already been considered in Part 11. Calderbank and Toor (1971) describe a variable bubble property version of the Orcutt models where bubble frequency is represented by
fo = foe8’ and bubble sizes are then obtained by assuming Toomey’s two-phase theory. As shown in Part I, neither eq 1 nor the two-phase theory was obeyed in the present work. Fryer and Potter (1972) used the linear correlation of Kobayashi as modified by Kat0 and Wen (1969) to represent the variation of bubble size with position in another variable property version of Orcutt’s models. Calculations showed that the single bubble size giving the same overall conversions as the variable property version corresponds to the local bubble size a t about 40% of the bed height, rather close to UBI, = H 2 and L ) B l y = H 2 used in the present work. Whjle the concentration profiles would be somewhat different due to variations in bubble size, the overall changes are expected to be much too small to give significant improvements since no change is made in original controlling factors-the interphase transport mechanism and flow distribution. Thus the extensions of Calderbank and Toor and Fryer and Potter were not pursued in the present work. (2) Modifications of the Kato and Wen Model
In Part I1 it was shown that the original model of Kat0 and Wen (1967) gives a good prediction of the bubble phase concentration profile but tends to overpredict dense phase profiles and outlet reactant concentrations for the conditions studied. One means of augmenting conversion is to include a wake with the cloud (Yoshida and Wen, 1970). Another modification which lowers the CO‘ profiles is to decrease the interphase transfer, hence providing more contact time for the reactant depletion in the dense phase. Profiles calculated using these modifications are plotted for one set of experimental conditions in Figure 1. The dotted lines represent the model predictions when wakes of volume equal to one-half the corresponding bubble volume are added to the original model. Compared with the original predictions (shown as solid lines), the bubble phase depletion of reactant is enhanced thus increasing the overall conversion, but the dense phase concentration is even further from the measured values. The dash-dotted profiles show the model predictions when the penetration theory, giving lower mass transfer rates, is substituted for the mass transfer correlation of Kobayashi. This procedure compensates for the undesirable effect of including the wake on the dense phase profile but makes the overall conversion worse. Hence it is clear that the interdependence of the above phenomena requires careful weighing of competing effects. Little overall improvement is achieved if the order of magnitude of modifications is conserved within reasonable physical limits (e.g., wakes cannot be so large that they occupy the entire dense phase). The curves in Figure 1 illustrate reasonable limits to which the Kat0 and Wen model can be extended. Thus the conclusions in Part I1 regarding this model remain essentially valid; i.e., the model gives reasonable predictions for bubble phase profiles but overpredicts dense phase concentrations and underestimates overall conversion for the reaction studied. (3) Modification of Rowe and Partridge Models
In Part I1 the models of Rowe (1964) and Partridge and Rowe (1966) could not be applied directly because the two-phase theory of Toomey and Johnstone (1952) seriously overestimates the visible bubble flow rate. Before dis-
10
I
\, I
_\
U = 12 7
cm 1 s e c
k : 020
sec-I sri;-,o
_____... ~ . . . ~ . .w c k e
rncle
ncluded
f.
:O 5
fw
=05
wake included
~
w i t h penetration t h e o r y
-.-I.-
~.
