way using adjoint variables. It is to be noted here that other types of dynamic optimization problems such as minimum time, maximum profit, etc., could also be formulated and solved using the algorithms mentioned above. Although it is desirable to maximize profit in any operation by solving a maximum profit problem, we have not done so at this stage as we plan to use the results of maximum conversion problem to develop a computationally efficient technique to solve such a problem at the later stage of this paper.
Table 2. Vapor-Liquid Equilibrium and Kinetic Data for Ethanol Esterification Vapor-Liquid Equilibrium acetic acid + ethanol S ethyl acetate + water (1) (2) (3) (4) K1 ) (2.25 × 10-2)T - 7.812 K1 ) 0.001 T e 347.6 log K2 ) - 2.3 × 103/T + 6.588 log K3 ) - 2.3 × 103/T + 6.742 log K4 ) - 2.3 × 103/T + 6.484
T > 347.6 K
Kinetic Data rate of reaction, gmol/(L min); r ) k1C1C2 - k2C3C4 where rate constants are k1 ) 4.76 × 10-4 and k2 ) 1.63 × 10-4 and Ci stands for concentration in gmol/L for the ith component
Example and Results of Dynamic Optimization In this work we consider the esterification of ethanol and acetic acid. The reaction products are ethyl acetate (main product) and water. The reversible reaction scheme together with the boiling temperatures are shown below:
acetic acid + ethanol S ethyl acetate + water (4) (3) (2) (1) bp, K
391.1
351.5
350.3
373.2
Ethyl acetate, the main product, has the lowest boiling temperature in the mixture and therefore has the highest volatility. Controlled removal of ethyl acetate by distillation will shift the chemical equilibrium further to the right and therefore will improve conversion of the reactants. This will also increase the yield proportionately. The data defining the column configuration, feed, feed composition, column holdup, etc., for the problem are given in Table 1. We use 4% of the total feed charge as the total column holdup for this example. Half of this holdup is taken as condenser holdup and the rest is equally divided for the plate holdups. Plate compositions, product accumulator compositions and reflux drum compositions (differential variables of the model equations) are initialized to the feed compositions for t ) 0 which ensure consistent initialization of the DAE system used in this work (for detail see Pantelides, 1988). Table 2 shows the vapor-liquid equilibrium data taken from Simandl and Svrcek (1991) and the kinetic data taken from Bogacki et al. (1989). Vapor and liquid enthalpies are calculated using data from Reid et al. (1977). It is to be empahsized that these data do not account for detailed VLE calculations and for any azeotropic formed.
Figure 2. Composition and reflux ratio profiles for ethanol esterification: 1, acetic acid; 2, ethanol; 3, ethyl acetate; 4, water.
The maximum conversion problem is solved for a fixed batch time and given product (ethyl acetate) purity. We solved a series of such optimization problems for different but fixed batch times tf (between 5 and 30 h) and for two given product purities, x*D ) 0.70 and x*D ) 0.80. Reflux ratio level was optimized over the batch time of operation. Typical plots of accumulated distillate and reboiler composition profiles for tf ) 16 h and x*D ) 0.80, achieved for the optimal reflux ratio operation, are shown in Figure 2. Figures 3-6 plot the maximum conversion, the corresponding amount of product, optimal constant reflux ratio, and heat load profiles for different batch times. Figure 3 also shows the maximum conversion profile achieved under total reflux operation (where no product is withdrawn). The latter approximates the conversion which would be achieved in the absence of distillation.
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2291
Figure 3. Maximum conversion vs batch time: curve 1, x*D ) 0.70; curve 2, x*D ) 0.80; curve 3, total reflux operation.
Figure 4. Amount of ethyl acetate vs batch time: curve 1, x*D ) 0.70; curve 2, x*D ) 0.80.
