Kobayashi, H.. Arai. F.. Chiba, T., Chem. Eng. Tokyo, 29, 858 (1965). Kobayashi. H.. Arai. F.. Chiba. T., Chem. Eng. Jpn.. 4, 147 (1966). Kunii. D . , Levenspiel, O.,"Fluidization Engineering," Wiley, New York, N.Y., 1969. Lockett, M. J . , Harrison, D., "Proceedings of International Symposium on Fluidization," A. A. H. Drinkenburg. Ed., Netherlands University Press, 1967. Pyle, D. L.,Adv. Chem. Ser., 109, 106 (1972). Pyle, D. L., Harrison, D., Chem. Eng. Sci., 22. 531 (1967). Rowe, P. M . , "Proceedings of 5th European/2nd International Symposium on Chemical Reaction Engineering," p A9-1, Amsterdam, 1972.
Toomey, R. D., Johnstone, H. P., Chem. Eng. Progr., 48. No. 5 , 220 (1952). Turner, J. C. R., Chem. Eng. Sci., 21, 971 (1966).
Received for review J u n e 14, 1974 Accepted F e b r u a r y 7,1975 A c k n o w l e d g m e n t i s m a d e t o t h e donors of t h e P e t r o l e u m R e search Fund, a d m i n i s t e r e d by t h e A m e r i c a n C h e m i c a l Society, for s u p p o r t of t h i s research.
Performance Analysis of a Fluidized Bed Reactor. II. Observed Reactor Behavior Compared with Simple Two-Phase Models Claude Chavarie and John R. Grace* Department of Chemical Engineering, McGill University, Montreal, Canada H3C 3G 1
The catalytic decomposition of ozone was monitored in a two-dimensional fluidized bed reactor. Experimental concentration profiles were obtained for both the dense phase and the bubble phase. Overall conversions were also measured. The experimental data were compared to predictions from six simple physical models which take account of the two-phase nature of gas fluidized beds, where one parameter, at most, was fitted. On statistical grounds, all of the models fail to represent the observed reactor behavior. A useful approximation of the reactor performance is, nevertheless, provided by the bubbling bed model of Kunii and Levenspiel.
Introduction The hydrodynamic behaviour of the fluidized bed used in this work was described in Part I. Part I1 analyzes the performance of the same fluidized bed as a chemical reactor and compares the reactor data with predictions from six of the more popular two-phase chemical reactor models. These models represent a broad range of basic assumptions (Grace, 1971). For example, clouds and/or wakes are assumed in some models and not in others; a single average bubble diameter is assumed in some models while others allow for changes in bubble size with height; some models rely on the two-phase flow distribution suggested by Toomey and Johnstone (1952) while others neglect the percolation of gas through the dense phase; five different interphase mass transfer mechanisms are employed. The fundamental assumptions of each model have been respected in the present paper, although minor modifications are necessary to adapt the models to two-dimensional systems. The objectives of the present paper are (i) to present experimental concentration profiles for the fluidized bed under conditions identical to those described in Part I, (ii) to observe how well the six representative models predict the concentration profiles and overall conversions, and (iii) to lay the grounds for model improvements to be discussed in Part 111.
nitrate provided a reliable catalyst. The activity of the catalyst was varied by changing the relative amounts of alumina particles and inert beads. The activity was measured before, during, and after each run by feeding samples of solids to a fixed bed reactor. The materials of the reactor and auxiliary equipment were carefully chosen to prevent attack by ozone or poisoning of the catalyst particles. The fluidizing gas consisted of air containing less than 1% ozone by volume generated by a Model 03B6-0 ozone generator manufactured by Ozone Research and Equipment Corporation. Bubble phase ozone concentrations were measured by passing a 254-pm uv beam and a reference beam through a pair of quartz windows (there were 54 pairs in all) and measuring the intensity of the beams with a DuPont 400 photometric analyzer connected to a high-speed recorder. The reference beam allowed correction for particle showering through the bubbles. Withdrawal of gas samples through porous disks located throughout the bed allowed dense phase ozone concentrations to be measured. Samples were withdrawn intermittently, between the arrivals of bubbles a t the sample point. Inlet and outlet concentrations were also measured using uv absorption. Full experimental details including a discussion of errors have been reported elsewhere (Chavarie, 1973; Chavarie and Grace, 1972).