____ . ~.
~
~
_-
~~~
cussing corrective action, it is well to discuss briefly the distribution of gas in fluidized beds between a cloud phase (including bubbles, wakes, and clouds) and an emulsion phase which incorporates the remainder of the dense phase. Even when the visible bubble flow, GB, is measured, there is no unambiguous way of assigning the remainder of the flow to either phase (Grace and Clift, 1974). This arises because bubble throughflow and dense phase interstitial flows are both invisible and unmeasured and because wake-to-bubble and cloud-to-bubble size ratios have been characterized for single bubbles but are virtually unknown for freely bubbling beds. The best one can do is to establish reasonable limits and then adjust the flow distribution within these limits subject to the constraint that the total gas flow remain UA. This process of adjusting the flow distribution will be termed “flow allotment”. Reasonable limits on the different parameters are listed in Table I. The flows through the individual phases are
while the cross-sectional area occupied by the bubble phase is
The problem in trying to apply the Rowe and Partridge models in strict accordance with the authors’ assumptions in Part I1 was that A( generally turned out to be greater thanA. (a) Simple Corrective Action. The first corrective action was t o use the measured GR distribution in place of the Toomey value. Because of the much lower value of GR, values of A( no longer exceed A , regardless of the values of f i t and R within the above ranges. Typical profiles predicted by the models with this modification are shown in Figures 2 and 3. Both the bubble and dense phase profiles as well as the outlet unconverted fraction of reactant are seriously overestimated. The larger cloud size used by Rowe seems to have little influence on the bubble phase and even though interphase transfer is nil in the latter model, the dense phase concentration predictions are hardly lower than for the Partridge and Rowe model. As seen from the dotted profiles of Figures 2 and 3, allowance for bubble wakes ( ~ L I = 0.5) lowers the estimated concentration profiles for both models, but the correction is not nearly enough to satisfy the data. The outlet unconverted fraction of reactant is rather insensitive to the inclusion of a wake since the bubble phase, though lower in Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
87
Table I Model
Assumptions Partridge model
Variable
Lower limit
Upper limit
Rowe model
GB
measured distribution (see P a r t 1)
Toomey value,
Toomey value
Toomey value
0 Davidson value
included within R Murray value
( U - Urnf)A
1.0 Davidson value (a + l)/(ff - 1)
0
fw
Murray value, a / ( f f- 1)
R
u:
1 6 5 cm / s e t
k: 0 2 0 s e C - 1
s i m u l a t e d GI a n d w a k e i n c l u d e d , f. ~ 0 . 5
-------
04-
I
001
0
20
40
60
80
l c m j ~ ~120
160
140
im
Figure 2. Concentration profiles and predictions from the modified Rowe model, U = 16.5cm/sec, k = 0.20 sec- l.
1
0
20
60
40
80 ' y ( C m lIO0
Id0
I40
160
IiO
Figure 4. Flow allotment factor d, and its dependence on superficial gas velocity and height. dense phase elements. Since such a high interstitial gas velocity appears unlikely on physical grounds, especially if one accounts for net solids downflow in the emulsion phase, it was decided to investigate other flow allotments. (b) Variable Flow Allotment. The time-average velocity of solids outside wakes may be written as
where the negative sign implies downward motion. The absolute gas velocity in the emulsion phase is then approximately L'LI Urnf/cmf so that the interstitial emulsion flow is given approximately by
+
0 01 0
A
20
40
60
80 Y
100
120
140
160
I 180
lcm)
Figure 3. Concentration profiles and predictions from the Partridge and Rowe model, U = 16.5 cm/sec, k = 0.20 sec-l. concentration, is weighted more heavily relative to the dense phase. Both models suffer from deficiencies in interphase transfer. Interphase exchange is zero in the Rowe model; while transfer is permitted in the Partridge model, its overall effect is almost cancelled by the assumption of smaller (Murray) clouds which give less conversion in cloud units. Mass transfer is also reduced greatly by the relatively high dense phase concentrations, particularly a t low bed levels. While the use of measured flow profiles removes inconsistency in the Ac values, the profiles are greatly influenced by the flow allotment adopted. The superficial dense phase percolation velocity corresponding to the flow allotment used above is 3 to 5 times Utnf at the lower levels. Thus it is not surprising that little depletion occurs in the dense phase near the distributor, so little that crossover of the profiles may even occur with depletion originally faster in the slower c l o d units than in the faster 88
