Ind. Eng. Chem. Res. 1996, 35, 4587-4596
4587
Some Design Aspects of Reactive Distillation Columns (RDC) Sanjay M. Mahajani*,† and Aspi K. Kolah Department of Chemical Technology, University of Bombay, Matunga, Bombay 400 019, India
The design approach of Doherty and co-workers (1988a,b; 1994) for reactive distillation columns (RDC) has been extended for the packed-bed column in which liquid phase backmixing is totally absent. The influence of various design parameters on the feasibility of design has been studied in detail for both kinetically controlled and equilibrium-controlled reactions. A hypothetical example of a three-component reactive system has been considered in the present exercise. Introduction Modeling for the design of reactive distillation columns (RDC) for several types of reactions has been the subject of investigation in the past few years. In a typical design problem the input and output compositions are known and one has to determine the optimal column dimensions by fixing the various design parameters such as reflux ratio, liquid holdup, etc. These problems are different from simulation problems, wherein the column dimensions are known and model equations are solved to obtain the output (top and bottom) compositions for the given reflux ratio, liquid holdup, etc. Barbosa and Doherty (1988a,b) have developed a design method for RDC with equilibrium-controlled reactions. These reactions are very fast, and while calculating the number of theoretical stages, it has been assumed that the vapor and liquid streams leaving any stage in the column are in phase as well as reaction equilibrium. Reactions in such systems are so fast that gas-liquid (G-L) mass transfer alone influences the performance of RDC. Reaction equilibrium is established in the reactive phase, either in the bulk or in both the liquid-side diffusional film and the bulk, depending on the availability of the catalysts in the respective region. In homogeneously catalyzed reactions, generally, reaction equilibrium is satisfied in both the liquidside film and the liquid bulk. Recently, Buzad and Doherty (1994) have studied the design problem for kinetically controlled reactions in RDC where the vapor and liquid streams leaving any stage are in only phase equilibrium, and since these reactions are very slow, reaction equilibrium may not be obtained on a given stage. Here, along with mass transfer, reaction kinetics also strongly influences the performance of RDC. Since the extent of reaction depends on the liquid phase volume, liquid holdup plays an important role in such problems. The design method proposed by Buzad and Doherty (1994) is based on the assumption that every stage in the column is equivalent to a gas-liquid contactor, wherein the liquid is perfectly backmixed and one can perform stage-to-stage calculations based on a finite difference method for a column to obtain composition profiles. Such an assumption is more appropriately valid for plate columns, wherein on a plate it is likely that there exists no concentration gradient in the liquid phase along the axial direction. However, in some cases, especially in the case of packed towers, where backmixing in the liquid phase is practi* Author to whom correspondence should be addressed. † Present address: Chemical Engineering Department, Monash University, Clayton, Victoria 3168, Australia.
S0888-5885(96)00304-1 CCC: $12.00
cally absent, this assumption may not hold well. Hence, it is necessary in such problems to formulate the model equations in the form of differential equations. Packedbed RDCs have been commercially exploited for the manufacture of fuel oxygenates such as methyl tertbutyl ether using ion-exchange resins as catalyst. Hence, design study of these columns finds commercial relevance. In the present work, a design method based on “Fixed Points”, initially proposed by Barbosa and Doherty (1988a,b) for equilibrium controlled, i.e., very fast reactions, and Buzad and Doherty (1994) for kinetically controlled reactions, has been extended for packed RDC in which backmixing is absent. The influence of various parameters on the feasibility of operation has been studied by taking the specific hypothetical example of a three-component system chosen by Buzad and Doherty (1994). A Typical Design Problem The system under consideration is a single-feed RDC with a reaction of the type shown in eq 1.