Experimental Apparatus
Experimental Profiles Figure 1 shows typical concentration profiles and illustrates the reproducibility of the data (pure error = 1 X the runs having been carried out on three different days. Profiles were obtained for superficial gas velocities, U , of 12.7, 16.5, and 22.8 cm/sec and catalyst activities, k , of 0.063, 0.09, and 0.20 and 0.4 sec -I. The height of the
The catalytic decomposition of ozone is particularly well-suited to the objectives of the present work since the reaction is very nearly first order, may be analyzed using uv absorption, and has appreciable rates of reaction at room temperature. A mixture of glass beads and porous alumina particles which had been impregnated with ferric
Ind. Eng. Chem., Fundam., Vol. 14, No. 2 , 1975
79
IO
IO
\
I
U:165cm/sec k = 0 2 0 sec-l
-DPPF model
....-... DPPM
~
model
00
0
20
40
60
80 100 y ( c m )
120
140
160
U
I cm /sec
)
Figure 1 . Concentration profiles and predictions from the Orcutt models, U = 16.5 cmisec, k = 0.20 sec-l.
Figure 2. Comparison of experimental outlet reactant concentrations with predictions from the Orcutt models.
bed a t minimum fluidization was 130 cm for all cases in which concentration profiles are presented herein. Hence the dimensionless reaction rate k‘ = k H / U , was always of order unity, well within the region where hydrodynamics play a very important role in determining the reactor performance (Grace, 1974). All the profiles show certain common features: the dense phase profile was lower than the bubble phase profile (as expected); for both profiles there was a pronounced concentration drop near the distributor followed by a gradual decrease in the center of the bed and a further concentration drop near the bed surface; the profiles were displaced upwards with increasing U and decreasing k . Some of the overall conversion results are given in Figure 2. Further concentration measurements are presented below when discussing the predictions from the various reactor models.
an arbitrary correlation factor, this diameter should clearly bear some relation to actual bubble sizes in the fluidized bed. Choice of an equivalent diameter has been discussed recently by Fryer and Potter (1972a,b). The mean bubble diameter at the mid-point between the distributor and bed surface was selected as the representative bubble size in the present work. (b) Dense Phase in Plug Flow (DPPF). A second model was suggested by Orcutt et al. where gas flow in the dense phase is now described by plug flow. With the boundary conditions Cg’ = CD‘ =- 1 at y = 0, the authors showed that
Comparison of Reactor Models with Experimental Profiles (1) Orcutt Reactor Models. The assumptions underlying the two reactor models devised by Orcutt et al. (1962) (see also Davidson and Harrison, 1963) are: (i) bubbles are of uniform size throughout the bed; (ii) reaction occurs only in the dense phase; (iii) interphase mass transfer is by two superimposed independent mechanisms, throughflow and molecular diffusion; (iv) all gas flow in excess of that required for incipient fluidization passes through the bed as bubbles. (a) Dense P h a s e Perfectly Mixed (DPPM). The particular assumption of the DPPM model is that the gas in the entire dense phase is perfectly mixed. Given the above assumptions, a mass balance on the bubbles and on the dense phase with the boundary conditions, Cp,/C1 = 1 a t y = 0, yields the concentration profiles
while the outlet concentration is given by
where $ is the fraction of the flow associated with the bubble phase and taken as (1 - Umf/U), Q is the overall volumetric rate of exchange between a bubble and the dense phase = g h,&, and X is the number of transfer units = QHIUgVg. The dimensional parameters for the original models and as modified to apply to two-dimensional reactors are given in Table I. The key parameter suggested by Orcutt et al. is the “equivalent bubble diameter.” In order to be more than
+
80
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
C,’
= Cle-mly
+
C2e-mzY
(4)
where ml,m2, CI, Cz may be expressed in terms of X , k’,
8, and H (Davidson and Harrison, 1963). (c) Comparison of Predictions with Experiment. Concentration profiles predicted using the Orcutt models are shown in Figure 1 where the top curve and blacked in symbols in each case refers to the bubble phase and the lower curve and open symbols to the dense phase. Predicted outlet concentrations have also been shown in Figure 2. It is clear that the DPPM model is too conservative in its estimate of overall conversion and that the concentration profiles are in no way similar to their experimental counterparts. Given this lack of agreement, one immediately suspects that the model is deficient, e.g., in the interphase mass transfer proposed, in assuming perfect mixing in the dense phase, or in assuming uniform bubbles throughout the entire bed. Overall conversions calculated using the DPPF model are in reasonable agreement with the experimental results (see Figure 2 ) . However, it is clear from Figure 1 and from other similar. comparisons (Chavarie, 1973) that agreement between measured and predicted concentration profiles is rather poor. While the DPPF model offers a fair estimate of btbble phase concentration profiles, predicted dense phase concentrations are far too high and too close to the bubble phase concentration. This suggests that the mass transfer rate in the Orcutt models is too high. To quantify the bias due to the model, the F test was applied to the data shown in Figure 1. The F values for the bubble and dense phases, with the experimental variance based on the results shown, yielded 8.0 and 60, respectively, compared with F o 05.21.13 = 2.45 and F o 05,18,11 = 2.7. While the bubble phase bias might be tolerated in view of the complexity of the system being modeled, the F value for the dense phase is so high that the model must be