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
where
q5 for fit = 0.5 and 1.0 and for the values of emf, UB,U m f , and tg given in Part I are plotted in Figure 4 . For
Values of
these values of 6, Gc values predicted by eq 6 are 140 to 200% of Gc predicted by eq 2 a t the bottom of the bed and 100 to 110% a t the top. This difference implies that void unit translation (defined by eq 2 ) is not able to account for all the gas flow associated with the bubble phase near the bottom of the bed, possibly because frequent coalescence of bubbles results ir. a significant increase in bubble throughflow. Excess dilute-phase flow has also been assumed implicitly by Kat0 and Wen who employed GL = UA in their mass balance but used a cloud size definition such that void unit translation accounts for only 60 to 70% of UA. The Partridge model was modified using this variable flow allotment. Typical profiles are plotted in Figure 5 for three different interphase transfer rates: (i) as proposed by Partridge and Rowe, (ii) a value chosen to minimize
the total sum of squared differences from the data (1.7 times the value in (i)), (iii) the penetration theory value. As noted by Litz (1973), changes in the flow distribution between phases implies further unidirectional corrective mass transfer from one phase to the other. Comparing Figures 5 and 3, we note a definite improvement in the predictions when variable flow allotment is used, especially for the dense phase, although the overall conversion remains almost the same. Nevertheless, the models cannot satisfy the initial marked concentration drop in both phases no matter what interphase mass transfer mechanism is assumed. The overall F statistic for the optimized Partridge and Rowe model is very nearly 10 compared with F l o 0 5 . 9 8 , 2 7 = 1.75. Thus, in spite of efforts to improve the description of the flow allotment and the interphase mass transfer, the models fail to give as a good representation of the data as the Kunii model. In order to follow the concentration profiles closely, it appears that a number of more or less arbitrary zones of different modus operandi may be required. (4) Kunii a n d Levenspiel Model
In Part I1 it was shown that the bubbling bed model of Kunii and Levenspiel (1968a, 1968b, 1969) gave a reasonable representation of dense phase and bubble phase concentration profiles and of overall conversions. None of the other models tested do as well, even when modifications or single-parameter fitting are allowed. For the conditions of the present work the reaction is relatively slow and the particles large enough that convective mass transfer dominates over diffusive transfer at the bubble surface. Hence it is reasonable to simplify the Kunii model by assuming perfect mixing between bubbles and cloud-wake regions. As seen in Part 11, the predicted profiles for this simplified model differ little from the more complete model for the conditions studied here. Following the procedure used in Part 11, the bubble diameter required to minimize the overall mean square deviation of the data for U = 16.5 cm/sec and different values of h was estimated for this simplified model. The value obtained was 8.7 cm with an F value of 6.2 (compared with 7.3 cm and 6.0 for the unsimplified model). The optimal bubble size is remarkably close to the measured D R ( ~ =2 H= 8.8 cm and the predicted overall conversion is in somewhat better agreement with the data. The simplified model is therefore reasonable, even though the optimum F value increases slightly. Further modifications of the Kunii model, e.g., use of variable bubble properties, result in vastly increased computational time which then puts the model outside the scope of the present study. (5) Type I or Arbitrary Two-Phase .Models
Not all models of fluidized had reactors are based on the properties of individual bubbles. Most early models considered a fluidized bed as two parallel reactors (one for the dense phase and one for the bubble phase) with crossflow (mass transfer) between them. These models are referred to as Type I and arbitrary models by Grace (1971) and Rowe (1972), respectively, who review the basic underlying assumptions. It is usual practice in such models to assume no reaction and piston flow in the bubble phase, and to estimate the flow distribution between phases from the Toomey postulate. With these assumptions, it remains only to specify the interphase transfer (or crossflow) and the axial mixing in the dense phase. Shen and Johnstone (1955) considered the limiting cases of perfect mixing and of plug flow in the dense phase. The latter case was also considered by Gomezplata and Shuster (1960). Thus the respective models are re-
08
