Conder, . R., Langer, S.H., Anal. Chem. 39, 1461 (1967). Doring, C. E., 2. Chem. 1, 347 (1961). Kwantes, A., Rijnders, G. W. A., “Gas Chromatography,” pp. 125-35, D. H. Desty, Ed., Academic Press, London, 1958. Kyle, B. G., Leng, D. E., Ind. Eng. Chem. 57 (2), 43 (1965). Langer, S. H., Purnell, J. H., J . Phys. Chem. 67, 263 (1963). Langer, S. H., Sheehan, R . J., “Progress in Gas Chromatography,” pp. 289-323, J. H. Purnell, Ed., Interscience-Wiley, New York, 1968. Lesteva, T. M., Gorodnikov, S. K., Zheleznyak, A. S., Z h . Prik. Khim. 39, 1628 (1966). Martire, D. E., “Progress in Gas Chromatography,” pp. 93-120, J. H. Purnell, Ed., Interscience-Wiley, New York, 1968. Martire, D. E., Pollara, L. Z., “Advances in Gas Chromatography,” pp. 335-61, J. C. Giddings, Ed., Marcel Dekker, New York, 1965. Porter, P. E., Deal, C. H., Stross, F., J . Amer. Chem. Soc. 78, 2999 (1956).
Porter, R . S., Johnson, J. F., Ind. Eng. Chem. 52, 691 (1960). Redlich, O., Kister, A. T., Ind. Eng. Chem. 40, 345 (1948). Rock, H., Chemie Ing. Tech. 28, 489 (1956). Sheehan, R . J., Doctoral thesis, University of Wisconsin, 1969. Sheets, M. R., Marchello, J. M., Petrol. Refiner 42 (12), 99 (1963). Sideman, S., Bull. Chem. SOC.Japan 37, 1565 (1964). Tassios, D., Chem. Eng. J . 76 ( 3 ) , 118 (1969). Treybal, R. E., “Liquid Extraction,” 2nd ed., McGrawHill, New York, 1963. Warren, G. W., Warren, R. R., Yarborough, V. A., Ind. Eng. Chem. 51, 1475 (1959).
RECEIVED for review August 13, 1969 ACCEPTED August 3, 1970 Supported in part by the Petroleum Research Fund, the E. I. duPont de Nemours and Co. (Fellowship for R . J. S.),and the Wisconsin Alumni Research Foundation.
Behavior of a Chromatographic Reactor Chieh Chu’ and louis C. Tsang2 University of California, Los A ngeles, Calif, 90024 The behavior of a chromatographic reactor was studied by use of the LangmuirHinshelwood kinetic model to account for the competitive adsorption on the catalyst surface. The effects of various parameters such as input wave form, reverse reaction rate constant, average reactant concentration in the feed, adsorption equilibrium constants, and active center concentration were investigated. Some limited study of the effect of longitudinal dispersion was also included.
M a n y chemical reactions cannot go to completion because of the existence of reverse reactions. In a chromatographic reactor, the reactants are introduced in pulses. If the products are adsorbed on the catalyst in the reactor to different extents and for different lengths of time, these products can be separated from one another, thus diminishing the rate of the reverse reaction. In this way, an ordinarily equilibrium-limited reaction can be carried t o completion or much nearer completion. That this is feasible has been shown experimentally by Roginskii et al. (1962) on the dehydrogenation of cyclohexane to benzene and by Semenenko et al. (1964) on the dehydrogenation of n-butene to divinyl. As for theoretical analyses, Roginskii and Rozental (1964) studied reaction kinetics under chromatographic conditions. Gaziev et al. (1963) investigated the effect of the reactant pulse shape and the order of chemical reaction on conversion. Magee (1963) used a mathematical model in terms of gas phase concentration. Gore (1967) improved on this by using the concentrations of adsorbed species instead, but the power law model was still retained. ’ T o whom correspondence should be addressed.
