Simplified Design of Batch Reactive Distillation ... - ACS Publications

4000

Ind. Eng. Chem. Res. 2004, 43, 4000-4011

Simplified Design of Batch Reactive Distillation Columns Maria E. Huerta-Garrido and Vicente Rico-Ramirez* Departamento de Ingenieria Quimica, Instituto Tecnologico de Celaya, Av. Tecnologico y Garcia Cubas S/N, Celaya, Gto., CP 38010, Mexico

Salvador Hernandez-Castro Facultad de Ciencias Quimicas, Universidad de Guanajuato, Col. Noria Alta S/N, Guanajuato, Gto., CP 36050, Mexico

This work proposes a simplified methodology for the analysis and design of reactive batch distillation columns based on the McCabe-Thiele method for reactive continuous columns and on the concept of a reactive difference point. To extend the application of the concept of a reactive difference point for reactive batch distillation columns, expressions for the McCabe-Thiele operating line and for the dynamics of the reboiler in the reactive case have been derived; two cases are considered depending upon the phase in which reactions occur (i.e., liquid or vapor). We also provide an approach to the derivation of an expression to calculate the molar turnover flow rate by considering each reactive distillation plate as a combined system of a conventional equilibrium stage linked to a chemical reactor (phenomena decomposition). Furthermore, we present our derivation of the Underwood equations (minimum reflux) for continuous and batch reactive distillation columns. Finally, five illustrative examples allow us to show that our results present reasonable agreement with those obtained through BatchFrac of AspenPlus. 1. Introduction Reactive distillation has recently emerged as a promising technology because of its integrated functionality of separation and reaction. The general advantages of reactive distillation are1 (1) it can achieve higher conversion rates for an equilibrium-limited reaction because of the continuous remotion of products being formed and (2) the heat of reaction can reduce the heat load of condensers or reboilers. Its most serious limitation, however, is that the temperature of the reaction must coincide with the distillation range.2 Despite this weakness, potential savings in terms of investment and operation costs of this synergistic unit operation have motivated considerable research effort aimed to generate efficient design strategies. Barbosa and Doherty3,4 developed reactive residue curves in transformed coordinates and then proposed an algorithm to calculate the minimum reflux on reactive distillation columns by using transformed variables. Espinosa et al.5 developed Ponchon-Savarit diagrams in a transformed enthalpy-composition space, and Perez-Cisneros6 proposed McCabe-Thiele diagrams for reactive distillation columns based on the element balance approach. Following these initial attempts, Westerberg’s group at Carnegie Mellon University developed the concept of the reactive difference point7 and used this concept to provide simplified design strategies and visual and graphical insights for reactive distillation columns.8-12 This reactive difference point is similar to the difference point for extractive distillation columns and can be used as a basis for performing numerical calculations and geometric visualizations of reactive columns. * To whom correspondence should be addressed. Tel.: +52(461)-6117575 ext. 156. Fax: +52-(461)-6117744. E-mail: [email protected].

Despite the extensive body of literature reported on the topic of reactive distillation, much work remains to be done. As yet, there are few works concerning the design of reactive batch distillation columns.13 Given the current applications of batch distillation in small-scale industries producing high-value-added specialty chemicals (pharmaceutical industry, fertilizers, etc.), attention should also be focused on that direction. Furthermore, very few reports provide results with respect to the theoretical limiting conditions of reactive distillation columns (such as minimum reflux, minimum number of plates, etc.). This paper intends to contribute to both of the topics mentioned above. First, we provide a simplified methodology for the analysis and design of reactive batch distillation columns based on the McCabe-Thiele method and on the concept of the reactive difference point. To extend the applications of the reactive difference point for reactive batch distillation columns, we propose a simplified approach to the calculation of the molar turnover flow rate. By using the idea of phenomena decomposition, we consider each reactive distillation stage as a combined system of a conventional distillation plate linked to a chemical reactor. Also, expressions for the dynamics of the reboiler and the McCabe-Thiele operating line (quasi-steady-state assumed) have been rederived for reactive batch distillation columns. With respect to the limiting operation conditions of reactive distillation columns, we have derived the Underwood equations (minimum reflux) for the reactive case (for both continuous and batch distillation). The paper has been divided into six sections. Section 2 presents some previous work relevant to our approach: (1) it briefly explains the graphical design of a continuous reactive distillation column using the McCabe-Thiele method9 and (2) it describes the McCabeThiele method for batch distillation columns for the nonreactive case.13 The contributions of the paper are

10.1021/ie030658w CCC: $27.50 © 2004 American Chemical Society Published on Web 06/08/2004

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 4001

to the left of the distillate composition for a given reaction molar flow rate. Then, as we move down the column, the accumulated sum of the reaction molar turnover flow rates increases and the intersection point also moves further to the left. These intersection points are the reactive difference points and can be calculated from eq 2.

xn ) xD -

Figure 1. McCabe-Thiele diagram for reactive distillation in the rectifying section.

