Ind. Eng. Chem. Fundam., Vol.
17, No. 1, 1978
1
The Chromatographic Reactor. A New Theoretical Approach Daniel Schweich’ and Jacques Villermaux Laboratoire des Sciences du Genie Chimique, C.N.R.S.-E.N.S.I.C.,Nancy, France
The coupling between a chemical reaction and chromatographic separation which occurs in a pulse-fed catalytic tubular reactor is investigated. For simplicity only dehydrogenation reactions are dealt with. A model which assumes a fast reaction rate compared with the residence times of the substances shows that the optimum operating parameters which maximize the yield depend only on the retention data obtained without any chemical reaction. The optimum yield can be obtained from experiments consisting of a single pulse injection of reactant. From these remarks, a simplified model describing the behavior of a single pulse injection of reactant is progressively set up. All but one of the parameters of the model can be predicted from knowledge of the retention data obtained in the case of no chemical reaction. The results of simulations of the simplified model are compared with experiments (dehydrogenationof cyclohexane), with the predictions of a rigorous model, and with published data. Good agreement is achieved when the unknown parameter is optimized. The value and the part played by this unknown parameter must be investigated in order to make an a priori prediction of the performance of a chromatographic reactor.
Introduction When the dehydrogenation of alkanes is carried out in a standard chemical reactor, thermodynamic equilibrium always limits the extent of reaction. If the products of the reaction are separated from each other, this limitation can be overcome. If a sample of alkane is fed into a chromatographic column packed with dehydrogenation catalyst, reaction and separation occur simultaneously. This is a chromatographic reactor in which reaction can be forced to completion as a consequence of Berthollet’s rules concerning the displacement of chemical equilibrium. The principle of the chromatographic reactor has been known for about 15 years, but only a few investigations dealing with industrial uses have been published by Matsen et al. (1965), Magee (1961), and Dinwiddie (1961). Most papers deal with analytical uses. Roginskii (1962), Hattori (1968),Langer (1972), and Tokehiko Furasawa et al. (1976) are interested in measurements of kinetic rates or determination of mechanisms. Magee (1963), Roginskii (1962 a,b), Chieh Chu (1971), Gore (1967), and Kocirik (1967) are interested only in the derivation of mathematical models. No conclusive quantitative comparison between model and experiments has been found. Two papers by Langer et al. (1969,1974) indicate the trend of research in this field. Statement of t h e Problem. General Assumptions The chemical engineer who thinks that a chemical reaction could be carried out in a chromatographic reactor has no simple mathematical tool to predict its performance from a small set of experiments. He must either perform further experimental investigations or use the mathematical models proposed in ihe literature. Unfortunately, most of them are difficult to use because of their mathematical structure. Our purpose is to propose a simple and quantitative descriptive method for the study of chromatographic reactors in order to predict their performance when physicochemical parameters, such as retention data, chemical equilibrium constant, and kinetic rate constant are known. Without reducing the generality of the results, only a simple reaction will be considered: AI ~1A2 A3. The performance of the reactor with this reaction can be measured by the improvement in yield in pulsed runs with respect to steady-state runs with the same feed. Other performance indices can be chosen such as the purity of the products or the yield of intermediate product in consecutive reactions.
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The following assumptions will be made throughout this paper in order to simplify the derivation of the models: infinite forward and backward reaction rates; instantaneous sorption rates; chromatographic model of Martin and Synge; dilute medium; product A2 is first eluted, then reactant AI and finally product As. Rigorous Model Two rules simplifying the investigation of the chromatographic reactor in repetitive pulsed runs will be laid down. These were established by simulations of a rigorous model. The term “rigorous model” is used only in the sense relative to the term “simplified model” used later. We shall suppose that adsorption isotherms are linear and that the mobile phase is incompressible. These two extra assumptions do not reduce the generality of the results but make them easier to obtain. The model is similar to that of Martin and Synge (1941). The column is regarded as being divided into J mixing cells of identical volume V / J (Figure 1).The choice of the mixing cell model instead of a continuous model is a consequence of the simplicity of the former. It gives rise to a set of ordinary differential equations instead of a pair of second-order partial differential equations which are more accurate but slightly more difficult to solve. The goal of this paper being to derive a simplified model, the accuracy of the mixing cell is sufficient. Moreover, it has been shown (Villermaux, 1972) that both models are equivalent in the representation of chromatographic processes where end effects are usually negligible. Let a, be the capacity factor of the columns and C, the concentration of substance A, in the mobile phase. In each mixing cell there are (C, ( V I J ) moles of A, in the mobile phase and (a,C,( V I J ) moles of A, in the stationary phase. The mass balance on component A, in cell number h is
The net chemical reaction rate r is written with a positive sign if the component A, is a reaction product and with a negative sign if Ai is a reactant. If reaction rate is omitted, the model reduces to that of Martin and Synge. Let us recall two important results which will be used later. The mean retention time of a chromatographic peak from an instantaneous feed is t ~ =j t o (1 a i ) ,where t o is the mean residence time of an unretained solute. The variance is uj2 = tR, 2/J.J is a parameter which accounts for broadening effects such as axial dispersion processes.
