Some aspects in the design of multicomponent reactive distillation columns including nonreactive...

Pergamon Chemical En#ineering Science, Vol. 50, No. 3, pp. 469 484, 1995 Copyright © 1995 Elsevier Science Ltd

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0009-2509(94)00258-4

SOME ASPECTS IN THE DESIGN OF MULTICOMPONENT REACTIVE DISTILLATION COLUMNS INCLUDING

NONREACTIVE SPECIES

JOSI~ ESPINOSA,* PIO AGUIRRE and GUSTAVO PI~REZ INGAR--Instituto de Desarrollo y Disefio-Avellaneda 3657-3000 Santa Fe, Argentina

(Received 27 September 1993; accepted in revised form 26 July 1994)

Abstract--Some aspects related to the design of reactive distillation columns are addressed in this paper. A new set of transformed composition variables is proposed for mixtures including one or more components that are inert under the process conditions. This set corresponds to an extension of that suggested by Barbosa and Doherty (1988a, Chem. Engng Sci. 43, 1523-1537) and allows to compute by means of any traditional procedure, the concentration profiles along the column and therefore, the minimum reflux ratio. The compositions of product streams of a reactive distillation column are subject to constraints of thermodynamic nature that can be determined before any attempt to design the column. These thermodynamic constraints do not have a counterpart in conventional distillation and become an essential piece of information in order to select the design variables and specify their values. A parametric analysis of the simultaneous chemical reaction and liquid-vapor equilibrium is suggested as a very useful instrument to select thermodynamically feasible design specifications, In this work, we also present an initial discussion about possible columns sequences to obtain the reaction product free of inert species. The selection and calculation of the relevant variables that are common in the first steps of the design and synthesis of reactive distillation processes are discussed through examples. Finally, throughout the entire paper several interest- ing physical and operational conclusions regarding inert components in reactive distillation are given. The main conclusion is that the inerts play a key role in the design of a reactive column.

l. INTRODUCTION

The design and synthesis of reactive distillation columns are the subject of recent papers (Barbosa and Doherty, 1988a; Doherty and Buzad, 1992 and Espinosa et al., 1993) for mixtures in which all species present participate in a single chemical reaction. For this type of system it was possible to generate a set of transformed composition variables (Barbosa and Doherty, 1988a). Consequently, the conservation equations are reduced to an identical form of that corresponding to nonreactive columns because the reaction terms in the mass balance equations are dropped out.

Furthermore, in the transformed field simple relationships between the dependent design variables can be derived. These relevant characteristics of the transformed model facilitate the design of reactive columns by using a boundary value method (Levy et al., 1985; Doherty and Caldarola, 1985). In our recent paper (Espinosa et al., 1993), the energy balance was in- cluded into the transformed model in order to consider nonideal mixtures, especially when the reaction enthalpy could not be neglected. For ternary reacting systems, the preliminary design can be carried out by a "Ponchon-Savarit" graphical method that considers the energy balances. The above-mentioned contribu- tions are first steps in the process of understanding the most important characteristics in reactive distillation.

tAuthor to whom correspondence should be addressed.

The purpose of this paper is to generate tools sup- porting the design of reactive distillation systems with feed streams containing nonreactive components.

Such problems were mentioned in Doherty and Buzad (1992) as typical process to be investigated. Frequently, the reacting mixture contains appreciable amounts of nonreactive species which do not participate in the reaction but greatly influence on both the simultaneous phase-reaction equilibrium and the column operating conditions.

As well as in the design problems with all the components participating in a single equilibrium reaction, it is possible to define a new set of transformed composition variables for systems including inert species. Simple relationships between the reboil and the reflux ratios can be obtained. However, the design specifications can be now defined in terms of the new variables.

The compositions of product streams of a reactive distillation column are subject to constraints of thermodynamic nature that can be determined before any attempt to design the column. These thermodynamic constraints do not have a counterpart in conventional distillation and become an essential piece of information in order to select the design variables and specify their values. The simultaneous phase-reaction equilibrium constrains the liquid compositions that could be present in a reactive distillation column. Therefore, a parametric analysis of the composite phase-reaction equilibrium is suggested as a very useful instrument to select thermodynamically feasible design specifica-

469

470

tions. As an important result of the parametric analysis, we can identify the components that show a maximum in the thermodynamically feasible compositions field. This maximum is not only useful to obtain optimal designs but also to prevent the selection of certain unfeasible component compositions as design variables. Despite the concepts mentioned above, the column feasibility will depend on another stronger fact, i.e. the possibility of obtaining an intersection between the rectifying and stripping profiles at least for one value of the reflux (or reboil) ratio. The residue curve maps obtained from the simple distillation of homogeneous reactive mixtures including nonreactive species can be used in determining the feasible product regions for a fixed feed composition. We present the residue curve maps for two reacting systems and suggest a further discussion regarding product composition regions in a next paper.

Since for both conventional and reactive distillations without inerts, the absolute minimum value of the reflux ratio is affected by the amount of heavy (light) components in the distillate (bottom) provided the feed pinch is placed on the stripping (rectifying) profile, we explore this issue for an ideal reacting mixture forming a maximum boiling reactive azeotrope (Barbosa and Doherty, 1988b; Doherty and Buzad, 1992).

Finally, we also consider the previous distillative separation of the inert from the reagents and we attempt to explain the reasons for which such task could not be attractive from both economical and operational standpoint.

2. THE M O D E L FOR REACTIVE DISTILLATION C O L U M N S

INCLUDING NONREACTIVE SPECIES

In this section we derive the design equations for reacting mixtures including nonreactive components. To derive these equations we suppose: (1)l iquid boiling feed, (2) heat losses are negligible, (3) the molar heat of phase change for the mixture is constant, (4) the heat of mixing is negligible, (5) the heat capa- city of the mixture is constant, (6) the reaction enthalpy change is negligible compared to the phase change enthalpy, (7) on each stage the equilibrium is attained for the leaving streams, (8) the operating pressure is constant, and (9) the column operates with a partial condenser.

The concepts developed in the next sections can be used in reactive distillation for a wide variety of multicomponent mixtures provided a single equilibrium reaction takes place in the liquid phase. However, in this work we demonstrate our approach by considering a ternary reacting system and one inert component 'T'. The reason for this is that the design problem for such systems is almost similar to the corresponding formulation for three-component distillative mixtures. The reacting species undergo an equilibrium reaction of type

vAA + vsB ~-- vcC. (1)

J o s e ESPINOSA et al.

Another important question is related with the first seven assumptions. As a consequence of these, the energy and mass balances can be decoupled and, hence, only the material balance and the composite phase-reaction equilibrium equations are used to calculate the composition profiles for the column. The overall material and energy balances around the column are used to give a relationship between the reflux ratio and the reboil ratio. Assuming a liquid boiling feed, the vapor flow rate remains constant in the column and, on the other hand, the liquid flow rate varies only due to changes in the total number of moles by chemical reaction (Barbosa and Doherty, 1988a).

2.1. Tray-by-tray equations. Stripping section In Fig. l(a), the column section below the feed point

is depicted. The mass balances around the envelope

(a)

vo

Y-I

1- 2

B

) X_B

(b)

D Y-D

Lm Lm~_ -~ R I v~_~

F JYm-I ~F

Fig. 1. Schematic representation of a reactive distillation column: (a) stripping section; (b) rectifying section.

