334
Ind. Eng. Chem. process Des. Dev. 1003, 22, 334-339
produced in that zone. Therefore, to determine the contactor dimensions, a compromise between temperature control and high efficiency must be found. In the particular instance of this work, it is suggested that the height of the lower zone must be the required one to obtain a gas conversion a t the range of 0.50.60. Nomenclature A, R, S = 2-ethylhexanol,olefins, and water De = axial dispersion coefficient d = particle diameter I$ = efficiency of the contactor, eq 1 E2= efficiency of the contactor, eq 2 FA0 = molar flow of A at the reactor entrance HiTj= height of the i-j zone i, J = points 1, 4, 5, or 6 in Figure 1 L = length of reactor M = ratio between sections of the upper and lower zones S = cross section of the upper zone s = cross section of the lower zone t = reaction time, min u = linear gas velocity u, = velocity at which the discontinuity in the bed disappears ui = gas velocity at point i u d = minimum fluidizing velocity ud,= = minimum fluidizing velocity in the upper zone u, = bed breaking velocity U R = reduced gas velocity (u/umf) W = weight of the catalyst Wi-j = catalyst weight in the i-j zone WlF, = space time, g-hlmol of A (X& = gas conversion at the outlet of the reactor at zero time
X
= gas conversion with gas piston flow 4 : = gas conversion with cylindrical fluidized bed X,,.= gas conversion with the new type of contactor
yi-, = weight fraction of the solid in the i-j zone
Literature Cited Claus, G.; Vergnes, V.; Le Goff, P. In "Fluldlzation Technology"; Keakns, D. L., Ed.; Hemisphere Publishing Corporation: Washington, 1976; Voi. 11, p 87. Corella, J.; Ellbao, R. Id.Eng. Chem. process Des. Dev. 1982, 21, 545. Cordla. J.; Bllbao, R.; Aznar, M. P. Proceedings of the 2nd World Congress of Chemical Englneerlng, Montreal, Canada, oct 1981; Voi. 111, p 36. Coreila, J.; Ellbao, R.; Lezaun, J.; Monzh, A. Proceedings Fowth IntemaUonal Conference on Fluidkatlon, KashlkoJlm, Japan, May 1983. Fujlkawa, M.; Kugo, M.; Sa@, K. I n "Fluldlzatkm Technology"; Kealrns, D. L., Ed.; Hemisphere Publishing Corporation: Washington, 1978; Vol. I. p 41. Furusaki, S. A I C N J . 1973, 19, 1009. Gulgon, P.; Large, J. F.; Eergougnou, M. A.; Eaker. C. G. J. In "Fluldlzetbn"; Davldson, J. F.; Keakns, D. L., Eds.; Cambridge University Press: London, 1978; p 134. HanJSOn, D.; Grace, J. R. In "Fluldhtbn"; Davideon, J. F.; Harrison, D. Eds.; Acedemlc Press Inc.: London, 1971; Chapter 13. Jodra, L. G.; Corella, J.; Aragon, J. M. Int. Chem. Erg. 1#7#r, 19, 654. Jodra, L. G.; Corella, J.; Aragon, J. M. Int. Chem. Eng. 1979b, 19, 664. Kelllor, S. A.; Bergougnou, M. A. In "Fluldkatkn Technology"; Keakns, D. L., Ed.; Hemisphere Publlshlng Corporation: Washington, 1978; Vd. 11, p 45. Laguede, C.; Mdinler, J.; Angellno, H. In "Fluldlzation and Its AppNcetions", Congress of Touiouse, oct 1973 Vol. 11, p 237. Levenspiel, 0. "Chemical Reaction Enginwrlng", 2nd ed.; Wiley: New York, 1972. Olmo, A. B.; Qarcla de la Eanda, J. F. An. Quim. (Spin) 1967, 63, 179. Rooney, N. M.; Harrison, D. I n "Fluldkatkn Technology", Keakns, D. L., Ed.; Hemisphere Publlshlng Corporatlon: Washington, 1976, Voi. 11, p 7. Venuto, P. E.; Hablb, E. T. Catel. Rev. Scl. Eng. 1978, 18, 15.