2B S A + C
(1)
The order of volatility is A > B > C. Components A-C are denoted by numbers 1-3, respectively. The schematic representation of the problem is shown in Figure 1. The design problem studied in the present work covers two extreme cases. The first is for the case when reaction is very slow, i.e., kinetically controlled, and the second is for the case when reaction is very fast, i.e., equilibrium controlled. General Assumptions. The following assumptions were made while formulating and solving the model equations. They hold good for both equilibriumcontrolled and kinetically controlled reactions. 1. Film theory of diffusion is the basis for accounting for mass transfer in both vapor and liquid phases. 2. The liquid composition in the bulk at a given position in the column is uniform in the radial direction. The typical profiles of the compositions involved are shown in Figure 2. The profiles may not be linear for the equilibrium-controlled reactions, wherein reaction takes place in the film. 3. Axial dispersion/diffusion is assumed to be negligible. 4. The internal reflux ratio remains unchanged throughout the column as the molar heat of vaporization of the mixture is assumed to be constant. 5. Feed enters into the column as a saturated liquid. 6. Heat balance in the column has not been considered; i.e., heat loss from the walls of the column, heat of reaction, heat of mixing, etc., are negligible. © 1996 American Chemical Society
4588 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Design of RDC with Kinetically Controlled Reactions. Kinetically controlled reactions are very slow, and knowledge of intrinsic kinetics of the reactions is a prerequisite for the design exercise. The design equations formulated here are based on the assumption that the reaction takes place only in the bulk of the liquid phase and not in the liquid-side diffusional film. Design Equations. The schematic representation of the column under consideration is shown in Figure 1. Material balance over the entire column for a particular component i can be written as
xF,i -
D/B 1 x x + RXNi ) 0 (2) D/B + 1 D,i D/B + 1 B,i
where RXNi is the extent of reaction for component i taking place in RDC. Writing eq 2 for i ) 1, 3 and solving simultaneously to eliminate the RXNi term, the following expression for D/B is obtained:
D (xF,1 - xF,3) - (xB,1 - xB,3) ) B (xD,1 - xD,3) - (xF,1 - xF,3)
(3)
Consider a differential element of length dz in both rectifying and stripping sections as shown in Figure 1. A differential mass balance across this volume element can be written as
Figure 1. Schematic representation of packed RDC for the ternary system studied in the present work.
Vapor phase mass balance for rectifying and stripping sections V
dyi ) -JiVaAc dz
i ) 1, 3
(4)
Liquid phase mass balance for the rectifying section L
dxi ) -JiLaAc + riAc dz
i ) 1, 3
(5a)
Liquid phase mass balance for the stripping section (L + F)
dxi ) -JiLaAc + riAc dz
i ) 1, 3
(5b)
where ri is the reaction rate of component i, per unit volume of RDC, and is given by the following expression:
(
ri ) lνik x22 Figure 2. Gas-liquid diffusional film and typical profiles of vapor and liquid compositions.
7. The column operates with a total condenser. 8. The reboiler is considered as a reactive stage unless otherwise mentioned. The reboiler is a reactive stage only in the case of homogeneously catalyzed reactions. The vapor and liquid streams leaving the reboiler are in phase equilibrium. 9. Liquid and vapor side mass-transfer coefficients, interfacial area, and liquid holdup remain constant throughout the column. Mass-transfer coefficients as well as diffusivities of all the individual components are assumed to be equal and independent of each other.
)
x1x3 K
i ) 1, 3
(6)
where l is the fractional liquid holdup. The extent of reaction in RDC can be calculated as
RXNiM )
∫0z ri dz + ∫0z ri dz r
s
i ) 1, 3
(7)
The equations for mass transfer at the G-L interface are
Vapor side mass transfer JiV ) CtVkV(yi - yiI) Liquid side mass transfer
i ) 1, 3
(8)
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4589
JiL ) CtLkL(xiI - xi)
i ) 1, 3
(9)
Since there is no accumulation at the G-L interface,
JiV ) JiL
i ) 1, 3
(10)
It is assumed that vapor and liquid compositions are in phase equilibrium at the interface. Hence,
yiI ) fi(xiI)
i ) 1, 3
(11)
Equations 4-6 and 8-11 form a set of 14 independent equations for both rectifying and stripping sections. They can be nondimensionalized to the following form:
Vapor side mass balance for rectifying and stripping sections ψi dyi )dξ HTUV
i ) 1, 3
(12)
Liquid side mass balance in the rectifying section
(
)
(
)(
)
ψi RF + 1 x1x3 2 dxi F RF + 1 2 )+ Da x 2 dξ V RF K HTUV RF i ) 1, 3 (13)
Rectifying section:
F 1 + B/D 1 + D/B ) ) V 1 + RF SF
(14)
Liquid side mass balance in the stripping section
(
)
)(
)
x1x3 F SF + 1 Da x22 L + F SF K
ξ ) 0; xi ) xi*; yi ) yi* ) fi(xB,i)
2
i ) 1, 3 (15)
1 + B/D 1 + D/B F ) ) F + L 1 + RF + B/D 1 + SF
yi - ψi ) fi(ψiβ + xi)
i ) 1, 3
x i* )
(17)
where the dimensionless numbers are
AcHCtLl (1/k)F
aAc 1 HkVCtV ) V V HTU ψi )
β)
JiV CtVkV CtVkV CtLkL
(18) (19)
(20)
(21)
where H is the length of the packed column zone for which concentration profiles are to be determined. Method of Solution. The design is said to be feasible if the following conditions are satisfied: (i) The composition profiles of both rectifying and stripping sections must intersect.