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01978 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
K
1
J
Figure 1. The Martin and Synge model. The column is regarded as being divided into J mixing cells (plates).The mobile phase concentration is uniform in each cell and in equilibrium with the stationary phase. 10
5
15
.
PERIOD
Figure 4. Yield improvement with respect to steady state as a function of pulse period. The shorter the pulse, the larger the yield improvement. The arrows point the approximate position of the maximum calculated with our formula (011 = 1,012 = 0, cy3 = 3, J = 100, Kt = 2500 L/mol, BCo/T = mol/L).
+ (1+ a2)J.G A = C2,kKt(1 + al)(l + ‘22) + (1+ %)[(I
+ a2) + C3,kKt(1 + ad] F = C2,k-1 - C3,k-1 - C2,k + C3,k G = CZ,k-I(KtC3,k-l + 1) - CZ,k(KtCB,k+ 1)
AT
Figure 2. Some pulsed feed shapes are shown in this figure. Duration of each pulse and time average feed concentration are fixed. The yield of the steady-state reactor fed with this time average concentration serves as a basis for yield improvement measurement (arbitrary units).
The boundary condition a t the inlet of the reactor is a periodic rectangular pulsating function as shown in Figure 2. This feed is defined by three parameters: 8, duration of the feed pulse; CO,reactant concentration in the feed pulse before chemical equilibrium is reached; and T , period of pulsations. As the reactor is not in steady-state conditions, the yield is defined as the ratio of the quantity of product A2 or A3 which flows out of the reactor during a period to the quantity of reactant fed during that same period. T o compare the yield reached during pulsed feed runs t o that reached in steadystate conditions, the average feed composition is kept constant: OCo/T = constant. Figure 2 shows some examples of such a feed. Choosing retention data (cy1 = 1,a2 = 0 , = 3, ~ J = 100) and the chemical equilibrium constant ( K , = 2500 L/mol), the set of eq 3 was numerically solved by the Euler method on a CII Mitra 15 computer. The time increment that ensured good results without requiring a lot of computer time was about 10-3to. The results are illustrated by the chromatograms (Figure 3) and yield improvements over the steady state (Figure 4). As pointed out by Chu, there is an optimum
When the forward and backward kinetic rates of the equilibrated reaction go to infinity, the net rate r goes t o a finite limit but it cannot be expressed in terms of the concentrations of the reacting species because the kinetic rate constants go to infinity. Thus, r must be eliminated from mass balance equations (1) by an appropriate linear combination, the missing equation being the thermodynamic equilibrium relationship. Clk = KtCPkC3k
(3)
(2)
After elimination of r and substitution of the expression for Clk, eq 1 and 2 lead to the following set of equations.
TIME
TIME
A
B
TIME
C
Figure 3. Simulated chromatograms. From top to bottom: reactant feed, reactant, product Az, product A3 chromatograms. A, Short period pulsed run at low feed concentration (T < Topt).Note the interferences between successive pulses which prevent zero concentration from being reached (simulation parameters same as in Figure 4, 0 = 0.25). B, Optimal pulse period ( T = Topt).Note the duplication of the reactant peaks. The rounded peak is the remaining part of the feed pulse. The sharp peak is reconverted reactant due to the reaction occurring between products of successive pulses. C, Low pulse frequency at high feed concentration ( T > Topt).Note the independence of successive pulse chromatograms as shown by the dotted vertical lines.