Received for review February 6, 1980 Revised manuscript received May 26, 1982 Accepted September 8, 1982
Model for the Gas Flow in a Fluidized Bed with Increase of the Cross Section in Its Upper Zone JOG Corella" and Rafael Bllbao Departemento de Qdmica TBCnlca, Facultad de Ciencias, Universidad de Zaragoza, Zaragoza, Spain
A gas flow model is developed to calculate the gas conversion at the exit of a reactor consisting of different beds in the same vessel, without any separating device, originated by a noticeable increase of the cross section in its upper zone. The calculation of the gas conversion is carrled out by adapting the known models for fluidized and flxed beds according to the different statm in which the contactor can operate depending on the gas velocity. The model Is tested by means of experiments in which the contactor dimensions, the particle diameter of the catalyst, and the gas velocity are varied. In these experhnents gocd agreement between the experimental and the theoreticai values predlcted by the model was obtained.
Introduction An enlargement of the cross section in the upper zone of a fluidized bed results in a reduction of the gas velocity in ita upper zone. The gas velocity in the lower zone of the contactor, Figure 1, is greater than ud, but in the upper zone it is less, equal to, or greater than umf. This contactor offers several advantages for the catalytic and noncatalytic gas-solid reactions and for drying processes. According to its fluidodynamic behavior (Corella and Bilbao, 1982a), when the system is discontinuous for the solid, the contactor can operate: (a) as a fixed bed, when u1 < u, (u, = bed breaking velocity), (b) as a fluidized/fiied bed with intermediate gas chamber, when u,
umf,=. For gas-solid reactions, the gas conversion at the contactor exit must be calculated in a different manner depending on the state of the contactor. If it is a fixed bed (u, < u,), the corresponding design equations are applied. If it operates as a fluidized/fiied bed with an intermediate gas chamber (u, < u1 < u,), the gas conversion can be calculated as two reactors in series: the f i i t one a fluidized bed and the second one a fixed bed.
Q19&43Q5/83/1122-Q334$Q1.50/Q 0 1983 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 335
'A51 A
Figure 1. Diagram of the contactor.
'"I
A
5
l
t ronconi cal zone (4-51
I
cl
16
12
$54
'A4
E 0.
bubbling
0
20
40
60
H,
100
H,
Figure 2. Bubble diameter along the contactor.
The problem of calculating the gas conversion arises when a clear separation between the two beds does not exist, when u1 > u, or u1 > uda, velocities under which the contactor shows its greatest advantages from the gas pressure drop aspect (Corella and Bilbao, 1982a). When u1 > u, or u1 > ud,=, the bubble size is very different in each zone of the contactor. As an example, the variation of bubble diameter along the contador obtained for a given case is shown in Figure 2. The contactor was a two-dimensional bed, the solid used as a-A1203,dp = 130 Fm, the gas was air, and the distributor was a sandwich type. It can be observed that the bubble size increases in the fluidized lower zone, but it decreases in the tronconical and upper zones. This diminution of the bubble size will originate an improvement of the gas-solid contact in the upper zona of the contador with respect to a conventional fluidized bed, with the advantages coming from this improvement (Corella and Bilbao, 1983). In this work a gas flow model to calculate the gas conversion at the contactor exit when u1 > u, or u1 > umf,= is presented. The models already known for fluidized and for fixed beds have been adapted and applied to the peculiar geometry of the new type of contactor. Gas Flow Model Proposed When u > u or When
(1-4)
Figure 3. Gas flow model.