Design of multicomponent reactive distillation columns

5" for the reacting components are

p = n dsp Ln+lXj,n+ 1 = Vyj, n -1- Bx j . B - Yj 2 d - t ,

p = l

j = 1, n c - 1 (2)

where the last term is the number of moles of component j generated by reaction over the entire section of the column within the envelope 5 e. For the inert component I (with index nc) it follows that

L . + lx.~,.+ 1 = Vy . . . . + Bx.~,n. (3)

By selecting the equation of an arbitrary reacting component k (with 1 ~< k ~< nc - 1) and solving eq. (2)

p=. for ~p= ~dep/dt we can eliminate this common term from the rest of eq. (2) by writing their reaction terms as a function of the liquid and vapor mole fractions of component k. It is clear that one variable and one equation are dropped out from the model. The result- ant equations are

\ Yj Vk / k Vj Vk /I

q_ B ( Xj'B Xk.B~

\ Vj 1~ k /¢

j = l , n c - 1 , j ¢ k . (4)

In Appendix A, the compositions as well as the vapor and liquid flow rates are rearranged to obtain the transformed variables

Xj X k

vj vk (5) X s = vk(l - x.c) - v,xk

y j Yk

Yj Vk YJ = Vk(1 -- y,~) -- v,yk (6)

j - n c - 1 v , = ~ vj (7)

j = l

L* = L.[vk(1 -- x.c,.) -- V, Xk,.] (8)

V* = V[VR(1 -- y.~,.) -- v,y~,.]. (9)

Hence, eq. (4) can be rewritten as

L * + I X j , . + I = V * Y j , . + B * X j , n ,

j = l , n c - 1 ; j # k . (10)

The transformed compositions defined in eqs (5) and (6) show properties similar to that of mole fractions, i.e. the sum of the transformed variables weighted with its stoichiometric coefficients is a constant (see Appendix A):

tic 1 1 v j X j = - (11)

j = 1 Yk ) ~k

. c - 1 1

Y v~ rj = - . (12) j = ! Vk j c k

471

This allows to derive an operating line, in the same manner as in nonreactive distillation, which relates the liquid-transformed compositions of stage n + 1 in terms of the vapor-transformed compositions of stage n and the bottom-transformed compositions

s* 1 X j. . +, = s* + 1 Y;'" + s * 7 ~ Xj .n ,

j = l , n c - l , j C k . (13)

The transformed reboil ratio is defined as

V* [Vk(1 -- Y.c..) -- vtyk..] S.* = ~ - = Sext [Vk( 1 __ Xnc,t~) - - Y t X k . B ] (14)

with

V Sex, = --. (15)

B

Normally, the bot tom compositions and the reboil ratio are given and the interest is in finding a feasible stripping profile. In Appendix B, details are given about a solving procedure for both eqs (3) and (13).

2.2. T r a y - b y - t r a y equations. Rec t i f y in9 sect ion

Considering now the component material balances around the envelope M above the feed [see Fig. l(b)], we can derive the operating line for the rectifying

section:

r* 1 YJ'=-' = r=* + ~ Xj.,. + r* +-~ Yj'>

.j = 1, nc - 1; j ~ k 116)

L * Lm[Vk(1 -- X . . . . ) -- V,Xk.,.] r* -= - - ; I17)

D* D[vk(l -- Y.c,o) - V,yk,O]

Vy,,~.,,,_ ~ = L,.x.~.. , + Dy.c.D. I18)

As in reactive distillation without inert components, the relationship between the transformed reflux ratio on any tray above the feed stage and the external one is more complex than in conventional distillation. Let us see how the transformed reflux ratio is related to the external reflux ratio, rext. The overall material balances in terms of transformed variables around envelopes :~ and ~g of Fig. l(b) yield

V* I = L * + D * (19)

V*. ~ = L* + D* . (20)

Solving eq. (19) for the vapor flow rate V, and replacing the result in eq. (20), after some rearrange-

ments we obtain

r * + 1 = Vk(1 -- Y.¢, , . -1) -- Vtyk,, .-1 (r~*xt + 1) (21) Vk(1 -- Ync,N- 1} -- VtYk,s- 1

re~xl - - L* LN[Vk(1 -- X,c,D) -- V, Xk,o]

D* D[Vk(1 - - Y . c . o ) - - v,yk.o] {22)

LN r,xt = - - . {23)

D

472

Equation (21) relates the transformed reflux ratio on tray m to the external reflux ratio. All the equations that have been deduced up to here become identical to those obtained by Barbosa and Doherty (1988a) when the inert mole fraction tends to zero.

Note also, that the operating line for the rectifying section is a nonlinear relationship for Y,,.

JosI~ ESP1NOSA et al.

in addition to the external reflux ratio. Let us assume that the product specification in mul t icomponent con- vent ional dis t i l lat ion is related to two hierarchical levels of distinct constraints:

2.3. Overall balances As pointed out by Barbosa and Doherty (1988a),

the compositions of the feed and product streams cannot be specified arbitrarily, since they are subject to the mass balance around the column. Furthermore, the overall mass and energy balances give the conditions that must be fulfilled by the external reflux and reboil ratios.

When the pivot component is assumed to be the reaction product C, then the transformed compositions for species A and B and their component material balances in terms of transformed variables are

F * X A, F : D * Y A.D + B * X A,B

F*Xs .F = D* Ys,o + B*XB, s . (24)

Multiplying each equation by its stoichiometric coefficient v~ and summing we obtain

F* = D* + B*. (25)

From both equations, the following relationship can be derived:

D* D[vc(1 - - YI,D) - - VtYc ,o] X A,B - - X A,F m _

B* B[vc(1 - XI.B) -- V, Xc.8] X A,V -- Ya,o

(26)

The relationship between the external reboil and reflux ratios can be obtained after dividing and multiplying eq. (26) by V and relating the ratio D / V to the transformed external reflux ratio from the material balance around the partial condenser

Sext[VC( 1 - - Yt , s -1 ) - v tYc ,N-1"] X A . B - - X A , F

(1 + r*xO[Vc(1 -- x1,B) -- VtXc,n] X a.v -- Ya.o

(27)

Finally, the inert component material balance around the column completes the system of equations

FxI,F = Dyl ,o + B x t , n . (28)

3. THERMODYNAMICALLY FEASIBLE DESIGN

VARIABLES: PARAMETRIC ANALYSIS O F THE

COMPOSITE EQUILIBRIUM

Before we can solve eqs (3), (13), (16) and (18) to find the composition profiles in the stripping and rectifying sections, we must first specify the values for the design parameters. In our case, the degrees of freedom for the ternary reactive system with one inert component correspond exactly to those of a ternary nonreactive system. Then, the feed composition (liquid boiling), the system pressure and three independent compositions in the product streams must be specified

(i) A first level of constraint is given by the overall mass balance. That is to say, there exists a limited set of product compositions for which the material balances around the entire column are satisfied.

(ii) A second level corresponds to the possibility, for the previously selected product compositions, of obtaining feas ib le composi t ions profiles into the column.