'u.16 5 t r n / s e c ' -----.Moss transfer a n d Rowe ---MOSS transfer fOClOf
of
k = 0 2 0 sec-'
o s per P o r t r i d y e augmented
by o
I 7
00 Y (cm)
Figure 5 . Predictions of Partridge a n d Rowe model with corrected flow allotment, U = 16.5 cm/sec, k = 0.20 sec
duced to one adjustable parameter, a transfer coefficient, which may be fitted to the data. It is clear, however, that the above approach is exactly equivalent (including the limiting cases of axial dispersion chosen for the individual phases) to the two Orcutt models. As shown in Part 11, one may fit the dimensionless transfer group but without any marked improvement in the model. In comparison with Type I models, Type I1 models at least provide grounds upon which one can make some a priori estimate of the magnitude of the interphase transfer parameter. It therefore becomes obvious that these limiting simple models offer no advantages over the bubble models considered in previous sections. It would, of course, be possible to adopt a multiparameter model where additional adjustable parameters accounted for such factors as axial dispersion in the dense phase, concentration of solids in the bubble phase etc. In particular, May (1959) and Mireur and Bischoff (1967) have had some success using a two-parameter model. However, the Mireur and Bischoff correlations for these parameters are only relevant to three-dimensional beds and it would not be justified to compare the fit obtained by fitting two parameters with the bubble models considered earlier where, a t most, one parameter and usually no parameters were fitted to the data. Thus, the Type I or arbitrary models do not appear to offer any improvement over the Type I1 or bubbling bed models in representing the measured concentration profiles and overall conversions. Discussion It is important at this point to recall the basis on which models have been analyzed in the present work. Only models able to give reasonable predictions of concentration profiles in both the bubble phase and dense phase for different flow rates and catalyst activities have been considered plausible. If overall conversion had been the sole discriminatory criteria, as is generally the case, then not only would the Orcutt DPPF model have competed strongly with the Kunii model, but the well-known result for a single-phase plug flow reactor also gives a good prediction of overall conversion as shown in Figure 6. This agreement would no doubt disappear under more severe operating conditions (Ishii and Osberg, 1965). A single phase model can be but a crude representation of a two-phase system and can in no way represent the behavior of the individual phases. Thus two-phase models like that of Kunii and Levenspiel, which gives comparable agreement with overall conversion data, are Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
89
IO
0 k k
0
conclude is that the proper controlling interphase transfer lies between the zero transfer of Rowe and the combined throughflow and diffusion mechanism of the Orcutt models and that the penetration theory provides the correct order of magnitude of transfer for this work. Other mechanisms or expressions are not ruled out. For example, a correlation based on the study of mass transfer from isolated bubbles in the same apparatus (Chavarie, 1973) leads to (&&, =6.4 (cgs units) (10)
= O I O sec-'
0
= O I 5 sec-1
3
k = 0 2 0 sec-1
-
DB
0
5
IO
15
20
25
30
U (cm/sec)
Figure 6. Reactant outlet concentration and prediction from a single phase plug flow reactor model.
likely to be more reliable for more extreme operating conditions and for situations where selectivity problems are important. The model which best describes the individual phase concentration profiles as well as the overall conversions among the many models entertained in the present work is the Kunii and Levenspiel model. The bubble size required to optimize this model is also encouragingly close to the measured mean bubble size half-way up the bed. It is important to scrutinize the reasons for the relative success of this model. Although other models considered do as well in predicting the bubble phase profiles, the relative success of the Kunii and Levenspiel model is largely related to its better predictions of the dense phase profiles. The assumptions which seem to be critical in leading to low enough dense phase concentrations are the negligible percolation rate of gas through the dense phase and the reduced rate of interphase mass transfer. The reduction in interstitial flow is consistent with the flow allotment considerations developed earlier. At the same time, depletion of reactant in the bubble phase is enhanced due to the inclusion of the cloud-wake region with little resistance between the bubble and the cloud-wake for