Ind. Eng. Chem. Process Des. Dev. 1980, 19, 24-30
24
Analysis of Complex Reaction Schemes in a Fluidized Application of the Kunii-Levenspiel Model' R. K. Irani, B. D. Kulkarni, and L. K. Doraiswamy" National Chemical Laboratory, Poona 4 11008, India
Based on the Kunii-Levenspiel model, a unified development for both conversion and product distribution in a fluidized bed reactor is presented for the successive reversible reaction scheme (Scheme I), reversible parallel scheme (Scheme 11), irreversible scheme (Scheme HI),and for the special cases thereof. The influence of bubble diameter on the maximum concentration and selectivity of intermediate R for successive reversible reactions is shown. A numerical example is presented to illustrate the procedure. Together with the earlier development for the Denbigh scheme (reaction I) and special cases thereof, this should provide a reasonably complete coverage of complex first-order networks of practical relevance.
Introduction Several models have been presented in the literature to account for the main features of a fluidized bed. These include homogeneous models in which the role of mixing is accounted for by a longitudinal dispersion coefficient, two-region models, and bubbling bed models. The last appear to be the most rigorous in that they explicitly recognize the role of the bubble in the analysis of the fluidized bed. Among the bubbling bed models the two basic models are those of Davidson and Harrison (1963) and Kunii and Levenspiel (1968a,b). These have been modified by several workers over the years, e.g., the Toor-Calderbank (1967) modification of the Davidson model (1961) and the Fryer-Potter (1972) modification of the Kunii and Levenspiel model. All these models are concerned with simple first-order reactions. T h e experimental results of Chavarie and Grace (1975a,b) on the ozone reaction (known to conform to the first-order scheme A R) in which concentration profiles obtained in the bubble, cloud, and emulsion phases were compared with those predicted by the various models, clearly showed that the Kunii-Levenspiel model best characterized the performance of a fluidized bed reactor. Extensions of the different models to complex first-order reaction schemes have also been proposed. Thus the model of Partridge and Rowe (1966) has been extended by Nashaie and Yates (1972) to a complex parallel reaction of the type
-
The Kunii-Levenspiel model has been the subject of extension to a few complex schemes (Kunii and Levenspiel, 1969; Carberry, 1976). Levenspiel et al. (1978) recently applied it to the first-order Denbigh reaction given by A-R-S
\T \U and obtained the performance equations thereof. Considering the fact that the Kunii-Levenspiel model has been reported to best represent the experimental data, the present study was undertaken to extend this model to a variety of first-order complex schemes and to present a unified development for both conversion and product distribution thereof. We first consider the rather general reaction scheme A
S
R
12
ke3
S
14
which, under special conditions, reduces to A
L
R
L
S
;
A -k i R S S ; k4
A & R - S ; kR 4 & S k2
k4
We then consider the reversible parallel reaction scheme
xA
R
s
R
S
which reduces to
and the compartment model of Kat0 and Wen (1969) has been extended to complex reaction schemes by Shaw et al. (1974). The model proposed by Miyauchi (1974a,b) has been extended by Miyauchi and Furusaki (1974) to cases such as
In addition, we consider a complex reaction scheme of the type A
t
'
R
L
S
L
T
which reduces to the special case A
NCL Communication No. 2287. 0019-7882/80/1119-0024$01.00/0
L
R
-
k3
S
L
k
T
Levenspiel et al. (1978) have accounted for the behavior of the fluidized bed by defining modified rate constants appropriate to the reaction steps involved. The fluid bed 0 1979 American Chemical Society
Ind. Eng. Chern. Process Des. Dev., Vol. 19, No. 1, 1980
Scheme 1 A
--
L RJk ,S
(all first order)
k4
k2
25
where I