presented in sections 3-5. Section 3 explains the proposed approach to the analysis and design of reactive batch distillation columns using the McCabe-Thiele method. It also shows how we have derived expressions for the dynamics of the reboiler, McCabe-Thiele operating lines, and the extent of the reaction so that the concept of the reactive difference point can still be applied. Section 4 presents our derivations for the Underwood equations for reactive distillation columns. Section 5 shows the results of the approach for several illustrative examples as well as a comparison to the results obtained through BatchFrac of AspenPlus. Finally, section 6 concludes the paper. 2. Previous Work This section summarizes the concepts that serve as a basis for the development of our approach to the analysis and design of reactive batch distillation columns. 2.1. McCabe-Thiele Method for Reactive Continuous Distillation Columns.2,9 Hauan et al.7 developed the idea of reactive fixed points arising from collinearity of reaction, separation, and mixing as individual phenomena; the authors also derived the concept of the reaction difference point characterizing the flow generated by reaction. Hence, by combination of the concept of a reactive difference point with the concept of a pseudofeed,12 reactive operating lines can be defined and used to construct McCabe-Thiele diagrams for reactive distillation columns. For example, for the case of an isomerization reaction taking place in the rectifying section of a column (so that 1 mol of heavy reactant R1 converts into 1 mol of light product P1), the expression for the operating line is given by

ξn L L yn+1 ) xn + 1 - xD V V V

(

)

(1)

where V and L are the vapor and liquid interstage flow rates, respectively, and ξn is the accumulated sum of the reaction molar turnover flow rates until the nth stage from the top. Equation 1 represents several operating lines parallel to each other because, for the same slope, the y intercept is changing in terms of the accumulated sum of the reaction molar turnover flow rates. Consider the McCabe-Thiele diagram shown in Figure 1. Starting from the top at the distillate composition, xD, the first operating line intersects the 45° line

ξn D

(2)

Lee et al.2 show the derivation of the operating line for the stripping section and the derivation for the q line and present an analysis of the performance of reactive columns depending upon where the reaction zone is located. In a related paper,9 the authors analyze the cases where changes in molar flow rates occur because of reaction; an analysis based on either mass (rather than molar) balance equations or changing-slope operating lines is suggested in that situation to construct McCabe-Thiele diagrams. A generalization of the use of the difference points in reactive and extractive cascades has also been reported.11,12 An important aspect of this approach to the design of reactive distillation is the knowledge of (or the ability to calculate) the molar turnover flow rate, ξ. As we explain later, an approach to the calculation of ξ based on phenomena decomposition is proposed in this paper. 2.2. McCabe-Thiele Method for Batch Distillation Columns.13 The unsteady-state nature of batch distillation implies solutionof a set of differential equations. Diwekar13 describes the two main operation policies for batch distillation: constant reflux and constant distillate composition. For any of them, the McCabe-Thiele method can be applied by considering only the reboiler and condenser dynamics and assuming a quasi steady state on the plates. The solution procedure involves an iterative scheme for the simultaneous determination of x(k) D (distillate composition of component k) and x(k) (still composition of component k) for B each time interval until the stopping criterion is met. Typical stopping criteria include reaching a specified average distillate composition, obtaining a given amount of distillate, or operating the column for a specified batch time. For a binary system, the McCabe-Thiele operating line for batch distillation columns presents the same basic form as that of continuous distillation, but it has to be applied for each time step:

yn+1 )

R 1 xn + x R+1 R+1 D

(3)

where the reflux ratio, R, is defined as R ) L0/D. Figure 2 shows the McCabe-Thiele diagram for two time steps in a column (three equilibrium stages) undergoing constant reflux operation. Initially, the still composition xB0 is known, and the composition of the distillate xD0 can be found iteratively from the specified number of stages and from the given slope of the operating line (constant reflux ratio). After the distillate compositions are calculated, the differential equations representing the dynamics of the reboiler can be integrated to obtain the amount of mixture remaining in the still (B1), the distillate product (D1), and xB1. Then, the McCabeThiele diagram is used again to calculate xD1 iteratively. This procedure continues for the following time steps in a similar way until the stopping criterion is met.

4002 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004

Figure 2. McCabe-Thiele diagram for batch distillation under constant reflux policy.

3. Simplified Method for Analyzing and Designing Reactive Batch Distillation Columns This section provides the derivation of the expressions needed to simulate the performance and/or to design reactive batch distillation columns using a simplified method. The solution strategy can be directly used for binary systems, but an approximation is also proposed for the case of multicomponent systems (see section 5.3). Our approach is based on the McCabe-Thiele method for reactive columns and on the concepts of the reactive difference point and molar turnover flow rate. We consider separately the cases in which reaction occurs either in the liquid phase or in the vapor phase, although the analysis could be integrated. As a simplified approach, the following are the assumptions made through our derivations for kinetically controlled reactive batch columns: 1. Heat of mixing, stage heat losses, and sensible heat changes of both liquid and vapor are negligible. 2. The pressure on the column remains constant through the operation. 3. Startup conditions will not be considered. 4. The stages of the column will be considered in quasi steady state. 5. Reaction takes place on all of the stages but the reboiler and condenser. 6. Reaction is irreversible because of the continuous removal of the product. If assumption 1 holds, then the molar flow rate of the phase in which reaction does not occur can be considered as a constant. Seeking simplicity, in the following derivations we will obviate most of the algebraic steps and provide only the main results. The analysis considers two cases depending upon the phase in which the reactions occur. The derivations for the liquid-phase reactions are presented first. 3.1. Reactive Batch Distillation Column: Reaction in the Liquid Phase. Consider the batch reactive distillation column represented in Figure 3. ξk,j represents the sum of the molar turnover flow rate for component k (reactant defined as the base component) in the stage j, the distillate product is represented as D, and the reflux ratio as R ) L0/D. The subtractions indicated in Figure 3 are used to emphasize the contributions of reaction on each stage because ξk,j is an accumulated sum. For a binary system involving components k (heavy reactant) and i (light product), the

Figure 3. Reactive batch distillation column.

