lnstationary Dispersed Plug Flow Model with Linear Rate Processes
The discussion of the analytical solution obtained by Hankel and Laplace transforms reveals that the application of the cumbersome two-dimensional dispersed plug flow model is unnecessary and the use of the simpler axial dispersed model is justified for all engineering purposes.
A variety of models has been developed-in the field of reactor design. The phenomenon of fluid mixing plays a prevailing part in all of them. Apart from the limiting cases of the well stirred tank and the plug flow tube reactor-along with combinations of them including dead regions and bypass flow-the dispersion model has become widespread. Although no examples of a priori design of commercially important reactors are known from the literature, the dispersion model is applied more and more frequently in the analysis of chemical reactors and mass transfer equipment. This is favored by the fact that dispersion coefficients are available in the literature for many different types of flow situations (Himmelblau and Bischoff, 1968; Levenspiel and Bischoff, 1963). According to the spatial dependence of the variables and parameters, different mathematical possibilities were evolved to describe chemical reactors. Taylor (1953, 1954a,b) and Aris (1956) and particularly Bischoff and Levenspiel (1962, 1963) presented a comprehensive picture of the different models with both transverse and longitudinal dispersion and the interrelationships among them. This paper will be concerned with the solution of the instationary two-dimensional dispersed plug flow model with linear rate processes. Including a first-order chemical reaction and considering the geometrical situation shown in Figure 1, the mass balance yields the following linear differential equation in dimensionless form (Levenspiel and Bischoff, 1963).
and the initial condition
As was shown by van Cauwenberghe (1966), the Danckwerts boundary conditions, eq 4 and 5, can only be applied in the transient situation if axial dispersion in the entrance and exit section is zero. These sections are not considered in our treatment. The solution was obtained by application of finite Hankel and twofold Laplace transforms (Sneddon, 1951). The method of residues was applied to invert the Laplace transforms. The complete solution is as follows.
where Nj, =
(I
+ q i ) 2 cos h P2 q i ( -~ x) + 2qi s i n h
x
The last expression on the right-hand side of eq 1 represents the source term. The reacting component A is fed to the reactor axis by an injection tube of finite size (diameter 2e) a t the axial position X O . To make the concentration dimensionless the concentration CA* is used which refers to the injection rate I b y the relation
I = i ~ E ~ t ' , C= ~ oiiR0 vCA* (3) Equation 1 has to be solved with the subsequent boundary conditions
pk
%(!' 2
- x) -
'[ x;
=--
PL
(1 - qi2) cos I zP4 q i ( o - x ) 2
+ PL(k' + d =
;Ye
+
2 ) + y]
xo
6 = x, - xo
(12) (13) (14)
The values of X R are obtained from the equations (Abramowitz and Segun, 1965)
Ind. Eng. Chem., F u n d a m . , Vol. 13, No. 4 , 1974
399
--------I r=' I I bulk flow
x=o F i g u r e 1.
x=xo
- - - - - - _I
x-x,
Flow s i t u a t i o n .
:.
I
0,6
I
1
I
r
U
0
Y e :IO
-
P, = 0,l
-
average concentration
0
S
IO
IS
20
- 0 F i g u r e 2. C o n c e n t r a t i o n a t reactor o u t l e t cs. t i m e a t d i f f e r e n t r a d i a l positions.
yR cot
x k x e = - p, 2 2
(k even)
and the ai are the solutions of Jl(ai) = 0 (Abramowitz and Segun, 1965). When the solution is evaluated the relation
is to be considered. Then the solution can be split in two terms
When this integral is evaluated the second term vanishes in eq 16 and the solution reduces to that of the axial dispersed plug flow model. This result has to be expected when considering the fact that all processes included in eq 1 (accumulation, dispersion, convection, source term, and reaction) are linear. Summarizing, it can be concluded that if only mean concentration is needed and linear processes occur (firstorder chemical reactions or mass transfer with linear rate expressions), the application of the cumbersome two-dimensional dispersed plug flow model is unnecessary and the use of the simpler axial dispersed plug flow model is justified for all engineering purposes. Hence there seens to be no urgency to determine radial dispersion coefficients in multiphase equipment such as bubble columns (Eissa, et al.,1971) if mass transfer rates are the limiting step as is frequently observed. Nomenclature ai = roots of Jl(ai) = 0 CA = concentration of the reacting component C A = concentration in injector C A * = fictive concentration CA = dimensionless concentration, c A / c A * DL = axial dispersion coefficient DR = radial dispersion coefficient E = radius of injector tube e = dimensionless injector radius, E/Ro I = injectionrate Jo,& = Bessel functions k = reaction rate constant k' = reaction rate group, kRo/v PL = axial Peclet number, uRo/DL PR = radial Peclet number, uRo/DR R = radial coordinate Ro = reactor radius r = dimensionless radial coordinate, R/Ro t = time 0 = dimensionless time, tu/Ro U = unit step function L? = axialvelocity us = velocity in injector X = axialcoordinate X , = reactor exit X O = injector position xo = dimensionless injector position, Xo/Ro x, = dimensionless reactor exit, X,/Ro x = dimensionless axial coordinate, X / R o 6 = Dirac delta function Literature Cited
Here the first term NiO,kO evaluated for a, = 0 does not depend on PR and r and corresponds to the solution of the one-dimensional problem, the axial dispersed plug flow model, if the additional condition xo = 0 is satisfied. This solution was already published elsewhere (Deckwer and Langemann, 1972). Figure 2 shows the concentration a t reactor exit calculated from eq 16 as a function of time 0 for a typical parameter combination a t various radial positions. The mean concentration a t the reactor exit is obtained from the integral
applying the relation
400
I n d . Eng. Chem., Fundarn., Vol. 13, No.4, 1974
Abramowitz, M., Segun, A . I., "Handbook of Mathematical Functions," Dover Publications, NewYork, N. Y . , 1965. Aris, R.. Proc. Roy. Soc.. Ser. A . 235, 67 (1956). Bischoff, K. B., Levenspiel, O., Chem. Eng. Sci.. 17, 245, 257 (1962). Deckwer. W., Langemann. H.. Chem. Ing. Tech.. 23, 1318 (1972) Eissa, H. S., El-Halwagi, M. M., Saleh, M. A,, Ind. Eng. Chem.. Process Des. Develoo.. 10. 31 119711 Himmelblau, d. M.;'Bisdhoff, 'K. E., "Process Analysis and Simulation,'' pp 118-119,317-322, Wiley, NewYork, N. Y., 1968. LevensDiel. 0 . . Bischoff. K. B.. Advan. Chem. Eno.. 4. 95 (19631 Sneddon, I . N . , "Fourier Transforms," McGraw-Hiil, New York. N. Y . , 1951. Taylor, G. I, Proc. Roy. Soc.. Ser. A . 219, 186 (1953). Taylor, G. I., R o c . Roy. Soc.. Ser. A . 223, 446 (1954a). Taylor, G. I.. Proc. Roy. SOC..Ser. A . 225, 473 (1954b). van Cauwenberghe, A. R., Chem. Eng. Sci.. 21, 203 (1966).
Institut fur Technische Chemie der Technischen Cnicersitcit Berlin lBerlin 12, Germany
Y a l c i n Serpemen Wolf-Dieter Deckwer*
Received for r e c i e u December 18, 1973 Accepted J u n e 17, 1974