conversion at the exit of this zone, X,, will correspond to the gas conversion at the contactor exit, X4. When u1> u,, the lower zone, 1-4, is a fluidlzed bed and this model considers that its regime is bubbling fluidization. Depending on the gas velocity, the gas flow in the tronconical and upper zones can vary from piston flow to bubbling bed flow. Therefore, in the model, the 4-5 zone will be considered as formed by two regions in parallel, with a bubbling bed flow in one region (I) and a piston flow in the other region (11). The model proposed for the gas is shown in Figure 3. The importance, in each zone, of these regions in parallel will depend on the gas velocity in the same zone. Once the gas velocity is known or has been established, the apparently complex model shown in Figure 3 can be simplified in some cases by eliminating in each zone one of the regions in parallel. In the tronconical and upper zones of the contactor, the molar flow rates at the inlet of each zone and the volume of the different regions have been related by the following equations: in the tronconical zone
FA,+
FA^ = FA,= Fb(1- XAJ
v(4-5)I
+
v(4-5)11
=
v4-5
(1)
(2)
in the upper zone
FA, + FAI,= F A I = Fb(1- XAJ
> umi,em
The contactor can be divided in three zones, as in Figure 1: a lower zone (from point 1 to 41, a tronconical zone (from 4 to 5), and an upper zone (from 5 to 6). It will be considered that each zone is acting as a different reactor in series with the other ones. Therefore, the gas conversion at the exit of the lower zone, X,, will be the gas conversion at the inlet of the tronconical zone. The gas conversion obtained at the exit of the tronconical zone, XAs,will be the gas conversion at the inlet of the upper zone. The gas
lower zone
XAO
140
H, c m
uI
l
V(5-6)I
+
v(5-6)II
=
v5-6
(3) (4)
The volumes through which the bubbles pass in the tronconical and in the upper zones can be calculated as
where 8,
v(4-5)I
=
v4-584-5
(5)
V(5d)I
=
v5-685-6
(6)
and 85-6 are the fraction of bubbles in the
336 Id.Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983
tronconical zone and in the upper zone. On the other hand, it is considered that the gas molar flow which goes through each zone of the contactor under a piston flow, (FAn, F i n ) can be calculated as
_- 1 - 64-5
(7)
FA4
-* i I
-1-6,
'A5
With eq 5,6,7, and 8, the number of parameters of the model for the 4-5 and 5-6 zones have been reduced to two: 84-5 and .6, According to Kunii and Levenspiel (1969), these parameters can be calculated in each zone by means of these equations
in which the average bubble velocity, iib, through the tronconical and upper zones has been calculated as abc5
= u4-5 - umf + 0a711(gdbe,)1/2
ub,
=
- umf + 0.711(gdbw)1/2
given solid and particle size, the bubble size increases with the gas velocity and along the fluidized bed db = f(u, H) (13) In this work, the gas conversion in the lower zone has been calculated applying a model which takes into account the assumptions of the K-L model on a number of phases, gas exchange coefficient, etc., but it considers the variation of bubble size along the bed and the change in gas volume to be due to chemical reaction (Corella and Bilbao, 1981, 1982b). It is considered that the change of gas volume due to chemical reaction causes a variation of the superficial gas velocity along the lower zone in accordance with u = ui(1 t A X A ) (14)
+
where tA is the expansion factor. On the other hand, unlike the analysis of Irani et al. (1980), it is assumed that this change of gas velocity contributes to the variation of the bubble diameter, so With these assumptions, the gas conversion along this lower zone is calculated by the equation