overall and component (i) balance equations around section 1 of Figure 3 are

yi,j+1 )

Lj νi ξk,j D xi,j + xi,D V V -νk V

V ) Lj + D - νTξk,j

(4) (5)

where ν represents the stoichiometric coefficients of reaction (negative for reactant k and positive for product i), νT is the total sum of stoichiometric coefficients, and (νi/-νk)ξk,j represents the production of i due to reaction of k on stage j. The last of the terms in eq 5 represents the overall change in the number of moles due to reaction. Notice that, although we are using a binary system as the basis for our derivations here, eqs 4 and 5 can actually be generally applied, and that is the basis for the extension of the approach to multicomponent systems. Because for this case the reaction is taking place in the liquid phase, the vapor flow rate can be assumed as constant, but the liquid flow rate might change depending on the stoichiometric coefficients. Changes in the liquid flow rate could be calculated as

Lj ) Lj-1 + νT(ξk,j - ξk,j-1)

(6)

Observe that if νi ) -νk, then νT ) 0 and Lj ) Lj-1, so that the assumption of constant molar flow will also hold for the liquid phase. Equations 4-6 can be used to obtain the McCabe-Thiele operating line for reactive batch distillation:

yi,j+1 )

Lj νi ξk,j 1 xi,j + xi,D V R+1 -νk V

(7)

and for the case in which there is no change in the liquid flow rate:

yi,j+1 )

νi ξk,j R 1 xi,j + xi,D R+1 R+1 -νk V

(8)

Also note that, if there is no change in the liquid flow


Figure 5. Stage on a reactive batch distillation column: reaction in the liquid phase.

Figure 4. Mass balance for the dynamics of the reboiler.

rate, νi ) -νk, eq 8 is basically the same as the operating line obtained for continuous distillation (eq 1). Furthermore, by comparing eqs 3 and 8 (nonreactive and reactive batch distillation), we see that the contribution of reaction is given through the term -(νi/-νk)(ξk,j/V), or -(ξk,j/V) if there is no change in the number of moles. 3.1.1. Dynamics of the Reboiler. To derive the dynamics of the reboiler, consider the overall and component balance for section 2 of Figure 4:

νTξk,T - D ) dB/dt

(9)

νi d ξ - Dxi,D ) (Bxi,B) -νk k,T dt

(10)

where ξk,T is the total molar turnover flow rate (k being the base component). The overall mass balance for the condenser is

V - L0 - D

(11)

By combining eqs 9-11, one can show that the differential equations representing the behavior in the reboiler and condenser are

dB V ) νTξk,T dt 1+R D)

{

(12)

V 1+R

[

(13)

] }

νi dxi,B 1 V ) (xi,B - xi,D) + - νTxi,B ξk,T dt B 1+R -νk

(14)

Observe again that, when compared to the nonreactive case for batch distillation, eqs 12 and 14 include an extra term to account for changes due to reaction. Further, notice that in eq 14 the separate effects of reaction and separation are well defined. 3.1.2. Extent of the Reaction: Molar Turnover Flow Rate. Hauan et al.7,11 provided a generalization of the concept of a difference point for reactive cascades. The authors assume that a reactive separation cascade

can be considered as a separation cascade linked to a chemical reactor having an equivalent conversion; such an approach is known as phenomena decomposition and has been explained in detail.7,11 Basically, Hauan et al.7 show that the phenomena occurring in some combined systems of separation and reaction (mixing, separation, and reaction) can be separated and analyzed independently from each other. Moreover, they show that, for the case of reactive cascades, given the extent of the reaction, the decomposition does not affect the mass balance equations.11 In a similar way, to derive an expression for the calculation of the molar turnover flow rate ξk,j, in this paper we assume that each separation stage of a reactive batch distillation column can be considered as a nonreactive equilibrium separation stage linked to a continuous stirred tank reactor (CSTR; see Figure 5). If we were able to know a priori the exact value of the molar turnover flow rate for each stage, then the mass balance equations obtained through the phenomena decomposition assumption would hold. However, the sequential approach (reaction-separation) applied here is, in fact, used for the calculation of such a value of the extent of the reaction. Hence, it is evident that in this case a sequential approach will not exactly represent a simultaneous phenomenon in nature and that the assumptions made, such as representing the reaction volume as a CSTR, are approximations that may lead to error. A CSTR is used because we are considering perfect mixing in the liquid and vapor phases. Also, the CSTR is placed above the separation stage just because of numerical implications in the solution scheme. The component mass balance equation for the reactor of Figure 5 is given by

υ(rk,j) ) Lj-1xk,j-1 - L′jx′k,j

(15)

where υ is the reaction volume (liquid holdup), rk,j is the rate of reaction of component k in stage j, L is the liquid flow rate, and x is the liquid composition. In principle, the calculation of υ would require the use of column hydraulic analysis, which could provide a relationship between Lj-1 and υ. The reactor outlet liquid stream, L′j, can be expressed in terms of the molar conversion fraction of the reactant, Cj:

L′jx′k,j ) Lj-1xk,j-1(1 - Cj)

(16)

so that eq 15 becomes

υ(rk,j) ) Lj-1xk,j-1Cj

(17)


yi,j+1 )

V1 V1 νi ξk,j R 1 xi,j + xi,D Vj+1 R + 1 Vj+1 R + 1 -νk Vj+1 (22)

where changes in the vapor flow rate due to reaction are given by

Vj+1 ) Vj + νT(ξk,j - ξk,j-1)

(23)

If there are no changes in the vapor flow rates, then eq 22 reduces to eq 8. 3.2.1. Dynamics of the Reboiler. The overall and component mass balances for section 2 of Figure 6 allow the derivation of the differential equations that represent the dynamics of the reboiler as described earlier:

Figure 6. Derivation of expressions for the case of reaction in the vapor phase.