‘Present address. Monsanto Co., St. Louis, Mo. 63166
Since chemical reactions in a chromatographic reactor take place on the surface of a catalyst with a limited number of active centers, it is important in analyzing the performance of a chromatographic reactor to account for the effect of competitive adsorption. This work uses the Langmuir-Hinshelwood kinetic model instead of the power law model. Furthermore, longitudinal dispersion is important in the chromatographic phenomenon (Gore, 1967; Magee, 1963; Roginskii and Rozental, 1964). This work also includes this phenomenon. With these two factors included, this work investigates the effects of various pertinent parameters on the behavior of a chromatographic reactor. These parameters are: input wave form, reverse reaction rate constant, average reactant concentration in the feed, adsorption equilibrium constants, active center concentration, and longitudinal dispersion. I t should be mentioned however, that, because of the particular numerical procedure being used, the effect of longitudinal dispersion was studied only t o a limited extent. Reaction Kinetics
The following reaction is considered: When reactant A enters the reactor, it is adsorbed on the catalyst surface inside the reactor, and dissociates into two products, one Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
47
of which enters the gas phase immediately and the other is desorbed a t a lower speed. The reaction is assumed to take place in three steps, with the surface reaction step controlling: Adsorption of reactant A on the catalyst surface
Decomposition of adsorbed A into adsorbed B and C in the gas phase k
A1
9B1+ C(g)
r = kCAl - k'CE,pc
(3)
Desorption of B from the catalyst surface
B1
ks ks
B(g) + 1
u =VO/L Equations 11 become:
Mathematical Model
We make the following assumptions: The process is under isothermal conditions. The superficial gas velocity is constant. Diffusion in directions perpendicular to bulk flow is negligible. A simple numerical value can be used for the effective diffusion coefficient of a gas component. A mass balance on each of the components leads to the following set of continuity equations along the reactor (Gore, 1967; Keulemans, 1959; Magee, 1963):
Equation 13 has the general form
(~YI~D T () a 2 y / a t 2+ ) U (aylat) = 7 where
?B
-@[e'$ + ( a d ' A / a 7 ) ] = p[g'$ - ( d d B / d T ) ]
YC
= BW
?A
=
The initial condition for each component is at T = 0 , y = 0 The boundary condition a t the reactor entrance is (Danckwerts, 1953)
at 5 = 0 , U y l 0 - = UyIo- - D ( d y d t ) l o -
After substitution of the expressions for r, CA1, and
C B I Equation , 1 becomes: 48
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
(17)
where y l o is a function of time. with a specified wave form. Because of the time dependence of y l o a t the reactor entrance, the zero-gradient condition (Danckwerts, 1953) which is normally specified a t the reactor exit is no longer expected t o hold true. The logical way of specifying the boundary condition a t the reactor exit is perhaps to con-
nect a hypothetical no-reaction section of infinite length to the reactor exit and then specify a zero-gradient condition at t: = a . This condition, being uneconomical in computation, is not used in this work. The boundary condition actually employed is still the zero-gradient condition.
+ bcyF.;i + ccy:.;
n - 1 acyc., - 1'
-ar));, -
+ b b j $ , - cry:
I
-
=
+ @ A T [ $ : 1 i + $:]
(26)
The initial condition is
at
T
= O? y ; = 0
(27)
The boundary condition a t the reactor entrance is
Solution Method
at 6 = 0, Uy, = Uyl - D ( d y ! d $ ) I
When Equation 14 describes the physical situation normally prevailing in the reactor, the first-order forced convection term is predominant over the second-order longitudinal dispersion term. Equation 14 is therefore essentially a hyperbolic equation of the form
+
( d y i d ~ ) U ( a y / d [ )= y
+ D(a2y/d(')
I
dY'd(1 1 = (-3yi
where avg denotes the average condition along the characteristic base curve connecting the points ( i + 1, n + 1) and [ i + 1 - U ( l ~ / l [n)],on the 5,r plane. For simplification, select l [ / l T = Equation 5 becomes
u.
