Selectivity Targets for Batch Reactive Distillation - Industrial

Ind. Eng. Chem. Res. 2000, 39, 1565-1575

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Selectivity Targets for Batch Reactive Distillation† Sagar B. Gadewar, Michael F. Malone,* and Michael F. Doherty Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003-3110

Reactive distillation has proved to be an important process alternative to the conventional reactor-separator configuration. Advantages of reactive distillation and flexibility of a batch process can be combined in batch reactive distillation. We present a simple method to estimate the advantage of using batch reactive distillation over conventional technology. For the examples studied, we determine yield and selectivity targets for a batch reactive distillation device. This method also determines the effect of operating parameters on yield and selectivity. We show that the advantage of using batch reactive distillation equipment is more significant for systems with fast side reactions. Comparison of these estimates with those for conventional reactors is useful for quick screening of process alternatives during process synthesis. Introduction In the production of low-volume, high-value specialty chemicals, batch distillation is one of the most common operations. For continuous processes, reactive distillation is a process alternative that can significantly reduce capital and operating costs,1-6 and batch reactive distillation (BRD) should be viewed as an important process alternative to conventional batch processing. Distillation with chemical reaction also offers advantages in overcoming the equilibrium limitation for reversible reactions and suppression of undesired byproducts.7 In a typical BRD process, reaction and separation occur simultaneously. The removal of one or more products from the reaction mixture not only increases the conversion of equilibrium-limited reactions but also increases or decreases the reaction vessel temperature if the product removed is lower or higher boiling, respectively. Although there is an extensive literature on batch distillation,8,9 relatively little has been published on batch distillation with reaction. No systematic design methods are available for BRD devices because most of the earlier work focused on detailed models for simulation and process optimization.10-12 Cuille and Reklaitis10 described a model for the simulation of a BRD system with reaction occurring in the reboiler, in the condenser, and also on the plates of the rectification column. Their emphasis was on developing a numerical method to solve the model and not on developing a design methodology. Nothing is known about the effect of operating parameters on the performance of a BRD system. The performance can be evaluated in terms of conversion, yield, and selectivity. A dynamic optimization of BRD with an objective to maximize conversion of the limiting reactant in the esterification of ethanol and acetic acid was described by Mujtaba and Macchietto.11 They used a detailed model including mass and energy balances and Raoult’s law for vapor-liquid equilibrium (VLE). They found the optimization using nonlinear program-

ming to be computationally very expensive. Therefore, using a dynamic optimization tool to determine operating conditions for improved performance is timeconsuming. Bollyn and Wright12 published a case study of the scale-up of a laboratory process involving BRD for the production of ethyl ester of pentenoic acid. The reaction occurred only in the still, and there was no reaction on the stages. For a parallel reaction system, they showed that significant improvement in selectivity can be achieved by using a BRD device over a conventional reactor. The model development involved extensive simulations and experimentation. Solving these models using existing techniques, however, is computationally expensive. Hence, there is a need to develop a computationally inexpensive model with applicability to systems with multiple reactions to provide quick estimates for the targets of yield and selectivity over different operating conditions. Recent work by Venimadhavan et al.13 demonstrated the utility of a simplified model for BRD in exploring alternative operating strategies for the production of butyl acetate at a reduced computational effort. Selectivity improvement leads to lower raw materials costs and a reduction of wastes. Prediction of selectivity targets for BRD assists in estimating its attractiveness as a process alternative to conventional technology. Because BRD is a very complex process, simulations are often difficult to converge;10,12,14 therefore, a simplified model is critical in rapid evaluation of process alternatives. This paper describes a rapid method for setting selectivity targets in BRD. This can be used as a screening technique for establishing targets for BRD so that expensive detailed design, simulation, and experimentation are performed only on promising candidate designs. First, we consider a simple system of reactions in series with equimolar stoichiometry. We then consider a simplified model of butane alkylation in which some of the assumptions are relaxed. Batch Reactor-Rectifier

†

This paper is dedicated to Professor J. D. Seader in recognition of his many outstanding contributions to process engineering. * To whom correspondence should be addressed. Present address: 154A Goessmann Laboratory, 686 North Pleasant Street, Amherst, MA 01003-3110. Telephone: (413) 545-0838. Fax: (413) 545-1133. E-mail: [email protected].

Figure 1 is a schematic of a BRD system. The system is similar to a conventional batch distillation column except for the reaction occurring in the reboiler. This system can be described as a “batch reactor-rectifier” because the products are obtained in the distillate. It consists of a reaction vessel (reboiler) and a fractionating

10.1021/ie990497p CCC: $19.00 © 2000 American Chemical Society Published on Web 03/24/2000

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Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000

as distillate. However, in mixtures with azeotropes, it is not always possible to meet the specified distillate composition; for these mixtures a VLE model is necessary to assess the feasibility of separation16-20 and also to determine the boiling point of the reaction mixture. In this example, feasibility is trivial, and because the reaction rate constants are temperature independent, we do not need a VLE model for implementing the procedure. (a) Balances around a Batch Reactor-Rectifier (Region I in Figure 1). Overall and component material balances are given as

dH ) -H, H(0) ) H0 dξ

(1)

dxA k1 H D 0 ) xA - xD,A - Da1xA , xA(0) ) 1 (2) dξ k1,ref H0 D k 1 H D0 dxB ) xB - xD,B + Da1xA dξ k1,ref H0 D k2 H D 0 , xB(0) ) 0 (3) Da2xB k2,ref H0 D