(11) (12)
Calculation of the Gas Conversion The superficial gas velocity along the contactor can be influenced by different factors: (a) the variation of the cross section of the vessel, 03) the change of the gas volume due to chemical reaction, (c) the expansion produced by the gas pressure drop, and (d) the change of temperature within the reactor. In this work, the model is developed to calculate the gas conversion for an irreversible reaction of first order, involving a change in gas volume. The effect of the change of temperature on the gas velocity is not considered, since it will be assumed that the reactor is isothermal, which is acceptable if u6 > umf. Also, the effect of the gas pressure drop is neglected, because the experiments have been carried out in a reactor with small height. If the reactor has considerable size, this effect can easily be taken into account and the model does not lose validity. Since in each of the three zones in series of the contactor the gas velocity and the bubble size, Figure 2, can be very different, the gas conversion calculation has to be done separately in each zone. Gas Conversion in the Lower Zone (1-4). The lower zone, 1-4, is a fluidized bed. Many fluidized bed reactor models with different assumptions have been proposed. They have been analyzed and classified by several authors (Grace, 1971, 1981; Horio and Wen, 1977). Of these models, one of the best known is the bubbling bed model of Kunii and Levenspiel (K-L) (1968). Chavarie and Grace (1975) proved that, in some cases, this model is the one which better characterized the results obtained in a fluidized bed reactor for simple first reactions. Also, Mori and Wen (1976) demonstrate that, for fast reactions, this model is better than other more sophisticated ones. The fundamental parameter of the K-L model is the bubble diameter, which is assumed to be constant in the bed. However, it is well known that bubble diameter varies along a fluidized bed, depending on the operation conditions, and several empirical correlations have been proposed for calculating this variation. In all these correlations, for a determined fluidized bed and working with a
where
1 " +
k, (Kce)b
+ -1 Ye
The resolution method of eq 16 has already been shown by Corella and Bilbao (1981,1982b). The gas conversion at the exit of the lower zone will correspond to the value calculated for H = Hi-4. Gas Conversion in the Tronconical Zone (4-5). According to Kunii and Levenspiel (1969), the gas flow in the emulsion can be upflow or downflow depending on the values of (u/umf)and of (1- 6 - a6)(1+ 1/aemf).In the tronconical zone, the gas velocity and the fraction of bubbles used to compare both values have been ii4-5,an average value in this zone, and 6., So, if ( l ~ ~ - ~ / u>, f(1 ) - 64-5 - (Y64-5)(1 l/atmf), the gas flow in the emulsion is downward, case 1, and if (ii4-5/umf)< (1- 64-6 - ( ~ 6 ~ - ~ ) ( 1 + 1/aemf)is upward, case 2. The average gas velocity of the tronconical zone, r.i4+, is calculated as an arithmetical average between ita values at the inlet and at the exit of this zone. Neglecting the gas pressure drop u4 = ui(1 + ~ A X A J (18)
+
+ CAXAJ/M ( M + 1) + EA(MXA, + XAJ u5 = ui(1
ii4-5
=
2M
(19) u1
(20)
where M is the ratio between the sections of the upper and lower zones. The gas conversion calculation at the exit of the tronconical zone, XA6,implies the use of an iterative process: a given value of XA6 is assumed; with this value ii4-5 is calculated, in eq 20. According to this value the gas flow in the emulsion will be upward or downward and X,, will be calculated in a different way.