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978 3
A3
Figure 5. Estimate of the optimum period: during optimum runs overall chromatograms are in contact with one another. Toptis estimated by means of the time required to elute the chromatogram resulting from a single pulse.
pulse period Toptleading to a maximum yield improvement. The shorter the pulse, the greater the yield improvement. The chromatograms show that elution curves of consecutive pulses are independent of each other provided that the period is greater than Topt.They begin to overlap when T = Topt. Matsen et al. found the same qualitative results by experiments. The first simplifying rule can be derived from these remarks: provided that T 3 Topt,a superposition principle can be applied for obtaining the yield in repetitive pulse experiments from the behavior of the chromatographic reactor in single-pulse experiments. Repetitive pulse chromatograms are no more than single-pulse chromatograms which have been suitably translated. The time required to elute all the components from a single pulse is of great interest. I t is estimated as follows. Three contributions are shown in Figure 5, namely: (1)the difference between shortest and largest retention times without any chemical reaction: t R 3 - t ~ =2 to(a3 - a2);(2) the duration of the pulse 8 ; and (3) the broadening of the first and last peaks, which are measured by the standard deviation oL = [ t o ( l a L ) ] / 4 7This . expression is
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to + u3 = (2 + a2 + a3) 47J
It is supposed that reaction does not contribute to the broadening of the reactant peak. This assumption is true provided that product peaks are on either side of the reactant peak. We are now ready to pass on the property of this elution duration. The optimum period is such that a reconverted peak of reactant begins to appear in the chromatogram as shown in Figure 3b. This goal is achieved when the pulse period is equal to the elution duration of the chromatogram from a single-pulse experiment which allows the second simplifying rule to be laid down: the optimum period Toptdepends on the chromatographic characteristics of the components without any chemical reaction. A good approximation is
In Figure 4,the small arrows point to the approximate optimum period. The good agreement between the approximation and the true maximum proves the validity of the assumptions. The consequence of these two simplifying rules is that the main features of the optimal repetitive pulsed runs may be determined from those of single-pulse experiments. Before dealing with this new problem, it is of interest to examine the decrease of the improvement in yield when the period is sufficiently large. I t is not an intrinsic property of the chromatographic reactor, but merely a consequence of our choice of the performance index. As the time average feed composition is fixed, an increase in the period results in an increase in the concentration of each feed pulse (see Figure 2 ) . Thermodynamic equilibrium displacement rules show that
CHROMATOGRAPHIC REACTOR
SEMIBATCH REACTOR
I
TIME
.
Figure 6. Principle of the simplified model. Only the small slice containing the reactant is considered. It behaves like a semibatch reactor out of which reaction products flow.
a concentration increase is not favorable to dehydrogenation reactions. If the part of the curves of yield improvement which increases with the period is a consequence of the chromatographic separation, the decreasing part arises from the unfavorable effect of the thermodynamics. This remark has been pointed out by Gore (1967) as a limitation to the use of the chromatographic reactor.
Simplified Model The two simplifying rules allow us to state the problem as follows: how can the behavior of the chromatographic reactor in single pulse experiments be predicted? The rigorous model answers this question, but the mathematical simulations are rather complicated and time consuming. Moreover, in order to describe the physical situation, pressure gradient and nonlinear isotherms must be taken into account. As we are interested in overall characteristics, such as yield, the shape of the chromatogram is not important. T o answer our question we shall set up a new model where concentration gradients will be neglected. Let us recall that Hattori (1968) showed that the shape of the feed pulse is of little importance. We shall only consider the small slice of fluid containing the reactant peak. This slice can be considered as a small semibatch reactor moving down the column, the volume of which increases due to peak boradening. Products and reactant differ in their flow velocities so that the products A2 and A3 escape from this reactor (Figure 6). The model which will be developed considers only the slice where the reaction takes place and assumes uniform concentrations. This small fluid plug is defined by four paramaters: ni is the quantity of component i in the mobile phase; Fi‘ is the flow rate of component i escaping from the slice; Vis the volume of fluid in the slice; and n~ is the overall amount of fluid in the slice. The behavior of these parameters depends on chromatographic data which will be discussed below. Nevertheless, if pressure drop has to be taken into account, parameters related to a chromatographic model in compressible medium must be chosen. Villermaux (1972) has solved this problem as follows. The column is regarded as being divided into J mixing cells containing the same mass of matter MIJ. It can be shown that retention time and variance are unchanged whatever the pressure drop.