Assuming an elemental irreversible reaction, νrR f νpP, the rate of reaction can be represented by an expression such as

( )

rk,j ) kj

L′jx′k,j υ0

νk

(18)

where the term inside the parentheses represents the molar concentration of the reactant in the reactor and kj is the kinetics parameter given by the Arrhenius equation. υ0 is the volumetric flow rate entering the stage. υ0 can be calculated from

(19)

where F is the density of the liquid stream. Knowing υ0 would also allow the evaluation of the residence time, τj ) υ/υ0; hence, if the liquid holdup is known or can be evaluated, the use of υ0 or τj is equivalent. When eqs 16-18 are combined, an implicit equation for the molar conversion of the reactant can be obtained:

[

]

Lj-1xk,j-1(1 - Cj) υ0

νk

(20)

Once eq 20 is solved to obtain Cj (analytically or numerically for νk greater than 2), one can finally calculate the molar turnover flow rate through the expression n

ξk,j )

(Lj-1xk,j-1Cj) ∑ j)1

[

(21)

3.2. Reactive Batch Distillation Columns: Reaction in the Vapor Phase. If the reaction takes place in the vapor phase, derivations similar to the ones presented in the previous subsection can be obtained. Hence, the component and overall mass balances for section 1 of the column presented in Figure 6 result in the operating line for reactive batch distillation columns (quasi-steady-state approximation):

] }

dxi,B 1 V1 νi - νTξi,B ξk,T ) (xi,B - xi,D) + dt B 1+R -νk

3.2.2. Extent of the Reaction: Molar Turnover Flow Rate. Our approach to calculate the molar turnover flow rate is based again on phenomena decomposition as represented in Figure 7. Calculations in the reactor are performed before the calculations on the equilibrium stage. The consideration of having a separation stage connected to a CSTR in the case of reaction in the vapor phase could seem inappropriate; however, more than the need of a CSTR, our approach works by assuming that the compositions of reactive and nonreactive streams are homogeneous as if the mixing on them was perfect. A derivation similar to that of section 3.1.2 results in the following expressions: n

υ0 ) Lj-1/F

Lj-1xk,j-1Cj ) υkj

{

V1 dB ) νTξk,T dt 1+R

ξk,j )

(

Cj

V′jy′k,j ∑ 1-C j)1

j

)

(24)

where the conversion fraction of the reactant can be obtained from the implicit eq 25:

[ ]

V′jy′k,jCj ) υkj

V′jy′k,j υ0

νk

(1 - Cj)

(25)

3.3. Solution Approach. The resulting model is a well-posed differential-algebraic system of equations (DAE). Table 1 presents a summary of the model specifications and the equations used for the calculations of the variables involved in each stage. The solution procedure for each time step of a reactive batch distillation column is similar to the procedure for solving the rectifying section of a continuous reactive column. The equations involved in the calculation could be solved either simultaneously with an appropriate numerical method or sequentially by using a method such as the direct substitution method. We describe next the sequential procedure for the case in which reaction occurs in the liquid phase (constant molar flow in the vapor phase) in all of the stages but the reboiler; an iterative procedure for the calculation of the distillate compositions has to be completed. Considering a constant reflux policy, we assume that the number of stages (N), the reflux ratio (R), and the initial conditions (feed load and still compositions) are


Figure 7. Extent of the reaction: reaction in the vapor phase. Table 1. Degrees of Freedom Analysis: Specifications and Computations on Stage j variable

specified/calculated from

N, R, V, B0, xB υ0, υ L0 Lj Cj ξk,j xk,j yk,j+1 x′k,jL’j

specified specified eq 5 eq 6 eq 20 eq 21 equilibrium (relative volatility) eq 8 eq 16 (plus summation of compositions) iteratively from independent equations used to calculate xB

xD

given. Also, a typical specification in this case is the vapor boilup flow rate (V), which will remain constant through the column. As one of the main limitations of our approach, assumed values of the liquid holdup, υ, and entering volumetric flow rate, υ0 (or residence time, τ), are used in this work (see section 5 for an explanation). Hence, we start by assuming the values of the distillate compositions. Then, by using a bubble-point calculation, the temperature of the distillate can be computed. When the temperatures of the reboiler (which can also be computed through a bubble-point calculation because the compositions in the reboiler are known) and the condenser are known, a linear temperature profile is assumed. The temperature of each stage is required because the computations of the kinetics parameter and the relative volatility of the mixture depend on it. Then, a mass balance in the condenser allows the calculation of L0; for the condenser ξk,0 ) 0 (no reaction in the condenser) and eq 5 reduces to V ) L0 + D, which is equivalent to L0 ) V[R/(R + 1)]. Also notice that the distillate compositions are equal to the compositions of the vapor flow entering the condenser (yk,1).With that, we proceed to perform the calculation on stage 1. Equations 20 and 21 are used for the calculation of C1 and the molar turnover flow rate, ξk,1, respectively. The liquid molar flow rate on stage 1, L1, is obtained from eq 6. Equation 16 and the summation equation for the compositions are used to find the values of the liquid flow (L′1) and the compositions (x′k,l) entering the equilibrium separation stage of Figure 5. Then, the equilibrium curve (equilibrium in terms of relative volatility) and the operating line (eq 8) allow the calculation of the rest of the compositions for stage 1 (xk,1 and yk,2). The procedure continues for the rest of the stages moving down the column until we evaluate the compositions in the reboiler. Those values provide the convergence criteria: If the computed values of the reboiler