For ideal three-component systems, both levels are quite trivial. However, in multicomponent and azeotropic multicomponent mixtures, the selection of product compositions leading to feasible column profiles is a difficult task. The use of residue curve maps in ternary azeotropic distillation allows to find for a given feed composition, the regions of the simplex that are candidates to be the products of a column with internal feasible profiles (Wahnschafft et al., 1992).

However, a new level of thermodynamic constraints, that does not appear in conventional distillation, can be considered in order to extend the above scheme to the case of specification of reactive distil la- t ion products. There exists a restriction for the liquid compositions that must be satisfied. This constraint arises by including the chemical reaction in the equilibrium model giving place to the composite equilibrium (Barbosa and Doherty, 1988b). This restriction is not trivial for some cases and prevents the selection of the concentration of some components as design parameters.

Two main implications are derived from the thermodynamic constraint related to the simultaneous phase-reaction equilibrium: First, not all the concentrations of the simplex can be considered as thermodynamically feasible. There exist upper bounds for the concentrations of some components according to the composite equilibrium.

Second, not all the components of a reacting mixture can be selected for specification without the risk of generating multiple solutions for the composite equilibrium. Furthermore, if a product composition is selected violating this thermodynamic constraint, the overall mass balance around a reactive distillation column could not be satisfied. A parametric analysis of the composite phase-reaction equilibrium is suggested as a very useful instrument to select thermodynamically feasible specifications, in the following we will show by means of an example how the parametric analysis can be used in order to overcome the above-mentioned difficulties.

We demonstrate our approach by considering an ideal reacting mixture forming a reactive azeotrope [for reactive-azeotrope definition see: Barbosa and Doherty, (1988b); Doherty and Buzad, (1992)]. The stoichiometric coefficients ofeq. (1) are set - 1 for the

Design of multicomponent

reagents and I for the reaction product. In Tables 1 and 2, the Antoine constants for the vapor pressure and the standard Gibbs free-energy change of reaction for the above-mentioned mixture are given. The most and least volatile components are the inert and the desired product C. Component C was chosen as the pivot element for the elimination of the reaction term and, hence, the decision parameters must be selected among the transformed variables for components A and B and the inert mole fraction. Let us select, for specification, two compositions in the bottom (XA.n and xl,n), one composition in the top (YA.O) and the column reflux ratio. In Appendix B an algorithm is presented for solving the overall balances for this particular set of design variables.

The graphics of Fig. 2 show the results obtained when the simultaneous liquid-vapor and chemical reaction equilibrium is solved for several fixed values of the pair (Xa, xt). The algorithm is presented in Appendix C and can be considered as a reactive bubble point temperature algorithm. Both Figs 2(a) and (b) are plotted for fixed values of the mole fraction of the nonreactive specie while Figs 2(c) and (d) show the Y vs X and temperature vs composition diagrams in the transformed field for the "pure ternary reacting mixture, (x~---0)". Since a reacting ternary mixture (x~ = 0) can be treated as a conventional binary one (Doherty and Buzad, 1992; Espinosa et al., 1993) it is possible to deal with graphics similar to that for nonreactive binary systems. Figures 2(c) and (d) ctearly show the existence of a maximum boiling reactive azeotrope for - X,4 = 0.427.

A key question is revealed from Fig. 2(a). In fact, from the analysis of this figure appears evident that some choices in the product compositions can produce thermodynamically infeasible streams. The compositions of the product streams are thermodynamically constrained for reacting mixtures unlike what happens in distillative mixtures. For each value of the inert mole fraction, a maximum in the mole fraction of the product C is encountered. A curve joining these maxima is also presented in Fig. 2(a). Each point in this curve of maximum product composition represents an upper bound for the mole frac-

Table 1. Coefficients in the Antoine equation, log psat =

C1-C2/(C3 + T) (Pa, K)

Component C1 C2 C3

A 8.99591 1221.901 - 49.980 B 10.19620 1730.630 - 39.734 C 9.51271 1533.313 - 50.851 I 9.22668 1244.951 - 55.279

Table 2. The standard Gibbs free-energy change of reaction for the hypothetical system

AG O = - 8314 J/mol, K ( T ) = e -aG°mr

reactive distillation columns 473

tion of component C for the corresponding fixed value of the inert mole fraction. In the limit when x~ = 0, the absolute maximum for xc is found. Note that we are specifying the product compositions by using the transformed variables (XA, xl) and we cannot equally well specify these compositions using the mole fractions of any component as was proposed in Barbosa and Doherty (1988a). In fact, if we try to fix a value for xc over the upper bound, the composite equilibrium could not be satisfied. Furthermore, if for a fixed value of x, we select a value for xc belo~ the corresponding upper bound, two different solutions for the composite equilibrium are possible. The mole fraction of the reaction product C should never be selected as a design parameter in combination with x~ because of the reasons given above.

Therefore, by means of the parametric analysis suggested, it is possible to reduce the search space in product composition specifications before any attempt to design the column have been done. Despite this necessary condition, the feasibility of the column will depend on the possibility of finding a value for the reflux (or reboil) ratio that produces an intersection between the rectifying and stripping profiles.

The parametric analysis of the composite equilibrium is also useful in the first calculation steps to supply transformed composition values according to the objectives of the designer. Since the most and least volatile components are the inert and the desired product C, therefore, both components can be separated in the same column avoiding a diminution in the product purity. To achieve high purity product in the bottom of the column, the equilibrium constant should be as great as possible. If the objective is to eliminate the inert component from the reaction product C; therefore, it becomes natural to select the liquid mole fraction for the inert component in the bottom as a design parameter. This composition should be very small if we want high purity product. If addition- ally, the interest also lies in sharp splits with high recovery of the reaction product, hence the compositions of the product C in the distillate stream can be specified at a small value. However, as shown, the mole fraction of C must not be selected as a design parameter. Figure 2(b) can be useful to select both the distillate and bottom transformed composition in order to obtain small values of Yc at the top and the maximum product composition at the bottom.

The maximum in composition of the reaction product C is placed on - X A = 0.5; this however, was independent of the nonreactive component mole fraction. It is important to note that such a maximum is reached when the reagents are in equimolar ratio: xA = xs. Considering other reactions we can conclude that the maximum in C depends on both the stoichiometry and the degree of deviation of the mixture from the thermodynamic ideality. In such cases the mole fraction of the reagent components can lie very far from the stoichiometric ratio.

The problem of fixing a small value for the concentration of the product C in the top of the column can

4 7 4 J O S E E S P I N O S A e t al .

x =

6 7 0

Curve of M a x i m u m Temlpermture

0.63 ~ v ~ ° f Max i r r t Jm Proc lL~t Composit ion

0 5 6

O.42

O. % I

0,2-8" •

0 . 1 4

0.O7

~ 1 ~ 0 . t0 0 2 0 ( ~ 0.'1113 ~ 0.60 0,70 0`80 0.90 1.00

X A

OJ -6 E "g_

._~

1.00 (b)

o.go

/ T = 3 7 2 . 4 8 K

0.70 T=37 , , 2K / 0.60

B A

O ~

OAO

O.3O

O2O

OlO ; ~ 0`00 ~ t I ii

0`00 0`10 0.20 0.30 0.40 0.50 0.60 0,70 0 ~ 0 O.CJO t 0 0

- X , ,

>_, I

1.00

O.gO

0`80

0.70

0.60

0.50

0.40

0..30

0.20

0.10

0.00

(cJ

- X

/ / Azeotropic comc~sit ions // XA = 0 . 1 1 4 0 7 2 7

/ / ~.: o~4o8o15

• L i i i i L L I I 0.00 0 . t0 0 2 0 0.30 0.40 0.50 0.60 0.70 0.80 O.gO 1.00

- X A

~ 5

F-

(cO

38o Az.