Figure 1. Schematic of a BRD device.

column on top. The vapors coming from the topmost plate of the column are condensed in a total condenser, and the distillate is collected in a receiver. We make the following assumptions in the model development: (a) A liquid-phase reaction takes place in the reaction vessel alone (e.g., with a heavy catalyst). (b) The separation system has sufficient stages and reflux to achieve the desired distillate composition. (c) There are no reactive or nonreactive azeotropes in the system. (d) A total condenser produces a saturated liquid product. (e) The holdup on the stages and in the condenser is negligible compared to the reaction vessel; i.e., the column is in a pseudo steady state. Model Development for Series Reactions Consider a liquid-phase system with the elementary, irreversible reactions in series k1

k2

A 98 B 98 C in an ideal solution. For the sake of simplicity, we assume that the rate constants are temperature independent. We will, however, relax this assumption later. We use activity-based rate expressions for reasons explained in Venimadhavan et al.15 Component B is the desired product while C is an undesired byproduct in the reaction system; B is the lightest, C is the heaviest, and A is intermediate boiling. Our aim is to develop a simple model to quantify the performance of a BRD device. The model does not consider stage-to-stage calculations for the rectification column. The model assumes that the specified distillate composition is always achievable for the given column configuration; e.g., for simple mixtures without azeotropes, the lightest boiling component can be removed

where the relationship between warped time and clock time is given as dξ ) (D/H) dt. The Damko¨hler numbers,21 Da1 ) H0k1,ref/D0 and Da2 ) H0k2,ref/D0 are the dimensionless ratios of the characteristic liquid residence time H0/D0 to the characteristic reaction times 1/k1,ref and 1/k2,ref, respectively. The quantities k1,ref and k2,ref are the values of the rate constants for the first and second reaction in the series reaction system at a reference temperature. Because the rate constants are temperature independent, k1,ref ) k1 and k2,ref ) k2. Here, xD,A and xD,B are the mole fractions of A and B, respectively, in the distillate. To solve this model, we must first specify a distillate rate policy. We choose a decreasing distillate rate proportional to the molar holdup as

D D0 ) H H0

(4)

We use it because (a) it implies that the instantaneous Damko¨hler numbers are approximately constant and also that the rate of product removal is kept in balance with the rate of production and (b) we get a set of autonomous differential equations with two operating parameters Da1 and Da2. Defining the dimensionless molar holdup and distillate rate as H ) H/H0 and D ) D/D0, respectively, where H0 is the initial molar holdup and D0 is the initial distillate rate, we can rewrite eqs 1-3 as

dH ) -H, H(0) ) 1 dξ

(5)

dxA ) xA - xD,A - Da1xA, xA(0) ) 1 dξ

(6)

dxB ) xB - xD,B + Da1xA - Da2xB, xB(0) ) 0 (7) dξ Based on the distillate rate policy and eq 5, we can write

Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1567

dD ) -D, D(0) ) 1 dξ

(8)

Equations 5 and 8 show that D ) H at any warped time ξ. (b) Balances around a Receiver (Region II in Figure 1). Overall and component material balances around the receiver give

(12)

xjB(ξ) ) HD,B(ξ) + H(ξ) xB(ξ)

(13)

The conversion of A is given as

X(ξ) ) 1 - xjA(ξ)

(14)

The yield, Y, is defined as the ratio of moles of B formed to the moles of A fed to the reboiler. Therefore,

dHD,A ) DxD,A, HD,A(0) ) 0 dξ

(9)

dHD,B ) DxD,B, HD,B(0) ) 0 dξ

(10)

where the cumulative amounts of components A and B collected as distillate in the receiver from warped time 0 to ξ are denoted by HD,A and HD,B, respectively, while HD,A ) HD,A/H0 and HD,B ) HD,B/H0 are the dimensionless holdups of components A and B in the receiver, respectively. The set of differential equations to be solved is given by eqs 5-10. We solve them numerically using the Livermore solver for ordinary differential equations (LSODES) which is a robust integrator for such equations. If the reaction rate constants are known, because the ratio of the Damko¨hler numbers, Da1/Da2, is equal to the ratio of rate constants, k1,ref/k2,ref, we can specify the value of only one of the Damko¨hler numbers independently. However, in this example, because the rate constants are arbitrary, we can specify a Damko¨hler number for each reaction independently. Apart from specifying the primary operating parameters Da1 and Da2, it is necessary to specify the distillate composition, which is a trivial matter here, though not in other examples. In this example, the lightest boiling component B is the unstable node and can be removed with almost 100% purity. These equations are then solved for instantaneous values of the dimensionless molar holdup, distillate rate, amount of distillate collected, and reaction vessel composition at different values of the warped time. Constraint. In this targeting methodology, we specify the Damko¨hler number for each reaction and the distillate composition of B. The target distillate rate is one that is proportional to the molar holdup in the reboiler. Because we desire a distillate stream with high purity of B, it is necessary to check if the production rate of B by reaction is sufficient to maintain the proportional distillate rate policy. This imposes an overall material balance constraint on the desired product B given as