conditions in the present study. While the equivalent bubble diameter required by the Kunii model is close to actual bubble sizes as noted above, Kunii and Levenspiel used a series of simplifications which led them to an expression for the fraction of bed occupied by bubbles (9) In the present work, eq 9 overestimates measured values of (CB)by 28 to 44%. However, a t least for the conditions of this work, overestimation of EB turns out to be a convenient way of allowing for excess throughflow of gas in the bubble phase (Chavarie, 1973) and indirectly decreases the rate at which reactant is depleted in the bubble phase. Because of the coupling of the wake and cloud, no conclusions can be drawn regarding the validity of the wake size ( f ~ 3 = 0.5) or cloud size (from Davidson, 1961) used in this work. It can only be said that the combined effects of these features, when defined as above, fit in well with the other assumptions of the Kunii model. The fact that neglect of resistance to mass transfer at the bubble/cloud interface has little effect on the model predictions confirms the controlling influence of the resistance across the cloud/emulsion interface for the conditions of this work. However, the range of bubble size encountered here is too narrow for us to be able to conclude that the penetration theory is the correct transfer mechanism. All that one can 90
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
Predictions using eq 10 are so close to those obtained using the penetration model that the two approaches cannot be distinguished. In the present work, end (entrance and exit) effects are very important. Indeed, the lack of fit consistently observed in the statistical analysis is in many cases attributable to failure of models to represent end effects. The Kunii and Levenspiel model is most representative mainly due to the fact that the order of magnitude of the predicted features and most of the hydrodynamic concepts used are compatible with experimental observations. The degree to which the model can again smooth out end effects to give a reasonable overall simulation for different systems is open to question. The nature of concentration profiles suggests that the bed might best be considered as three zones in series. (i) Near the distributor the concentration drops rapidly from the inlet condition, or the occurrence of backmixing or downflow ensures that the initial dimensionless concentration does not start from unity in one or both phases. It was impossible to measure concentrations very near the distributor in the present work so the true nature of the lower region remains unexplored. Different profiles appear likely if other types of distributors are used (Walker, 1970). (ii) In the middle section of the reactor the relatively flat experimental concentration profiles for both phases appear to result from large rapid bubbles which coalesce infrequently. The rate of interphase gas exchange is probably somewhat less than proposed by the Kunii model. (iii) In the top portion of the reactor, a reproducible dip in bubble phase concentration was observed. A plausible explanation for this dip is that bubbles are diluted by a net flow of gas from the dense phase (where the reactant concentration is lower). Such dilution is consistent with the observed tendency for bubbles to expand rapidly as they reach the bed surface (Pyle, 1965). Continuity of dense phase gas may be ensured by entrainment of gas from the freeboard and/or by dense phase gas drawn from below. None of the models considered in the present work makes any allowance for radial property or concentration gradients. Consequently, most of the concentration measurements were for quartz windows or sampling ports near the axis of the two-dimensional column. What measurements were taken (Chavarie, 1973) indicate that there is no detectable concentration difference between different horizontal positions in the lower part of the bed, but there are indications that radial gradients might be more serious a t higher levels in the bed. This is in accord with hydrodynamic considerations (see Grace and Harrison, 1968; Park et al., 1969; Werther, 1973) which show that bubble distributions become less and less uniform with increasing distance from the distributor. While it is tempting to suggest that future work on reactor models for fluidized beds should take account of these radial gradients as well as the axial zones discussed above, it is not clear whether the added model complexity would be justified.