+ 43'n - 3 ' 1 ! 3 1
31
0
% 30 W W
2 W z
k!W
29
a
28 0.000001
0.00001
0.0001
0.01
0.001
0 , (FT‘ISEC)
Because of the unrealistic boundary condition used a t 1, the validity of the solutions is questionable when the diffusion coefficient increases. However, the constant feed conversion remains the same when Dn changes from 0.000001 to 0.01. The numerical results showed nonzero gradients a t the boundary where { = 1. This suggests that the effective diffusion coefficient of 0.01 ft’isec is still too small to affect the condition a t that boundary, and the unrealistic boundary condition used to help obtain solutions does not affect the solution appreciably. The data presented in Figure 13 might therefore be valid after all.
{ =
Conclusions
Improvement of chromatographic reactor conversion over constant feed conversion is realized with a delayed sine function input. This improvement is higher with an increased reverse reaction rate constant. When the reverse reaction rate constant is high, increase in the average reactant concentration in the feed brings about larger improvement in conversion. A certain combination of adsorption equilibrium constants gives the maximum benefit owing to the chromatographic effect. A certain active center concentration can also be found to give maximum improvement. A large longitudinal dispersion in the reactor, however, tends to offset the chromatographic effect. Acknowledgment
The authors express their gratitude toward the Campus Computing Network of UCLA for the use of the computer. Nomenclature
A = component A avg = average condition along the characteristic base curve B = component E, $ c = component C, or concentration of a component, lb mole/cu f t D = diffusion term, dimensionless D = effective diffusion coefficient, ft’isec 1 = spatial index I = number of spatial node points along reactor k = forward reaction rate constant. sec-l k’ = reverse reaction rate constant, sec-l atm-’ K = adsorption equilibrium constant, atm-’ K’ = dimensionless adsorption equilibrium constant 1 = active center L = reactor column length, f t n = time index
P = partial pressure of a component, atm = average partial pressure of A in the feed, a t m r = reaction rate, lb mole/ft3-sec r, = adsorption rate, lb moleiftj-sec r d = desorption rate, lb mole/ft3-sec R = gas constant, a t m ft3/lb mole-0 Rankine active center concentration, lb moleif? t = time, sec T = temperature, O Rankine dimensionless gas velocity gas velocity, ft/sec x = distance along reactor column, f t y = dimensionless concentration of a component
PA0
s =
u=
v=
GREEKLETTERS P = product of S , R and T , atm Y = reaction term, dimensionless A =
incremental sign
8 = time per cycle, sec
dimensionless distance along the reactor column dimensionless time @ = concentration term, atm-’ i c = reaction term, sec -’/ atm
E =
T
=
literature Cited
Danckwerts, P. V., Chem. Eng. Sci., 2, 1 (1953). Gaziev, G. A., Filinovskii, V. Yu., Yanovskii, M. I., Kinet. Katal., 4, 688 (1963). Gore, F. E., Ind. Eng. Chem. Process Des. Develop., 6, 10 (1967). Keulemans, A. I . M., “Gas Chromatography,” 2nd ed., pp 134-5, Reinhold, New York, 1959. Magee, E. M., Ind. Eng. Chem. Fundam., 2, 32 (1963). Hildebrand, F. B., “Advanced Calculus for Engineers,” pp 386-99, Prentice-Hall, Englewood Cliffs, S . J., 1949. Roginskii, S. Z., Rozental, A. L., Kinet. Katal., 5 , 104 (1964). Roginskii, S. A., Yanovskii, M. I., Gaziev, G. A , , ibid., 3, 529 (1962). Semenenko, E . I., Roginskii, S. Z., Yanovskii, M. I., ibid., 5 , 490 (1964). Thomas, L. H., “Elliptic Problems in Linear Difference Equations Over a Network,” Watson Scientific Computing Laboratory, Columbia University, New York, 1949. Tsang, 0. C., M. S.thesis, University of California. Los Angeles, Calif., 1969. RECEIVED for review December 8, 1969 ACCEPTED August 31, 1970 Work supported in part by UCLA Academic Senate Grant l o . 2330. Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 1, 1971
53