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Ind. Eng. Chem. Fundam., Vol. 17,’No. 1, 1978
Thus, chromatographic parameters are the same as in isobaric chromatography. When chromatography involves nonlinear isotherms, a, and J measured on an experimental curve vary with the amount of substance injected. Even if no physical meaning is linked to these data, they still describe chromatographic peaks. They will thus be used provided that nonlinearities are weak enough. Let us write the mass balance in the slice. The reactant is considered to be confined in a batch reactor -rV = (1
+ a l ) dndt
1
(4)
~
Products A2 and A3 have escaping flow rates which account for chromatographic separation rV = (1
dni + ai) + Fi’ at
nT = FT At
The flow rate F,’ is measured with respect to the moving reactant peak. It can be expressed with the aid of the flow rate F , measured with respect to the column wall. Let FT be the overall molar flow rate and n,lnT the mole fraction of component i. By definition
F, = F T -n, FT n , ( l + ai) (6) 1+a, nT Let us suppose that the whole mass is in the mobile phase. The mole fraction of component i is [ n l ( l + a , ) ] / nwhich ~ is the ratio of the overall amount of A, to the amount of fluid. F T / ( ~+ a,) is thus the apparent overall flow rate carrying component A,. For example, F T / ( 1 + a1) is the apparent overall flow rate carrying the reactant, that is the slice. I t follows that the apparent overall flow rate carrying A, with respect to the moving slice is =-
-FT -__
1+a,
1
+
“1
I t is the elution duration of a chromatographic peak flowing out of a column whose length is x. In chromatography, this is measured by the standard deviation g: At = A - a ; A is a constant of the order of 2 to 4. For example, if the peak is a rectangular pulse straightforward calculations lead to A = 2 d 3 . Its true value will be specified later. If reaction does not increase peak broadening, the expression for is known. I t is composed of two contributions. (1)The first, due to the column, is q 2 = t z / k for a component flowing out of the hth mixing cell at time t after the sample injection. t and k are not independent since the residence time in a section of column is proportional to the number of mixing cells the component has flowed through. Thus
+‘
nT
which can be written
F,
The main part of the work is now complete, but in order to solve the set of eq 9 some transient parameters, such as V and nT, must be given as functions of elution time. The rigorous model showed us that reaction does not increase the reactant peak broadening provided that products are eluted on both sides of the reactant. If broadening due to chromatography is negligible, the peaks do not get out of shape. Consequently, the quantity of fluid in the slice, n~ is constant while the volume V increases due to decrease in pressure along the column. Unfortunately, such an idyllic situation is rare. To account for band broadening, consider any point inside the column. The distance between this point and the inlet is taken to be X . Let us suppose the reactant slice takes a time A t to flow past this point, and the observation is made at time t after the sample injection. The amount of matter n~ is proportional to the apparent overall flow rate and to the flow duration
k=J
L
t o ( 1 + ad (2) The second r22 due to the shape of the injection is, for a rectangular pulse lasting 0 02 g22 = -
12
FT
and so we get
1+q
and the flow rate F,’
FT F,’ = n, (1 + (7) l + a , 1 FT tal nT Absolute values are introduced to get the right sign in the mass balance equations. So we obtain
I
(10)
Thermodynamic equilibrium is expressed by
Dividing nT by the overall concentration CT leads to the volume of the slice, V . The expression for CT is deduced from Darcy’s law describing the flow of compressible fluid through porous media; Le., pressure drop is proportional to flow rate
This relationship allows us to substitute the expressions for r and nl and leads to the following pair of ordinary differential equations
dP -=-AQ dx Assuming the fluid is an ideal gas PQ=FTRTorP=CTRT we get X
cT(X)
= c T E ~ ‘ 1 -(1 - 42) -
L
4 = pE/pS To obtain the expression for CT as a function of retention time t , note that
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
Table I. Dehydrogenation of C6H12. Experimental Conditions
5
'6 __ H12
Column Internal diameter: 2.17 mm Catalyst Platinum: 0.3% by weight on 7,-alumina Weight: 2.05 g Physical Conditions Temperature: 186 "C Inlet pressure: 2.36 bars Outlet: 1.00 bar Equilibrium constant Length: 1 m
TIME (SECONDS)
Figure 8. Cyclohexane dehydrogenation performed at 186 "C. Behavior of the overall chromatogram as a function of the amount of reactant injected. No cyclohexane was detected in the 0.72-pL pulse, thus showing improvement above equilibrium.