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983 337
Case 1. (a,/ud) > (1-,8 - a64d)(l+ l/aed). In this case, it is only considered one region in the tronconical zone, the one with a bubbling flow (region I). The K-L bubbling bed model is applied using the following conditions: gas conversion at the inlet, X,;_average gas velocity, ii4-5; height, H4-5;bubble diameter, db db,. As can be seen in Figure 2, the bubbg size decreases in the tronconical zone. However, since in the reactors used in this work the height of the 4-5 zone is small, it is considered that the bubble diameter in the tronconical zone is constant and equal to that at the inlet of this zone, db,. This assumption simplifies the method of calculation. Case 2. (iiM/umf)< (1- 64a - a6,)(1+ l/aemf). In this case, in the tronconical zone two regions in parallel are considered (Figure 3). The molar flow which passes as bubbles would be FAIand through the emulsion FA,. These two values must be calculated with eq 1and 7, for which the previous calculation of 64-5, eq 9 and 11, is required. The gas conversion at the exit of each of the two regions is calculated in a different way. (i) For the gas which passes as bubbles, FA,, the gas conversion at the exit, X is calculated in the same way as in case 1already indica&: (ii) For the gas which passes through the emulsion, FA,, it is considered that the emulsion in under minimum fluidization conditions and a gas piston flow is assumed in it. In the tronconical zone, with a gas conversion at the inlet equal to X ,the space time of the gas under piston flow would be [%(&5)11/ FA. (1- 64-5)], where w(4-S)II= v(4-5)11&(1 - emf) = v4-5(1 - 64-5)&(l - €mf) (21) With the above space time, the gas conversion at the exit of the region I1 of the tronconical zone, X,, is calculated. After obtaining X , and X , values, the gas conversion a t the exit of the tronconical zone can be calculated by a simple balance FA,,(^ - X A ~=) F&64-6(1 - X b d + Fb(1- 64-5)(1 - xhu) (22) which can be transformed into
Gas Conversion in the Upper Zone (5-6). The gas enters in this zone with a conversion XA6 and a velocity u5,given by eq 19. Since in this zone there is no variation of the cross section and the change of gas volume due to chemical reaction is small, the gas velocity taken as a representative value of this upper zone is up Depending on this value, three different situations can be found out (1) u5/umf < 1 The upper zone is a fixed bed and a gas piston flow is assumed. Applying the corresponding design equation for a gas conversion at the inlet equal to X b and a space time equal to (W,/FAo), the gas conversion a t the exit of the contactor, X,, is calculated. (2) u5/umf > (1-,6 - a6,)(1 + l / a e m f )
In this case, one region (I) is only considered in the upper zone. The K-L bubbling bed model is applied. The bubble diameter will decrease in this zone, Figure 2, but for a major simplicity of the calculation it is considered to be a constant bubble diameter equal to db6. The conditions used for the calculation have been: gas conversion at the inlet, XA5;gas velocity, u5;height, H,; and bubble diameter, d,. (3) 1 < u5/umf < (1 -,6 - a6,)(1 + l/CYemf)
Figure 4. Experimental assembly: 1,pressure supply; 2, feed supply; 3, capillary flowmeter; 4, preheater; 5, reactor; 6, external fluidized bed; 7, condensers; 8, liquid product collector.
Two regions in parallel, Figure 3, are considered. Part of the gas passes as bubbles with a molar flow equal to FAl and the other part passes through the emulsion with a molar flow equal to FA,,. These two values must be calculated with eq 3 and 8, for which the previous calculation of ,,6 eq 10 and 12, is required. The conversion of the gas which passes as bubbles, X,,, is calculated in the same way as in case 2. The conversion of the gas which passes through the emulsion, X , is calculated as in case 1. Once X+I and X,, values%ave been calculated, the gas conversion at the exit of the reactor will be