(23)
x* is obtained by solving the following material balance equation for reboiler
(16)
Vapor-liquid equilibrium at interface in terms of the bulk compositions
Da )
ξ ) 0; xi ) xD,i; yi ) yD,i xD,i ) yD,i: total condenser (22) Stripping section:
ψi SF + 1 dxi )+ dξ HTUV SF
(
(ii) The extent of reaction in RDC calculated by eq 2, i.e., RXNi, should match with that calculated by model, i.e., RXNiM (eq 7). The composition profiles were obtained by the following procedure: (i) Equations 12, 13, and 15 are ordinary differential equations and were solved simultaneously using the Runge-Kutta-Merson method for the vapor and liquid composition profiles in rectifying and stripping sections. The equations were solved until the composition profiles cross the boundaries of the ternary diagram. This exercise was performed for various values of Da, and a range of Da values for which two profiles intersect was determined. (ii) The ψi values required to solve these equations were obtained by solving eq 17 (for i ) 1, 3) simultaneously using the Newton-Raphson technique for multiple nonlinear algebraic equations. (iii) D/B was calculated using eq 3 on the basis of input and output compositions. Volatilities of components 1 and 2 were taken as 5 and 3, respectively; the value of the reaction equilibrium constant (K) equal to 0.25 was used for all the computations. Boundary conditions to solve the differential equations are as follows:
SF
y* +
SF + 1 F
1 n
νiDa*
L+F
xB,i -
SF + 1
(
xj,22 ∑ j)1
)
xj,1xj,3 K
i ) 1, 3 (24)
Da* is the Damkohler number corresponding to the reboiler, and for all the present calculations its value is taken as 0.115. (iv) Once it is assured that the profiles intersect, one has to check whether the extent of reaction calculated by the overall material balance represented by eq 2 should match with that calculated by eq 7. It should be noted here that there is a range of Da values which satisfy the intersection condition. However, there is only one value of Da in this range, for which the condition of material balance (i.e., eq 7) is satisfied. Since intersection of profiles is a prerequisite for a feasible design, much of the attention is given to satisfy this particular condition. In most of the cases, i.e., when the column has sufficient height of reactive zone, condition (2) of material balance can be made to satisfy by manipulating the value of Da in the desired range. This value was found by trial and error. Results and Discussion for Kinetically Controlled Reactions. The results of the model can be visualized with the help of residual maps expressed over a rectangular ternary diagram for the present system as shown in Figure 3. It is a graph of compositions of
4590 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Figure 3. Rectangular ternary diagram. The dashed line represents the reaction equilibrium curve.
Figure 4. Composition profile for the rectifying section for the specified conditions with different values of Da.
the most volatile compound vs that of the least volatile compound. The dashed line represents the reaction equilibrium curve. The upper portion shown in Figure 3 is the negative reaction zone, whereas the lower part is the positive reaction zone. For a feasible design, one should manipulate the design parameters in such a way that the profiles of both rectifying and stripping sections lie in the positive reaction zone and intersect. (a) Fixed Points. In a residual map of any nonreactive or reactive distillation process, there exist certain points in the vicinity of which composition profiles behave in such a peculiar manner that compositions do not change significantly as one moves along the height of the column. These points are called fixed (or pinch) points, and based on the knowledge of these points, one can study the feasibility of the design. In a RDC with kinetically controlled reactions, Buzad and Doherty (1994) have shown that for the given input and output compositions and reflux ratio there exists a critical value of the Damkohler number at which the corresponding profile nearly touches the fixed point and changes its course. In the present work, composition profiles were obtained by solving the model equations, at different values of Da for given input-output compositions mentioned in the respective figures, reflux ratio, HTUV, and β. The profiles so obtained for rectifying and stripping sections are shown in Figures 4 and 5, respectively. At low values of Da composition profiles
Figure 5. Composition profile for the stripping section for the specified conditions with different values of Da.