(Average value from Stull (1969) and Roginskii (1962)) Volume flow rate at the inlet: 0.364 cm3/s Mean residence time of an unretained solute: 11.1s
The amount of reactant in the gas phase will be our approximate measure of the amount observed a t the outlet of the chromatographic reactor, and the yield can be calculated as follows
-
I
g t
I
:
ir' 1
z
3
4 kICROLlTRES
5
6
7
a*
INJECTED
Figure 7. Cyclohexane dehydrogenationperformed at 186 "C. Yield as a function of the amount of reactant injected. Points are experimental results; the curve is the prediction of the simplified model.
Thus, with the aid of a simple model, a new investigation method allow us to describe the system with simpler and less numerous equations than does the rigorous method. In spite of the rather coarse assumptions used, this simplified model will be shown to be in rather good agreement with experimental and simulated as well as published data.
Comparison with Experiments a n d O t h e r Models In order to test the accuracy of the simplified model, deF r r / [ n C ~ ( xbeing ) ] the velocity of the fluid. Furthermore hydrogenation of cyclohexane to benzene experiments were performed in a chromatographic column. Experimental results t=0; x = o were compared with those which were simulated. The chrox =L t = to( 1 a1); matographic reactor consists of a one meter length of %-in. stainless steel tubing packed with a platinum on alumina Combining these relationships with CT(X) leads to catalyst which acts as the chromatographic adsorbent. The column was fitted in a Varian 2760-30 chromatograph with to(1 + 4 an injector, oven, and detector a t the same temperature. Liquid cyclohexane was injected with the aid of a syringe, and and pure helium was used as the carrier gas. The components were detected by a thermal conductivity cell and trapped in liquid air for analysis. Table I gives the experimental conditions. On a freshly activated column, catalyst deactivation was observed during the first few pulses. Steady-state activity was reached after pulsating about 0.1 cm3 of liquid cyclohexane through and the column. Then the quantitative evaluation of the yield was t performed. Results of experiments are represented in Figure k=J 7 as the yield vs. the amount of cyclohexane injected. Figure t o ( 1 + a11 8 shows some typical overall chromatograms. The poor chroNow the set of eq 9 only requires the initial conditions in matographic separation caused by the chemical reaction can order to be solved: a t time t = 0, (1 al)nomoles of reactant be seen. are fed into the slice. Adsorption and chemical equilibriums In order to perform simulations with the simplified model, are reached, which lead to the conditions chromatographic data related to the retention behavior of (1 a2)ndO) = (1 + a:dndO) = (1 + a d n o - n1(0)) cyclohexane, benzene, and hydrogen must be determined. Experiments show that these had rather nonlinear adsorption isotherms. A detailed study of the retention and broadening of the cyclohexane peak was made in a chromatographic colThe amount of reactant injected is related to the duration umn packed with noncatalytic alumina. In Figure 9 retention of injection: (1 al)no = B C ~ F TT. o obtain this relationship, time and variance are plotted as a function of the amount of it is supposed that the feed only consists of reactant and inert cyclohexane injected. Average values of tRI and a L 2have been fluid. The more unfavorable the thermodynamics, the more chosen to account for such nonlinearity. Consequently cyI and accurate the approximation. J no longer have a real physical meaning. Table I1 lists the The set of eq 9 is integrated from t = 0 t o t = t o (1 q). values chosen.
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Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
MlCROLllRES INJECTED
80tazir~~
9
rl
:/"
00
ao
e 0'
02
03
208
0 4 05 06 07 PULSE DURATION
oa
09
10-
Figure 10. Comparison between the rigorous model (solid line) and ,MlCROLllRES
NJECTEO"
Figure 9. Retention time and variance of the cyclohexane peak as a function of the amount injected. Physical parameters are those of Table I.
