Equilibrium theory and nonlinear waves for reactive distillation columns and chromatographic reactors

Chemical Engineering Science 59 (2004) 901 – 918

www.elsevier.com/locate/ces

Equilibrium theory and nonlinear waves for reactive distillation columns and chromatographic reactors S. Gr'unera , A. Kienleb; c;∗ a Institut

fur Systemdynamik und Regelungstechnik, Universitat Stuttgart, Pfaenwaldring 9, D-70550 Stuttgart, Germany fur Dynamik komplexer technischer Systeme, Sandtorstra'e 1, Magdeburg D-39120, Germany c Lehrstuhl f ur Automatisierungstechnik, Otto-von-Guericke-Universitat, Universitatsplatz 2, Magdeburg D-39106, Germany b Max-Planck-Institut

Received 2 April 2003; received in revised form 29 October 2003; accepted 24 November 2003

Abstract A general framework for analyzing and understanding the dynamics of reactive separation processes is developed. The theory is based on the assumption of simultaneous phase and reaction equilibrium. It makes use of transformed concentration variables, which were 4rst introduced by Doherty and co-workers for the steady state design of reactive distillation processes (Proc. Roy. Soc. London A 413 (1987a) 459). It is shown that these transformed variables can be directly generalized to the dynamic problem considered here. Further, they can also be applied to other reactive separation processes like 4xed bed as well as countercurrent chromatographic reactors, for example. They provide profound insight into the dynamic behavior of these processes and reveal bounds of feasible operation caused by reactive azeotropy. It is shown that reactive azeotropy, which is a well-known phenomenon in reactive distillation (Proc. Roy. Soc. London A 413 (1987b) 443) may also rise under very similar conditions in other reactive separation processes like chromatographic reactors for example. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Reactive separation processes; Reactive distillation; Chromatographic reactors; Equilibrium theory; Nonlinear waves; Reactive azeotropy

1. Introduction The dynamic behavior of many chemical processes is governed by the evolution of characteristic spatio-temporal patterns also termed as nonlinear waves (Marquardt, 1990). Typical examples are traveling concentration and temperature fronts in 4xed bed reactors and distillation processes as illustrated in Figs. 1 and 2, or traveling concentration pulses in chromatographic processes. Fig. 1 shows the transient behavior of a 4xed-bed reactor, which is used for the catalytic combustion of volatile organic compounds in waste air. Usually such a reactor is operated under steady-state conditions. An inlet heating is used to raise the temperature of the feed to reaction conditions. If the heating is switched o> a traveling reaction zone evolves, which is moving downstream as the reactor undergoes a transient from an ignited to an extinguished steady state. During the transient the temperature in the interior may rise ∗ Corresponding author. Tel.: +49-391-6110-369; fax: +49-391-6110-515. E-mail address: [email protected] (A. Kienle).

0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2003.11.021

signi4cantly due to the heat accumulation in the reaction zone. A detailed analysis of traveling reaction zones in 4xed bed reactors is given elsewhere (Gilles, 1974; Mehta et al., 1981). Similar patterns of behavior can be observed in nonreactive distillation processes. A very simple binary separation problem is illustrated in Fig. 2. At steady state the concentration and temperature pro4les inside the columns show steep fronts. After a disturbance these fronts start moving until 4nally a new steady state is established. A detailed analysis of traveling wave fronts in distillation processes was given by Marquardt (1988, 1990), Hwang (1995) and Kienle (1997, 2000a). Although the behavior in Figs. 1 and 2 looks fairly similar, the physico-chemical causes for nonlinear wave propagation are completely di>erent in both cases. The distillation process is dominated by convective transport and mass transfer between the vapor and the liquid phase. In contrast to this, nonlinear wave propagation in the 4xed bed reactor is primarily due to heat conduction and chemical reaction. For a comparison of nonlinear wave propagation in separation and reaction processes we refer to Marquardt (1990).

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S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

x 10

traveling reaction zone

700

steady state

600 500

t0

t1

t2

t3

t4

t5

400 300

−3

2.5

Concentration [−]

Temperature [K]

800

0

0.2

0.4

0.6

0.8

1

z [m]

2

t0

t1

t2

t3

t4

t5

1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

z [m]

Fig. 1. Traveling reaction front in a catalytic-4xed bed reactor after switching o> the inlet heating.

1

Methanol Propanol

Height

Methanol

0 0

1 Concentrations

Propanol Fig. 2. Traveling concentration fronts in a nonreactive distillation column after a stepwise increase of the reFux.

During the past decade combined reaction separation processes received a lot of attention due to potential economic bene4ts compared to conventional technologies where reaction and separation are carried out in di>erent devices (see e.g. Sundmacher and Kienle, 2002; Kulprathipanja, 2002). Depending on the separation principle di>erent types of reactive separation processes can be distinguished. Typical examples are reactive distillation processes, which combine chemical reaction with distillation, or chromatographic reactors, which combine chemical reaction with chromatographic separation. These reactive separation processes show sometimes very similar patterns of behavior like nonreactive separation pro-

cesses. An example for an industrial reactive distillation column is shown in Fig. 3. Clearly, the dynamic behavior illustrated in Fig. 3 is governed by nonlinear wave propagation. However, this kind of wave propagation dynamics cannot be observed for all reactive distillation processes. Therefore, the question rises for which reaction systems and for which operating conditions these nonlinear wave propagation phenomena will occur. First investigations in this direction for reactive distillation processes were made by Marquardt and co-workers (Gehrke et al., 1998). They considered a reversible reaction A + B
C for di>erent Damk'ohler numbers. For high Damk'ohler numbers the column is close to chemical

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

903

condenser 1

A

2 3

10

Chemical Reactions:

25

. . .

B+C
A+D B+D
A+E

Tray

B, C

. . . .

20

30

40

k-1 k k+1

. .

50

NS-1

60

NS

E

325

reboiler

330

335

340

345

350

355

360

Temperature [K]

Fig. 3. Traveling temperature fronts in an industrial reactive distillation column after a stepwise increase of the heating rate.