Table I. Parameter Definitions for Orcutt Models Parameter Equivalent bubble diameter, De Bubble velocity,
Three-dimensional system Guessed or assigned
0.71(gD,)1/2 + (U
-
Reference for modification
Two-dimensional system y=H/2
Umf)
O.48(gDe)’/* + (V - Umf)
see Part I
2DeUinf
Davidson and Harrison (1963)
u, Bubble gas throughflow, q Diffusive mass transfer coefficient for bubble, iz,, Interfacial area of bubble, S, Bubble volume,
3/47D,2 [Jmf
’
0.9 7 5 aD,
a
’
(q/ 0,)
0.6 D,’
TD,2
aD, = perimeter
~D,3/6
7 D e 2 / 4 = area
judged inadequate. The same conclusion is, of course, readily apparent from visual observation of figures like Figure 1. The dense phase data shown in Figure 1 were nonlinearly fitted to determine the most likely value of the mass transfer parameter X.A value of X = 0.11 (with 95% confidence range of -0.02 5 X 5 0.24) was found, confirming that X is overestimated by Orcutt’s model. Similar estimates of X were obtained by using other dense phase profiles at different catalyst activities but for the same flow conditions. The above value of X corresponds to such an unrealistic bubble diameter, D B = 30 cm compared with D g l , = H 2 = 8.8 cm, that the controlling interphase transfer mechanism of superimposed throughflow and molecular diffusion used in the Orcutt models appears to be inappropriate for the conditions investigated in the present work. However, it is clear that the lack of agreement between the Orcutt models and experiment is due to more than simply a question of an improper transfer mechanism. For example, for seven runs a t different h but under identical flow conditions X values between 5 and 10 were required to minimize the bubble phase deviations between the DPPF model and predictions while only values very close to zero would satisfy the dense phase requirements. A value of X = 2.4 was found to minimize the overall deviation between experimental data and concentrations predicted from the DPPF model a t U = 16.5 cm/sec. The global F test value of 23.8 compared with F 0 . 0 5 98.27 1.75 shows further that the model is inappropriate, even if the interphase parameter X is fitted to the data rather than based on the hydrodynamics of the system. (2) Rowe and Partridge Reactor Models. While still relying on Toomey’s two-phase theory to estimate the visible gas flow, the models of Rowe (1964) and Partridge and Rowe (1966) interpret the flow distribution in terms of void units, i.e., bubbles and associated gas clouds, thus modifying the flow distribution of the previous models. With the inclusion of the gas cloud, it also follows that some reaction occurs within the bubble (or cloud unit) phase due to gas-solid contact there. The rate of interphase mass transfer proposed by these authors is much lower than suggested by Orcutt et al. Finally, these models allow for bubble size variations within the bed. (a) Partridge Model. (i) The visible bubble flow is given by the “two phase theory” of Toomey and Johnstone (1952). (ii) Gas flow occurs by translation of bubble and cloud units and by percolation through the dense phase. (iii) Dense phase gas is in piston flow while perfect mixing always exists between each bubble and its cloud. (iv) The
Walker (1970)
(g/0,)‘I 4
cloud size is obtained from a semiempirical modification of an expression given by Murray (1965). (v) Reaction occurs both in the dense phase and in the clouds. (vi) Interphase mass transfer occurs at the outer cloud boundary as for rigid spheres a t the same Reynolds number. The gas flow is divided into two streams. The flow due to translation of void units may be written Gc = GB(l
+
(R- 1 ) ~ ~ ~ )
(7)
where Gg is the visible bubble flow obtained from (i) above, R is the ratio of bubble and cloud volume to bubble volume, and the voidage in the cloud is assumed to be Cmf (see Lockett and Harrison, 1967). The remainder of the flow is accounted for by percolation through the dense phase, i.e. G I = U A - G, ( 8) The cross-sectional area occupied by void units is A, = RAE, = RG,/U,
(9)
An approximate expression for R, also valid for two-dimensional beds (Clift, 1970)~ is R=-
cy
a-1
Rate constants for interstitial gas and for the well-mixed bubble phase are
k i = k ; k, = k(R - 1)/R (11) respectively. Additional parameters are given in Table 11. Differential equations can be obtained by writing a material balance on each phase of the reactor (see Partridge and Rowe, 1966). These equations can be integrated numerically using the same boundary conditions as for the Orcutt DPPF model. (b) Rowe Model. This model is identical with the Partridge and Rowe model except that no interphase mass transfer is allowed and the cloud size is based on Davidson (1961). Thus Shc = 0 while for two-dimensional bubbles