The simplified model is easily made suitable for reactions such as A1 ~t 3A2 As. The corresponding equations were integrated with the data of Tables I and I1 by a fourth-order Runge-Kutta method with 1000 time steps. The unknown parameter A is optimized to fit experimental data. I t was found that A = 2.9. This optimum value allows us to draw the theoretical curve of Figure 7 which is in good agreement with experimental data shown by points. Another way of improving the validity of the simplified model is to compare the results with those of the rigorous model. In Figure 10 single pulse yield curves are given as a function of pulse duration for different feed concentrations. Continuous curves are predictions from the rigorous model, and dotted curves are from the simplified model, with A = 1.6. The difference between the two values of A has no importance, but can probably be explained by the use of linear chromatographic parameters in nonlinear chromatography. Finally, we compared our results with those of Magee et al. These authors used two further assumptions: components A1 and A2 are not separated (a1 = ( ~ 2 and ) broadening does not take place ( J = m). In such a situation, it can be shown that the set of eq 9 has an analytical solution. When unfavorable chemical equilibrium is considered (large K t ) the integration leads to
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This relationship which expresses the residence time necessary for a given yield allows us to explain Matsen's experimental results, namely: the yield does not depend on carrier gas flow rate; if a column twice as long is used, twice as much reactant must be pulsed to give the same yield. These results are easily explained. First, the flow rate appears neither explicitly nor implicitly in eq 14 because t o and 0 are both inversely proportional to it. Secondly, if the column is twice as long, t o is twice as large. To keep eq 14 unchanged, K Owhich , is proportional to the amount of reactant in the feed pulse, must also be twice as large. There is a limiting case which is of great interest. What happens when X is sufficiently large, say greater than 0.9? An error of less than 30% is made when substituting log (1- X ) for X log (1- X) in eq 14. Thus it is found that even though an infinite reaction rate is assumed, the part of reactant which remains decreases exponentially with residence time as does an irreversible first-order reaction. Fourteen years ago Roginskii supposed that reactions become irreversible in a
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the simplified model (dotted line). Hachured area corresponds to the prediction of the simplified model when A is varied within the limits of f1W. Yield as a function of pulse duration ( t o is the unit of time). a1 = 2, a2 = 0, a3 = 6, J = 100,Kt = 1000 L/mol. From top to bottom CO= 0.025, 0.050,0.100,0.200 mol/L.
Table 11. Average Chromatographic Data; Pulse Smaller Than 7 p L Component A1 = C6H12 tRi,
S
uz, 52 ai
J
45 40
3.15 50
A2
= H2
11 0 -
A3
= CsHs
125 10 -
chromatographic reactor as a consequence of the separation of the products. Thus, assuming dehydrogenation is a firstorder reaction he derived a linear model leading to the same exponential decrease. But his experiments are inconsistent with his model because yield depends on the amount of substance injected. Our simplified model explains these experimental results because our "pseudo-first-order kinetic constant" depends on the sample size. Discussion In the previous section we have shown that the simplified model is a useful tool to investigate the behavior of a chromatographic reactor. Nevertheless, the obtainment of quantitative results depends on the value of the broadening parameter A , the precise value of which is unknown. Thus, it is of prime importance to know the sensitivity of the model to this parameter. The first method is to adjust A in order to fit the yields calculated from the rigorous model for various a,, J , Kt, and 0, and to compare the values of the optimized broadening parameters. The result is that A lies between 1.4 and 2.0 and that it seems independent of Kt and 8, but varies slightly with a, and J . This poor sensitivity of A to the physicochemical parameters conversely shows that any determination of such parameters from experiment would give suspicious values highly dependent on A . The second method used to test the sensitivity of the model to A is to compare the predicted yields for various values of A . In Figure 10 the hachured regions corresponds to the yield prediction of the simplified model when A is varied within the limits of 10%. Once more a poor sensitivity is observed, which allows us to compute a fairly good approximation of the yield without having an accurate value of A . Thus the simplified model is quantitatively efficient in yield prediction but not in parameter estimation. Nevertheless, it would be of practical interest
Ind. Eng. Chem. Fundam., Vol. 17, No. 1, 1978
to know A with a better accuracy. We think that aLand J are the main parameters affecting the value of A , but further experimental and theoretical investigations are needed. Another way of improving the efficiency of our model is to compare the computer time requirement of the simplified and rigorous approaches. As it can be easily seen on the set of eq 3, the time increment ensuring the convergence and the stability of the integration method must decrease with J because of the expressions in the right-hand side of the equations. Moreover the duration of integration increases with the number of equations, namely with J . As a consequence the computer time requirement of the rigorous model increases as 5 2 . On the other hand, the time requirement of the simplified model reveals to be independent of J . This partly explains that the rigorous computation of each data point of Figure 10 requires several minutes (up to 30 min for large Co by the Euler method), while the simplified computation requires about 15 s (by fourth-order Runge-Kutta method). Finally it is important t o note that the computer core requirement of the simplified model is very small. Conclusions Only two main assumptions need be kept in mind. (1) Chromatography may be described with the aid of the Martin and Synge or a similar model. (2) The concentrations can be considered as uniform in the slice containing the reactant. These conditions allow us to draw four important conclusions. (1) Repetitive pulse feed runs may be predicted from single-pulse experiments. (2) A simplified model describing the behavior of a single pulse is proposed. It may be easily extended to the case of finite reaction rate, simultaneous reactions, or more complicated situations. It may be necessary to modify the expressions for V and T ~ ifT reaction increases the reactant peak broadening (3) The simplified model is easier to implement than the rigorous one though one parameter is unknown. (4) Finally, as an answer to our main problem, time average yield, or yield improvement above steady state in pulsed feed experiments can be predicted with the aid of the simplified model and with simple algebraic calculations relating feed pulse parameters to steady-state feed. These calculations will not be discussed here. Thus, our simplified model can be a valuable tool to allow us to examine industrial reactions to find those for which the chromatographic reactor would be economical. However, behind this study of chromatographic reactors new questions arise. As pointed out, the simplified model is easier to use than the rigorous model. Are there any general methods leading to simplified models? This is not a simple problem because it requires a quantitative definition of the simplicity of a model. What is this definition? We think that research in this field would be of great interest. Acknowledgments The authors thank MM. Graulier and Brunelle of RhonePoulenc S A . for their assistance with problems in catalysis.