DxD,Bδξ e (Da1xA - Da2xB)Hδξ + DxB

xjA(ξ) ) HD,A(ξ) + H(ξ) xA(ξ)

(11)

Note that the left-hand side represents the amount of B collected in the receiver over a time interval δξ, while the sum of the production of B over the same time interval and the residual amount of B in the reboiler is given by the right-hand side. If the constraint is violated, it is necessary to decrease the distillate rate to satisfy the constraint (11). Yield and Selectivity. The time average mole fractions based on the cumulative amount of A and B in the receiver and the reboiler at any time, determined using material balances, are given as

Y(ξ) ) xjB(ξ)

(15)

The selectivity, S, is defined as the ratio of moles of B formed to the moles of A consumed. Therefore,

S(ξ) )

xjB(ξ) 1 - xjA(ξ)

(16)

Note that yield and selectivity are simply related, Y(ξ) ) S(ξ) X(ξ). Solution Algorithm. Our target-setting methodology for BRD consists of specifying the distillate composition of B and the Damko¨hler number for each reaction and solving the model equations to determine conversion, yield, and selectivity at each instant of warped time. The algorithm is as follows: 1. Specify the distillate composition. Select the lightest component as the distillate and assume that it can be removed with high purity (approximated as 100% purity). 2. Specify a value for each of the Damko¨hler numbers. 3. Integrate eqs 5-10 numerically for a small time interval δξ. 4. Check the material balance constraint, eq 11. If the constraint is not satisfied, do the following: (a) Decrease the distillate rate D to satisfy eq 11; this decreased distillate rate is given as

D)

(Da1xA - Da2xB)Hδξ xD,Bδξ - xB

(b) Recalculate the values of all dimensionless holdups and mole fractions using this decreased distillate rate. Use eqs 1-3 instead of eqs 5-7 in the recalculation. (c) Use the new values for further calculations. 5. Calculate the conversion, yield, and selectivity using eqs 12-16. 6. Repeat from step 3 until the mole fraction of the limiting component in the reboiler xA ≈ 0. It is relatively easy to determine the yield and selectivity profiles for conventional reactors such as plug-flow reactor (PFR), continuous stirred tank reactor (CSTR), etc.; the solutions can be found in any standard chemical reaction engineering text.22-24 We compare the yield and selectivity profiles of BRD with those of conventional reactors. For this example, an isothermal PFR will always give a higher yield of B than a CSTR.22 Also, because a PFR gives higher selectivity than the CSTR at all conversions for this reaction system, we only compare BRD with an isothermal PFR. The yield and selectivity for a PFR can be found without knowledge of the individual rate constants because the profiles depend only on the ratio of the two rate constants, k2/ k1, which is equal to the ratio of the two Damko¨hler numbers, Da2/Da1.

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Figure 2. Yield and selectivity plots for reactions in series A f B f C at Da1 ) 1 and Da2 ) 1.

Figure 3. Yield and selectivity plots for reactions in series A f B f C at Da1 ) 2 and Da2 ) 2.

The solution algorithm for BRD can be implemented for different sets of Damko¨hler numbers for any specified distillate composition. Based on the ratio Da2/Da1, profiles for the PFR can be determined. Figure 2 shows the yield and selectivity for the desired product, B, for the condition Da1 ) Da2 ) 1. We find that BRD has the potential for essentially perfect selectivity of 100% at all conversions, while in the case of a PFR, the selectivity decreases with increasing conversion. Figure 3 shows the yield and selectivity for Da1 ) Da2 ) 2. Even though BRD achieves selectivities lower than 100%, the profiles for the yield and selectivity are much better than those for a PFR at all but the lowest conversions where the PFR approaches the BRD performance. Parametric Screening. The model equations for BRD are autonomous and can be solved after specifying values for the Damko¨hler numbers. We use a constant distillate composition and proportional distillate rate policy if the constraints are satisfied. However, when the material balance constraint is violated, we must

reduce the distillate rate to be less than proportional to the molar holdup. Because the Damko¨hler number is a ratio of the characteristic liquid residence time to the characteristic reaction time, a higher Damko¨hler number means a relatively higher liquid residence time and vice versa. Figures 2 and 3 show that dramatic changes in the yield and selectivity can be expected with changes in Damko¨hler numbers. We can evaluate the potential advantage gained by BRD over conventional reactors at different combinations of Damko¨hler numbers. A plot of selectivity vs conversion is useful for this evaluation because it gives an estimate of raw materials lost to byproduct formation. Figure 4 shows the comparison of selectivity profiles for different Damko¨hler numbers. Low Da1 and Low Da2. Figure 4 shows that a BRD device gives excellent selectivity to the desired product. Comparatively, a PFR gives poor selectivity. The performance of a PFR depends on the ratio k2/k1 (or Da2/


Figure 4. Plot showing the effect of changes in Damko¨hler numbers on selectivity for a system of reactions in series A f B f C.