Experimental Verification of the Model In order to verify the proposed model, experiments with chemical reaction have been carried out. The reaction chosen is the gas phase dehydration of 2-ethylhexanol to olefins and water with a silica-alumina catalyst, L.A. 300 type, supplied by Akzo Chemie. The installation used is shown in Figure 4. All the experiments have been carried out at 240 "C, the feed of the reactor is 100% pure alcohol at a total pressure of 740 mmHg. Under these conditions, the formation of the aldehyde has not been detected with 2,4-dinitrophenylhydrazine.In addition, analyzing the products of the reaction by gas chromatography, only traces of ether formed by intermolecular dehydration have been observed. Therefore, the reaction responds to the model A + R + S. The liquid product of the reaction was analyzed for determining the experimental gas conversion (X,)exptl. This was done by measuring the concentration of water with Karl Fischer reagent, which offered good reproductivity (Olmo and Banda, 1967). Different Pyrex glass reactors have been used. These reactors differ in their characteristic dimensions (Figure 1). All of the reactors have a diameter of 16 mm in their lower zone. Their specifications include three parameters: length of the lower zone (H14, in cm), length of the tronconical zone (Hk5,in cm), and the ratio between sections (M= S/s). So the reactors designated R-12/2/4, R-6/2/4, R-6/2/2, R-6/4/6.25, and R-10/2/3 have been used. In
938
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 2, 1983
Table I. Comparison between the Experimental and Theoretical Gas Conversionsfor d p = -300 + 200 gm (umf = 3.0 cm/s) and W I F A ~= 5 g.h/mol of A XA
reactor R-121214 R-121214 R-121214 R-61214 R-61214 R-61212 R-61212 R-61416.25 R-6/416.2 5 R-101213 R-101213
u , Iumf 4 6 8 4 6 4 6 4 6 4 6
(XA,)exptl
0.954 0.923 0.860 0.962 0.931 0.946 0.909 0.966 0.946 0.949 0.915
(XA,)theor 0.803 0.659 0.490 0.621 0.476 0.621 0.476 0.621 0.476 0.780 0.619
order to guarantee the isothermal condition inside the reactors, their walls were maintained at a constant temperature. For this purpose, the preheater-reactor was submerged in an 80 mm diameter fluidized bed with corundum particles of -250 +lo0 pm (Figure 4). In the different reactors, experiments have been carried out varying the particle size, the reduced gas velocity ( u l / ~ and ) , the space time. In each run the experimental gas conversion at zero time, (Xl\l)aptl,was obtained from the representation of gas conversion against reaction time and by extrapolation to the origin. The (X,),,tl values obtained for d = -300 +200 pm (umf= 3.0 cm/s) and W/F& = 5 g hT(mo1 A) are shown in the third column of Table I. To apply the model proposed, the following data have been determined beforehand by the conventional methods in each case: kinetic equation a t 240 OC,calculated with the results obtained in an isothermal fixed bed with gas piston flow, -rA' = 11.6CA; diffusion coefficient of 2ethylhexanol in the reaction mixture at XA = 050,240 O C and 1 atm, DAm= 0.042 cm2/s; expansion factor,.t+ = 1; solid particle density, ps = 0.73 g/cm3; porosity of " m u m fluidization, emf = 0.50; wake fraction in the bubble, a = 0.30. For the lower zone, among the different empirical correlations proposed for the calculation of bubble diameter along the fluidized bed (Mori and Wen, 1975; Rowe, 1976) the correlation of Kobayashi et al. (1965) has been used db
= d,
+ 1.4pSd,(u/umf)H
(25)
which has working conditions similar to those employed in this work and verified by different authors (Jodra et al., 1979; Zabrodsky et al., 1973). The gas distributor used was a porous plate, so that d, = 0. Solving eq 16, the theoretical gas conversion at the exit has been calculated. The of the lower zone, (XA4)theor, values for the different experiments are shown in the fourth column of Table I. Applying the method of calculation above described for the tronconical and upper zones, the gas conversions at the exit of the reactor, (X,)wr, have been calculated. These values are shown in the fifth column of Table I. Comparing the third and fifth columns of Table I, one can observe that the experimental results are in agreement with those calculated theoretically and based on the model. Thus, in the sixth column of Table I, the absolute error or mean deviation, the difference between the theoretical and experimental values, is shown. One can also observe, in the seventh column of Table I, that the relative error, the ratio between the absolute error and experimental conversion, is always less than 3%.