equilibrium and behaves very similar to a binary nonreactive distillation column. This is due to the fact that the chemical equilibrium condition reduces the dynamic degrees of freedom by one. Further insight into the dynamic behavior in the limit of reaction equilibrium can be gained by using transformed composition variables (Kienle, 2000b). These transformed composition variables were 4rst introduced by Barbosa and Doherty (1987a) for the steady-state design of reactive distillation processes with a single reversible reaction. Later, the results were generalized for multireaction systems by Ung and Doherty (1995). This approach will be further elaborated in this paper for reactive distillation processes and will be also applied to chromatographic reactors. First attempts towards a rigorous mathematical analysis of chromatographic reactors are summarized in the book of Rhee et al. (1989). Focus, is on a simple reversible reaction of the form A
B in the solid phase. During the last years, interest in 4xed and in particular moving bed chromatographic reactors has increased. In practice, a continuous movement of the solid phase is not possible. Therefore, simulated moving bed technology was developed to approximate a continuous movement of the solid phase by cyclic switching of the in- and outlets in a multiport plant (Ruthven and Ching, 1989). Emphasis of recent work in the 4eld of reactive chromatography is on feasibility and the development of new processes (see e.g. Meurer et al., 1997; Falk and Seidel-Morgenstern, 1999; Lode et al., 2001; Fricke and Schmidt-Traub, 2003). Major applications are concerned with esteri4cation and isomerization processes. However, a thorough understanding of the dynamics and the inherent limitations of chromatographic reactors for certain types of reaction mixtures is missing to a large extent. To close this gap and to develop a more general approach towards the understanding of reactive separation processes

we will proceed in the following way: 4rst, focus is on distillation processes. A suitable model for the mathematical analysis is derived and some results for the nonreactive case are brieFy summarized and illustrated. Afterwards, extensions for reactive distillation processes in the limit of chemical equilibrium are given. It is shown that the corresponding dynamic problem can be reduced to a nonreactive problem in a reduced set of transformed composition variables. Hence, results for nonreactive separation processes can be directly applied to the transformed problem. This gives useful insight into the dynamic behavior of reactive distillation processes. Afterwards the same procedure is applied to chromatographic processes. This reveals some striking parallels between the behavior of chromatographic reactors and reactive distillation columns not known so far. Finally, extensions to other reactive separation processes are discussed. 2. Distillation processes 2.1. Model equations The following analysis is based on a continuous nonequilibrium model for a single section of a distillation column as depicted in Fig. 4. Thus, it applies strictly only for packed columns. But it is a well known fact that this model is also a good approximation for staged columns provided the number of stages is suHciently large (Marquardt, 1988). The continuous model is chosen here, because it allows an analytical approach as will be shown subsequently. The column section shown in Fig. 4 consists of a homogeneous liquid phase (’) with compositions x(z; t) and a homogeneous vapor phase (”) with composition y(z; t). The function j(x; y) describes the mass transfer rate between the two phases. Compared to earlier studies on nonreactive

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S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

xi (z = 1; t) = xi; in (t);

(1) N



r
1 @yi @yi + = ji (x; y) + 

ij rj

(y); @t A @z

j=1

yi (z = 0; t) = yi; in (t); i = 1(1)Nc − 1;

(2)

z ∈ [0; 1]

with the dimensionless time and spatial coordinates z; t de4ned by zN z= ; l

t=

tNJin
ln


(3)

and the dimensionless quantities =

Fig. 4. A single section of a (reactive) distillation column.

distillation (Marquardt, 1988, 1990; Kienle, 1997, 2000a) the model is extended by suitable reaction rate vectors r
(x) and r

(y) accounting for potential chemical reactions in each phase. It should be noted that the nonreactive case or the case where reaction is only occurring in one phase are included as special cases in this very general setting. In reactive distillation the main reaction is typically taking place in the liquid phase. For uncatalyzed or homogeneously catalyzed processes it is taking place in the bulk of the liquid phase. For heterogeneously catalyzed reactions it is taking place at the surface of a heterogeneous catalyst. In the latter case the reaction rate r
(x) can be viewed as a quasihomogeneous rate expression. Di>usional transport resistances inside a porous catalyst are therefore not covered by this approach. In addition, (unwanted) reactions can take place in the vapor phase. A typical example is dimerization of carboxylic acids in esteri4cation processes (see e.g. Barbosa and Doherty (1988) and references therein). Assuming • • • •

constant molar Fow rates and holdups in both phases, negligible axial dispersion, constant pressure and thermal equilibrium,

the dimensionless dynamic model of the column section follows from the material balances of Nc − 1 linearly independent species in both phases N


r
@xi @xi − = −ji (x; y) + 
ij rj
(x); @t @z

j=1

n

; n


A=

Jin
; Jin



(4)

which are the ratios of the molar holdups and the molar convective Fow rates in both phases. A further simpli4cation of models (1) and (2) is possible if phase equilibrium is assumed. In phase equilibrium the composition yi of phase (”) can be calculated from the composition xi of phase (’) and the given pressure. Further, the mass transfer rates ji become indeterminate and are therefore eliminated from the model equations by summation of Eqs. (1) and (2). The resulting pseudohomogeneous model reads   @ 1 @ ( yi (x) + xi ) + yi (x) − xi @t @z A


=

Nr
j=1






ij rj
(x)

+

Nr




ij rj

(y):

(5)

j=1

In the remainder, application of the pseudohomogeneous model will be discussed for nonreactive as well as reactive distillation processes. 2.2. Nonreactive distillation Let us 4rst focus on nonreactive distillation. Without chemical reactions the pseudohomogeneous model reads   @ @ 1 ( yi (x) + xi ) + yi (x) − xi = 0: (6) @t @z A This represents a homogeneous set of quasilinear partial differential equations of 4rst order. For this type of problem an almost complete mathematical theory based on the method of characteristics is available (see e.g. Smoller, 1994; Rhee et al., 1986, 1989), which was rigorously applied to distillation processes (Marquardt, 1988; Kienle, 1997). In particular, it can be shown that each concentration and temperature

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

pro4le consists of di>erent fronts. The number and type of these fronts depends on the number of components in the system and the properties of the equilibrium function y(x). For ideal and moderately nonideal mixtures the concentration and temperature pro4les in each column section will consist of Nc − 1 fronts. Additional fronts are possible for highly nonideal, azeotropic mixtures (Kienle, 1997). Under steady-state conditions, at most one of these wave fronts can be located at the middle of the column section, whereas the others are located at the boundaries, where they can overlap. For ideal and moderately nonideal mixtures, these concentration fronts are of the constant pattern type, i.e. after a disturbance some of these fronts—depending on the sign and magnitude of the disturbance—will travel through the column with initially constant shape and velocity. Finally, they are stopped by the repelling inFuence of the system boundaries and settle down to a new steady state. Such a situation, with a single wave in each column section was illustrated for a binary problem in Fig. 2. Illustrating examples for the nonreactive multicomponent case are given in (Kienle, 2000a). 1 For nonideal mixtures, so-called combined waves are possible where only a part of the wave front represents a constant pattern wave, whereas the other part has a variable shape (Kienle, 1997). In the binary case, all types of wave propagation phenomena can be studied graphically in the McCabe– Thiele-diagram (Marquardt, 1988). A stable wave pinch at the bottom of a distillation column is the lowest boiling component or azeotrope in the system. A stable wave pinch at the top of the column is the lightest boiling component or azeotrope in the system. An isolated constant pattern wave is represented by a time invariant straight line connecting the feed composition with the respective pinch composition at the bottom or the top. Hence, for ideal and moderately nonideal mixtures all waves are of the constant pattern wave type. A typical situation for a rectifying column is illustrated schematically in Fig. 5a. For nonideal mixtures, inFection points may occur. Here, the wave solution is represented by the convex hull of the equilibrium function (Kienle, 1997). A typical situation is illustrated in Fig. 5b. Here the solution is a combined wave. The straight line connecting the feed with the tangent point represents a constant pattern wave and the second part along the equilib1 It should be noted that constant pattern wave solutions of the continuously distributed equilibrium model (6) are discontinuities or so-called shock waves. Any mass transfer resistance, axial dispersion or discretization of the underlying PDEs will results in constant pattern waves with 4nite slope. Throughout the paper equilibrium cell models are used for the numerical simulation of distillation (Figs. 2–7) and chromatographic columns (Figs. 10–15). These cell models can be interpreted as discrete approximations of the corresponding continuously distributed equilibrium models and therefore lead to constant pattern waves with 4nite slope. They approach discontinuities as the number of cells or stages tends to in4nity.