It is to be noted that if there is a large column holdup, total reflux conversion will not approximate the conversion achieved in the absence of distillation. For each purity specification it is seen from Figure 3 that maximum conversion increases with increasing batch time. This is expected because as the product species are withdrawn by distillation and there is more time available, the reaction goes further to the right. This increase is very sharp at the beginning as it is easier to shift equilibrium by eliminating the plentiful product at the given purity. The curve is flattened near the end of the graph as it becomes progressively more difficult to remove the product at the given purity. The product amount curves (Figure 4) support this observation. Figure 5 shows that the reflux ratio falls initially as the available batch time increases. When only a short time is available to complete the batch, only little product is produced by the reaction and separating it in the distillate requires a high reflux ratio. With larger batch times, more product is produced by the reaction and separation becomes easier (hence lower reflux ratio). Finally as the batch time is increased further, less and less reactants in the column are available to react to give products, and therefore high reflux ratio
Figure 5. Reflux ratio vs batch time: curve 1, x*D ) 0.70; curve 2, x*D ) 0.80.
Figure 6. Reboiler heat duty vs batch time: curve 1, x*D ) 0.70; curve 2, x*D ) 0.80.
is required again to achieve products at the given purity. The total heat load curve (Figure 6) follows a linear trend with respect to batch time, since in practice the maximum heat load is used at all times during a batch. Figures 3 and 4 also show that for each batch time, the conversion and the yield for x*D ) 0.70 are higher than those for x*D ) 0.80. A higher reflux ratio (Figure 5) is required to produce ethyl acetate product at higher purity with consequently lower rate of reaction and lower conversion. Figure 7 shows the percent improvement in conversion (compared to the conversion achieved under total reflux operation) achieved for different fixed batch times. The results show that about 40% more conversion is possible when the column is operated optimally compared to total reflux operation. Computationally, the solution of the dynamic optimization problem is time-consuming and expensive. The number of “function” and “gradient evaluations” for each maximum conversion problem is between 7 and 9. A fresh solution requires approximately 600 cpu s in a SPARC-1 Workstation. Subsequent solutions for different but close values of tf can take advantage of the good initialization values available from the previous solutions.
2292 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997
Figure 7. Percent improvement in conversion vs batch time: curve 1, x*D ) 0.70; curve 2, x*D ) 0.80.
Figure 8. Profit vs batch time: curve 1, x*D ) 0.70; curve 2, x*D ) 0.80. Table 3. Cost Parameters for Ethanol Esterificationa
The profit of the operation can also be calculated using the results presented above. A profit function for this esterification problem can be defined as follows
CD1D1 - CB0B0 - ChQR P) t
C1 ) cost of acetic acid, $/kmol C2 ) cost of ethanol, $/kmol C3 ) CD1 ) price of ethyl acetate at x*D ) 0.70, $/kmol C3 ) CD1 ) price of ethyl acetate at x*D ) 0.80, $/kmol C4 ) cost of water, $/kmol CB0 ) raw material cost at the feed composition, $/kmol Ch ) steam cost (at 100 psig), $/kJ
(23)
where D1 is the amount of product (kmol), B0 is the amount of raw material (kmol), QR is the total heat input (kJ), t is the batch time (h), CD1 is the product price ($/kmol), CB0 ($/kmol) is the raw materials cost, and Ch ($/kJ) is the energy cost for heating (operating cost). It is to be noted that for a fixed operation time, t in eq 23 above, the profit will increase with the increase in the distillate amount and a maximum profit optimization problem translates into a maximum distillate optimization problem (Mujtaba and Macchietto, 1993; Diwekar, 1992). However, for any reaction scheme (some presented by Barbosa and Doherty, 1988) where one of the reaction products is the lightest in the mixture (and therefore suitable for distillation) the maximum conversion of the limiting reactant will always produce the highest achievable amount of distillate for a given purity and vice versa. This is true for a reversible or irreversible reaction scheme and is already explained in the introduction section. Therefore, the results (in terms of the amount of distillate, recovery, energy consumption, etc.) of the maximum conversion optimization problem can be used to evaluate the maximum achievable profit (for a given operation time) using eq 23. This means that for reactive batch distillation the maximum conversion problem and the maximum distillate problem can be interchangeably used in the maximum profit problem for fixed batch time. For a nonreactive distillation system, of course, the maximum distillate problem has to be solved. Figure 8 shows the maximum achievable profit at different batch times, calculated using the profit function defined in eq 23, the cost parameters given in Table 3, and the values of distillate and heat load obtained from the solution of the maximum conversion problems. For x*D ) 0.70, Figure 8 shows that the optimum batch time lies between 12 and 14 h where the profit is the maximum for the entire range of operation.