R=-
c y + 1
(2-1 ( c ) Comparison of Predictions with Experiments. A serious mechanical incompatibility arose when the measured flow characteristics of Part I were used to obtain solutions for the above models. While Partridge and &we suggest that N should be based on measured bubble volumes and velocities, they did not use the same data to evaluate the bubble flow, GB, believing that Toomey’s simple two-phase theory would convey the same informaInd. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
81
Table 11. Parameters for Partridge and Rowe Model Parameter Average bubble velocity, U , Relative velocity, VR Dense phase velocity, U , Bubble phase velocity, UC Bubble diameter, DB Cloud diameter, Dc Cloud Reynolds number, Re, Schmidt number, Sc Sherwood number, Sh, Mass transfer correlation for Sh, Interphase transfer per unit volume. of void unit,
Three-dimensional system Two-dimensional system
ncmfD,D,Shc VBR
Reference for modification
Geometry
QE
Table 111. Parameters for Kunii and Levenspiel Model Parameter
Three-dimensional system
Bubble diameter, De Bubble velocity, UB Fraction of bed occupied by bubbles, E , Cloud and bubble volume/ bubble valume, R Bubble wake fraction, f, Bubble/cloud coefficient of mass transfer, (KBD)b Cloud/emulsion coefficient of mass transfer, (&D)b
Fitted O.71l(gDe)’” + (U - U m f )
DBIy;H/2
(u- U m f ) / U B
(U - u m f ) / C ’ B
(a + 2 ) / b - 1) 0.47
(a + I)/(@- 1) 0.5
4.5h
2.6% + 2.4 D. 4.5 < m f D G U B
De
+ 5.9 (-) DG2g
tion. As shown in Part I, actual GB measurements invariably fall short of the Toomey theory with the largest discrepancy a t lower bed levels. In order to adhere to the philosophy of the models, measured G B values were first disregarded and only the experimental bubble size and velocity expressions derived in Part I were fed into the computer program. Equation 11 then gave A( values larger than the total bed cross-sectional area, particularly a t lower bed levels. This was even more pronounced for the Rowe model due to the larger cloud size estimate. Thus concentration profiles and related features obtained from the above models have no physical meaning due to this inherent physical incompatibility. This finding underlines the strictness of the test to which reactor models are submitted in this work. The parallel analysis of the reactor chemical and physical properties has shown that one of the fundamental hypotheses of the model may create such an unrealistic situation that even reliable order of magnitude calculations are impossible. Modification of these models to remove the incompatibility is treated in Part 111. (3) Kunii a n d Levenspiel “Bubbling Bed Model”. The bubbling bed model of Kunii and Levenspiel (1968a, 1968b, 1969) may be considered representative of a group of models often referred to as back-mixing or dense phase flow-reversal models (see also Van Deemter, 1967; Latham et al., 1968; Fryer and Potter, 1972). When clouds and wakes are included with the bubble phase, then, on average, solids must be moving downward in the rest of the bed to compensate for net upward solids motion in the bubble phase. If one makes the reasonable assumption that the velocity of interstitial gas in the dense phase re82
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
Two-dimensional system Or D B l y = H m f / 2
0.48(gD,)‘/2
(+)
Reference
+ (U - Umf)
(e) ‘I4
lii
(Part I)
Davidson (1961) See comments Chavarie (1973) Chavarie (1973)
mains approximately Umf/cmfrelative to particles, then it follows that the downward flux of solids in the dense phase must reduce the absolute upward percolation of gas there and may even entrain gas downward. The onset of the flow reversal has been found to occur a t U/U,,f = 3 (Latham and Potter, 1970). Thus it is reasonable to apply the Kunii and Levenspiel model, with the assumption of negligible percolation of gas through the dense phase, to the flow conditions of the present work. Another peculiarity of the model is the allowance for two distinct resistances to mass transfer around a bubble, one located between the bubble and its cloud-wake, the other between the cloud and the rest of the dense phase (called the emulsion). This results in three zones of different concentration, Le., the bubble, the cloud-wake, and the emulsion. The controlling influence of the cloud to emulsion phase transfer resistance results in a considerable reduction of the overall interphase transfer compared to the transfer proposed by Orcutt et al. The assumptions in applying the Kunii and Levenspiel model are: (i) bubbles are assumed to be uniform throughout the bed; cloud sizes are calculated using the model of Davidson (1961); (ii) gas flow in the emulsion phase is neglected; (iii) transfer between a bubble and its cloud-wake occurs by molecular diffusion and by bulk flow as outlined by Orcutt et al; (iv) transfer between the cloud-wake and emulsion occurs by molecular diffusion according to the Higbie penetration theory; (v) reaction due to catalyst particles raining through the bubbles is neglected. With these assumptions, a balance on the reactant leads to the following expressions for the concentration
LOh
.
I
I
U
:
12 7 c m
/
~~
~
U
sec
y._ \
k -020 sec-'
-
... . . -. FLII
model