905

y+

x+

1

1

y−

y

0 (a) 0 1

z

x−

x+

0 1

x

0 x−

x

0 x−

x

x+

1

0 x−

x Az x x+

1

1

1

y+ y− z

y

0 (b) 0

x−

x+

0 1

x

1

1 y+

y

x−

0 (c)

z

y−

0

x + xAz x

0 1

Fig. 5. Possible wave solutions for a binary rectifying column: (a) ideal mixture, (b) nonideal zeotropic mixture with inFection point, (c) nonideal azeotropic mixture with minimum azeotrope.

rium line represents a dispersive or spreading wave, which changes its shape. A third scenario is found in azeotropic mixtures and is illustrated in Fig. 5c. Here, the stable pinch at the top is the light boiling minimum azeotrope. Since the inFection point of the equilibrium line and the azeotropic point usually do not coincide, wave solutions are likely to be of the combined wave type as illustrated in the 4gure. In all cases the propagation velocity of the constant pattern wave is directly related to the slope of the straight line Qy=Qx, the internal Fow rate ratio A and the holdup ratio in the column section according to A(1 + w) Qy = : 1 − A w Qx

(7)

This equation can be interpreted as a global material balance across the constant pattern wave (Marquardt, 1988). The same sort of construction applies to a stripping column. Here, wave solutions are represented by the convex hull connecting the feed point with the heavy boiling component or azeotrope.

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S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

A is intermediate boiling 1

feed

z [−]

Y [−]

0

0 0

(a)

feed

1

1

X [−]

0

X [−]

1

A is light boiling 1

feed

z [− ]

Y [−]

0

0

feed 0

(b)

1

1

X [−]

0

X [−]

1

A is heavy boiling 1

z [−]

Y [−]

0

0

(c)

feed 0

X [−]

1

1 0

feed X [ −]

1

Fig. 6. Equilibrium function Y (X ) for reactive vapor liquid equilibria with a chemical reaction 2A
B + C in the liquid phase (left). Characteristic wave patterns in a reactive distillation column in transformed variables (right). (a) component A has intermediate volatility, (b) component A has highest volatility, (c) component A has lowest volatility.

The same sort of construction also applies to a distillation column with rectifying and stripping section. Additional complexity will rise from the mutual interaction of the waves located in the stripping and the rectifying section as was shown in Fig. 2, for example. In the multicomponent case the construction of wave solutions is more involved and relies on the representation of time-invariant wave solutions in the concentration state space (Kienle, 1997), which is also frequently called the hodograph space. We will come back to this point in the third section of this paper on chromatographic processes.

2.3. Reactive distillation For reactive distillation processes the general model (5) has to be considered. In contrast to the nonreactive case this represents a nonhomogeneous set of quasilinear partial di>erential equations of 4rst order. For this type of equations not much theory is available for our purpose. However, in many cases the chemical reactions are rather fast compared to the other processes in such a column, so that in addition to phase equilibrium, chemical equilibrium may be assumed. Typical processes, which are operated close to

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

907

A is intermediate boiling 0

0

z [−]

0

1 0

(a)

1

1

0

1

1

0

1

0

1

A is light boiling 0

0

z [−]

0

1 0

(b)

1

1 0

1

1

A is heavy boiling 0

0

z [−]

0

(c)

1

0

x1 [−]

1

1

0

x2 [−]

1

1

0

x3 [−]

1

Fig. 7. Wave patterns in a reactive distillation column corresponding to the three di>erent cases in Fig. 6. Physical concentration variables.

chemical equilibrium are some esteri4cation and etheri4cation processes (Doherty and Buzad, 1992). In general, the conditions for chemical reaction equilibrium are Nc


i

ij = 0;

∀j = 1; : : : ; Nr
;

(8)

i



ij = 0;

∀j = 1; : : : ; Nr

:

(9)

i=1 Nc
i=1

Therein i
and i

are the chemical potentials of phase (’) and (”). The above equilibrium conditions represent Nr
+Nr



algebraic constraints, which reduce the dynamic degrees of freedom of the system to Nc − Nr
− Nr

− 1. In an open system in chemical equilibrium the kinetic rate expressions for the reaction rates r
and r

are indeterminate. The balance equations (5) and the equilibrium conditions (8) and (9) represent a system of Nc − 1 + Nr
+ Nr

equations to determine Nc −1 independent mole fractions xi and the Nr
+ Nr

unknown reaction rates in both phases. However, since the balance equations (5) are linear in the unknown reaction rates, these can be easily eliminated by linear combinations of the balance equations. Let us illustrate this with a simple example. Consider a ternary system with a single reaction in the liquid phase with reaction rate r
. The material balances for

908

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

the two independent components are   @ @ 1 ( y1 + x1 ) + y1 − x1 = 
1 r
; @t @z A   @ @ 1 ( y2 + x2 ) + y2 − x2 = 
2 r
: @t @z A

(10)

which is again a homogeneous partial di>erential equation. Upon introducing new transformed concentration variables X = x2 −

x1 ;

Y = y2 −


2 
1

y1

Eq. (12) is obtained as   @ 1 @ ( Y + X ) + Y − X = 0: @t @z A

Parameter

(a) A has intermediate volatility

1 2 3 K xfeed Number of stages Feed stage Feed Fow rate ReFux/destillate Vapor Fow rate

3.0 5.0 5.0 3.0 1.0 1.0 1.0 x1 = 1:0, x2 = 0:0, x3 = 0:0 40

(11)

To eliminate the unknown reaction rate, the 4rst equation is multiplied by 
2 =
1 and subtracted from the second      @ 

y2 − 2
y1 + x2 − 2
x1 @t 1 1      @ 1 
2 
2 y2 −
y1 − x2 −
x1 + = 0; (12) @z A 1 1


2 
1

Table 1 Parameters for the reactive distillation column examples shown in Figs. 6–8

20 2.0 1.25 2:25 → 2:3

(b) A has highest volatility

1:5981 → 1:4 3:2942 → 3:1925

(c) A has lowest volatility 1.0 5.0 3.0

3:5 → 3:9 3:2942 → 3:2

The last two rows show the initial step changes of the operating parameters used to illustrate the transient behavior in the above 4gures.