375

370

365

T.= ~6.1o K \ \ = ?-.= 355 A

3~0 I 1 i i i i i i i 0.0<3 0 . t0 0.20 0.30 0.40 0,50 0.60 0.70 O.BO 0`gO 1.00

- × A / - Y A

Fig. 2. Parametric analysis of the phase-reaction equilibrium: (a) curves for 0.001 < x, < 0.85; (b) curves for x, = 0.15; (c) diagram - Y vs - X for the "pure reacting mixture"; (d) diagram temperature vs

composition for the "pure reacting mixture".

be handled by using Fig. 2(b) newly. It can be stated that for both small and high values of the transformed variables (XA or Xn), any mixture will contain the reaction product C in small amounts. The value of the transformed compositions of A in the distillate will be close to 0 or 1 according to the values of Xa in the feed stream and the bottom composition. In particular, for the maximum value ofxc in the bottom, - XA.B = 0.5, feed streams with A (B) in excess will give a distillate with A (B) in excess too; therefore, - Y A , O ~ 1

( - - Y A m ~ 0) .

Summarizing, the following aspects based on the composite equilibrium analysis must be considered in order to obtain thermodynamically feasible products specifications:

(i) The mole fraction of some components should not be treated as design parameter if upper bounds for their concentration or multiple equilibrium solutions are found.

(ii) The reaction products can show maximum concentration values that can be recognized and evaluated in terms of the transformed variables of the reagents.

(iii) If the component whose composition cannot be selected as design variable is the desired product; then, there exist simple ways of indirectly fixing their relative quantity in the product streams by specifying the values of the transformed variables for the reagents at the top and bottom of the column, and the bottom mole fraction for the nonreactive specie.


It is important to remark that a separation by ,~o reactive distillation is feasible if the overall mass bal-

O.9O ance is satisfied and if there is at least one path of calculation accounting for material and energy bal- o~o ances and equilibrium relationships describing the individual trays of a column from the distillate to the aTo bottom. The parametric analysis only gives thermodynamically feasible product streams.

Az.

4. M I N I M U M - R E F L U X C A L C U L A T I O N S

The design specifications discussed above lead to a feasible column only when the stripping and the rectifying profiles contain a tray with the same liquid composition; this corresponds to the feed stage. To specify a feasible column, not only the product compositions but also the reflux ratio or alternatively the number of trays must be selected. The new variables are also of great help in this case, because a simple relationship between the external reflux and the external reboil ratios can be derived and the order of the equations system is reduced.

In order to obtain the stripping profile, the procedure to solve the equations system is similar to that proposed in Barbosa and Doherty (1988a). We compute the operating line beginning from the bottom at stage I and going upward until finding an end pinch, for which no further progress in the liquid composition is noted. Since at this composition, the downcom- X~

ing liquid and the vapor rising from below have reached vapor-liquid and chemical reaction equilibrium, an infinite number of trays would be required for the stripping profile to pass on. Therefore, the very nature of this pinch or stationary point lies on the fact that the driving force for mass transfer between the phases has become zero. In a similar manner, the rectifying operating line is solved from the distillate downward until an end pinch is found. However, this operating line could be computed beginning with a guess feed tray composition and moving upward up to the distillate. In this way we have always at each stage the liquid-transformed composition and the algorithm would be the same as in the stripping section, For this case, we would solve the composite equilibrium as a reactive bubble point temperature algorithm. Due to the feed tray composition is unknown whereas the distillate composition can easily be calculated, a little modification is needed to proceed by computing from the distillate downward using at each stage a reactive dew point temperature algorithm to z

solve the simultaneous phase-reaction equilibrium. A feasible column design is found when the profiles

of each half of the column intersect. Thus, by solving the tray-by-tray model, the search for a feasible column design involves changing the external reflux ratio until an intersection is obtained.

Furthermore, as in the ternary nonreactive distillation, which corresponds to the reactive system under consideration, the minimum reflux condition de- mands that either the rectifying or the stripping profile or both together end with their end pinchs just on Fig.

Composi t ion Feed Dist i l la te Bo t t om

A - 0 . 6 - 1 . - 0 . 5 5

I 0 .20 0 . 6 9 1 8 0.001

rN=2 .75

475

ID 0`60

0*50

0.40

0-30

0.20

0.10

OJO0

- X A

o , , _ _ 0`40 0`45 0.50 055 0.60 0.65 0.70 0.75 0JB0 0~15 0.9(3 o.g5 1~O

lJ0O

Composi t ion Feed Dist i l la te Bo t t om

x

(b)

0,90 A - 0 . 6 - 1. - 0 . 5 5

o~o t 0 .20 0 .6g 18 0 .00 1

r~=3 .2 0.70

Feed pinch 0~50

0.40 (330

O=2O

0 . tO

AZ. O.00 v I I ~

0.40"0.45 0.50 0,55 0`50 0.55 0,70 0.75 OJE]O OB5 o.go 0.95 1,o()

- X ~

1.00

0`cjO

OJBO

(c) C.ompositpon Feed Dist i l la te Bottocn

A - 0 . 6 - 1. - 0 . 5 5

I 0 .20 0 .6g 18 0 .00 1

0.70 r=~=5.00 [3 .

0.60

0.50

0.40

030

O.20

0.10

AZ- 8 0~00 V L

0.40 0.45 0.50 O.55 0.6O O.65 0.70 O.75 O~BO 0~5 O.gO ~,95 t O O

- X A

3. Composition profiles as a function of the reflux ratio.

476

the feed tray (Barbosa and Doherty, 1988a; K6hler et al., 1991). A reactive column under minimum reflux operation will require an infinite number of stages because of the reasons explained above. In Fig. 3, the composition profiles are shown for a reactive distillation column fed with a stream with x~.r = 0.20 and (Xa/XB)F = 1.50. According to the parametric analysis, in order to produce thermodynamically feasible product streams, the inert composition in the bottom was selected to be 0.1%, XA,bot tom = --0.55 and YA,distillat¢ = - - 0 . 9 9 9 9 9 . Different reflux ratios were chosen to demonstrate that by approximately a value of 3.2, the reflux ratio is the minimum to make the column feasible. In this case the stripping profile contains an infinite number of trays close to the feed stage composition (feed pinch).

Let us examine the composition and reaction profiles (see Fig. 4) for a feasible column of 46 theoretical trays working at a reflux ratio of 5.14 and fed at stage

JOSE ESP1NOSA et al.

39. The design specifications correspond to those of Table 3. From the graphics of Fig. 4, the following conclusions can be obtained:

(i) In the stripping section two regions can be iden- tified: immediately below the feed, the inert component is completely stripped from the liquid phase whereas the composition of C and B remain approximately constants. In the other region, near the bottom of the column, the product composition (xc) in- creases in a few number of trays up to their design specification.