could then be treated in a manner similar to that for a plug flow reactor. In the present communication a mathematical analysis for successive reversible and irreversible reactions is presented. A numerical example is also given to illustrate the method involved. The various reaction schemes and their special cases presented here, together with the Denbigh reaction scheme analyzed earlier (Levenspiel et al., 1978), serve to present a comprehensive picture of the various commonly encountered complex reaction schemes that can be analyzed by the Kunii-Levenspiel model. First-Order Reversible Reaction System Successive Reversible Reactions. Consider the reaction scheme (Scheme I). For any differential bed height dl the following material balance equation for any species can be written overall disappearance = reaction in bubble + transfer to cloud-wake transfer to cloud-wake = reaction in cloud + transfer to emulsion transfer to emulsion N reaction in emulsion (1) In writing the material balance equations, the transport coefficients Kbc (for bubble-cloud) and Kce (for cloudemulsion) can be assumed to be identical for all the species A, R, and S (Levenspiel et al., 1978). The conservation equations for the three species then become dCAb - U b x
=
Ybki[CAb
- cA*l
1
hi
1
-Kbc+
1 +
-
and
4’ =
ki
1
Kce
Ye
-+-
[ %((
)
”” Kce Yek3Kce + rek3 x +
+ Yckl[CAc - cA*l + Yekl[cAe - cA*l
Kbc[CAb
- CAcl
rckl[CAc
Kce[CAc U
dCRb b T
=
- cA*l
+ Kce[CAc
- CAel = Yekl[CAe
r b k l [ C A b - cA*l - Ybk3[CRb
-
(2)
CAel
-
cA*l
- cR*l
+
Kbc[CRc - CRbl Kbc[CFk - CRbl
=
Y&l[cAc
- c A * l - ?’ck3[CRc - c R * l
+
Kce[CRe - CRcl Kce[CRe - CRcl U
dCSb b T
= =
Yekl[CAe Ybk3[CRb
Kbc[CSc - CSbl
=
- c A * l - ?eh3[CRe - cR*]
7ck3[CRc
-
cR*l
(3)
+ Kbc[CSc - CSbl
- cR*l
+ Kce[CSe
(12) Equations 5 and 6 can be written in dimensionless form
- CScl
(4) Neglecting the effect of solids dispersed in the bubbles ( Y b = 0), eq 2 to 4 can be simplified further. This assumption of Y b equal to zero is reasonable for all but extremely fast reactions. I t should also be noted that the various rate constants appearing in eq 2 to 4 have the dimensions of inverse time. Since, for solid catalyzed heterogeneous reactions the rate constants are usually expressed as cm3/(g of catalystas), they may be suitably modified by multiplying by the catalyst density. Combining and expressing the cloud and emulsion phase concentrations CAc,Ch, Ch, and CRein terms of the bubble phase concentrations C A b and C R b , eq 2 and 3 can be rewritten as K c e [ C s e - (3scI
=
Yek,[C~e- cR*l
(5)
as
where the caret denotes the dimensionless parameter and the double subscripted letters are defined as Ki,
= K,
UO -
ErnflUbr
(i =1, 3, or 4)
26
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
The concentration profiles for the species A and R can now be obtained by integrating eq 13 and 14 subject to the initial conditions
ki
(16)
= 0 at 1 = 0
= 1;
Simplified Cases Consecutive Reaction Scheme IA. k3
A-R-S
(IN
e,*
with the result
For this case of a consecutive reaction both and 6,. are zero and the various parameters in eq 23 become
K1’ p = -K3 -.p = -Pz; P3 = 0 K1’ - [K3- K,]
(27)
Substituting these in eq 20 and 23 gives
and
[
exP{-K335
1) - 1
The concentration of the species S can be obtained from the material balance equation
e,,
ki
+ e R b + e,,
=
In P
=-
K33 - K11
w lmax -
k
A - R ~ S
(19)
=1
The distance at which maximum production of R occurs can be obtained by setting eq 14 to zero. Using the profiles given by eq 17 and 18 this can be rearranged to give 7max
which are the well known expressions for a consecutive reaction carried out in a fluidized bed reactor. Reaction Scheme IB. ki
For this scheme in eq 23 become
(IB)
e,* is zero and the various parameters
(20)
UOPS
Substituting these in eq 20 and 23 gives
W -
r=-1
w-
UOPS
7max
The maximum concentration of R is then (?R,max
=
[PPI + Pz]PK31(K3-K1) + P3
= - 1max =
In
[
Ki’K3
+ K 3 e ~ * [ K-3 K,] K
= 30cm
04
0 2
O0
I O
20 30
T
40
5 0
60 7 0
ism I
Figure 1. Concentration profiles with varying bubble diameters for the reaction scheme A ==R + S. 0 45
I
I
I
may be obtained by substituting for PI,Ps,P3, and P, as given above, in eq 23. Reversible Parallel Reaction System The reaction scheme (Scheme 11),which is a reversible parallel scheme, can (for the purpose of analysis) be regarded as a variation of the consecutive reversible reaction scheme considered earlier, with A going to R going to S, while in fact R is the reactant species and A and S are the products of the reaction. Following the procedure outlined in the earlier section, the following concentration profiles can be obtained O 3 O I
0.25
* 14
20
26
2
db(cm)
[K3 - KII
Figure 2. Variation in scheme A + R 5 S.