Da1); if it is greater than or equal to unity, selectivity drops very sharply even at low conversions. Therefore, BRD provides a significant advantage over a PFR when both Damko¨hler numbers are low. High Da1 and Low Da2. At the values Da1 ) 10 and Da2 ) 1, both BRD and PFR give good selectivities. The selectivities drop sharply at high conversions for both alternatives. Therefore, no significant improvement in selectivity is achieved by using a BRD device. High Da1 and High Da2. In this region of the parametric chart, both BRD and PFR perform poorly. There is a sharp decrease in selectivity for both alternatives even at low conversions. It is not preferable to use either a BRD device or a PFR in this region of operating conditions. Low Da1 and High Da2. When the selectivity profiles are compared, it is found that BRD offers a much higher selectivity over a PFR at all conversions. In this region, because the ratio Da2/Da1 and, hence, k2/k1 are much greater than unity, the selectivity profiles for a PFR show a sharp decline with increasing conversion. A significant advantage can be achieved by using a BRD device. This analysis is useful in the selection of process alternatives and also in estimating the operating pa-

rameters for improving selectivity in an existing BRD setup. To simplify the screening of alternatives, we can specify a minimum acceptable value for the selectivity achieved at a given conversion and then determine the domain of operating conditions where using a BRD is advantageous over conventional reactors (PFR, CSTR etc.,), as well as the operating conditions where it is not. As a representative example, assume that the selectivity is good if it is g90% at conversions e40%. The regions where this objective is satisfied are given in Figure 5. The dark-shaded region is the domain of operating parameters where either a BRD or a PFR will satisfy the objective of good selectivity. No significant advantage is gained by using a BRD over a PFR for operating parameters in the dark-shaded region. The light-shaded region is the domain where only BRD will satisfy the defined objective. It is, therefore, preferable to use a BRD device when the operating parameters lie in the light-shaded region, i.e., where the byproduct reaction is relatively fast. Alkylation of Butane The acid-catalyzed alkylation of isobutane with butenes is an important process in the production of high octane

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Figure 5. Area of operating parameters for reactions in series A f B f C, where selectivity is g90% for conversions of e40%.

gasoline. The product of this process is a complex mixture of C5-C7 compounds, octanes, and some amount of higher boiling compounds. We consider a simplified version of the process reactions involving only the following parallel steps: k′1

2-butene (1) + isobutane (2) 98 isooctane (3) (17) k′2

2-butene (1) + isooctane (3) 98 dodecane (4) (18) The rate of reaction of 2-butene is given by

Figure 6. Schematic of a batch reactor-stripper for the butane alkylation system.

tropic mixtures8,14,26 or when the desired component is a higher boiler. To use the formalism developed in the previous example, it is necessary to convert the concentrationbased rate expressions to those based on mole fractions. The reaction rates in mole fraction based rate expressions are

r1(c) ) k′1c1c2

(19)

r2(c) ) k′2c1c3

(20)

r1(x) ) k1x1x2

(23)

where c1, c2, c3, and c4 are the concentrations of 2-butene, isobutane, isooctane, and dodecane, respectively. The rate constants are reported as25

r2(x) ) k2x1x3

(24)

( (

) )

-7828.65 L/(gmol‚s) (21) k′1 ) 1.657 × 109 exp T (K) k′2 ) 4.229 × 1012 exp

-9785.83 L/(gmol‚s) (22) T (K)

The reactions are catalyzed by a homogeneous sulfuric acid catalyst. We perform all of the calculations on an acid-free basis. We assume that the liquid reaction mixture is ideal and the reactions are irreversible. At a system pressure of 7.8 atm, the lowest boiling component is the reactant isobutane (56.78 °C) and the highest boiling component is the byproduct dodecane (326.5 °C). The reactant 2-butene (73.13 °C) and desired product isooctane (193.22 °C) are intermediate boilers. Unlike the previous example, we use a stripper for removing the desired product. A schematic diagram of the apparatus is shown in Figure 6. This arrangement can be described as a “batch reactor-stripper”. The batch tank acts as the reflux drum as well as the reactor, while the products are withdrawn as bottoms from the reboiler. Such an arrangement is uncommon but also offers some important advantages for separating azeo-

The reaction rate per unit volume r(c) and the reaction rate per mole of liquid mixture r(x) are related by the molar density F, given as r(c) ) Fr(x). The relationship between concentration and mole fraction of each component can be written as ci ) Fxi, where i ) 1, ..., 4. We can easily relate the rate constants for concentrationbased rate expressions to those for mole fraction based expressions as k1 ) Fk′1 and k2 ) Fk′2.27 The above equations show that k1 and k2 vary with the molar density; hence, k1 and k2 depend not only on temperature but also on the composition of the reaction mixture. The Damko¨hler number is defined as Da ) k1,refH0/B0. Here k1,ref is the rate constant of the first reaction at a reference temperature, which we take as the boiling point of the lowest boiling component in the reaction mixture (Tref ) 56.78 °C at P ) 7.8 atm). H0 is the initial reaction vessel holdup while B0 is the initial bottoms rate. Assumptions. (a) The liquid-phase reactions occur only in the reflux drum. (b) The separation system has sufficient stages and reboil ratio to achieve the desired bottoms composition. (c) There are no reactive or nonreactive azeotropes..