(ii) In the column section above the feed stage, the inert component is separated from the reactive which is in excess (A) and from the two heavy components B and C.

(iii) The reaction occurs in a great degree only at the feed stage and in minor degree at the bottom where the conditions for the occurrence of the reac-

l

-I

1.00

o_qO

0.70

°t O.40

O.20

0.10

0

I~ttom

A

I I I I I

4 8 1 2 1 6 20 24 28

plate

30

25

20 ~>

15

- 6

- 1 0

I J - 2 0 32 36 40 44 48

T o p

<

J

Fig. 4. Composition and reaction profiles for the feasible reactive distillation column operating at r=, = 5.14.

Table 3. Stream results for an operating reflux ratio, r=, = 5.14 (compositions in mole fractions)

Feed stream (39) Distillate Bottom

A 0.48 B 0.32 C 0.00 I 0.20

Flow rate (kmol/h) 100.00 Temperature (K) 359.70

0.30818 0.29272 6.85 × 10 -7 0.13577 2.4 × 10- 6 0.57052 0.69182 0.00100

28.84 45.31 351.99 375.35


tion are reinstated because of the presence of reactant in excess A in large amount.

In Fig. 5 another example is considered in which the feed stream contains an excess of B. When the bot tom specification was made near the absolute maximum in C, the profiles no longer intersect independently of the value of the reflux ratio. On the other hand, we encountered a feasible column by selecting the bot tom practically at the reactive-azeotrope composition. In a next paper we propose a method for determining the feasibility of a desired separation at both finite reflux and total reflux without being concerned about the details of the design. All we need to determine such feasibility are the residue curve maps obtained from reliable thermodynamical data. At the end of this paper, we include the residue curve maps for two mixtures and briefly comment the main characteristics of these systems.

477

tions will show composit ion profiles that pass close to the two saddle pinch and consequently a greater number of steps are necessary in both regions.

Finally, as in Barbosa and Doherty (1988a) the lower bound for the minimum reflux ratio (maintain- ing the optimal bot tom composition) is obtained when the rectifying profile passes close to the saddle pinch. In our example, according to the terminology adopted in the mentioned paper, the rectifying profile is the saddle pinch profile while the stripping profile is the feed pinch profile. As was mentioned in Barbosa and Doherty (1988a), the above fact has important implications on how to specify the compositions of the column products in order to obtain rational designs.

In Table 4 different minimum reflux values are shown when the specification of the heavy components in the distillate is varied. The lower bound for rmj, is found to be 3.12.

4.1. The minimum reflux value.for the sharp reactive distillation

For sharp reactive distillation, the minimum reflux structure shown in Fig. 3(b) defers to that encountered by Barbosa and Doherty (1988a) for reactive quaternary systems. At minimum reflux there are three active pinch zones; one pinch occurs in the rectifying profile, which is located on - YA,O = 1. The others are located in the stripping profile (the saddle pinch on Xa axis and the feed tray pinch). The stripping saddle pinch occurs because the composit ion of the lighter component tends to zero in order to obtain the maximum mole fraction of the reaction product. On the other hand, the rectifying saddle pinch appears when the amount of the heavy components in the distillate is made as low as possible ( - Ya,o = 1). Hence, feasible columns satisfying the sharp specifica-

4.2. The minimum reboil ratio as function of the.feed inert composition

The minimum energy demand of a reactive distillation system is one of the most important decision variables in the design step. The influence of the inert mole fraction in the feed stream over the minimum reflux (or reboil) ratio has to be considered in order to obtain optimal designs. Let us present the following example with two different compositions of the inert in the feed stream, In order to generate comparable alternatives we select the following specifications:

(i) (xa..'xn)v = 1.15 in both cases, (ii) - )[4.~ = 1 in both cases.

(iii) The bottom composition in both cases corresponds to the optimal concentration of C and hence becomes identical for the examples considered.

1J30

0.B0 D r,~=4.25

Uclfeasible stripping profile with -Xsa = > - X ~

0.643 /

feasible 8tripping profile

0 . ~ I 0.GC 0.t0 020 0-30 0.40 0.50 0.60 0.70

--X A

Fig. 5. Composition profiles showing both feasible and unfeasible design specifications.

The performance variable we suggest to evaluate these columns is the reboil ratio. Since both alternatives show the same bottom composition, it follows that the reboil ratio can express the vapor flow rate that must be generated in the boiler of the reactive column for each mole of the withdrawn product C

In Table 5 the molar flow rate, the mole fraction of the column streams and the minimum reboil rat~o Sm~n are presented for case I with Xl.r = 0.2 and for

Table 4. Minimum reflux ratio for different amounts of heavy components in the distillate ( X4R = 0.55,

x~.R = 0.001)

- - YB, D rmin YB.D YC',D

l x l 0 - l ° 3.12 6.8 x 10 -12 2.4 x 10 ~1 l x l 0 -5 3.20 8.0x 10 7 2.3 x 10 ~ l x l 0 -4 3.25 6.6x I0 -~ 2,42 x 10 5 1×10 -3 3 .40 6.87×10 -s 2.401 x 10 4 l x l 0 -2 3.60 7.028 x 10 -4 2.4347 x 10 3

478 Josl~ ESPINOSA et al.

Table 5. Stream results for a column with feed streams containing 20 and 10% of the nonreactive component

Feed stream Distillate Bottom

Case 1 (Sml, = 2.09) A 0.44 B 0.36 C 0.00 1 0.20


Case 2 (Sm~, = 1.35) A 0.495 B 0.405 C 0.000 1 0.100


0.2461543 0.2206773 6.41 x 10 -12 0.1888682 1.82 x 10-11 0.5904535 0.7538457 0.0000010

26.53 46.19 351.62 377.23

0.4235307 0.2206773 7.46 x 10-12 0.1888682 3.49 x 10-11 0.5904535 0.5764693 0.0000010

17.35 51.97 352.69 377.23

case II with xl,r = 0.1. The corresponding composition profiles at minimum reflux are depicted in Fig. 6. The lower the feed inert concentration, the lower the minimum reboil ratio needed to satisfy the column operation. In the limit when x~.v = 0, the value obtained for smi, = 0.573. Figure 7 shows in the Y vs X diagram, the operating lines at minimum reflux for the "pure reacting mixture".

The changes observed in s,~i. when the inert component mole fraction varies, indicates that the alternative of the separation by nonreactive distillation (NRD) of the inert component, before performing the reactive distillation (RD), could be attractive at least in energy terms.

5. ANALYSIS OF TWO ALTERNATIVES FOR OBTAINING

THE REACTION PRODUCT FREE OF INERTS

In this section we compare two alternatives that allows to obtain the reaction product, which in turn is the heaviest component, and recover the reactant in excess. To simplify the treatment we will not consider possible recycles of the excess reactant. The structures of the reaction-separat ion trains to analyze are as follows.

Case I: React ion-separat ion of the reactive mixture containing the nonreactive component, followed by a conventional column to recover the reactant in excess from the inert species.