Figure 8. Reactive distillation column with an infinite number of stages.

compositions are equal to the known values, then we proceed to the next time step; otherwise, we modify our guess for the distillate compositions and repeat the calculations in the column. Notice that, with respect to the number of degrees of freedom, the model is square because the distillate compositions can be indirectly found from the independent equations used to calculate the still compositions (stage-by-stage procedure). Before one moves on to the next time step, integration of eqs 1214 is needed to find the values of the amount of mixture in the still, the distillate collected, and the bottom compositions for the following time step. N, R, and V remain as constants so that the simulation continues for the next time steps and ends when a specified stopping criterion is met. 4. Limiting Conditions for Reactive Distillation Columns In this section we derive the expressions to calculate the minimum reflux (Underwood equations) for continuous and batch reactive distillation columns; furthermore, a discussion regarding the Fenske equation (minimum number of stages) for continuous reactive distillation columns is also provided. 4.1. Underwood Equations. To derive the expression for the minimum reflux in the case of continuous reactive distillation columns, a system such as the one shown in Figure 8 is assumed. Our approach is based on the derivation for the nonreactive case presented by King.14 The section with an infinite number of stages is placed in the rectifying section above the feed plate. A further assumption in our derivation is that there is no reaction in the section with an infinite number of plates (to avoid depletion of the reactant). The mass balance equation for component i around section 1 of Figure 8 is


Vyi,∞ ) L∞xi,∞ + Dxi,D -

νi ξ -νk k,T

(26) NC

where the index ∞ has been assigned to the streams coming from the section with an infinite number of plates. The overall mass balance for the condenser is

V ) L0 + D

xi,∞ ) yi,∞/Ki,∞

(

)

νi ξk,T -νk D 1 L∞ Ri,∞ Kr,∞ V

(30)

Here a new variable, φ ) (1/Kr,∞)(L∞/V), called the absorption factor, is introduced. Also, because the summation of the compositions in the vapor stream going into the section with an infinite number of stages has to be 1, eq 30 reduces to eq 31. where NC is the

D

)

∑ i)1

[

(

Ri,∞ xi,D -

)

νi ξk,T -νk D

Ri,∞ - φ

]

NC

Rmin + 1

∑ i)1

[

(

-νk D

Ri,∞ - φ

]

(32)

V h B

NC

)

[

R′i,∞xi,B

∑ i)1 R′

i,∞

]

- φ′

(33)

Combining eqs 31 and 33 and taking into account that φ ) φ′ and Ri,∞ ) R′i,∞ result in

(35)

-φ

which is the second of the Underwood equations. Equation 35 can be used for calculating φ and eq 32 for calculating Rmin for continuous reactive distillation columns. 4.1.1. Extension of the Underwood Equations to Reactive Batch Distillation Columns. A direct extension to the Underwood equations derived above can be done for the case of reactive batch distillation columns. An approach similar to the one proposed by Diwekar13 serves that purpose (Figure 9). In Figure 9, a reactive batch distillation column is represented as a continuous column where the feed enters the column at the bottom stage with a temperature equal to its bubble point and with a composition equal to xi,B. Such a representation allows rewriting of eqs 32 and 35 as

[ ] ( ∑

NC

Ri,∞xi,B

NC

which is the first of the Underwood equations for calculating the minimum reflux. Following a similar procedure for section 2 (nonreactive stripping section) of Figure 8, one can obtain

-

i,∞

(31)

)

[ ] Ri,∞zi,F

∑ i)1 R

[

i,∞

νi ξk,T

(34)

νi ξ + Bxi,B -νk k,T

one can get

∑ i)1 R

number of components. Finally, the substitution of eq 27 and the definition of the reflux ratio in eq 31 result in

Ri,∞ xi,D -

Fzi,F ) Dxi,D -

(29)

Ri,∞ xi,D -

]

)

and the component mass balance equation for the whole column

1-q)

When the definition of relative volatility (with respect to the key component r) is incorporated:

NC

Ri,∞ - φ

NC

νi ξk,T -νk D V yi,∞ ) D 1 L∞ 1Ki,∞ V xi,D -

V

[

ξk,T + Bxi,B -νk

V-V h ) F(1 - q)

(28)

When eq 28 is substituted into eq 26 and with rearrangement:

∑ i)1

νi

Finally, by introducing the definition of q

(27)

It is also considered that the liquid-vapor equilibrium on the last stage of section 1 is given by

V yi,∞ ) D

V-V h )

(

Ri,∞ Dxi,D -

Rmin + 1 )

i)1

-φ

)0

Ri,∞ xi,D -

)

νi ξk,T -νk D

Ri,∞ - φ

]

(36)

(37)

which correspond to the Underwood equations for reactive batch distillation columns. Observe that the minimum reflux computation not only depends on the distillate and bottom compositions, as it does for a nonreactive column, but also depends on the extent of the reaction. Therefore, a value of the extent of the reaction has to be given in order to make eq 37 useful in practical terms. It might be difficult to provide an approximation to the extent of the reaction a priori, and a parametric search might be necessary to obtain a feasible value. For a specific reaction-separation system, a reasonable value of the extent of the reaction can be estimated, for instance, by assuming that the molar turnover flow rate is equally distributed among the reactive stages of the column and by defining the amount of product obtained through the reaction (using a value of the molar conversion fraction based on the kinetics of the reaction involved and the amounts of the reactants initially fed to the column).