Ind. Eng. Chem. Res., Vol. 39, No. 6, 2000 1571 Table 1. Antoine Coefficients for Butane Alkylationa component

normal boiling point (°C)

a

b

c

2-butene isobutane isooctane dodecane

3.8 -11.6 99.3 216.4

15.8171 15.5381 15.685 16.1134

2210.71 2032.73 2896.28 3774.56

-36.15 -33.15 -52.41 -91.31

a Antoine equation: ln(Psat) ) a - b/(T + c). Psat is in mmHg and T in K.

(d) The holdup on the stages and in the reboiler is negligible compared to the reflux drum; i.e., the column is in a pseudo steady state. Model development for this system is analogous to that of the previous example, with the only difference being that the desired product is removed as a bottoms stream. A general model for systems with multiple reactions is described in Appendix A. (a) Balances around a Batch Reactor-Stripper (Region I in Figure 6). The overall and component material balances are given as

dx1 ) x1 - xB,1 + (x1 - 1)Daκ1x1x2 + dξ (x1 - 1)Daκ2x1x3, x1(0) ) x01 (25) dx2 ) x2 - xB,2 + (x2 - 1)Daκ1x1x2 + dξ x2Daκ2x1x3, x2(0) ) βx01 (26) dx3 ) x3 - xB,3 + (x3 + 1)Daκ1x1x2 + dξ (x3 - 1)Daκ2x1x3, x3(0) ) 0 (27) where xB,1, xB,2, and xB,3 are liquid-phase mole fractions of 2-butene, isobutane, and isooctane in the bottoms, respectively, β is the molar feed ratio of isobutane to 2-butene, κ1 ) k1/k1,ref, and κ2 ) k2/k1,ref. The Damko¨hler number is defined as Da ) H0k1,ref/B0. In the previous example, because the reaction rate constants were arbitrary, we had the freedom to specify a Damko¨hler number for each reaction. However, in this example, the rate constants for each reaction are known; therefore, we must formulate the model in terms of a single Damko¨hler number. The VLE is modeled using Raoult’s law (see Table 1 for vapor pressure relations28) to calculate the reaction vessel temperature at each instant of warped time ξ, so that the temperature-dependent reaction rate constants can be calculated. The y vs x data generated using the VLE model are needed only for checking the feasibilty of separation. The reactions in this system are not equimolar; hence, the molar density and therefore the rate constants change as the reactions proceed. Assuming the excess molar volume to be negligible and noting that molar density is the reciprocal of molar volume, we can determine the liquid-phase molar density of the reaction mixture using the following expression for an ideal mixture:

1

F)

∑i

i ) 1, ..., 4

(28)

xi/F0i

where F0i is the liquid-phase molar density of component i at the reaction conditions.

The value of the dimensionless liquid molar holdup in the reaction vessel can be determined using the following equation:

dH ) -H - (Daκ1x1x2 + Daκ2x1x3)H, H(0) ) 1 (29) dξ In the previous example, we had D ) H. In this case, we have

B)H

(30)

where B ) B/B0 and H ) H/H0. (b) Balances around a Receiver (Region II in Figure 6). The values of the dimensionless holdup of each component in the receiver can be determined by applying a differential material balance over the receiver

dHB,i ) BxB,i, HB,i(0) ) 0, i ) 1, ..., 4 dξ

(31)

Constraint. In this targeting methodology, we specify the bottoms composition of isooctane and the Damko¨hler number for the primary reaction (17). The target bottoms rate is one that is proportional to the molar holdup in the reflux drum. We desire a bottoms stream with high purity of isooctane; therefore, it is necessary to check if the production rate of isooctane by reaction is sufficient to maintain the proportional bottoms rate policy. This imposes an overall material balance constraint on the desired product isooctane given as

BxB,3δξ e (Daκ1x1x2 - Daκ2x1x3)Hδξ + Bx3

(32)

Note that the left-hand side represents the amount of isooctane removed as bottoms over a time interval δξ, while the sum of the production of isooctane over the same time interval and the residual amount of isooctane in the reflux drum is given by the right-hand side. Yield and Selectivity. The conversion of the limiting reactant 2-butene is given as

X)

x01 - x1(ξ) H(ξ) - HB,1 x01

Y) S)

x3(ξ) H(ξ) + HB,3 x01 x3(ξ) H(ξ) + HB,3

x01

- x1(ξ) H(ξ) - HB,1

(33)

(34)

(35)

where Y and S are the yield and selectivity to the desired product isooctane, respectively. Solution Algorithm. Our aim is to determine the conversion of 2-butene, the yield and selectivity to isooctane at each instant of warped time by specifying the Damko¨hler number, and the bottoms composition of isooctane. The solution algorithm is as follows (see Appendix A for a more general solution algorithm): 1. Specify a value for the Damko¨hler number of the first reaction. 2. Determine the bottoms composition using the following equation (see Appendix B):

xB,4 ) Daκ2x1x3

H B

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3. Integrate eqs 25-31 numerically for a small time interval δξ. 4. Check the material balance constraint, eq 32. If the constraint is not satisfied, do the following: (a) Decrease the dimensionless bottoms rate B so that the left-hand side in eq 32 equals the right-hand side. This decreased bottoms rate is given as