enter into the negative reaction zone, whereas at higher values of Da the profiles remain in the positive reaction zone. There exists a value of Da for each section for which the corresponding profile just touches the equilibrium curve. This value is called the critical Damkohler number (Dac), and the point at which it touches the equilibrium curve is nothing but the fixed point for that particular section. Figures 4 and 5 clearly show the effect of Da on the composition profiles and the existence of the fixed points. If the fixed points are wide apart, the profiles cover a major portion of the positive reaction zone and the probability of intersection of the profiles increases. The fixed points obtained by this model match that proposed by Buzad and Doherty (1994) for the same reflux ratio and input-output compositions. The fixed points can be determined a priori without solving the model equations. The fixed points always lie on the reaction equilibrium curve, and they satisfy the following equations:
(y1 - y3) -
(x1 - x3) -
RF 1 (x - x3) (x - xD,3) ) 0 RF + 1 1 RF + 1 D,1 (25a) SF 1 (y - y3) (x - xB,3) ) 0 SF + 1 1 SF + 1 B,1 (25b)
(
x22 -
)
x1x3 )0 K
(26)
Hence, by solving eqs 25a and 26 and eqs 25b and 26 simultaneously, fixed points for rectifying and stripping sections are obtained, respectively. The conditions given in eqs 25a,b and 26 can be derived with the help of the present model by equating the extent of reaction and mass transfer equal to zero. It should be noted here that, at critical Da, the values of ψ for all the components become nearly equal to zero as we approach the fixed points. Figures 6 and 7 show the profiles of ψ at the Dac of rectifying (Da ) 0.046 128) and stripping (Da ) 0.042 58) sections, respectively, for the specified conditions. The abscissas in both the figures represent the integration stages (n) proportional to column length. The column length in terms of integration stages can be given by n[V/(HTUVaAckVCtV)] units. (b) Minimum Reflux Ratio. Equations 25a,b and 26 show that the position of the fixed points is influ-
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4591
Figure 6. Plot of dimensionless fluxes vs integration stages for the rectifying section at critical Da.
Figure 7. Plot of dimensionless fluxes vs integration stages for the stripping section at critical Da.
enced by the value of the reflux ratio. At high reflux ratios, the fixed points of both the sections are away from each other, thereby increasing the probability of the intersection of profiles. As the reflux ratio decreases, the fixed points approach each other along the reaction equilibrium curve. The value of the reflux ratio at which fixed points cross each other is nothing but the minimum reflux ratio, below which there is no possibility of obtaining a feasible design. The value of the minimum reflux ratio so obtained for the inputoutput compositions specified in Figure 6 is 1.4. (c) Significance of Different Parameters. The model equations were solved for different values of HTUV with fixed values of all the other parameters, and critical values of Da were determined for both rectifying and stripping sections. As HTUV increases, the value of Dac decreases, and a plot of HTUV vs Dac(HTUV) so obtained indicates that there is no change in Dac(HTUV) with the change in HTUV. It was also observed that, though the total height required to perform the given duty is highly influenced by HTUV, the path followed by the composition profiles on the ternary diagram remains unchanged as long as the product Dac(HTUV) is constant. This led us to the conclusion that, in the case of kinetically controlled reactions accompanied by slow mass transfer in RDC, the dimensionless number Dac(HTUV) carries a special significance and plays an important role in deciding the feasibility of a design. It is a dimensionless number which consolidates two
Figure 8. Plot of β vs Dac(HTUV) for rectifying and stripping sections at different reflux ratios.