K3
The concentration of the species S can be obtained from the material balance given by eq 19. For maximum production of A, eq 34 suggests a very long reactor with t 4 , m a x = 6.4. (38)
First-Order Irreversible Reaction System For the complex irreversible scheme of the type in Scheme 111,an analysis similar to that outlined above leads t o the results summarized in Table I. The results for a special case ki
k3
A R S are also included in the table. +
k4
T
--+
cH,mru. with bubble diameter for the reaction
Discussion For establishing the effects of variables in accordance with the various equations developed, we shall consider the most general case A R S and compute the profiles of the various species for different conditions. The following data will be used: hl = 10.0 s-l; h2 = 1.0 s-’; h3 = 1.0 h4 = 0.1 p s = 2.0 g/cm3; W / U ,= 12.0 g s/cm3; umf= 3 cmjs; uo = 30 cm/s; emf = 0.4; De = DA = 0.2 cm2/s; V,/ v b = 0.3; d b = 8-30 cm. The results of the calculations are presented in Figures 1-3, while the calculation procedure for a representative bubble diameter [dh = 15 cm] is summarized in the Appendix. Figure 1 shows the concentration profiles with varying bubble diameter. As a general observation it can be said that the conversion of A drops with increase in bubble diameter, in analogy with the physical situation. Also, the fractional height of the fluidized bed a t which the maximum concentration of R occurs in the reactor shifts to the
28
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
Table I. Summary of Model Equations for the Complex Reaction Scheme 111 and Its Special Case concentration profile conservation eq (and boundary conditions)
( a t 1 = 0; C A b = 1)
where
following equations = [ k , + k,l
L
We also obtain the expressions
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980 29 Table I (Continued) concentration eq (and boundary conditions)
concentration profile k
k
k
A--$R-%S-%F
A
K", = K 1 b 4 ;K",, = K,$4
u0
Ernflubr
where
This is a simplified case of Scheme 111.
0
,
1
larger bubble diameter corresponds to a greater deviation from plug flow conditions. The fluid bed performance is compared with that of a plug flow reactor (PFR) for ki/'k3 =. 10 in Figure 3, which shows the variation of selectivity with conversion for different diameters of bubble. For the same extent of conversion higher selectivities can be realized at lower bubble diameters in the case of a fluid bed reactor. From the nature of these curves it is apparent that operation a t higher conversions in a fluid bed is feasible without serious loss of selectivity for the case when k l / k 3 > 1, except in the vicinity of the maximum conversion possible. In this sense the analysis of the fluid bed reactor based on the Kunii-Levenspiel model is similar to that of a PFR. It has been found (figure not shown) that for k l / k 3 C 1operation at lower conversion in a fluid bed is preferable as higher conversions would entail a significant reduction in selectivity, again in analogy with a PFR. Acknowledgment The fruitful discussions with Professor S. Z. Hussain of the Indian Institute of Technology, Bombay, and the financial assistance from Indian Petrochemicals Corporation Ltd., Baroda, are gratefully acknowledged. Appendix The values of the various parameters of the model are calculated as follows
From the details of the model we obtain Ubr = 0.711(gdb)''2 = 86.20 cm/s
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 1, 1980
30
Ub
=
UO - Umf
K,, = 6.78(
+ Ubr = 113.20 cm/s
7) = 0.351 s-l 112
%nfDeUb
0.351
Substituting these values in eq 7 gives K1 = 1.025 s-l K3 = 0.438 s-l From eq 9 we obtain $1
= 0.85
and from eq 8 K1' = 0.652 s-l Substitution in eq 11, 12, and 10 gives us respectively q52 = 14.23 s-l
d 3 = 14.76 s-l @l'
= 0.49 s-l
From eq 15, we obtain KI1 = 0.594 s-l; Kll' = 0.378 s-l; K,, = 0.254 s-] The concentration profiles as well as eR,max and then calculated as follows. From eq 17 and 18 = 0.909 exp(-0.594T) + 0.091 eRb
= -1.01 exp(-0.5947)