B)

(Daκ1x1x2 - Daκ2x1x3)Hδξ xB,3δξ - x3

(b) Recalculate the values of all mole fractions and dimensionless holdups using this decreased bottoms rate. Equations for butane alkylation equivalent to eqs 1-3 should be used for this recalculation. (c) Use the new values for further calculations. 5. Calculate the conversion, yield, and selectivity using eqs 33-35. 6. Repeat from step 3 until the mole fraction of limiting reactant 2-butene in the reaction vessel x1 ≈ 0. In the previous example, the desired product was the lowest boiling component. In this case, however, the desired component is an intermediate boiler. Therefore, it is not possible to remove the desired component isooctane with 100% purity in a single still. It is desirable to remove dodecane completely from the reflux drum because its accumulation will increase the boiling point of the reaction mixture. There will be increased byproduct formation at a higher temperature because the side reaction is more sensitive to temperature. We now study the effect of the Damko¨hler number on the product distribution, represented by the yield and selectivity to the desired product. The molar feed ratio is an important variable for this system. The molar feed ratio, β, is defined as

β)

moles of isobutane in the feed moles of 2-butene in the feed

Industrially, a value of β ) 5-20, and preferably β ) 10, is used.29 The reaction is usually carried out in CSTRs arranged in series. This arrangement enables the profiles to mimic those of a PFR. The usual temperature of operation is between 0 and 40 °C. The preferred pressure of operation is between 4.5 and 11 atm so that the reactants are in a liquid phase.30 We evaluate the yield and selectivity profiles for a BRD device for β ) 10 at a pressure of 7.8 atm. For comparison, we evaluate the profiles of an isothermal PFR operating at 10 °C. The following equations can be used in conjunction with eqs 19-22 to determine the yield and selectivity for a PFR at different conversions:

conversion )

c01 - c1 c01

yield ) c3/c01 selectivity )

c3 c01

- c1

(36) (37) (38)

Figure 7 shows the yield and selectivity profiles for Da ) 2. A BRD has the potential for almost 100% selectivity at all conversions at the given operating conditions. For an isothermal PFR at 10 °C, the selec-

Figure 7. Yield and selectivity plots for the butane alkylation system at 7.8 atm pressure, β ) 10, and Da ) 2.

tivity decreases sharply with conversion. Therefore, a significant advantage can be achieved by using a BRD device, especially at higher conversions. The boiling point of the reaction mixture in BRD varies from 61 to 59 °C as the reaction proceeds. Because isobutane is in large excess, there is no significant change in the boiling point of the reaction mixture. The yield and selectivity profiles at Da ) 14 are shown in Figure 8. The molar feed ratio and system pressure are the same as above. In this case some byproduct is formed in BRD (i.e., S < 100%), but nevertheless BRD achieves good selectivities (S > 95%) even at high conversions. When the profiles for an isothermal PFR are compared, significant selectivity improvement can be achieved by using BRD at all conversions. Figure 9 shows the yield and selectivity profiles for Da ) 20. The selectivities drop with increasing conver-




sion for both BRD and isothermal PFR, and both are poor at higher conversions. Accumulation of isooctane in the reaction vessel results in increased byproduct formation in BRD. No potential selectivity improvement is possible by using a BRD over an isothermal PFR at these conditions.

estimated. For the butane alkylation system, because the rate constants are known, the model is formulated in terms of a single Damko¨hler number. The yield and selectivity targets are determined by specifying a value for the Damko¨hler number and the bottoms composition of isooctane. At low values of Da, significant selectivity improvement can be achieved by using a BRD over conventional processes. Therefore, the Damko¨hler number is a key design parameter. The practical meaning is that the distillate rate (bottom product rate) relative to the holdup in the reactor is a critical design parameter. Therefore, the heating policy corresponding to this distillate rate (bottoms rate) is important. The interpretation of batch experiments to screen reactive distillation depends on the distillate rate (bottoms rate) relative to holdup that was used in the experimental protocol. We have also developed a parametric screening procedure for the quick selection of operating parameters for BRD using selectivity as the performance

Conclusions We have developed a simplified model to quickly determine yield and selectivity profiles for a BRD system with minimum computational effort. For a hypothetical example of reactions in series, A f B f C, where the rate constants are arbitrary, the model is formulated in terms of the Damko¨hler number for each reaction. Yield and selectivity targets are determined by specifying the values for the Damko¨hler numbers and the desired distillate composition of the product, B. Also, the effect of operating parameters on these targets is

1574


indicator. This procedure also identifies promising candidate BRDs for which detailed design, simulation, and experiments could be justified. We have developed a general methodology for determining yield and selectivity targets for BRD applicable to any reaction system with one or more reactions. Although the examples considered in this work involve ideal mixtures with no distillation boundaries, this analysis can be implemented for nonideal mixtures with azeotropes and distillation boundaries.