parameters: Da, which decides the extent of reaction, and HTUV, which influences the distillation efficiency. For a given design problem, profiles intersect if and only if the value of these numbers (for rectifying and stripping sections) fall in a specific range. The width of this range depends on the relative positions of the fixed points of both sections. If the fixed points are close enough, the range is narrow and one should design a column with a certain value of Dac(HTUV) which falls in this particular range. If the value of Dac(HTUV) is beyond the desired range or, in other words, if either reaction (i.e., Da) or distillation (i.e., HTUV) dominates over the other, profiles would not intersect and feasibility would be disturbed. The dimensionless number β, which is the ratio of vapor phase to liquid phase mass-transfer coefficients, is also an important parameter in the design of RDC in the present case. The model equations were solved for different values of β at constant reflux ratio and inputoutput compositions. A plot of β vs Dac(HTUV) was obtained for both rectifying and stripping sections and is shown in Figure 8. It can be seen that the curves corresponding to each section intersect at a certain value of β (i.e., βI). It has already been mentioned that, for a feasible design, it is necessary that the values of Dac(HTUV) for both sections should be close enough. Hence, at β ) βΙ there is a certainty that the profiles would intersect to give a possible feasible design. Figure 9 shows an example of such a feasible design when the value of β (i.e., 0.2) is close to βI (i.e., 0.129). As we move away from βI in any direction, values of Dac(HTUV) depart from each other and the chances of obtaining a feasible design become less. Figures 10 and 11 show two such cases where the profiles do not intersect as the values of β (β ) 0.1 and 1.0) are considerably away from βI. The effect of the reflux ratio on the value of βI was studied, and it was observed that at sufficiently high values of reflux ratios βI was insensitive to the change in the reflux ratio. Hence, design feasibility could not be altered by increasing the reflux ratio. In such cases one can exploit an additional design parameter, i.e., the height of nonreactive zones, in either the rectifying or stripping section. (d) Nonreactive Sections. It has been shown by Buzad and Doherty (1994) that in some cases an unfeasible design can be made feasible by introducing nonreactive stages in either the rectifying or stripping section. In the present case also, the profiles obtained
4592 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Figure 9. Composition profiles for rectifying and stripping sections for the specified conditions (feasible design).
Figure 10. Composition profiles for rectifying and stripping sections for the specified conditions without nonreactive zones (infeasible design).
Figure 12. Composition profiles for rectifying and stripping sections for the specified conditions with nonreactive zones of two integration stages (4% of the height of the reactive rectifying section) in the rectifying section (feasible design).
Figure 13. Composition profiles for rectifying and stripping sections for the specified conditions with nonreactive zones of ten integration stages (4% of the height of the reactive stripping section) in the stripping section (feasible design).
to 10[V/(HTUVaAckVCtV)] units (40% of the height of the stripping section) was provided in the stripping section (see Figure 13) to make the design feasible. (e) Systems with Linear Phase Equilibria. If the vapor-liquid equilibria at the interface can be expressed in the form of linear relationship, i.e., yi ) kiVLxi, it is known that the model equations can be simplified and the flux equations can be expressed in terms of the bulk compositions as follows:
JiV ) JiL ) kOVCtV(yi* - yi)
Figure 11. Composition profiles for rectifying and stripping sections for the specified conditions without nonreactive zones (infeasible design).
with β ) 0.1 and 1.0, as shown in Figures 10 and 11, can be made to intersect by providing nonreactive zones of certain heights in the rectifying or stripping section. For instance, when β < βI (i.e., for β ) 0.1), a nonreactive zone corresponding to 2[V/(HTUVaAckVCtV)] units (i.e., 4% of the height of the rectifying section) was provided in the rectifying section (see Figure 12), and when β > βI (i.e., for β ) 1.0), a nonreactive zone corresponding
(27)
The calculation procedure becomes simpler, and one need not solve for ψ values and interface concentrations as was done earlier in the case of nonlinear phase equilibria. The profiles were obtained by solving the model equations, and the results obtained were similar to the earlier ones. HTUOV combines the effect of both β and HTUV, and the dimensionless number Dac(HTUOV) plays the same role as that of Dac(HTUV). Design of RDC with Equilibrium Controlled Reactions As mentioned earlier, Barbosa and Doherty (1988a) published the first paper on the design of RDC. The
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4593
reactions considered by them were very fast, and reaction kinetics had no significance in the design exercise. They introduced new variables for reactive distillation, called transformed variables, which reduce the dimensionality of the reactive systems. The presentation of behavior of the concentration profiles of various components along the distillation path, i.e., the height of the column, becomes extraordinarily simpler by the use of these variables. In the present work, this approach has been extended for the design of packed RDC. It has been assumed here that since the reaction is instantaneous, it takes place at the G-L interface and even in the liquid side film. The law of mass action is satisfied everywhere, i.e., in the film, at the G-L interface and in the bulk liquid. Design Equations.