Case II: Separation of the inert in a conventional column followed by the reaction-separat ion process.

Since the costs involved in the evaporat ion and condensation of the streams inside the towers (as those corresponding to columns diameters) can be related in a direct way with the external reboil ratio, the estimation of the minimum vapor flow rate is useful in comparing the alternatives. We assume that all the towers can operate at the relatively low temper- atures and pressures habitually encountered in distillation (i.e., when the column operate at atmospheric

1.oo

0,90

0.80

0.70

(a) Coml:x)s i t ion F e e d D is t i l l a te B o t t o m

A - 0 . 5 5 - 1. - 0 . 5 1

I 0 . 2 0 0 . 7 5 3 8 0 . 0 0 0 0 0 1 D

s . . - - 2 . 0 9

030 Feed pitch

0`1o ~ . B

o % - £ , 0`~ 0`,, o~o: 0`,, a~o 0.;, £o 0;, 0.;0 o;~ , ~

- X ,

%00

0.90

O.8O

0.70

0.6O

X- 0.5O

(b)

~ i t i o n Feed D i s t i l l a t e Elot tom

A - 0 . 5 5 - 1 . - -0 .51

I O. 10 0 . 5 7 6 5 O . 0 0 0 0 0 1

s,~= 1 .35

O.lO

0.00 ~ i . . . . . . . . i i i i 0,40 0`45 0.50 0.55 0.60 0.65 0.70 0.75 0.B0 0.85 0.90 0.95 t.00

--X A

Fig. 6. Influence of the feed ~tream inert mole fraction on minimum reboil ratio.


1.OO

O.OO EoJili

0,85

0,80

0.6.60.70 ~ / ~ " / Deslgtl s~oecJflcations / X,...=-0.55

x..=

o.55 I ~ b ° t t o m Y~=-I

0.50 ~ 0.50 0.55 0.150 0.65 0.70 0.75 0`80 0.155 O.gO O.g5 1.00

Fig. 7. Minimum reflux profiles for the "pure reacting mixture".

pressure and use cooling water in the condenser). The results of the two schemes proposed are presented in Tables 6 and 7. As a consequence of the results obtained, the first alternative seems to lead to lesser

479

Costs, which can be explained by the following reasons: Although in case II, the reactive column

operates at a substantially lower vapor flow rate than the corresponding in case I, in both cases the distillative separations require the greatest vapor flow rates and, hence, they determine the optimal sequence. In fact, the first alternative lead to lesser costs since the distillative task is alleviated by the reactive tower. In the reactive column, most of the light components are

eliminated and also a volumetric contraction occurs because of the reaction.

The column sequencing ideas developed in this paper must be taken as first steps for a more specific analysis that consider all the reactive column economical aspects and even the design of others possible sequences (i.e., those considering multiple-feed or col-

umns with a reacting core).

6. RESIDUE CURVE MAPS FOR REACTING MIXTURES

INCLUDING NONREACTIVE SPECIES

In the remainder of this paper we present the residue curve maps for two reacting mixtures. As in conventional distillation, these curves contain the main information to be able to assess the range of product composi t ions achievable by individual reactive distillation columns. We suggest a further discussion about this topic in a next paper. However, ,at

Table 6. Overall stream results for the first alternative (RD + NRD)


RD, Sm~, = 2.09, 1~° = 96.5 kmolh A 0.44 0.2461543 0.2206773 B 0.36 ~ 0 0.1888682 C 0.00 ~ 0 0.5904535 l 0.20 0.7538457 0.00000 t 0

Flow rate (kmol/h) 100.00 26.53 46.19 Temperature (K) 360.36 351.62 377.23

NRD, Rmi., = 0.43, Vmi . = 149 kmol"h 1 (z~t, A = 1.206) 0.7538457 1 ()

A (:t4..4 = 1 .000 ) 0.2461543 0 I Flow rate (kmol/'h) 26.53 2000 6.53 Temperature (K) 351.62 350.13 356.10

Table 7. Overall stream results ['or the second alternative (NRD + RD)


NRD, Rmi n = 16.28, Vmi n =: 346 kmol/h 1 (~t.A = 1.21) 0.20 1.0 0.00 A (ct4.. 4 = 1.00) 0.44 0 0.55 B (ctn.a = 0.53) 0.36 0 0.45

Flow rate (kmol/h) 100.00 20.00 80.00 Temperature (K) 360.36 350.13 363.52

RD, Stain = 0.573, ~%i, = 26.4 Kmol/h A 0.55 1.0 0.2206773 B 0.45 ~ 0 0.1888682 C 0.00 ~ 0 0.5904535

Flow rate (kmol"h) 80.00 6.53 46.19 Temperature (KI 363.52 356.10 377.23

CES 50-3-I

480 Josl~ ESPINOSA e t a l .

this point we shall concentrate our attention in describing some characteristics regarding inert components in reactive distillation from the operating conditions point of view.

Figure 8 shows the residue curve maps in the transformed field for the ideal mixture forming a reactive azeotrope considered along the paper. The transformed variables used in deriving these residue curves slightly differ from those for the design problem. More details are given in a next paper. The edges of the composition triangle in Fig. 8 represent:

(i) the "pure reacting system, xl = 0", (ii) the nonreactive mixtures between any of the

reagents and the inert.

As can be seen, the ideal system forming a maximum boiling reactive azeotrope shows a unique distillation region at total reflux since all the trajectories start at the nonreactive component vertex (an unstable node) and end at the reactive azeotrope (a stable node). In the vicinity of each of the reagents, some trajectories approach the critical point while others move away from it; therefore, such points are called "saddle points". Note that the residue curve maps resemble those for homogeneous azeotropic distillation (Doherty and Caldarola, 1985). In fact, the nonreactive component does not produce an internal distillation boundary allowing that the pure reagents or the maximum in composition of C could be obtained. The pure reactants shall be obtained when the feed promotes the "indirect reaction", while the maximum in C arises when the "direct reaction" is en-

forced. In short, the inert acts as a conventional en- trainer. This fact, in addition to those mentioned in the previous section highlights the influence of the nonreactive components on both the design and synthesis of reactive distillation processes.

Finally, we consider the highly nonideal mixture forming a conventional azeotrope; namely, the system isobutene-methanol-MTBE-butane (Fig, 9). For this highly nonideal system, in contrast with what happens for the ideal system, there exists two funda- mentally different types of residue trajectories. In the first type, the residue curves emanate from regions rich in isobutene (unstable node) while in the second type, the residue curves have their starting point in the minimum boiling azeotrope between butane and methanol (unstable node). Both types of composition trajectories ends at pure methanol (stable node).

These different types of residue curves give rise to two different distillation regions at total reflux. The corresponding separatrix begins at the pure butane vertex (saddle node) and terminates at the methanol vertex (stable node). Similar behavior was encountered by Jacobs and Krishna (1993) for systems having n-butene instead of n-butane as nonreactive component.