Greek Symbols

Acknowledgment

0 ) initial 1 ) component 2-butene 2 ) component isobutane 3 ) component isooctane 4 ) component dodecane B ) bottoms B,i ) component i in bottoms D ) distillate D,i ) component i in distillate i ) component i j ) reaction j

We are grateful to the sponsors of the Process Design and Control Center, University of Massachusetts, Amherst. Financial support was provided by the National Science Foundation (Grant No. CTS-9613489). Nomenclature A, B, C ) generic chemical species B ) bottoms flow rate (mol/time) B ) dimensionless bottoms flow rate B0 ) initial bottoms flow rate (mol/time) c ) number of components ci ) concentration of component i (mol/volume) c0i ) initial concentration of component i (mol/volume) D ) distillate flow rate (mol/time) D ) dimensionless distillate flow rate D0 ) initial distillate flow rate (mol/time) Daj ) Damko¨hler number for reaction j H ) liquid holdup in the reaction vessel (mol) H ) dimensionless liquid holdup in the reaction vessel H0 ) initial liquid holdup in the reaction vessel (mol) HB,i ) amount of component i accumulated in the receiver as bottoms (mol) HB,i ) dimensionless amount of component i accumulated in the receiver as bottoms HD,i ) amount of component i accumulated in the receiver as distillate (mol) HD,i ) dimensionless amount of component i accumulated in the receiver as distillate kj ) forward reaction rate constant for reaction j (1/time) kj,ref ) forward reaction rate constant for reaction j at reference temperature (1/time) k′j ) concentration-based forward reaction rate constant for reaction j (volume/mol/time) N ) total number of independent chemical reactions rj(c) ) concentration-based rate of reaction j (mol/volume/ time) rj(x) ) mole fraction based rate of reaction j (1/time) R ) normalized reaction rate (dimensionless) S ) selectivity to the desired component (mol/mol) t ) clock time Tref ) reference temperature X ) conversion of the limiting reactant x ) vector of mole fractions in the reaction vessel xB ) vector of mole fractions in the bottoms xD ) vector of mole fractions in the distillate xi ) liquid-phase mole fraction of component i in the reaction vessel x0i ) initial liquid-phase mole fraction of component i in the reaction vessel xB,i ) liquid-phase mole fraction of component i in the bottoms xD,i ) liquid-phase mole fraction of component i in the distillate xji ) time average liquid-phase mole fraction of component i Y ) yield to the desired component (mol/mol)

β ) molar feed ratio for the alkylation system (mol/mol) κj ) dimensionless ratio, kj/kref F ) molar density of the reaction mixture (mol/volume) F0i ) molar density of pure component i at standard state (mol/volume) vij ) stoichiometric coefficient of component i in reaction j vTj ) sum of stoichiometric coefficients of reaction j ξ ) warped time (dimensionless) Subscripts and Superscripts

Appendix A: General Procedure for Setting Selectivity Targets In this appendix we present a general target-setting procedure for selectivity, which is applicable to systems with multiple reactions. We assume that the desired product is removed as the distillate, extending this formulation to the case where the desired product removed in the bottoms stream is straightforward. For a system with N independent reactions and c components, we can write the following overall and component balances for a BRD device:

Overall balance dH dξ

N

rj(x) , H(0) ) 1 kj

DavTjκj ∑ j)1

) -H + H

(39)

where vTj is the sum of the stoichiometric coefficients), rj(x) is the reaction rate per mole of mixture for reaction j, and κj ) kj/kref.

Component balances dxi dξ

N

) xi - xD,i +

rj(x) , kj i ) 1, ..., c - 1 (40)

Da(vij - vTjxi)κj ∑ j)1

Also, we can write the overall balances on the receiver.

Receiver balances dHD,i ) DxD,i, HD,i(0) ) 0, i ) 1, ..., c dξ

(41)

The above equations are solved for the instantaneous values of the dimensionless holdups and mole fractions. We then determine the values of conversion of the limiting reactant, yield, and selectivity at each time instant. A solution algorithm for the model is as follows: 1. Specify a value of the Damko¨hler number for one of the reactions (preferably the main reaction). 2. Specify the initial conditions for eq 40; molar feed ratios usually relate these initial conditions.


3. Specify the distillate composition of the desired product. 4. Check the feasibility of separation.16-20 Manipulate the distillate composition and repeat this step if the separation is not feasible. 5. Calculate the boiling point of the reaction mixture and evaluate the rate constants at this temperature. 6. Integrate eqs 39-41. 7. Check the material balance constraint on the desired product. If the constraint is not satisfied (a) manipulate the distillate rate to satisfy the constraint, (b) recalculate the values of all dimensionless molar holdups and mole fractions, and (c) use the values for further calculations. 8. Calculate the conversion, yield, and selectivity. 9. Repeat from step 5 until the limiting reactant is completely reacted. Appendix B: Estimation of Bottoms Composition In the butane alkylation example, it is desirable to remove dodecane as fast as it is formed to avoid a large increase in the reaction temperature, because more byproduct is formed at higher temperature. Because the production rate of the byproduct changes with time, its mole fraction in the bottoms stream will change. Therefore, we must evaluate the bottoms stream composition at each instant of warped time. Setting up a material balance between the amount (in moles) of dodecane produced and the amount removed over the time step δξ, we get

BxB,4δξ ) Hk2x1x3δξ + Hx4

(42)

Because there is no dodecane at zero time and because it is removed as fast as it is produced (with a suitable reboil ratio and number of stages in the column), there is no dodecane at the beginning of each time step, and we substitute x4 ) 0 in the above equation:

BxB,4 ) Hk2x1x3 Rearranging and substituting the dimensionless bottoms rate and liquid holdup in the reaction vessel, we solve for the mole fraction of dodecane in the bottoms stream,

k2 H xB,4 ) Da2 xx k2,ref 1 3B

(43)

The mole fraction of isooctane in the bottoms stream can then be given as

xB,3 ) 1 - xB,4

(44)

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(6) Smith, L. A. Method for the preparation of methyl tertiary butyl ether. U.S. Patent 4,978,807, Dec 18, 1990. (7) Buzad, G.; Doherty, M. F. New Tools for the Design of Kinetically Controlled Reactive Distillation Columns for Ternary Mixtures. Comput. Chem. Eng. 1995, 19, 395-408. (8) Bernot, C.; Doherty, M. F.; Malone, M. F. Feasibility and Separation Sequencing in Multicomponent Batch Distillation. Chem. Eng. Sci. 1991, 46, 1311-1326. (9) Barolo, M.; Guarise, G. B. Batch Distillation of Multicomponent Systems with Constant Relative Volatilities. Trans. Inst. Chem. Eng. 1996, 74A, 863-871. (10) Cuille, P. E.; Reklaitis, G. V. Dynamic Simulation of Multicomponent Batch Rectification with Chemical Reactions. Comput. Chem. Eng. 1986, 10, 389-398. (11) Mujtaba, I. M.; Macchietto, S. Efficient Optimization of Batch Distillation with Chemical Reaction Using Polynomial Curve Fitting Techniques. Ind. Eng. Chem. Res. 1997, 36, 22872295. (12) Bollyn, M. P.; Wright, A. R. Development of a Process Model for a Batch Reactive DistillationsA Case Study. Comput. Chem. Eng. 1998, 22, Suppl., S87-S94. (13) Venimadhavan, G.; Malone, M. F.; Doherty, M. F. A Novel Distillate Policy for Batch Reactive Distillation with Application to the Production of Butyl Acetate. Ind. Eng. Chem. Res. 1999, 38, 714-722. (14) Bernot, C.; Doherty, M. F.; Malone, M. F. Patterns of Composition Change in Multicomponent Batch Distillation. Chem. Eng. Sci. 1990, 45, 1207-1221. (15) Venimadhavan, G.; Buzad, G.; Doherty, M. F.; Malone, M. F. Effect of Kinetics on Residue Curve Maps for Reactive Distillation. AIChE J. 1994, 40, 1814-1824. Correction: AIChE J. 1995, 41, 2613. (16) Wahnschafft, O. M.; Koehler, J. W.; Blass, E.; Westerberg, A. W. Product Composition Regions of Single-Feed Azeotropic Distillation Columns. Ind. Eng. Chem. Res. 1992, 31, 2345-2362. (17) Wahnschafft, O. M.; Westerberg, A. W. Product Composition Regions of Azeotropic Distillation Columns. 2. Separability in Two-Feed Columns and Entrainer Selection. Ind. Eng. Chem. Res. 1993, 32, 1108-1120. (18) Fidkowski, Z. T.; Malone, M. F.; Doherty, M. F. Feasibility of Separations for Distillation of Nonideal Ternary Mixtures. AIChE J. 1993, 39, 1303-1321. (19) Rooks, R. E.; Julka, V.; Doherty, M. F.; Malone, M. F. Structure of Distillation Regions for Multicomponent Azeotropic Mixtures. AIChE J. 1998, 44, 1382-1391. (20) Venimadhavan, G.; Malone, M. F.; Doherty, M. F. A Bifurcation Study of Kinetic Effects in Reactive Distillation. AIChE J. 1999, 45, 546-556. (21) Damko¨hler, G. Stro¨mungs und Wa¨rmeubergangsprobleme in Chemischer Technik and Forschung. Chem. Ing. Tech. 1939, 12, 469-480. (22) Levenspiel, O. Chemical Reaction Engineering, 2nd ed.; Wiley: New York, 1972; Chapter 7. (23) Fogler, S. H. Elements of Chemical reaction Engineering, 2nd ed.; McGraw-Hill: New York, 1992; Chapter 9. (24) Smith, J. M. Chemical Engineering Kinetics, 3rd ed.; McGraw-Hill: New York, 1981; pp 84 and 85. (25) Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988; p 161. (26) Bernot, C.; Doherty, M. F.; Malone, M. F. Design and Operating Targets for Nonideal Multicomponent Batch Distillation. Ind. Eng. Chem. Res. 1993, 32, 293-301. (27) Nisoli, A. Private communication, 1998. (28) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. (29) Kramer, G. M. Admantyl Carboxylic and Sulfonic Acid Catalysed Paraffin-Olefin Alkylation. U.S. Patent 4,426,545, Jan 17, 1984. (30) Estes, J. H.; Cole, E. L. Isoparaffin Alkylation. U.S. Patent 3,867,475, Feb 18, 1975.

Received for review July 13, 1999 Revised manuscript received October 15, 1999 Accepted October 18, 1999 IE990497P

Selectivity Targets for Batch Reactive Distillation - Industrial

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