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Nomenclature A = broadening parameter Ai = component i (i = 1 , 2 , 3 ) Ci = concentration of component i, mol/L Ci,k = concentration of component i in the kth mixing cell, mol/L Co = concentration of reactant in the feed, mol/L CT = overall concentration, mol/L CTE = overall concentration a t the inlet, mol/L Fi = molar flow rate measured with respect to the column wall, mol/s Fi' = molar flow rate measured with respect to the reactant peak, mol/s FT = overall molar flow rate measured with respect to the column wall, mol/s J = number of mixing cells in series Kt = chemical equilibrium constant L = column length, cm M = mass of fluid in the column, g ni = number of moles of component i in mobile phase TZT= overall number of moles of matter in mobile phase P E = inlet pressure, atm P S = outlet pressure, atm Q = volume flow rate, cm3/s r = rate of reaction, mol/(cm3 s) t = time, s t o = residence time of an unretained solute, s t ~ =i retention time of component i, s T = feed period, s Topt= optimal feed period, s V = volume, cm3 X = yield AX = yield improvement x = length, cm Greek Letters ai = capacity factor @ = pressure drop parameter Q = free cross section, cm2 ai = standard deviation, s 0 = pulse duration, s Literature Cited Chieh, C., Tsang, L. C., Ind. fng. Chem. Process Des. Dev., IO, 47 (1971). Dinwiddie, J. A,, U.S. Patent 2 976 132 (1961). Gore, F. E., Ind. fng. Chem. Process Des. Dev., 6, 10 (1967). Hattori, T., Murakami, Y., J. Catal., IO, 114 (1968). Kocirik. M., J. Chromatogr., 30,459 (1967). Langer, S. H.,Yurchak, J. Y.. Patton, J. E., Ind. fng. Chem., 61,10 (1969). Langer, S.H., Patton, J. E., J. Phys. Chem., 76,2159 (1972). Langer, S. H.,Patton, J. E., "Chemical Reactor Applications of the Gas Chromatographic Column," in "New Developments in Gas Chromatography," pp 294-373, H. Purnell, Ed., Wiley, New York, N.Y., 1974. Magee, E. M., Canadian Patent 631 882 (1961). Magee, E. M., Ind. fng. Chem. Fundam., 2, 32 (1963). Martin, Synge, Biochem. J., 35, 1358 (1941). Matsen, J. M., Harding, J. W., Magee, E. M., J. Phys. Chem., 69, 522 (1965). Roginskii, S.Z., Rozental, A. L., Dokl. Akad. Nauk. SSSR, 146,152 (1962a). Roginskii, S. Z., Yanovskii, M. I., Gaziev, G. k . , Kinet. Katal., 3, 529 (1962b). Stull, R. D., Westrum, E. F.,Sinke. G. C., "The Chemical Thermodynamics of Organic Compounds," Wiley. New York, N.Y., 1969. Tokehiko, F., Motoyuki, S.,Smith, J. M., Catal. Rev., 13,(1976). Villermaux, J., Chem. fng. Sci., 27, 1231 (1972).
Receioed f o r reuieu: November 19, 1976 Accepted October 18,1977