(13)

(14)

This equation is completely analogous to the nonreactive problem (6). An extension of this procedure to multireaction systems is straight forward. Hence, in chemical equilibrium the reactive problem is equivalent to a nonreactive problem in a reduced set of Nc − Nr
− Nr

− 1 transformed concentration variables. These transformed concentration variables were introduced by Doherty and co-workers for the steady-state design of reactive distillation processes (Barbosa and Doherty, 1987a). An extension to the multireaction case was given by Ung and Doherty (1995). As shown above, the transformed concentration variables can be directly generalized to the dynamic problem considered here. In analogy to the nonreactive case, solutions of this homogeneous partial di>erential equation will now depend on the properties of the nonlinear function Y (X ), which contains information on the combined phase and reaction equilibrium. Unfortunately, this function is only implicitly de4ned due to the nonlinear equilibrium relations (8) and (9). Only in special cases an explicit expression can be found. However, it is always possible to calculate this function numerically. Afterwards, it can be used for the construction of wave solutions the very same way as the equilibrium function yi (x) in the nonreactive case. Let us illustrate this with a simple example. Consider an ideal ternary mixture with components A, B and C with constant relative volatilities i  i xi yi = (15) Nc −1 1 + k=1 (k − 1)xk undergoing a single equimolar reaction in the liquid phase according to 2A
B + C. In the remainder component A

has index ‘1’, component B has index ‘2’ and component C has index ‘3’. For an ideal mixture the chemical equilibrium condition (8) reads x 2 x3 K= 2 : (16) x1 From Eq. (13) the following de4nitions of the transformed variables are found: x1 y1 X = x2 + ; Y = y2 + : (17) 2 2 A value of the transformed variables of zero corresponds to pure product C, a value of 1 to pure product B and a value of 0.5 to pure reactant A. The transformed equilibrium function is computed from Eqs. (15)–(17) and is illustrated in Fig. 6 for three di>erent cases. In Fig. 6a component A is the intermediate boiling component, in Fig. 6b component A is the heavy boiling component and in Fig. 6c component A is the light boiling component. The 4rst case corresponds to an ideal binary nonreactive system, the second to a binary system with a minimum azeotrope and the third to a binary system with a maximum azeotrope. It should be noted that the phase equilibrium is ideal. Consequently, the azeotropic points are so-called reactive azeotropes where reaction and distillation compensate each other (Barbosa and Doherty, 1987b). Like in a nonreactive column these azeotropes can not be crossed at steady state in a fully reactive column. Consider for example a single feed fully reactive distillation column with pure reactant A as feed, corresponding to a value of the transformed variable X of 0.5. The corresponding wave patterns in transformed variables for the three di>erent cases are illustrated in the right column of Fig. 6 while Fig. 7 shows the same pro4les in physical coordinates. The physical parameters for this example are given in Table 1. Like in the nonreactive binary case the composition pro4les consist of a single front in each section. Only in the

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

A is light boiling

A is intermediate boiling

Y[ ]

Y[ ]

Y[ ]

1

−

−

−

0

(a)

A is heavy boiling

1

1

0 0

1

X [−]

(b)

909

0 0

1

X [− ]

(c)

0

1

X [ −]

Fig. 8. InFuence of chemical equilibrium constant K on the reactive vapor liquid equilibria of Fig. 6. K = 10−3 ; 10−1 ; 100 ; 101 ; 103 . The arrows point in the direction of increasing K.

4rst case total conversion is possible and pure components can be obtained at the bottom and the top. In the second case, only pure component C can be obtained at the bottom, whereas always a mixture of all three components is obtained at the top and vice versa in the third case. Due to the occurrence of an inFection point, which does not coincide with the azeotropic point in cases 2 and 3, combined waves occur. From the equilibrium functions shown in Fig. 6 it can be seen that the shock cord connecting the feed point with the point of tangency di>ers only little from the equilibrium line and thus the sharpening tendency of the corresponding shocks is weakly pronounced. This explains the Fat initial pro4les in the top and bottom sections for cases 2 and 3, respectively. As a consequence the expansive part of the combined waves are hardly visible in Figs. 6 and 7. At the end of the transients illustrated in these 4gures, the pro4les are pushed against the boundary of the corresponding column sections and steep up. In Fig. 8 the combined equilibrium functions for the three di>erent cases are plotted for 4ve di>erent values of the chemical equilibrium constant K. The arrows point in the direction of increasing K. In case (a), where the educt is intermediate boiling component, no reactive azeotrope occurs. For very low values of K the equilibrium line is bulged and touches the bisection line for the limiting case K → 0. And therefore also admits combined wave solutions. For high values of K the equilibrium function is convex and thus admits only constant pattern wave solutions. For the other two cases, i.e. the educt is the highest or lowest boiling component, reactive azeotropy occurs for all values of K. A rigorous proof of that was already given by Barbosa and Doherty (1988). In the second case the reactive azeotrope is always a minimum azeotrope, which moves from the pure reactant A (X = 0:5) to pure product B (X = 1) as K is increased. In the third case it is always a maximum azeotrope which moves from the pure reactant A to pure product C (X = 0) as K is increased. In summary we 4nd, that chemical reactions can introduce inFection points and reactive azeotropy (selectivity reversal) in the simultaneous phase and reaction equilibrium of an

ideal mixture, and that this has severe inFuence on steady state as well as dynamic transient behavior. Extensions to the multireaction case and to systems with more than one degree of freedom are straight forward. However, these will not be explicitly discussed here. Instead, we will have a look at corresponding problems in reactive as well as nonreactive chromatographic processes in the next section. This discovers some striking parallels to reactive distillation, chemical engineers, to our knowledge, were so far not aware of and therefore opens the view for a uni4ed approach to reactive separation processes with simultaneous phase and reaction equilibrium. 3. Chromatographic processes 3.1. Model equations In this section a moving bed chromatographic process is considered as depicted in Fig. 9. The system consists of a Fuid phase with molar concentrations ci , which is moving with interstitial velocity vf , and a solid phase with molar concentrations qi , which is moving counter-currently with velocity vs . The special case of 4xed-bed chromatography is included for vs = 0. Both phases exchange mass across the phase boundary. In addition, chemical reactions in both phases are taken into account. As a special case the nonreactive case is included. In the reactive case the main reaction can take place in each of both phases. For uncatalyzed or homogeneously catalyzed processes the reaction will occur in the Fuid phase. A practical example is the separation of binaphtol enantiomers (Baciocchi et al., 2002), where unwanted dimerization reactions occur mainly in the Fuid phase. In other cases the adsorbens can simultaneously catalyze the chemical reaction, which is then taking place in the solid phase. Practical examples are esteri4cation processes catalyzed by ion exchange resin (see e.g. Lode et al., 2001). Additional side reactions can occur in each phase. It is worth noting that the con4guration in Fig. 9 is in principal completely equivalent to the

910

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

and the dimensionless quantities =

1− ;

=

1 − vs ; vf

(21)

which are the volume ratio and the volumetric Fow rate ratio of both phases. The tilde in Eqs. (18) and (19) indicates that mass transfer and reaction rates are now in volumetric units. The corresponding pseudohomogeneous model is @ @ (qi (c) + ci ) + (qi (c) − ci ) @t @z