Vapor phase mass balance V
Conversion of these equations to the form of transformed variables leads to
J1TV ) CtVkVψ1T
(37)
ψ1T ) Y1 - Y1I
(38)
Y1I ) (y1I + y2I/2)
(39)
where
Liquid side mass transfer L
[ ]
L
Ji ) -Ct DGL
dxif ds
(40a)
i.e.
dyi ) -JiVaAc dz
i ) 1, 2
(28)
J1TL ) -CtLDGL
Liquid phase mass balance of rectifying section dxi L ) -JiLaAc + riAc dz
i ) 1, 2
(29a)
Liquid phase mass balance stripping section (L + F)
i ) 1, 2
s)0
dxi ) -JiLaAc + riAc dz
i ) 1, 2 (29b)
Equation 6, which gives the magnitude of extent of reaction (ri), is not applicable here as it does not depend on the liquid phase holdup because the reaction is instantaneous. The above equations can be reduced to the form of transformed variables of compound 1 as follows:
[ ] dX1f ds
s)0
(40b)
It should be noted here that the composition profiles in the G-L diffusional film are not linear as the reaction also takes place in the film, and, hence, the form of the liquid side flux equation used for kinetically controlled reactions (eq 9) is not applicable here. s in eq 40 represents the coordinate across the G-L film as shown in Figure 14. Figure 14 shows the G-L diffusional films in which a differential element of length ds is considered. A mass balance of component i across this element can be written as
DGLCtL
d2xif ds2
i ) 1, 2
) -rf,i
(41)
dY1 ) -J1TVaAc dz
(30)
L
dX1 ) -J1TLaAc (rectifying section) dz
(31)
(L + F)
dX1 ) -J1TLaAc (stripping section) dz
(32)
By solving eq 42 with appropriate boundary conditions, the following expression for transfomed flux in the form of transformed composition variables can be easily obtained.
X1 ) (x1 + x2/2)
(33)
∴ J1TL ) kLCtL(X1I - X1)
Y1 ) (y1 + y2/2)
(34)
V
where
TV
J1
V
V
) (J1 + J2 /2)
(35a)
J1TL ) (J1L + J2L/2)
(35b)
Since the formulation has been done in this form, dimensionality or degrees of freedom of the problem have been reduced by 1 and the equations have to be solved for only one compound. The equations for masstransfer fluxes at the G-L interface can be written as
Vapor side mass transfer JiV ) CtVkVψi
i ) 1, 2
(36)
Equation 41 can be written in the form of transformed variables as
DGLCtL
d2X1f ds2
)0
(42)
(43)
The form is similar to that of the actual mass-transfer fluxes when the concentration profile in the G-L film is linear. Since there is no accumulation at the G-L interface,
JiV ) JiL
i ) 1, 2
(44)
and hence
J1TV ) J1TL
(45)
Equations 30-32 can be simplified with the help of eqs 37 and 45 and can be reduced to the following nondimensionalized form as
4594 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996
Stripping section ξ ) 0; xi ) xB,i; yi ) yB,i*
Figure 14. Liquid-side diffusional film.
dYi ψiT )dξ HTUV
(46)
dXi ψiT RF + 1 )dξ HTUV RF
)
(47)
dXi ψiT SF + 1 )dξ HTUV SF
)
(48)
(
(
where RF, SF, HTUV, and β have the same definitions (eqs 19 and 21) as those given in the earlier section on kinetically controlled reactions. Equilibrium Relations. In this system, along with the phase equilibria at the interface, there exist reaction equilibria everywhere in the liquid phase.