5. Numerical Examples and Analysis of the Results This section describes the numerical examples performed to illustrate the application of the proposed methodology for reactive batch distillation columns. Because of the dynamic nature of batch columns, the compositions of the distillate product change with respect to time for a constant reflux operation policy. Hence, the results of the simulations are reported in terms of the average distillate composition profiles of the key component versus time. The average distillate composition xji,D is calculated through integration as follows:

∫0tDxi,D dt xji,D(t) ) ∫0tD dt Figure 9. Reactive batch column as a continuous column.

4.2. Comments on the Fenske Equation for Reactive Distillation. The derivation of the Fenske equation for nonreactive continuous columns assumes operation at total reflux. Interesting conclusions can be drawn with respect to the corresponding derivation in the reactive case. Let us assume that an irreversible isomerization reaction (binary mixture) is taking place in the column and that the product of the reaction is the most volatile component. The case of total reflux is then the first one that we analyzed for the reactive case, as shown in Figure 10a. A problem arises, however, in such a scheme. Because the reaction takes place regardless of whether products are being obtained or not, as time and the reaction progress, the reactant will get depleted and the column behaves just like a chemical reactor. Hence, the minimum number of stages in a reactive distillation column does not necessarily correspond to total reflux operation. An outlet product stream, such as the distillate stream in Figure 10b, makes the problem worse because the light product will be continuously removed and the column will become empty if no inlet stream is incorporated; a feed to the column is therefore required. Figure 10c shows a column similar to the scheme considered for reactive batch distillation columns in the derivation of the Underwood equations because the feed enters the column just above the reboiler. We believe that such a configuration could be used for obtaining the Fenske equation in reactive distillation. However, our experience of using the methodologies for the derivation of the Fenske equation reported by King,14 Henley and Seader,15 and Hohmann16 is that the appearance of the molar turnover flow rate and the reflux ratio makes the equation more complicated as we move down the column. Also, most of the reasonable assumptions used to simplify it introduce significant errors in the numerical calculation. Moreover, the summations involved do not seem to correspond to a converging series of any kind. This topic should be the subject of further investigation. Sundmacher and Kienle17 provide a simple expression for calculating the minimum number of stages for the case of a reaction taking place in the reboiler of the column.

(38)

When possible, the results were compared to those obtained through the use of BatchFrac from AspenPlus. The Antoine parameters (for the vapor-liquid equilibrium) were taken from Reid et al.18 To keep the design method simple enough, we did not include column hydraulic equations in our approach; that means that values for the liquid holdup have to be assumed. Furthermore, the kinetic parameters (frequency factor and activation energy) were provided and adjusted to achieve an accumulated value of the extent of the reaction equal to the one obtained through AspenPlus simulations. Note that our goal in this work was not to perform realistic calculations but to show that our simulation results present reasonable agreement with those of the state of the art process simulators; as a matter of fact, the assumed values for the liquid holdup and the kinetic parameters were the same as the ones used in BatchFrac. Finally, we also assumed a constant value of the volumetric flow rate υ0; however, such a simplification can easily be removed by evaluating the liquid density on each stage as a function of temperature and composition and then using eq 19. 5.1. Reaction Causes No Change in the Number of Moles. For a scenario in which the assumption of constant molar flow holds for both the liquid and vapor phases, two cases were solved. In the first one, the reaction takes place in the liquid phase; in the second example, the reaction is assumed to take place only in the vapor phase. 5.1.1. Reaction in the Liquid Phase. The example corresponds to an isomerization reaction in the liquid phase: o-xylene (C8H10) f ethylbenzene (C8H10). The lightest component is ethylbenzene. The data used for this example are shown in Table 2. The McCabe-Thiele diagram for the initial time step of this example is shown in Figure 11. As expected, the contribution of the reaction is greater for stages closer to the reboiler (larger reactant concentrations). Figure 11 does not show the last of the operating lines because the difference point is negative. Figure 12 presents three instances of the calculation of the average distillate composition by changing the value of the reflux ratio. Because the reaction shows high conversion values, the effect of the reflux ratio is not significant. Observe also that, because of reaction, the average distillate composition increases with time; this result is contrary to the behavior observed in nonreactive batch distillation columns. This effect can be explained by looking at the


Figure 10. Fenske equation for reactive distillation.

Figure 12. Average distillate composition profiles of ethylbenzene in example 1.

Figure 11. McCabe-Thiele diagram for the initial step in example 1. Table 2. Data for the Reaction-Separation System o-Xylene-Ethylbenzene parameter

value

no. of stages feed feed composition of ethylbenzene vapor boilup flow rate column pressure batch time order of reaction activation energy frequency factor volumetric flow rate liquid holdup

4 + reboiler 100 kmol 0.1 140 kmol h-1 1 atm 1.4 h 1.0 30 000 kJ kmol-1 1.6 × 109 h-1 2 m3 h-1 4 × 10-4 m3

right-hand side of eq 14. Because xB < xD, the first term is always negative; the still composition (and, therefore, the distillate composition) will only increase if the second term (reaction) of eq 14 is positive and larger than the first one (separation). Finally, Figure 13 shows a comparison between the results of our approach and the results obtained through BatchFrac of AspenPlus for a reflux ratio equal to 2 and a production of 73.83 mol of ethylbenzene. An excellent agreement is observed.