=

Nr







ij r˜
j (c)

+

Nr


j=1



ij r˜

j (q);

(22)

j=1

which is completely equivalent to Eq. (5). In the pseudohomogeneous model qi is the equilibrium composition of the solid phase, which can be calculated from the composition of the Fuid phase by means of the adsorption isotherms. 3.2. Nonreactive chromatography In the nonreactive case, Eq. (22) reduces to a set of homogeneous quasilinear partial di>erential equations of 4rstorder according to

Fig. 9. Moving bed chromatographic reactor.

reactive distillation process illustrated in Fig. 4. The solid and the Fuid phase in Fig. 4 correspond to the vapor and the liquid phase in Fig. 4. To work out the analogy between both processes the orientation of the spatial coordinate in Fig. 9 is chosen in the direction of the solid phase. Assuming • isothermal operation, • constant velocities vf and vs and • negligible axial dispersion the model equations follow from the material balances of the Ns independent solutes in both phases N


r
@ci @ci − 
ij r˜
j (c); = −j˜i (c; q) + @t @z

j=1

ci (z = 1; t) = ci; in (t);

(18) N



r
@qi @qi 

ij r˜

j (q);  + = j˜i (c; q) + @t @z

j=1

qi (z = 0; t) = qi; in (t); i = 1(1)Ns ;

(19)

z ∈ [0; 1]

with the dimensionless time and spatial coordinates z; t de4ned by tNvf zN (20) z= ; t= l l

@ @ (qi (c) + ci ) + (qi (c) − ci ) = 0: @t @z

(23)

For this type of problem an almost complete mathematical theory with application to 4xed bed and moving bed chromatographic processes was given by Rhee et al. (1986, 1989). Equivalent results were obtained with a di>erent approach based on physical insight by Hel>erich and Klein (1970). A nice review of the latter approach was given more recently in a series of articles (Hel>erich and Carr, 1993; Hel>erich and Whitley, 1996; Hel>erich, 1997). Let us brieFy summarize some of the main results, we shall need in the remainder. Emphasis in the books and papers cited above is on Langmuir systems, i.e. systems, whose sorption equilibrium can be described by the competitive Langmuir isotherm qi =

1+

a i ci Ns

k=1

bk ck

;

∀i = 1(1)Ns :

(24)

For simplicity, consider 4rst a 4xed-bed chromatographic column. For Langmuir systems, any concentration step change introduced at the entrance of an initially unloaded chromatographic column will be resolved in a sequence of at most Ns shock waves si as illustrated in the middle diagram of Fig. 10 for a two solute system. A reverse step change is introduced, when a totally loaded column is purged with pure solvent. This reverse step change is resolved in at most Ns dispersive or spreading waves ri as illustrated in the bottom diagram of Fig. 10. Combined waves are not possible for Langmuir systems.

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

911

0.4 s

c

2

1

s

0 0

r2

2

r1 0.4

c1 [−]

0.35 s1

s2

c

s

1

0 1

0 z [−]

0.35

c

r2

r2

r

1

0 1

0 z [−]

Fig. 10. Wave solutions for Langmuir systems. Step changes of the concentration in the inlet. Top: wave solutions in the hodograph space. Center: concentration pro4les for the loading of an initially empty bed. Bottom: purge of a completely loaded bed with pure solvent. Solid line—c1 , dashed dotted line—c2 .

These solutions are easily constructed in the concentration phase space or the so-called hodograph space. Both types of wave solution represent Ns time-invariant curves in the hodograph space. For Langmuir systems shock and spreading wave curves are straight lines in the hodograph space and therefore coincide. They do not depend on the operating conditions. They are given by the right eigenvectors rk of the Jacobian matrix of the phase equilibrium relation @q rk = !k rk ; @c

∀k = 1(1)Ns :

(25)

The initial condition and the boundary condition of a step change represent points in this hodograph space. Any wave solution can be constructed from the path grid of the

eigenvectors by connecting the initial point with the boundary point as illustrated at the top of Fig. 10. The construction is done in the following way: We start at the point, which represents the boundary condition, and follow the path belonging to the smaller eigenvalue of Eq. (25). The paths belonging to the smaller eigenvalues are indicated by the dotted lines. Afterwards, we switch to the solid line running through the point, which represents the initial condition. The solid lines represent the paths belonging to the larger eigenvalue of Eq. (25). If we follow the path in the direction of the arrows, the corresponding wave solution is a spreading wave and otherwise a shock wave. It is worth noting that the same procedure also applies to multicomponent distillation processes with constant relative

912

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

0.35

c

s2

s2 r

r2

2

r1

0

s1

s1

r1

1

0 z [−]

Fig. 11. Wave solutions for Langmuir systems. Two snapshots at di>erent times after a pulse injection are shown. At the left: shortly after injection. The front has resolved into two shocks, while the back is just about to resolve into two rarefaction waves. Further to the right: the pulse has resolved into two separate pulses containing pure components. Solid line—c1 , dashed dotted line—c2 .

Table 2 Parameters for the nonreactive chromatography examples shown in Figs. 10 and 11 Parameter

Value

a1 = b 1 a2 = b2 cfeed Number of stages Bed length Pulse length (Fig. 11)

3.0 1.0 c1 = 0:25, c2 = 0:25 400 100.0 length units 25.0 time units

volatilities discussed in the previous section (Kienle, 1997). Here, the stable pinch at the top is given by the lightest boiling component and the stable pinch at the bottom is given by the heaviest boiling component. Any pulse injection to an unloaded bed can be interpreted as a sequence of the two step changes discussed above. This situation is illustrated in Fig. 11 and gives rise to Ns shock waves in the front and Ns spreading waves in the rear. According to the elementary rules of wave interactions these are 4nally resolved into Ns pure component pulses with a shock in front and a spreading wave in the rear (Rhee et al., 1989). The parameters for Figs. 10 and 11 are given in Table 2. Additional complexity will rise for systems with more complicated sorption isotherms, involving inFection points and adsorptivity reversal (azeotropy). For these type of mixtures spreading waves and shock waves are usually represented by curved lines in the hodograph space and will there-

fore not coincide. Spreading waves follow from the eigenvectors of the phase equilibrium and do not depend on the operating conditions. In contrast to this shock waves follow from the global material balances across the shock and will therefore depend on operating conditions. However, as proposed by Hel>erich and Klein (1970) at least a qualitative picture of the overall situation can be gained from the path grid of the eigenvectors. The reason for that is that the shock wave curves and the eigenvectors are always tangent at the starting point. New phenomena compared to Langmuir systems are: the existence of combined wave solutions for adsorption isotherms with inFection points and limitations on feasible product composition due to adsorptivity reversal similar to azeotropic distillation. Examples for the latter were treated by Basmadjian and Coroyannakis (1987) and Basmadjian et al. (1987a,b), among others. 3.3. Chromatographic reactors For chromatographic reactors the nonhomogeneous equation (22) has to be considered. For suHciently fast chemical reactions, reaction equilibrium can be assumed in addition to phase equilibrium and we can proceed as in the last section on reactive distillation. The unknown reaction rates are eliminated by linear combination of the balance equations (22) and a reduced set of Ns −Nr
−Nr

homogeneous partial di>erential equations in transformed variables is obtained. Consider for example a three solute system with a single reaction in the Fuid phase with reaction rate r˜
. The balances