K)
x1I(1 - x1I - x2I)
(at interface)
(x2I)2
K)
x1(1 - x1 - x2) (x2)2
(in bulk)
(49)
(50)
Vapor-liquid equilibrium at the interface is given by the following relation:
yi )
Rixi
∑j Rjxj
i ) 1, 2
Rectifying section ξ ) 0; xi ) xD,i; yi ) yD,i
(ii) The solution of these equations requires the knowledge of transformed fluxes. The interface concentrations at a particular height were calculated by solving eqs 33, 34, 36, 43, and 49-51 simultaneously with the help of either Newton-Raphson or the iteration technique for the known values of X1 and Y1. Once the interface concentrations are determined, transformed fluxes can be calculated very easily. Results and Discussion for Equilibrium Controlled Reactions. It has been shown in the model formulation that the nature of the operating curves for RDC with very fast reactions, if expressed in the form of transformed composition variables, resembles to that of nonreactive distillation. Hence, the whole behavior of the system can be viewed mathematically as normal packed-bed distillation with an additional constraint of reaction equilibrium for the liquid phase compositions, both at the interface and in the bulk. The liquid holdup (or Da) has no direct influence on the performance of the column or feasibility of design. The reflux ratio and HTUV are the two important dimensionless numbers involved in the model, and for a given problem only the reflux ratio determines the feasibility of design, i.e., the possiblity of intersection of the profiles of both rectifying and stripping sections. (f) Minimum Reflux Ratio. For any value of RF, HTUV, and β there exist stable nodes on the concentration profiles of both the sections. The compositions of the components near these nodes (pinches) do not change significantly and remain practically constant along the length of the column. Figure 15 shows the different profiles obtained at different values of the reflux ratio under otherwise similar conditions. For low values of RF, as shown in Figure 15a the stable nodes are reached before the profiles intersect and, hence, the feasibility is not obtained. At high values of RF (Figure 15c) profiles intersect and feasible design is possible. There exists a value of RF at which the profiles just touch each other and stable nodes match (Figure 15b); this is nothing but the minimum reflux ratio for the given problem. Its value can be determined a priori without actually solving the differential equations for the operating curves. At the stable nodes, the values of the transformed fluxes are zero. These values are substituted in eqs 38 and 47, and if the equations are solved simultaneously, coordinates of the stable nodes are obtained and RF can be determined by the following expression: st D RF + 1 Xi - Xi ) st RF Y -YD
(51)
Method of Solution. (i) The differential equations for the rectifying section (eqs 46 and 47) and for the stripping section (eqs 46 and 48) were solved simultaneously with the help of the Runge-Kutta-Merson method. The following boundary conditions were used:
(52)
(53)
i
(54)
i
HTUV and β do not influence the possibility of intersection; however, these factors certainly play an important role in deciding the height of the column, which, in turn, decides the extent of reaction taking place in RDC. Conclusions The methodology for the design of the reactive distillation column (RDC) has been developed for a ternary system for both kinetically controlled and equilibrium-
Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4595
a
b
In the kinetically controlled reactions “Fixed Points” on the ternary diagrams have been identified and the influence of their relative positions on the feasibility of design was studied. The positions of the fixed points depend on the reflux ratio. The procedure to calculate the minimum reflux ratio was illustrated. It was shown that the new dimensionless number Dac(HTUV) carries a special significance in the design of packed RDC. For a feasible design the values of this number for rectifying and stripping sections should be close enough, and the allowable difference between these values is again decided by the positions of the fixed points. The design of RDC with equilibrium-controlled reaction can be simplified by transforming the model equations to the new variables called transformed composition variables. It allows one to make an analogy between the normal distillation and reactive distillation. The behavior of the transformed composition variables is similar to that of the mole fractions in nonreactive distillation. It was observed that the value of the minimum reflux ratio for equilibrium-controlled reactions was much lower than that for kinetically controlled reactions under otherwise similar conditions. For the given input-output composition and reflux ratio, the required height of RDC for the kinetically controlled reaction is obviously greater, if the design is worked out by considering the same reaction as the equilibriumcontrolled reaction. In short, the approach of Doherty and co-workers for the design of tray columns has been extended in the present work for packed-bed RDC. The overall methodology and some important results (e.g., value of minimum ratio) are not different from the similar problem solved for the tray column. The present work widens the scope of the design method to packed-bed reactive distillation columns which are more important on a commercial scale. The future work in this field should be aimed in the direction to extend the method to various systems involving more than three components including inerts and for multiple reactions. Application of the theory of “Fixed Points” to practical systems and design of RDC for selectivity of a desired product in the case of multiple reactions will attract the attention of researchers in the future. Nomenclature
c
Figure 15. Profiles of transformed composition variables for different values of reflux ratios: (a) RF < RFmin; (b) RF ) RFmin; (c) RF > RFmin. HTUV ) 2.0; β ) 1.0; R1 ) 1.25; R3 ) 1.1. feed distillate bottoms X1 0.5 0.999 0.001 X2 0.0 0.000 0.000 X3 0.5 0.001 0.999 (A) Stable node for the rectifying section. (B) Stable node for the stripping section.