) 32.01 ) 17.87 ) 80.0b ) 85.0b ) 0.0c ) 22.45 ) 0.32d
a
Prices taken from Chemical Marketing Reporter, October 1992. Assumption based on quoted price for product purity xD ) 0.85. c Assumption. d Taken from Peters and Timmerhaus (1980) and adjusted for inflation. b
Alternative to above, a maximum profit problem can be formulated as follows
P2 subject to:
Max tf,C,rf(t),D1
P
(24)
product purity constraints, DAE model equations, etc.
using the profit function P defined by eq 23, the solution of which will automatically determine the optimum batch time (tf), conversion (C), reflux ratio (rf), and the amount of product (D1). However, as the cost parameters (CD1, CB0, etc.) can change from time to time, it will require a new solution of the dynamic optimization problem P2 (as outlined in Mujtaba and Macchietto, 1993, 1996), to give the optimal amount of product, optimal batch time, and optimal reflux ratio, which is computationally expensive. Polynomial Based Optimization Framework: A New Approach A new and an alternative technique is proposed in this section which permits very efficient solution of the maximum profit problem using the solutions of the maximum conversion problem already calculated. This is detailed using again the ethanol esterification example presented in the previous section.
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2293
Figure 9. Curve fitting for maximum conversion profile (x*D ) 0.70).
For a specified purity of the product, the results presented in Figures 3-6 are only dependent on the batch time (t) but are independent of the cost function P and cost parameters used to compute the profit. They are also quite well behaved with respect to batch time. These results can be easily represented by mathematical functions leading to algebraic equations as follows:
(a) maximum conversion:
C ) g1(t)
(25)
(b) optimum distillate:
D1 ) g2(t)
(26)
(c) optimum reflux ratio:
rf ) g3(t)
(27)
(d) total reboiler heat load:
QR ) g4(t)
(28)
Figure 10. Curve fitting for product profile (x*D ) 0.70).
where, g1(t), g2(t), etc., are polynomial functions. The profit function in eq 23 can now be written as
P)
CD1g2(t) - CB0B0 - Chg4(t) t
Figure 11. Curve fitting for reflux ratio profile (x*D ) 0.70).
(29)
which is a function of only one variable (t) for a given set of values of (CD1, CB0, Ch). The dynamic optimization problem P2 now becomes a single variable (the only variable to be optimized is batch time t) algebraic optimization, and the solution of the problem no longer requires full integration of the model equations. Under frequently changing market prices of (CD1, CB0, Ch) this method will solve the maximum profit problem very cheaply and will thus determine new optimum batch time for the plant. The functions represented by eqs 25-28 can then be used to determine optimal values of C, D1, rf, QR, etc. The above technique is illustrated with the example already presented ealier. For example, at product purity x*D ) 0.70 the data presented in Figures 3 and 5 are fitted very well by a fifth order polynomial while the data in Figure 4 are fitted by third order and those of Figure 6 by first order polynomials, respectively. The resulting curves and polynomial equations are shown in Figures 9-12. Now, the term D1 and QR in the maximum profit problem (problem P2) can be simply replaced by the time dependent polynomial equations presented in Figures 10 and 12. For a given set of cost parameters the only
Figure 12. Curve fitting for reboiler heat duty profile (x*D ) 0.70).