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

A has intermediate adsorptivity 0.4 1

0 r 0 1

r2

s2

2

r1 0 0

0.4 C1

C

r

2

2

2

s2

C

A has lowest adsorptivity 0.4 s1 s

s1 C

2

s

A has highest adsorptivity 0.4

r2 0 r 0 1

0.4

0.4

0.4

C

s2

s1

s1

s1

s2

s

2

0 1

0

0.4 C1

pure educt fed to unloaded columns

0.4

913

0 1

0

0 1

0

pure solvent fed to completley loaded columns 0.4

0.4

0.4

r

r

2

r2

C

2

r1

r1

0 1

0 z [−]

(a)

0 1 (b)

r1 0 z [−]

0 1 (c)

0 z [−]

Fig. 12. Wave solutions for Langmuir systems with reaction 2A
B + C in transformed variables. Step changes of the concentration in the inlet. Top: wave solutions in the hodograph space. Center: concentration pro4les for the loading of an initially empty bed. Bottom: purge of a completely loaded bed with pure solvent. Solid line—C1 , dashed dotted line—C2 . (a) component A has intermediate adsorptivity, (b) component A has highest adsorptivity, (c) component A has lowest adsorptivity.

equations for the three independent solutes are @ @ (qi (c) + ci ) + (qi (c) − ci ) @t @z = 
i r˜
(c);

i = 1; 2; 3:

(26)

Let component ‘1’ be the reference component. The reduced set of equations is obtained by multiplying the 4rst equation by 
2 =
1 or 
3 =
1 and subtracting it from the second and the third equation to eliminate the unknown reaction rate. This yields @ @ (27) (Qi + Ci ) + (Qi − Ci ) = 0; i = 1; 2; @t @z which is completely equivalent to a nonreactive problem in a reduced set of transformed variables 

C1 = c2 − 2
c1 ; C2 = c3 − 3
c1 ; 1 1 Q1 = q2 −


2 q1 ; 
1

Q2 = q3 −


3 q1 : 
1

(28)

Let us further illustrate this. In analogy to the previous section on reactive distillation a ternary system with compo-

nents A, B, C undergoing a chemical reaction 2A
B + C in the Fuid phase will be considered. 2 The sorption equilibrium is given by the Langmuir isotherm (24) and the chemical equilibrium condition is given by c2 c3 (29) K= 2 : c1 In comparison to the corresponding problem in reactive distillation the problem has two dynamic degrees of freedom. The two transformed variables (26) for the speci4c reaction are c1 c1 (30) C1 = c2 + ; C2 = c3 + ; 2 2 q1 q1 Q1 = q2 + ; Q2 = q3 + : (31) 2 2 Solutions for simultaneous phase and reaction equilibrium will now depend on the properties of the transformed equilibrium functions Q1 (C), Q2 (C). These can be calculated 2

The corresponding problem, where the reaction is taking place in the solid phase is discussed in Appendix A.

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918 A has intermediate adsorptivity 0.4

C

r2 s2

(a)

0

s2

r

s1

2

r1

s

1

r1 1

0 A has highest adsorptivity

0.4

r

2

C

from Eqs. (31), (30), (29) and (24). Their properties are illustrated by means of the path grid of the eigenvectors in the hodograph space of the transformed variables C1 and C2 in Fig. 12. In this 4gure, pure reactant A is represented by the bisection line, pure reactant B by the horizontal axis and pure reactant C by the vertical axis. Again three di>erent cases are distinguished in Fig. 12. In the 4rst column A has intermediate adsorptivity, in the second column highest and in the third column lowest adsorptivity. Like in reactive distillation the 4rst case is almost completely equivalent to a nonreactive problem with ideal behavior, in this case a two solute Langmuir system as illustrated for example in Figs. 10 and 11. The only di>erence to the nonreactive Langmuir system is the existence of inFection lines indicated by the dashed dotted lines in hodograph plots. These inFection lines may give rise to combined waves. The second case always has a line of reactive adsorptivity reversal below the feed line, whereas the third case always has such a line of reactive adsorptivity reversal above the feed line no matter what parameter values are considered. A detailed mathematical analysis is given in Appendix A. As indicated in Fig. 12 with the shaded region, these adsorptivity reversals result in similar product limitations as reactive azeotropy in reactive distillation. This is further illustrated for a chromatographic 4xed bed reactor in Fig. 13. At the beginning a pulse of pure reactant A is injected to the column. In all three cases the 4nal pattern will consist of two pulses like in a nonreactive two solute problem. However, only in the 4rst case, where A has intermediate adsorptivity, total conversion is possible and two pulses with pure products B and C are obtained. In the second case, where A has highest adsorptivity, only pure component C can be obtained in the front pulse of the 4nal pattern, whereas the second pulse will always contain a mixture of azeotropic composition, no matter how long the column is. In the third case, where A has the lowest adsorptivity, only pure component B can be obtained in the rear pulse of the 4nal pattern, whereas the front pulse will always contain a mixture of azeotropic composition, no matter how long the column is. The corresponding parameters for Figs. 12 and 13 are given in Tables 3 and 4. In principal, another nonreactive chromatographic column section can be added to overcome these limitations on product purities. In the nonreactive column section all three components are separated and the remaining reactant can be recycled to the reactive column section. However, we doubt that this is of much advantage compared to a conventional process, where reaction and separation are carried out in di>erent processing units. Similar patterns of behavior can be observed for a counter current moving bed chromatographic reactor as illustrated in Fig. 15. Those patterns can easily be deduced from the pathgrids in Fig. 12. In Fig. 15a the educt has intermediate adsorptivity. From the pathgrid shown in Fig. 12a it can be seen that both products can be obtained as pure components since the waves from and to the feed point always connect

s

1

s

s

1

r

s

r2

2

r

2

1

1

(b)

0

1

0 A has lowest adsorptivity

0.4 s

1

r

2

C

914

s r

1

2

r1

r2

s1

s

2

0 (c)

1

0 z [−]

Fig. 13. Pulse patterns in a 4xed bed chromatographic reactor for the di>erent cases in Fig. 12. Transformed concentration variables: Solid line—C1 , dashed dotted line—C2 . Table 3 Parameters for the chromatographic reactor examples shown in Figs. 12–14 Parameter

(a) A has intermediate adsorptivity

(b) A has highest adsorptivity

a1 = b 1 a2 = b 2 a3 = b 3 K cfeed Number of stages Bed length Pulse length

3.0 5.0 5.0 3.0 1.0 1.0 1.0 c1 = 0:5, c2 = 0:0, c3 = 0:0 2000 80.0 length units 20.0 time units

(c) A has lowest adsorptivity 1.0 5.0 3.0

it to the pure component axis and the following waves go to the origin. Candidate product compositions are indicated by the small arrows in Fig. 12a, the precise values depend on the mixing in the feed node and hence on the Fow rates. In Fig. 15b the educt has highest adsorptivity and as already pointed out for the pulse experiment there is a fraction

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918 Table 4 Parameters for the moving bed chromatographic reactor results shown in Fig. 15 (a) A has intermediate adsorptivity

(b) A has highest adsorptivity

Fluid velocity zone I Solid velocity Feed rate RaHnate rate Extract rate Number of stages

5.0 3.2 1.0 1.0 0.375 0.15 1.825 1.25 2.6 1.1 100 in each zone

A has intermediate adsorptivity

(c) A has lowest adsorptivity

c

Parameter

915

5.0 1.0 1.055 1.2425 3.4

(a)

A has highest adsorptivity

c

Physical parameters and feed conditions are taken from Table 3.