T,,
are
+ 0.887 exp{-o.254Tj + 0.123
These equations are plotted in Figure 1 for db = 15 cm. For a bed of infinite height, Le., T C A b = 0.091 = &A* +
= 0.123
E
eR*
The discrepapcy between the value of e R b a t infinite bed height and CR* is due to the assumption made in the theoretical analysis that Y b = 0. As can be seen, however, this does not result in any appreciable variation in the concentration a t the top of the bed. From a material balance we obtain = 1.0 + eRh]
[e,,
From eq 21, 24, 25, and 26, respectively, we have P = 0.373 P] = -1.01 Pz = 0.887 P3 = 0.123 Thus eq 20 and 23 give 2.88 s CR,man = 0.368 rmax=
The results of the computations are summarized in Figures 1 to 3. Nomenclature A, R, S, T = reaction components CAO= inlet concentration of A, g-mol/cm3 CA,CR, Cs, CT = concentration of A, R, S, and T, g-mol/cm3 CA*,,CR*,= eguilibrium concentration of A and R, g-mol/cm3 C A , CR, Cs, CT = fractional concentration of A, R, S, and T, dimensionless cA*,CR* = fractional equilibrium concentration of A and R, dimensionless dh = effective bubble diameter, cm De, DA= molecular diffusivity for gas, cm2/s g = acceleration due to gravity, cm/s2 k i = rate constant for a first-order reaction, s-l Khc = gas interchange coefficient between bubble and cloud-wake, s-l K,, = gas interchange coefficient between cloud-wake and emulsion, s-] Ki,Kii, K,', K,,', etc. = modified rate constants as defined individually in the test, s-l 1 = height, cm & = height of a bubbling fluidized bed, cm 1 = 1/L = fractional height of fluid bed, dimensionless P, P,, P2,P3 = terms as defined in the text, dimensionless ub = velocity of a bubble, cm/s Ubr = velocity of a bubble with respect to the emulsion phase, cm/s umf = superficial velocity at minimum fluidization, cm/s uO= superficial entering gas velocity, cm/s uO = volumetric flow rate of gas, cm3/s v h = volume of a gas bubble, cm3 V , = volume of bubble wake, cm3 W = weight of solids in the bed, g Greek Letters y i = volume of solids in region i/volume of bubbles in bed, dimensionless b = volume of bubbles/volume of bed, dimensionless t d = void fraction in a bed at minimum fluidizing conditions, dimensionless ps = density of solids, g/cm3 T = residence time, s 4L,4i',4Li'= terms as defined in the text, s-l + i , $I,+,) = terms as defined in the text, dimensionless Subscripts b = bubble phase c = cloud phase e = emulsion phase max = value of the parameter corresponding to maximum concentration of the intermediate species mf = minimum fluidizing conditions L i t e r a t u r e Cited Carberry, J. J., "Catalytic and Chemical Reaction Engineering", p 575, McGraw Hili, New York, N.Y., 1976. Chavarie, C., Grace, J. R., Ind. Eng. Cbem. Fundam.. 14, 79 (1975a). Chavarie. C., Grace, J. R., Znd. Eng. Cbem. Fundam., 14, 85 (1975b). Davidson, J. F., Trans. Inst. Cbem. Eng., 39, 230 (1961). Davidson, J. F., Harrison, D., "Fluidized Particles", Cambridge University Press, New York, N.Y., 1963. Fryer, C., Potter, 0.E.. Ind. Eng. Cbem. Fundam., 11, 338 (1972). Kato, K., Wen, C. Y.. Cbem. Eng. Sci., 24, 1351 (1969). Kunii. D., Levenspiel, O., Ind. Eng. Cbem. Fundam., 7, 446 (1968a). Kunii, D., Levenspiel, O., Ind. Eng. Cbem. Process D e s . Dev., 7, 481 (1968b). Kunii, D., Levenspiel, O., "Fluidization Engineering", pp 249-251, Wiley, New York, N.Y., 1969. Levenspiel, O., Baden, N., Kulkarni, E. D., I&. Eng. Chem. frocess Des. Dev., 17, 478 (1978). Miyauchi, T., J . Cbem. Eng. Jpn., 7, 201 (1974a). Miyauchi, T., J . Cbem. Eng. Jpn., 7, 207 (1974b). Miyauchi, T., Furusaki. S., AZCbE J . , 20, 1087 (1974) Nashaie, S., Yates, J. G..Cbem. Eng. Sci., 27, 1757 (1972). Partridge, 9. A,, Rowe, P. N., Trans. Inst. Cbem. Eng., 44, 335 (1966). Shaw, 1. D., Hoffman, T. W., Reilly, P. M., AZCbE Symp. Ser., 70, No. 141, 41 (1974). Toor, F. D., Caiderbank, P. U., "Proceedings, International Symposium on Fluidization", Netheriands University Press, Amsterdam, 1967.
Received f o r reuierv March 21, 1978 Accepted May 11, 1979