(XA,)theor 0.965 0.930 0.883 0.970 0.934 0.946 0.908 0.973 0.954 0.960 0.924
std dev
re1 error, %
0.001 0.007 0.023 0.008 0.003 0 0.001 0.007 0.008 0.011 0.009
1.15 0.761 2.67 0.83 0.32 0 0.11 0.72 0.84 1.16 0.98
Nomenclature A, R, S = 2-ethylhexanol, olefins, and water C A = concentration of A DAm= diffusion coefficient d b = bubble diameter d, = particle diameter e = parameter defined by eq 17 F b = molar flow of A at the contactor inlet FAI = molar flow at region I of the tronconical zone Fk, = molar flow at region I of the upper zone H = height (K&,, (K$b = coefficient of gas interchange between bubble and cloud-wake region, and between cloud-wake region and emulsion phase, respectively k, = rate constant based on volume of solids M = ratio between sections of the upper and lower zones rA' = reaction rate based on volume of solid u = gas velocity u1 = gas velocity at the inlet of the contactor iii-j = mean gas velocity between i and j points ib," mean bubble velocity between i and j points u, velocity at which the discontinuity in the bed disappears u d = minimum fluidizing velocity = minimum fluidizing velocity in the upper zone E,"g bed breaking velocity V,-L= bed volume between i and j points W - weight of the solid W / F b = space time X A = gas conversion XA, = gas conversion at i point (XJerpd= experimentalgas conversion at the reador exit for
-
t = O
(X Itheor, (X&heor = theoretical gas conversions at the exit
3 the lower zone and of the reactor, respectively
Greek Symbols a = ratio of wake volume to bubble volume yb, yo ye = ratio of solids in bubbles, in the cloud-wake, and in the emulsion, respectively 6 = fraction of bubbles cA = expansion factor cd = minimum fluidization porosity pa = solid density Subscripts 1-4,4-5,5-6,1-6 = lower, tronconical, upper, and overall bed, respectively i, j = points 1, 4,5, or 6 in Figure 1 I = bubbles region I1 = emulsion or region under piston flow s = reactor exit 0 = reactor inlet
Literature Cited Chevarie, C.; Grace, J. R. Ind. Eng. Chem. Fundam. 1975. 14, 79. Corella. J.; Bllbao, R. Proceedings of JownCs EuropCnnes sur la FluMisatlon, Toulouse, France, Sept 1981. p 20-1. Corella, J.; Bltbao, R. Ind. Eng. Chem. Process Des. Dev. 1082a, 21, 545.
Ind. Eng. Chem. Process Des. Dev. 1903, 22, 339-347 Corelia. J.; Bllbao, R. An. Ouim. (Spain) 1982b, 78. 231. Cor&, J.; Bibao, R. I n d . Eng. Chem. Recess Des. Dev. 1983, companbn paper in this issue. Grace, J. R. AIChESymp. Ser.1971, 67, 159. Grace, J. R. ACSSymp. Ser. 1981. 168, 3. Horio, M.; Wen, C. Y. AIChESymp. Ser.1977, 73, 9. Iranl, R. K.; Kukarny, B. D.; Doralswamy, L. K. Ind. Eng. Chem. Fundem. 1980, 19, 424. Jodra, L. 0.; Aragon, J. M.; Corella, J. Int. Chem. Eng. 1979. 19, 654. Kobayashi, H.; Aral, F.; Chlba, T. Chem. Eng. Jpn. 1961, 2 9 , 658. Kunii, D.; Levensplei, 0. Ind. Eng. Chem. Fu&m. 1988, 2 , 446. Kunll, D.; Levenspiel, 0. “FiuMlzation Engineering”; Wiley: New York, 1969.
339
Mori, S.; Wen, C. Y. In “Fluldizatlon Technology”; Keairns, D. L., Ed.; Hemisphere Publishing Corporation: Washington, 1976; Voi. I , p 179. Morl, S.; Wen. C. Y. A I C M J . 1975, 2 1 , 109. Olmo. A. B.; Garcia de ia Banda, J. F. An. Ouim. (Spain) 1987, 63, 179. Rowe, P. N. Chem. Eng. Sci. 1978, 3 1 , 285. Zabrodsky, S . A.; Borodulya, V. A.; Dikalenko, V. I. In “Fluidization and Its Applications”; Congress of Touiouse, France, Oct 1973; Vol. 11, p 121.