Another important characteristic is that, for feed streams richer in methanol than in isobutene, the product stream rich in MTBE would be the distillate or the bottom depending whether the butane is absent or not in the feed stream, respectively. Effectually, since the methanol is the heaviest component, when we consider the "pure reacting mixture", feed streams with an excess of methanol will give a distillate rich in

I 350.13 K C3 -"

-.. 0

BO 0.2 04g~ 06 08 1/5 ̀375.70 K max boilin X A ~ 3,56. I0 K

azeotrope 578 70 K

Fig. 8. Ideal reacting mixture showing a reactive azeotrope: residue curve maps.


n - C 4 H JO 3 0 5 . 6 0 4 K

min boiling azeotrope ""-----~ '

303 880 K

\ -j// o

CH nH J 0 3'-' 0 0 2 0 4 06 0 8 I i-C4H8

368.36:5 K X i- C,~He--" 298 .177 K

Fig. 9. Highly nonideal reacting mixture showing a binary azeotrope residue curve maps.

481

MTBE and a bottom rich in the excess reagent (Doherty and Buzad, 1992; Espinosa et al., 1993). On the other hand, feed Streams containing some amount of butane (or butene, or both of them) give rise to a bottom rich in the product of reaction and a distillate containing a binary mixture methanol-butane.

From the above concepts, it is evident that the inerts play a key role in the design and synthesis of reactive distillation columns.

7. SUMMARY AND CONCLUSIONS

Some aspects related to the design of reactive distillation columns with feed streams containing nonreactive components are the subject of this paper.

The design equations of a reactive column are derived by using a new set of transformed composition variables which allow to find a simple relationship between the reflux and reboil ratios.

A parametric analysis of the simultaneous phase- reaction equilibrium is suggested for equilibrium limited reactions in order to select the design variables and specify their values. A system forming a maximum boiling reactive azeotrope is studied. In such system the product streams leaving the reactive column are strongly constrained by the physicochemical equilibrium. Two main implications are derived from the thermodynamic constraint related to the simultaneous phase-reaction equilibrium: First, not all the concentrations of the simplex can be considered as thermodynamically feasible. There exist upper bounds for the concentrations of some components according to the composite equilibrium. Second, not all the components of a reacting mixture can be selected for specification without the risk of generating multiple solutions for the composite equilibrium. The

parametric analysis of the composite equilibrium is also proposed in the first calculation steps, to supply thermodynamically feasible transformed composition values according to the objectives of the designer. Furthermore, the maximum concentration value of the desired component that could be obtained from the reactive column can be determined in terms of the transformed variables of the reagents.

The liquid profiles in the transformed variables of several feasible designs are given. Also unfeasible design specifications, for which the profiles no longer intersect independently of the value of the reflux ratio, are mentioned. As Levy et al. (1985) and Barbosa and Doherty (1988a) showed for distillative and "pure reacting" mixtures, respectively, the absolute minimum value for the reflux ratio is obtained when the amount of heavy (light) components in the distillate (bottom) is made as small as possible if the feed pinch is located on the stripping (rectifying) profile. How- ever, for sharp reactive distillation, the minimum reflux structure for the system under consideration- de- fers to that encountered by Barbosa and Doherty (1988a) for reactive quaternary systems. That is, at minimum reflux there are three active pinch zones. We observe for the mixture under consideration that the previous elimination of the inerts in a distillative column could be attractive at least in energy terms.

An analysis of two alternatives to obtain the reaction product free of nonreactive components is performed. Despite the results obtained in the previous section, we conclude that the previous distillative separation of the inerts is not necessarily a good synthesis policy. The reasons we claim are: Since the distillative separations require the greatest vapor flow rates, they determine the optimal sequence. As a consequence of

482 Jose ESPINOSA et al.

this, the (RD-NRD) sequence leads to lower costs Xj since the distillative task (NRD) is alleviated by the reactive tower (RD). In the reactive column, most of yj the light components are eliminated and also a volumetric contraction occurs because of the reaction. Yj

Finally, residue curve maps for reacting mixtures including inerts are considered. Two systems are ana- lyzed. An ideal mixture forming a maximum boiling reactive azeotrope and the highly nonideal mixture isobutene-methanol MTBE-butane that in turn, form a binary azeotrope between the inert and meth- ?j anol. From an inspection of both residue curve maps several conclusions are reported. The ideal system e,p forming a maximum boiling reactive azeotrope shows vj a unique distillation region at total reflux since all the v, trajectories start at the nonreactive component vertex and end at the reactive azeotrope. On the other hand, Subscripts

for the highly nonideal system, there exists two funda- A, B, C, I mentally different types of residue trajectories. These B different types of residue curves give rise to two differ- D ent distillation regions at total reflux. F

From the concepts developed throughout the entire m paper, it is evident that the inerts play a key role in the min design and synthesis of reactive distillation column, n Our first goal was to determine feasible product stream specifications in reactive columns containing Superscripts inerts. Once these thermodynamic feasible specifica- * tions were obtained, we could easily extend the conventional minimum reflux calculations to reactive systems. However, we carried out the product selection by trial and error in order to obtain feasible column profiles. Therefore, the set of feasible products for a given feed specification leading to feasible column profiles, becomes an essential piece of information not only for design but also for synthesis of reactive column systems. In subsequent papers we will analyze these topics on the basis of the residue curve maps for reactive mixtures including inerts.

Acknowledgements We are indebted to Millie Atdjian for her valuable help in the translation of this paper to the English language. The authors gratefully acknowledge the financial support of CONICET (Consejo Nacional de Inves- tigaciones Cientificas y T~cnicas de Argentina).

A,B,C,I B

D F K L nc Py' P, text

Sext

t

T V xj

NOTATION

generic chemical species bottom molar flow rate distillate molar flow rate feed stream molar flow rate reaction equilibrium constant internal liquid molar flow rate number of components vapor pressure of pure component j column pressure external reflux ratio external reboil ratio time temperature internal vapor molar flow rate molar fraction of component j in the liquid phase

transformed composition of component j in the liquid phase mole fraction of component j in the vapor phase transformed composition of component j in the vapor phase

Greek letters ~j,nK relative volatility between component

j and heavy key liquid phase activity coefficient of component j extent of reaction on tray p stoichiometric coefficient of component j quantity defined by eq. (6)

components A, B, C, I bottom distillate feed generic tray above the feed tray minimum generic tray below the feed tray

transformed molar liquid and vapor flow rates

REFERENCES

Barbosa, D. and Doherty, M. E., 1988a, Design and minimum-reflux calculations for single-feed multicomponent reactive distillation columns. Chem. Engng Sci. 43, 1523 1537.

Barbosa, D, and Doherty, M. F., 1988b, The influence of equilibrium chemical reactions on vapor-liquid diagrams. Chem. Engn9 Sci. 43, 529-540.

Doherty, M. F. and Buzad, G., 1992, Reactive distillation by design. Trans Instn Chem. Engng Part A, 70.

Doherty, M. F. and Caldarola, G. A., 1985, Design and synthesis of homogeneous azeotropic distillations. 3. The sequencing of columns for azeotropic and extractive distillations. Ind. Engng Chem. Fundam. 24, 474-485.

Espinosa, J., Scenna, N. and Phrez, G., 1993, Graphical procedure for reactive distillation systems. Chem. Engng Comm. 119, 109 124.

Jacobs, R. and Krishna, R., 1993, Multiple solutions in reactive distillation for methyl tertbutyl ether synthesis. Ind. Engng Chem. Res. 32, 1706-1709.