Conclusions The main overall conclusions of the present work may be stated as follows. 1. Simultaneous bubble and dense phase concentration profiles, overall conversions, and hydrodynamic features have been measured in a fluidized bed reactor. 2. These measurements provide an exacting reactor model discrimination and evaluation procedure of considerable general value despite the fact that the study is limited to a two-dimensional reactor, simple kinetics, and a limited range of operating variables. 3. A number of relatively simple two-phase reactor models were tested. None was found acceptable on statistical grounds even when a n equivalent diameter or interphase mass transfer coefficient is treated as an adjustable parameter. 4. However, some models provide a useful approximation of the reactor behavior. In particular, the model of Kunii and Levenspiel gave a reasonable simulation of the fluidized bed reactor. The relative success of this model seems to be mainly due to the use of an interphase mass transfer coefficient having the right order of magnitude and to the allowance for negligible percolation of gas in the dense phase. 5 . The results show the importance of end effects in fluidized bed reactors. They also indicate that radial gradients may become important with increasing height above the distributor. 6. Allowance for axial variations in bubble properties does not seem to be of over-riding importance in representing the reactor performance. In fact, the best model found is one which assumes uniform bubble properties throughout the reactor. Nomenclature A = cross-sectional area of bed Ac = cross-sectional area occupied by the bubble phase (including clouds) B = constant in eq 1 C, C,, C, = reactant concentration, inlet, outlet CB' = dimensionless bubble reactant concentration CD' = dimensionless dense phase reactant concentration DEI = circular area equivalent bubble diameter f, fo = bubble frequency, initial frequency f u = wake fraction, ratio of wake volume to bubble volume F = F statistic GB = visible bubble flow rate Gc = gas flow associated with bubble phase G I = gas flow associated with dense phase H = mean expanded bed height
H,f = bed height a t minimum fluidization k = reaction rate constant based on unit volume of dense phase a t t m f
k' = dimensionless reaction rate constant, kH,,f/U (KcD)= ~ coefficient of gas interchange between cloudwake region and emulsion phase
R = ratio of volume occupied by a bubble and its cloud to volume occupied by bubble itself
U = superficial gas velocity UB = bubble rising velocity Umf = superficial minimum fluidization velocity U D = time-average velocity of solids outside wakes y = distance from distributor
Greek Letters = ratio of bubble velocity to interstitial gas velocity at minimum fluidization C B = fraction of the bed occupied by bubbles t m f = dense phase void fraction at minimum fluidization 6 = correction factor on the dense phase percolation gas velocity defined by eq 6
(Y
Literature Cited Calderbank. P. H., Toor, F. D., in "Fluidization," J. F. Davidson and D. Harrison, Ed., Academic Press, New York, N.Y , 1971 Chavarie, D.. Ph.D. Thesis, McGill Universitv. 1973. Davidson, J. F., Trans. Inst. Chem. Eng., 39; 230 (1961). Fryer, C., Potter, 0. E., Powder Tech., 6, 317 (1972) Gomezplata, A . , Shuster. W. N.. AlChEJ., 6, 454 (1960). Grace,J. R.,A.I.Ch.E. Symp. Ser., 67, No. 116, 159 (1971). Grace, J. R., Clift, R . , Chem. Eng. Sci., 29, 327 (1974). Grace, J. R., Harrison, D . , 1. Chem. E. Symp. Ser., No. 30, 105 (1968) Ishii,T., Osberg. G. L., AIChEJ., 11, 279 (1965). Kato, K., Wen, C. Y., Chem. Eng. Sci., 24, 1351 (1969). Kunii, D., Levenspiel. 0 . Ind. Eng. Chem., Fundam., 7, 446 (1968a) Kunii, D., Levenspiel, O.,Ind. Eng. Chem., Proc. Des. Dev., 7, 481 (1968b). Kunii. D., Levenspiel. 0.. "Fluidization Engineering." Wiley, New York. N.Y., 1969. Litz, W., Chem.-/ng.-Tech., 45-6, 382-387 (1973). May, W. G., Chem. Eng. Prog., 55 (12), 49 (1959). Mireur, J. P., Bischoff, K. E., AICh'EJ., 13, 839. (1967). Orcutt, J. C., Davidson, J. F.. Pigford, R . L., Chem. Eng. Prog. Symp. Ser.. 58, No. 38, 1 (1962) Park. W. H., Kang, W. L., Capes, C. E., Osberg, G. L., Chem. Eng. Sci., 24,851 (1969). Partridge. B. A., Rowe. P. N., Trans. Inst. Chem. Eng.. 44, 335 (1966). Pyle, D. L.. Ph.D. Thesis, Cambridge University, 1965 Rowe, P. N., Chem. Eng. Prog.. 60 ( 3 ) , 75 (1964) Rowe, P. N , "Proceedings of the 5th European/2nd International Symposium on Chemicai Reaction Engineering," A9-1, Amsterdam, 1972. Shen. C. Y . . Johnstone, H. F., AIChEJ., 1, 349 (1955). Toorney. R . D., Johnstone, H. P., Chem. Eng. Prog., 48 i s ) , 220 (1952). Walker, B. V., Ph.D. Thesis, Cambridge University, 1970. Werther, J.,AIChESymp. Ser., 70, No. 141, 53 (1974). Yoshida, K.,Wen, C Y., Chem. Eng. Sci., 25, 1395 (1970).
Received f o r reuierc J u n e 14, 1974 Accepted F e b r u a r y I, 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 w o r k .
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