Figure 13. Our results as compared to those of BatchFrac for ethylbenzene.

5.1.2. Reaction in the Vapor Phase. The isomerization reaction n-butane (C4H10) f isobutane (C4H10) takes place in the vapor phase; isobutane is the lightest component. Table 3 shows the parameters used in the simulation for this example. The McCabe-Thiele diagram for a simulation time of 0.6 h is shown in Figure 14. Once again, the last of the difference points is negative, and the last of the operating lines has not been drawn. The diagram corresponds to a reflux ratio equal to 2. Figure 15 shows the behavior of the average distillate composition of isobutane; the trend is similar


Figure 14. McCabe-Thiele diagram for example 2.

Figure 15. Average distillate composition profiles for isobutane in example 2. Table 3. Data for the Reaction-Separation System n-Butane-Isobutane parameter

value

no. of stages feed feed composition of n-butane vapor boilup flow rate column pressure batch time order of reaction activation energy frequency factor volumetric flow rate reaction volume

4 + reboiler 100 kmol 0.1 140 kmol h-1 1 atm 1.4 h 1.0 40 000 kJ kmol-1 2 × 103 h-1 2 m3 h-1 4 × 10-4 m3

to the one on the previous example because the average distillate composition increases with time because of reaction. Also observe in Figure 15 that the reflux ratio has very little effect on the average product composition. Comparison of this case to BatchFrac was not possible because BatchFrac does not support the case where reaction occurs in the vapor phase in a kinetically controlled column. 5.2. Reaction Causes a Change in the Number of Moles. This situation is illustrated with the system involving the reaction isobutyl-isobutyrate (C8H16O2) f 2-isobutyraldehyde (C4H8O) in the liquid phase. The lightest component is isobutyraldehyde. Kinetics and equilibrium data are shown in Table 4. Figure 16 shows the McCabe-Thiele diagram for the operation time of 0.6 h. The distillate composition is very close to 1. Changes in the number of moles present an impact on the difference points as well as on the slope of the

Figure 16. McCabe-Thiele diagram for example 3.

Figure 17. Average distillate composition for isobutyraldehyde in example 3. Table 4. Data for the Reaction-Separation System Isobutyl Isobutyrate-Isobutyraldehyde parameter

value

no. of stages feed feed composition of isobutyraldehyde vapor boilup flow rate column pressure batch time order of reaction activation energy frequency factor volumetric flow rate liquid holdup

3 + reboiler 100 kmol 0.1 140 kmol h-1 1 atm 1.4 h 1.0 30 000 kJ kmol-1 1 × 103 h-1 2 m3 h-1 4 × 10-4 m3

operating lines. Observe that the movement from one difference point to another is very notorious as we move to the left from stage 2 to stage 3. Figure 17 shows the average distillate composition profiles for three values of the reflux ratio; because the liquid flow rates are changing from one stage to another, the reflux ratio has a significant effect in the composition. Also, in this example separation predominates over reaction and the average distillate composition of isobutyraldehyde decreases with time. Finally, Figure 18 compares our results to those obtained through BatchFrac for a reflux ratio of 2. 5.3. McCabe-Thiele Method for Multicomponent Mixtures. The McCabe-Thiele method as described in section 3 of this paper can also be applied for multicomponent mixtures (NC components) because the equations developed for the reaction either on the liquid phase or the vapor phase remain valid. To solve the problem for multicomponent mixtures, a procedure


Figure 18. Comparison in example 3 for the composition of isobutyraldehyde. Table 5. Data for the Reaction-Separation of the Multicomponent Mixture of Example 4 parameter

value

no. of stages feed feed composition of benzene feed composition of o-xylene vapor boilup flow rate column pressure batch time order of reaction activation energy frequency factor volumetric flow rate liquid holdup

2 + reboiler 100 kmol 0.006 0.485 140 kmol h-1 1 atm 1h 1.0 30 000 kJ kmol-1 8 × 102 h-1 2 m3 h-1 4 × 10-4 m3

consisting of the simultaneous calculation of NC - 1 components is needed. A further assumption is that the rate of reaction can be expressed in terms of the concentration of just one of the reactants (if more than one). Hence, the molar turnover flow rate for the rest of the components in terms of the base component is given by a stoichiometric relationship

ξi,j )

νi ξ νk k,j

(39)

Also, equilibrium can be calculated in terms of the equilibrium constant instead of relative volatilities as follows.

xi,j ) Ki,jyi,j

(40)

An example for the reaction-separation of multicomponent mixtures is illustrated with the system involving the reaction (no change in the number of moles) in the liquid phase: 2-toluene (C7H8) f benzene (C6H6) + o-xylene (C8H10), where the order in relative volatility is benzene > toluene > o-xylene. The data used for this case are shown in Table 5. The McCabe-Thiele solution approach was used for benzene and toluene. Figure 19 shows the average distillate composition of the three components when the reflux ratio is equal to 2. This is an interesting case because reaction presents a higher contribution than separation for benzene and o-xylene, so that their distillate compositions increase with time. However, for the case of toluene, the trend is different and the composition decreases with time. 5.4. Minimum Reflux Calculation. The calculation of the minimum reflux ratio by using the Underwood equations derived in this work (eqs 32 and 35) for

Figure 19. Average distillate composition for a reflux ratio equal to 2 in example 4.