optimization variable remaining in problem P2 is the batch time, which can be obtained with extremely little effort. Once the optimal batch time is obtained, the corresponding optimal reflux ratio and maximum achievable
2294 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 4. Summary of Results for Maximum Profit Problem case
CB0 ($/kmol)
CD1 ($/kmol)
Ch ($/kJ)
profit ($/h)
batch time (h)
D1 (kmol)
reflux ratio
max conv
QR (kJ)
1 2 3 4 5 6
22.45 22.45 22.45 15.00 35.00 22.45
80.0 150.0 70.0 80.0 80.0 80.0
0.32 0.32 0.32 0.32 0.32 0.45
4.29 15.78 2.80 7.03 0.59 4.18
14.79 11.66 15.83 12.53 19.96 14.79
2.25 1.99 2.30 2.08 2.40 2.25
0.936 0.927 0.939 0.923 0.945 0.936
0.729 0.694 0.739 0.705 0.766 0.729
13.36 10.54 14.31 11.33 18.04 13.36
conversion are calculated directly using the polynomial equations presented in Figures 9 and 11, respectively. If the cost functions are changed, the same methodology can be applied without any difficulties to find the optimal batch time, so as to achieve maximum profit. This approach is certainly very efficient and could be used for calculating on-line the optimal batch time for processes involving batch distillation with frequently changing market prices, energy costs, etc. As an example, for a product purity (x*D ) 0.70), the maximum profit problem was solved for a number of cost parameters using the efficient technique described above. The results are presented in Table 4. For each case Table 4 also gives the optimal batch time, amount of product, reflux ratio, total reboiler duty, and maximum conversion, calculated using the polynomial equations. Case 1 of Table 4 is the base case and shows the optimization results using the cost parameters presented in Table 3. The maximum profit and optimal batch time obtained by optimization are in very close agreement to those shown in Figure 8. The maximum profit in Figure 8 lies between 3.99 and 4.126 ($/h), which occurred within a batch time of 12-14 h. Each of the optimization problem presented in Table 4 requires three to four iterations and about 3-4 cpu s in a SPARC-1 Workstation. It is to be emphasized that the maximum profit problem solved using the techniques presented in this paper will be close to rigorous profit optimization solution (using techniques presented in Mujtaba and Macchietto, 1993, 1996) only if the polynomial approximations are very good as in the case for the example presented in this paper. Conclusions Dynamic optimization framework and solution with detailed model are presented in this work for reactive batch distillation. Because of the inherent dynamics of the process, the solutions of the optimization problems using existing techniques are computationally expensive. In this work we proposed a new technique to optimise reactive batch distillation with very little computational effort. Dynamic optimization problems (say maximum conversion problem) independent of the cost functions are solved parametrically, a-priori, for a wide range of operation times. For a given purity, the optimal product yield, optimal reflux, and optimal heat load profiles thus obtained are fitted with polynomial equations which are dependent only on operation time. Using these algebraic equations, the maximum profit problem for any cost functions can then be solved very easily, without a full solution of the dynamic optimization problem. The new approach is computationally efficient and provides a framework which can be used for on-line optimization of batch distillation within scheduling programs for batch processes. The technique presented above is general and can be applied to any reaction
schemes with proper VLE, kinetic data, and information of possible azeotropes. The method can be easily extended to nonreactive batch distillation and also to nonconventional batch distillation with or without chemical reaction (Mujtaba and Macchietto, 1994). Finally, it is to be noted that all the results presented in this paper are based on the model used in this work. If a short-cut simple model (Diwekar, 1992) or more rigorous models (Cuille and Reklaitis, 1986; Bosely and Edgar, 1994) are used, the techniques proposed in this paper may give different optimum results. Therefore, all models must be validated using plant data before the proposed techniques are implemented in a real plant. Acknowledgment This work was carried out at the Centre for Process Systems Engineering, Imperial College, and was funded by SERC/AFRC, whose support is gratefully acknowledged. Nomenclature B0, xB0 ) initial amount (kmol) and composition (mole fraction) of the reboiler charge D, xiD ) amount and composition of the main reaction product Ha, Hc, Hj ) product accumulator, condenser, and plate holdup, respectively (kmol) L, V ) liquid or vapor flow rate (kmol/h) LD ) distillate and bottom product flow rate, respectively (kmol/h) QC, QR ) condenser or reboiler duty (kJ/h) T, P ) temperature (K) and pressure (bar) K ) vapor-liquid equilibrium constant kr ) array of reaction rate constants rf ) internal reflux ratio t ) batch time (h) x, y ) liquid or vapor composition x, y ) array of liquid or vapor composition xa ) accumulated distillate composition xDi ) instant distillate composition r ) reaction rate (1/h) ∆n ) (1/h) Superscript and Subscript i ) component number (1, nc) j ) stage number (1, N)
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Received for review September 18, 1996 Revised manuscript received February 20, 1997 Accepted February 21, 1997X IE960573D
X Abstract published in Advance ACS Abstracts, April 15, 1997.