Received for review April 28, 1980 Revised manuscript received July 12, 1982 Accepted September 8, 1982
COMMUNICATIONS Maximum Mass Transfer to the Wall or Immersed Objects in Liquid Fluidized Beds
A correlation to predict the maximum mass transfer from a liquid fluidized bed to the column wall or to an immersed surface has been proposed on the basis of available data in the literature. In addition, an equation is presented to determine the optimum fluidizing velocity at which the bed should be operated.
Introduction Transport phenomena in gas-fluidized beds have been the subject of innumerable investigations; mainly, the heat transfer characteristics, which cause considerable enhancement of the value of heat transfer coefficients, have been examined exhaustively. It is well recognized that, among other parameters, the heat transfer coefficient depends on bed porosity, passing through a maximum value. Since it is desirable to operate the fluidized bed in optimal conditions of transfer, many efforts have been made in order to find widely applicable correlations for maximum heat transfer coefficients and the corresponding optimum fluidizing velocities. Heschel and Klose (1971), Heyde and Klocke (1979), Martin (1980), and Grewal and Saxena (1981), for instance, have done excellent work in this field. In contrast to the numerous investigations dealing with gas-fluidized beds, little information is available for heat and mass transfer in liquid fluidized beds. Richardson et al. (1976) and Pate1 and Simpson (1977) analyzed the existing data and models for wall to bed heat transfer, adding new results. Kato et al. (1981) presented a general equation which correlates the heat transfer coefficient data for solid-liquid and gas-liquid-solid systems. Mass transfer between the liquid and a wall or an immersed surface in fluidized beds has been reviewed by Storck and Coeuret (1980) and by Riba and Couderc (1980). Tonini et al. (1981) added to the long list of correlations an equation to correlate mass transfer results obtained with Newtonian and non-Newtonian solutions. However, the resulta obtained in all these studies led to no general conclusions, probably because of the complexity of the process and the dissimilar geometries used, thus referring to different hydrodynamic flow characteristics. Although most of the papers point out the existence of a maximum transfer coefficient at a given porosity of the fluidized bed (which depends on the other involved parameters), no expression has been proposed to predict this optimum transfer rate. In the present paper the conditions 0198-4305l831 1722-Q339$Q1.50/0
leading to the maximum mass transfer coefficient will be analyzed and correlations of the type used in gas-fluidized beds will be deduced. Table I summarizes the investigations of previous workers and gives the relevant physical properties of the solid particles and the fluidizing medium. (When the values of the physical properties were not available, values reported by other authors were used for calculating the dimensionless groups). Unfortunately, the magnitude of the density difference between the solid particles and the fluid varies only slightly, since the liquid density was almost the same (about lo00 kg/m3) in all studies, and only two authors present a few data with materials other than glass or sand. Maximum Mass Transfer Coefficient The majority of the correlations for the prediction of the maximum heat transfer coefficient in gas-fluidized beds consider the maximum value of Nusselt number to be a function of the Archimedes number. Consequently, the Sherwood number calculated with the maximum mass transfer coefficient will be utilized as function of the Archimedes number, for evaluating the literature data presented for mass transfer in liquid-fluidized beds. Figure 1shows a plot of the maximum Sherwood number against Archimedes number together with the equation Sh,,
= 0.761Ar0.41
(1)
obtained by least-squares treatment of the data. This regression analysis excludes the experimental values of Sh, showing a deviation of more than *20% compared with the values predicted by eq 1, which reduces the number of points from 53 to 45. The result is quite satisfactory bearing in mind the wide variation shown by the correlations summarized in Table I, which relate the mass transfer rate to the Reynolds number, porosity, and in some cases to geometric parameters. Equation 1 shows that the maximum mass transfer rate only depends on the Archimedes number, i.e., on the relation between the drag 0 1983 American Chemical Society