continuous reactive distillation columns was tested for the case of the isomerization reaction isobutane f n-butane. The system considered is a column with nine stages where the feed enters the column as a saturated liquid in stage number 8. The column pressure is 1 atm. The feed is 1000 kmol h-1 with a composition of 0.04 of n-butane; the distillate composition of n-butane is 0.98. It is assumed that 380 kmol h-1 of n-butane are produced in the column and that such a molar turnover flow rate is equally distributed in the stages of the column. Solving eqs 32 and 35 results in Rmin equal to 0.518. After the solution approach using the McCabeThiele method is used for a range of values of the reflux ratio, a value of 0.51 was found for the minimum value of the reflux ratio that allows the reaction and separation specified. Such a result indicates that the Underwood equations derived in this work present a reasonable agreement with the simulations. 6. Concluding Remarks This paper provides the mathematical derivations needed for the simplified analysis and design of reactive batch distillation columns. In our opinion, the main contributions of the paper are as follows: (1) The development of the theoretical and mathematical tools that allow the use of a McCabe-Thiele strategy for reactive batch distillation columns. This includes the derivation for the dynamics of the reboiler and the operating line under the assumption of quasi steady state in the plates. Furthermore, an approach to the computation of the molar turnover flow rate of reactive distillation stages has been proposed. Such an approach is based on phenomena decomposition and is needed in order to apply the concept of the reactive difference point to the case of batch distillation. (2) The derivation of the Underwood equations for continuous and batch reactive distillation columns. The many assumptions considered in this work have been extensively explained, and we expect that our numerical examples involving various reactive systems show the scope and confirm the usefulness of our approach. Through our analysis and numerical examples we have (1) shown that our results present a reasonable agreement with those of the state of the art process simulators for reactive batch distillation columns, (2) emphasized the advantages of the combination of separation and reaction for batch distillation, where the product purity can even be improved as time progresses, and finally (3) proved that the results obtained for Rmin by the derived Underwood equations


for reactive columns are approximated enough to the results obtained through parametric search. Acknowledgment M.A.H.-G. and V.R.-R. are thankful for the financial support provided by the Consejo del Sistema Nacional de Educacion Tecnologica (CoSNET) and by the Consejo Nacional de Ciencia y Tecnologia (CONACYT). V.R.-R. and S.H.-C. also thank the Consejo de Ciencia y Tecnologia del Estado de Guanajuato (CONCyTEG). Literature Cited (1) Lee, J. W.; Westerberg, A W. Visualization of Stage Calculations in Ternary Reacting Mixtures. Comput. Chem. Eng. 2000, 24, 639. (2) Lee, J. W.; Hauan, S.; Westerberg A. W. Graphical Methods for Reaction Distribution in a Reactive Distillation Column. AIChE J. 2000, 46, 1218. (3) Barbosa, D.; Doherty, M. F. A New Set of Composition Variables for the Representation of Reactive Phase Diagrams. Proc. R. Soc. London 1987, 413, 459. (4) Barbosa, D.; Doherty, M. F. Design and Minimum-Reflux Calculations for Single-Feed Multicomponent Reactive Distillation Columns. Chem. Eng. Sci. 1988, 43, 1523. (5) Espinosa, J.; Scenna, N.; Perez, G. Graphical Procedure for Reactive Distillation Systems. Chem. Eng. Commun. 1993, 119, 109. (6) Perez-Cisneros, E. S. Modelling, Designing and Analysis of Reactive Separation Processes. Ph.D. Thesis, Technical University of Denmark, Lyngby, Denmark, 1997. (7) Hauan, S.; Westerberg, A. W.; Lien, K. M. Phenomena Based Analysis of Fixed Points in Reactive Separation Systems. Chem. Eng. Sci. 1999, 55, 1053.

(8) Lee, J. W.; Hauan, S.; Lien, K. M.; Westerberg, A. W. A Graphical Method for Reaction Design Reactive Distillation Columns. 1. The Ponchon-Savarit Method. Proc. R. Soc. London 2000, 456, 1953. (9) Lee, J. W.; Hauan, S.; Lien, K. M.; Westerberg, A. W. A Graphical Method for Reaction Design Reactive Distillation Columns. 2. The McCabe-Thiele Method. Proc. R. Soc. London 2000, 456, 1965. (10) Lee, J. W.; Hauan, S.; Westerberg, A. W. Circumventing an Azeotrope in Reactive Distillation. Ind. Eng. Chem. Res. 2000, 39, 1061-1063. (11) Hauan, S.; Ciric, A. R.; Westerberg, A. W.; Lien, K. M. Difference Points in Extractive and Reactive Cascades. I. Basic Properties and Analysis. Chem. Eng. Sci. 2000, 55, 3145. (12) Lee, J. W.; Hauan, S.; Lien, K. M.; Westerberg, A. W. Difference Points in Extractive and Reactive Cascades. II. Generating Designing Alternatives by the Lever Rule for Reactive Systems. Chem. Eng. Sci. 2000, 55, 3161. (13) Diwekar, U. M. Batch Distillation; Taylor and Francis: Washington, DC, 1995. (14) King, C. J. Separation Processes; McGraw-Hill: New York, 1980. (15) Henley, E. J.; Seader, J. D. Equilibrium Stage Separation Operations in Chemical Engineering; Wiley: New York, 1981. (16) Hohmann, E. C. Analytical McCabe-Thiele Method (Smoker’s Equation). AIChE J. 1980, B1.8, 42. (17) Sundmacher, K.; Kienle, A. Reactive Distillation; WileyVCH: Weinheim, Germany, 2003. (18) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.

Received for review August 11, 2003 Revised manuscript received May 5, 2004 Accepted May 5, 2004 IE030658W

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