A has intermediate adsorptivity 0.4

(b)

c

c

A has lowest adsorptivity

(a) 01

0 A has highest adsorptivity

0.4

(c)

extract

feed

raffinate

z [ −] c

Fig. 15. Characteristic steady-state pro4les in a moving bed chromatographic reactor for the di>erent cases in Fig. 12. Physical concentration variables: Solid line—component A, dashed dotted line—component B, dashed line—component C.

(b) 01

0 A has lowest adsorptivity

c

0.4

−

0 1

(c)

0

z[ ]

Fig. 14. Pulse patterns in a 4xed bed chromatographic reactor for the di>erent cases in Fig. 12. Physical concentration variables: Solid line—component A, dashed dotted line—component B, dashed line—component C.

that can only be obtained as azeotropic mixture. In this instance it is obtained at the extract port. Again in Fig. 12b candidate product compositions are marked by arrows. The fact that not both products can be obtained as pure components follows directly from the pathgrid since now one of

the two waves from and to the feed point does not hit the pure component axis but the line of the reactive selectivity reversal. In Fig. 15c the educt has lowest adsorptivity and now the azeotropic mixture is obtained at the raHnate port. In the pro4les in Section II, between extract and feed port the e>ects of a combined wave are evident. Besides getting only one product pure, the existence of a combined wave is also obvious from the pathgrid in Fig. 12c. The wave intersects a dashed dotted inFection line just above the arrow indicating the product composition on the C1 axis and since the wave up to that point is a spreading wave it must turn beyond that point into a combined wave. Note that having either the azeotropic mixture as extract or raHnate is in perfect analogy to the reactive processes discussed in the previous section. 4. Conclusion In this paper a new approach for understanding the dynamic behavior of reactive distillation processes and

916

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

chromatographic reactors was developed. The approach is based on the assumption of simultaneous phase and reaction equilibrium. Under these assumptions the reactive problem is equivalent to a nonreactive problem in transformed concentration variables and all results available on the dynamics and steady-state behavior of nonreactive separation processes can be directly applied. Additional complexity is introduced by the fact that chemical reactions can introduce inFection points and reactive adsorptivity reversals, which limit the feasibility of reactive separation processes similar to the nonreactive case. In nonreactive separation processes these phenomena are characteristic for highly nonideal phase equilibria. For illustration purposes, focus in this contribution was on simple model systems with ideal phase equilibria and a single reaction in one of the phases. The results for these simple model systems are somewhat obvious from the physical point of view. Therefore, it is important to note that the formalism developed in this contribution is also valid for multireaction systems and may uncover limitations, which are by no means obvious. Further work in this direction is in progress. The approach proposed above is easily extended to other reactive separation processes consisting of two phases with simultaneous phase and reaction equilibrium. Further examples are reactive extraction processes or sorption enhanced reaction processes, for example. It extents results and theoretical concepts originating from reactive distillation processes to other reactive separation processes, where similar patterns of behavior are found and thereby provides unifying concepts for reactive separation processes. An extension to processes with 4nite mass transfer kinetics is straight forward. As long as simple linear driving force rate laws are considered the reactive problem can be transformed to a nonreactive problem in transformed concentration variables with 4nite transfer rates. Therefore a similar inFuence of mass transfer kinetics like in the nonreactive case can be expected. Future work will mainly focus on 4nite reaction rates. From reactive distillation it is known that 4nite kinetics can introduce new patterns of behavior, like the appearance or disappearance of reactive boundaries (Venimadhavan et al., 1999; Chadda et al., 2001) or the appearance of new wave patterns and characteristics (Kienle and Marquardt, 2002).

ji j˜i K l n Nc Ns Nr qi Qi

Notation

Acknowledgements

A

The authors would like to thank Michael Mangold for providing Fig. 1. The 4nancial support of Deutsche Forschungsgemeinschaft (DFG) within the joint research project SFB 412 on computer aided modeling and simulation of chemical processes for process analysis, synthesis and operation is greatly acknowledged.

ci Ci J

ratio of the molar Fow rates in both phases, dimensionless molar concentration in the Fuid phase, kmol=m3 transformed concentration variable of the Fuid phase, kmol=m3 convective Fow rates, mol/s

rj r˜j tN t v w xi Xi yi Yi zN z

mass transfer rate, dimensionless mass transfer rate, kmol=m3 chemical equilibrium constant, dimensionless length, m molar holdup per length unit, kmol/m number of components number of solutes number of reactions molar concentration in the solid phase, kmol=m3 transformed concentration variable of the solid phase, kmol=m3 reaction rate, dimensionless reaction rate, kmol=m3 time, s time, dimensionless velocity, m/s wave velocity, dimensionless mole fraction of the liquid phase, dimensionless transformed concentration variable of the liquid phase, dimensionless mole fraction of the vapor phase, dimensionless transformed concentration variable of the vapor phase, dimensionless spatial coordinate, m spatial coordinate, dimensionless

Greek letters i  i  ij

constant relative volatility, dimensionless ratio of the molar holdups in both phases, dimensionless void fraction of the bed, dimensionless volumetric Fow ratio of solid and Fuid phase, dimensionless chemical potentials, dimensionless volume ratio of solid and Fuid phase, dimensionless stoichiometric coeHcients, dimensionless

Sub- and superscripts i; k j in








species reaction input Phase (
) Phase (

)

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

Appendix A. Reactive azeotropy in chromatographic reactors

In the remainder it is shown that the above reaction may introduce reactive azeotropy—i.e. a reactive adsorptivity reversal of the transformed concentration variables— depending on the order of the adsorptivities of components A–C. Focus will be on systems described by the competitive Langmuir isotherm Eq. (24). It should be noted that for this class of adsorption isotherms no adsorptivity reversal is possible in the nonreactive case. For the reactive case it will be shown that • no reactive adsorptivity reversal occurs if reactant A has intermediate adsorptivity, • reactive adsorptivity reversal always occurs if reactant A has highest or lowest adsorptivity. Further, some characteristic features of the reactive adsorptivity reversals in case 2 are given and it is shown that the behavior is completely equivalent to ternary reactive distillation with the same type of chemical reaction, although the number of degrees of freedom is increased by one in reactive chromatography compared to reactive distillation. In the remainder component A has index 1, component B has index 2 and component C index 3. From de4nitions (30) and (31) of the transformed concentration variables we 4nd c3 = C2 − c1 =2:

(A.1)

A (reactive) adsorptivity reversal of the transformed concentration variables is de4ned by 1; 2 =

Q1 =C1 =1 Q2 =C2

(A.2)

or Q1 Q2 = ; C1 C2

(A.3)

respectively. Upon substitution Eq. (31), the Langmuir isotherm (24) and Eqs. (A.1) into Eq. (A.3) an explicit relation between the concentration variable c1 of the reference component and the transformed concentration variables is obtained c1 =

2C1 C2 (a2 − a3 ) : C2 (a2 − a1 ) + C1 (a1 − a3 )

(A.4)

Finally, substitution of (A.4) and (A.1) into the chemical equilibrium condition (29), yields the following implicit equation for the adsorptivity reversal in the hodograph space of the transformed concentration variables Ci 4KC12 C22 (a2

Upon substituting % = C1 =C2

(A.6)

a quadratic equation for the unknown % is obtained

A.1. Reaction 2A
B + C in the >uid phase

c2 = C1 − c1 =2;

917

% 2 + K1 % + K 2 = 0

(A.7)

with coeHcients 4K(a2 − a3 )2 K1 = −2 + (a1 − a3 )(a2 − a1 ) and

(A.8)

K2 = 1:

(A.9)

Hence we conclude, for a given reaction mixture with given ai ; K the unknown % = C1 =C2 is also a constant. Therefore adsorptivity reversals according to Eq. (A.6) represent straight lines in the C1 ; C2 -plane through the origin with slope 1=%. Because of the quadratic equation (A.7), in principal, two di>erent slopes %1 and %2 are possible  −K1 ± K12 − 4 %1; 2 = : (A.10) 2 The solutions are symmetric according to %1 = 1=%2 :

(A.11)

However, as will be shown below at most one of these solutions is feasible with c1 ¿ 0. In the remainder three di>erent cases are distinguished according to the adsorptivities of the components (1) reactant A has intermediate adsorptivity according to a2 ¿ a1 ¿ a3 , (2) reactant A has highest adsorptivity according to a1 ¿ a2 ¿ a3 , (3) reactant A has lowest adsorptivity according to a2 ¿ a3 ¿ a1 . Without loss of generality, it is further assumed that product B has always higher adsorptivity than product C. For case (1), it is found from Eq. (A.8) that K1 ¿ − 2 for arbitrary equilibrium constants K ¿ 0. Hence, Eq. (A.10) has no feasible positive real solution. Consequently, a reactive adsorptivity reversal will never occur for feasible C1 ; C2 ¿ 0. Vice versa, it can be shown that in cases (2) and (3) K1 ¡ − 2 for arbitrary equilibrium constants K ¿ 0. Hence, Eq. (A.10) has two positive real solutions. One solution is greater than one and the other—the reciprocal according to Eq. (A.11)—is smaller than one. With regard to Eq. (A.4) it is found that for case (2) % ¿ 1 is the only feasible solution with c1 ¿ 0. In contrast to this, for case (3) % ¡ 0 is the only feasible solution with c1 ¿ 0. Therefore, in both cases, a reactive adsorptivity reversal will always take place. A.2. Reaction 2A
B + C in the solid phase

2

− a3 ) − (a1 − a3 )(a2 − a1 )

×[C12 − C1 C2 ][C22 − C1 C2 ] = 0:

(A.5)

The above relations are also valid for a corresponding chemical reaction in the adsorbed phase. The only di>erence

918

S. Gruner, A. Kienle / Chemical Engineering Science 59 (2004) 901 – 918

is that the reaction equilibrium has to be replaced by q 2 q3 (A.12) K= 2 : q1 Upon substituting the Langmuir isotherm (24) this relation can be written in the form a2 c2 a3 c3 K= : (A.13) (a1 c1 )2 Introducing K˜ = Ka21 =(a2 a3 ) Eq. (A.13) can be brought in analogous form to Eq. (29) c 2 c3 (A.14) K˜ = 2 : c1 Hence, all results from the previous section with reaction in the Fuid phase directly apply to the present case with reac˜ Hence, tion in the adsorbed phase by substituting K by K. no reactive adsorptivity reversal will occur, if reactant A has intermediate adsorptivity. Moreover, reactive adsorptivity reversal will always occur if reactant A has highest or lowest adsorptivity. The slope of the adsorptivity reversal will be altered by the factor of a21 =(a2 a3 ) in the de4nition of K˜ compared to the case with reaction in the liquid phase. References Baciocchi, R., Zenoni, G., Mazzotti, M., Morbidelli, M., 2002. Separation of binaphtol enantiomers through achiral chromatography. Journal of Chromatography A 944, 225–240. Barbosa, D., Doherty, M.F., 1987a. A new set of composition variables for the representation of reactive phase diagrams. Proceedings of the Royal Society of London A 413, 459–464. Barbosa, D., Doherty, M.F., 1987b. Theory of phase diagrams and azeotropic conditions for two-phase reactive systems. Proceedings of the Royal Society of London A 413, 443–458. Barbosa, D., Doherty, M.F., 1988. The inFuence of equilibrium chemical reactions on vapor-liquid phase diagrams. Chemical Engineering Science 43, 529–540. Basmadjian, D., Coroyannakis, P., 1987. Equilibrium-theory revisted— isothermal 4xed-bed sorption of binary systems. 1. Solutes obeying the Langmuir isotherm. Chemical Engineering Science 42, 1723–1735. Basmadjian, D., Coroyannakis, P., Karayannopoulos, C., 1987a. Equilibrium-theory revisted—isothermal 4xed-bed sorption of binary systems. 2. Non-Langmuir solutes with type I parent isotherms— Azeotropic systems. Chemical Engineering Science 42, 1737–1752. Basmadjian, D., Coroyannakis, P., Karayannopoulos, C., 1987b. Equilibrium-theory revisted—isothermal 4xed-bed sorption of binary systems. 3. Solutes with type I, type II and IV parent isotherms— Phase- separation phenomena. Chemical Engineering Science 42, 1753–1764. Chadda, N., Malone, M.F., Doherty, M.F., 2001. E>ect of chemical kinetics on feasible splits for reactive distillation. A.I.Ch.E. Journal 47, 590–601. Doherty, M.F., Buzad, G., 1992. Reactive distillation by design. Transactions of the Institute of Chemical Engineers 70, 448–458. Falk, T., Seidel-Morgenstern, A., 1999. Comparison between a 4xed-bed reactor and a chromatographic reactor. Chemical Engineering Science 54, 1479–1485. Fricke, J., Schmidt-Traub, H., 2003. A new method supporting the design of simulated moving bed chromatographic reactors. Chemical Engineering Progress 42, 237–248.

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