controlled regimes. The formulation can be extended easily to other multicomponent systems and even for the systems with multiple reactions. The assumption of backmixing in the liquid phase has been relaxed, and the model can be very well applied for the design of packed-bed RDC.
A, B, C ) chemical species Ac ) cross-sectional area of the column, m2 a ) interfacial area per unit volume of dispersion, m2/m3 B ) bottoms product molar flow rate, kmol/s Ct ) mixture molar density, kmol/m3 D ) distillate molar flow rate, kmol/s Da ) Damkohler number given by eq 18 Dac ) critical value of the Damkohler number DGL ) diffusivity of components in the G-L film, m2/s F ) feed molar flow rate H ) height of the column zone, m HTUV ) height of the transfer unit defined by eq 9 JV ) molar diffusion flux for the vapor side, mol/m2 s JL ) molar diffusion flux for the liquid side, mol/m2 s K ) chemical equilibrium constant k ) forward reaction rate constant kL ) liquid-side mass-transfer coefficient, m/s kOV ) overall mass-transfer coefficient, m/s kV ) gas-side mass-transfer coefficient, m/s l ) fractional liquid holdup L ) liquid molar flow rate
4596 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 N ) number of integration stages RF ) reflux ratio RXNi ) extent of reaction in RDC given by eq 7 ri ) reaction rate of component i, per unit volume s ) cooridanate in the direction across the G-L diffusional film SF ) reboil ratio V ) vapor molar flow rate, kmol/s xi ) mole fraction of component i in the liquid phase (bulk) X ) transformed composition variables of liquid phase compositions yi ) mole fraction of component i in the vapor phase (bulk) Y ) transformed composition variables of vapor phase compositions z ) differential height of the column, m Greek Letters R ) relative volatility β ) CtVkV/CtLkL ) characteristic time for mass transfer in the liquid phase/characteristic time for mass transfer in the vapor phase νi ) stoichiometric coefficient of component i ξ ) dimensionless height of the column section ψi ) dimensionless flux ) JiV/CtVkV δ ) length of the G-L diffusional film Subscripts B ) bottoms product D ) distillate f ) composition or reaction in the G-L film F ) feed i ) components r, s ) rectifying and stripping sections
1, 2, 3 ) reactants A, B, and C, respectively Superscripts I ) gas-liquid interface D ) distillate L ) liquid M ) model st ) stable nodes T ) transformed variable TV ) transformed composition variable for vapor phase compositions or fluxes TL ) transformed composition variable for liquid phase compositions or fluxes V ) vapor
Literature Cited Barbosa, D.; Doherty, M. F. Design and minimum-reflux calculations for single-feed multicomponent reactive distillation columns. Chem. Eng. Sci. 1988a, 43, 1523. Barbosa, D.; Doherty, M. F. Design and minimum-reflux calculations for double-feed multicomponent reactive distillation columns. Chem. Eng. Sci. 1988b, 43, 2377. Buzad, G.; Doherty, M. F. Design of three-component kinetically controlled reactive distillation columns using fixed-point methods. Chem. Eng. Sci. 1994, 49, 1947.
Received for review May 31, 1996 Revised manuscript received September 12, 1996 Accepted September 13, 1996X IE960304N X Abstract published in Advance ACS Abstracts, November 1, 1996.