K6hler, J., Aguirre, P. and Blass, E., 1991, Minimum reflux calculations for nonideal mixtures using the reversible distillation model. Chem. Engng Sci. 46, 3007-3021.

Shacham, M., 1989, An improved memory method for the solution of a nonlinear equation. Chem. Engng Sci. 44, 1495 1501.

Wahnschafft, O. M., Koehler, J. W., Blass, E. and Wester- berg, A. W., 1992, The product composition regions of single-feed azeotropic distillation columns. Ind. Engng Chem. Res. 31, 2345-2362.

APPENDIX A: TRANSFORMED VARIABLES

If we define the transformed compositions [see eq. (4)] as

X j X k

X j - ~'~j ~'k A

whereA~A(xi, v~). j = l , n c - 1 ; j g : k (A1)

Design of mult icomponent reactive distillation columns

it will be possible to choose an adequate functionality for A in such a way that the sum of the transformed variables (or a linear combination of these) will be a constant. The manner of obtaining this is through the use of eq. (A2). i.e.

m

vj = I . (A2) j l

If we replace in eq. {A2) the component mole fractions in terms of the transformed variables and keep in mind that [i) the unknown fraction A does not depend of mole fraction of component ) and fib eq. (71 can be used to reduce the resulting expression. Then, the sum of the transformed compositions weighted with its stoichiometric coefficients can be expressed as

no- 1 v j X i - - Vk(l - ' q ) - VtXk (A3)

i - I "4Vk j * k

It is evident that for the left-hand side of eq. (A3) be a constant, A must be defined as

A = vdl .xll - v,x k. (A4)

A P P E N D I X B: A L G O R I T H M S FOR S O L V I N G T H E O V E R A L L B A L A N C E S A N D T H E S T R I P P I N G S E C T I O N P R O F I L E S

As we have mentioned in the main body of this paper, before we attempt to solve both overall balances and column profiles, we must first specify the values for the design parameters. The transformed composition variables suggested as design variables must be thermodynamically feasibles according to the parametric analysis of the phase-reaction equilibrium. Hence, in order to find a feasible column, values for r~, must be proposed until an intersection between the profiles of each half of the column is obtained.

The main objective of this section is concerned with the solution of both overall balances and column profiles by using the transformed composition variables. For the sake of simplicity, we will present the algorithms for solving the overall balances and the stripping profiles. However, it is important to note that the methodology to compute the rectifying profile is slightly different to the corresponding for the stripping section when the profiles are calculated beginning from the distillate; therefore, a little modification must be performed. In the remainder of this section we assume known values for

X . 4 , B , X l , l t , Y 4 . O , t e x t .

O~ erall balances (a) Calculate the transformed variables (flow rate and

composition) for the feed stream using eqs (5) and (8). (b) Compute D* and B* from eqs (25) and (26). (cl Once the bot tom compositions (XAm, Xl.n) are known,

the phase-reaction equilibrium can be solved (see Ap- pendix C) and the bot tom stream is completely specified; therefore, compute B from eq. (8).

(d) Compute D and 3'~.~ from the following iterative procedure: (dl) assume a value for Y~m- (d2) solve the phase-reaction equilibrium (as a reac-

tive dew point temperature algorithm). {d3l compute D from eq. (9l. (d4} compute },',,o from eq. (28) and check with the last

value in (d 1). If convergence is achieved. Stop. If not, Go to (d 1).

The iterative procedure for solving .v,., (and consequently the distillate flow rate) can be performed by using any of the methods that deals with a single iteration variable. We suc- cessfully used the improved memory method (Shacham, 1989l.

Stripping section pr(~liles The procedure to solve eqs (3) and (13} is similar to that

proposed in Barbosa and Doherty (1988a). We compute the

483

operating line beginning from the bottom at tray 1 and going upward until finding an end pinch, for which no further progress in the liquid composition is noted. The steps that must be performed in order to solve the equations system are:

Ca) Once the overall balances are solved, it is possible to obtain the compositions and vapor flow rate for the stream feeding the partial condenser. These calculations are easily performed because the transformed reflux ratio is knm~n at this stage. In short, the vapor flow rate is obtained from mass balances around the partial condenser. Therefore. calctdate s,,~, from eq. {27t.

(b) Since XA and .x~ are known at stage , , the liquid and vapor mole fractions and the temperature for the leaving streams can be evaluated at this tray. This is done by solving the phase-reaction equilibrium as a reactivc bubble point temperature algorithm.

(c) Calculate the transformed rehoil ratio from eq. 114).

[d) Calculate X , . 1 from eq. (13). (e) Calculate L**~ from the overall mass bahmce in

the transformed variables. (f) Calculate L,+ l and xt.~+l from eqs {3) and (81. (g) If no further progress in the liquid composition is

noted. Stop. Otherwise. Go to (bJ.

A P P E N D I X C: R E A C T I V E BUBBLE P O I N T T E M P E R A T t RE

ALGORITHM

In this section we deal with the problem to solve the composite phase-reaction equilibrium. As in conventional distillation, the algorithm for solving the simultaneous vapor liquid and chemical reaction equilibrium when the transformed liquid composition are given differs from that

for which x. v and T arc obtained from known wdues of the transformed vapor compositions. We have termed such algorithms as "reactive bubble point temperature" and "reactive dew point temperature", respectively.

We will demonstrate how a "'reactive bubble point temperature'" problem can be easily solved, for tile ideal mixture forming a reactive azeotrope, by means of an iteralive procedure.

Let us assume that A 4 and Xn as well as the inert mole fraction x~ are given; the liquid and vapor mole fractions in addition to the phase temperature are still unknown, The defining equations for -\'A and X,~ can be written as

\'4 ¥ ( "

V 4 I ' ( X4 ((;ll

vdl - x~) - v~x{

X B X C

I' B Y(, X 8 - tC2)

The liquid compositions must satisfy the reaction equilibrium constant:

/ n c - 1

K ( T ) = ~ ( x i , ' ) " . (('31 j l

If a temperature value is supposed, these three nonlinear equations can be solved for all the independent mole fractions as an inner loop, being the temperature correction the outer loop. In the example selected, we can derive a simple expression by setting the activity coefficients to I. After the rearrangement of eqs (C 1)-(C3) we have

ax~ + bx, + c : 0 (('4)

with the parameters a, b and c defined as

a = K(T)X4X~ t('5)

h - K(T)[(1 xt)(2X~X n -- l l ] - 1 i('6)

c = (I - x i )Zx4XnKIT) . 1('7)

484 Josl~ ESPINOSA et al.

Once the mole fraction of component C is calculated, we are able to obtain all the remaining mole fractions using the defining equations of the transformed compositions. Only one of the roots of eq. (C4) has physical sense and produce thermodynamically feasible solutions.

The temperature is corrected in an outer loop by means of the vapor-liquid equilibrium relationship as in a nonreactive bubble point temperature routine:

f ( T ) = ) ' y j - 1 (C8)

with j= 1 po

y~ = p---~ x j, j = 1, nc. (C9)

The improved memory method of Shacham (1988) was implemented for the solution of the outer loop [eq. (C8)].

Some aspects in the design of multicomponent reactive distillation columns including nonreactive...

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