838
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bwtlon, The Combustlon Institute, Pittsburgh, Pa., 1974,p 1303. Barooah, J. N.; Long, V. D. Fuel 1976, 55, 116. Benson, S. "Thermochemical Kinetics"; Wiiey: New York, 1968. Berkowitz, N. Fuet 1960, 39, 47. Cheong, P. ti. Ph.D. Thesis, California Institute of Technology, Pasadena,
1976. Havens, J. A.; Weiker, J. R.; Sliepcevich, C. M. J. fire Flammsbi/#y1971, 2 ,
321. W e , R.; Jwves, T. A. Assoc. Comput. Mach. J . 1961, 8 , 212. Howell, J. A. M.S. Thesis, Kansas State University. Manhattan, 1979. Jahnke. E.; Emde, E.; Losch, F. "Tables of Hlgher Functions", 6th ed.; McGraw-Hill: New York, 1960.
1981, 20, 636-640
Mea, P. S.; Bailie, R. C. Combust. Sci. techno/. 1973. 6 , 1.
Pitt, G. J. Fue/1962, 41, 267. Raman, K. P.; Walawender, W. P.; Fan, L. T. Prepr ACS Div. Fuel Chem., 1980, 25(4),233. Snedecor, G. W.; &&ran, W. G. "Statistical Methods";Iowa State University Press, Ames, Iowa, 1978. Whistler, R. L.; Smart, C. L. "Polysaccharide Chemistry"; Academlc Press: New York, 1963.
Received for review August 5, 1980 Accepted June 22, 1981
Formation of Drops and Bubbles in Flowing Liquids Yoshlnorl Kawase and Jaromlr J. Ulbrecht' Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260
A model of the process when bubbles and drops are formed at a nozzle submerged vertically in a continuous phase flow has been developed by simulating the influence of the continuous phase flow by a virtual inclination of the nozzle. Predictions were compared with available experimental data and the agreement was found to be very satisfactory. The model has two versions, one for low flow rates of the dispersed phase and the other for high flow rates. The model developed in this work provides a simple but a reasonably accurate means of estimating the diameter of drops and bubbles formed in a flow normal to the nozzle axis.
Introduction Most of the literature dealing with the formation of fluid spheres (drops and bubbles) is concerned with the fluid release from a nozzle or an orifice surrounded by a liquid at rest (see the review by Kumar and Kuloor, 1970). Fluid spheres, however, are in most cases formed in liquids moving past nozzles or orifices of various types of equipment, such as perforated tray column, valve-cap tray column, bubble aerator, mixer settlers, fermentation reactors, and others. In spite of this fact less attention has been given to the formation of fluid spheres in flowing liquids and very little is known about the effect of the motion of the liquid on the formation of fluid spheres. Chuang and Goldschmidt (1970) and Sada et al. (1978) considered bubble formation in co-flowing streams, i.e. for the case when the streamlines of the continuous phase are parallel with the axis of the nozzle. The effect of the velocity of a liquid flowing past a horizontal, submerged orifice on the formation of air bubbles was investigated, and dimensionless empirical equations were proposed by Sullivan et al. (1964). Kumar and Kuloor (1970) suggested that the reduction in bubble size due to the momentum transport from the moving continuous phase could be attributed to an extra upward drag force which adds to the bubble's buoyancy and they presented an equation to estimate the final bubble volume. Itoh et al. (1979a) investigated drop formation in a uniform flow and compared the experimental data with their empirical model. In the case of co-flowing streams, their model provided an acceptable agreement with the experimental results. However, in the case where uniform stream flows normal to the nozzle axis, their model was not in satisfactory agreement with the experimental data. The formation of drops in a stirred tank was analyzed by Itoh et al. (1979b) using their empirical model for drop formation in a uniform flow mentioned above. The agreement between the predicted and the measured drop sizes was reasonable.
The purpose of this work is to develop a new model for the formation of drops and bubbles in a uniform stream flowing normal to the nozzle and to compare the predictions of this model with available experimental data. Formation of Drops and Bubbles in Flowing Liquids. Consider a drop being formed at a nozzle in a flow normal to its axis. This is recognized as being a fundamental flow situation for the formation of drops in a plate column or in a stirred tank. In the following, we shall initially follow the line of thoughts proposed by Kumar and Kuloor (1970). This model, which assumes that the bubble is formed in two stages, the expansion stage and the detachment stage, has been used to investigate the formation of a drop in a quiescent liquid and it agrees well with experimental data (Kumar and Kuloor, 1970). The expression for the final drop volume may be written as the sum of the volumes obtained from the two stages. Thus v = VE + Qt, ( 1) where VE is the force-balance drop volume and Qt, is the volume entering the drop during detachment. There are four major forces which act on a drop during the process of formation at a nozzle submerged in a quiescent liquid: the buoyancy force due to the density difference between the two fluids and the inertial force associated with fluid flowing out of the nozzle which acta to separate the drop from the nozzle, while the interfacial tension force at the nozzle tip and the drag force due to the upward movement of the expanding drop act to keep the drop attached to the nozzle. The expansion stage is assumed to end when the upward forces are equal to the downward forces. (buoyancy force) + (inertial force) = (interfacial tension force) + (viscous drag force) (2) From this equation, the force-balance drop volume VE is determined. Substituting the quantitative expressions for
om-4305iaiii 120-0636~01.2sio 0 1981 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981
637
various forces into eq 2 and arranging, we obtain
Q P & ,-!- TdNY COS 4 (3) The motion of the expanding drop during the detachment stage can be quantitatively expressed by Newton’s second law of motion.
6=tan-’(F,/F,) 4
@ = tan-’ [(uv/a)/(oh)]
b)
Figure 1. Model of drop formation in a flowing fluid at low dispersed phase flow rates: a, low continuous phase flow rate; b, high continuous phase flow rate.
This equation is solved to determine the detachment time tc with appropriate boundary conditions. Modeling in the Region of Low Flow Rates of the Dispersed Phase. In the case of extremely small dispersed phase flow rate (Q = 0), the first three terms of the right-hand side of eq 3 vanish. In the second stage, Q being vanishingly small, the value of Qtcalso becomes negligible. Hence, V = VE and we finally obtain the following equation TdNY COS 4 V= J, (5) APg where J, is a correction factor suggested by Harkins and Brown (1919). This factor is introduced to take account of a residual drop which remains at the nozzle when detachment occurs and causes the volume of the detached drop to be less than the volume at which the net buoyancy force exactly balances the interfacial tension force. Heertjes et al. (1971) presented a recalculated HarkinsBrown correction factor based on the physical properties of the system and the diameter of the nozzle used. 4 is the angle between the nozzle and the vertical. As described above, eq 5 is obtained from the forcebalance exerted on a drop, that is, the equality of the upward buoyancy force with the downward interfacial tension force. In the presence of a continuous phase flow, the size of the drop growing at the tip of the nozzle will depend on an additional force due to the surrounding flow and, therefore, this additional force must be considered in the force balance equation. The force F3 due to a continuous phase flow normal to the axis of the nozzle is, however, perpendicular to the upward buoyancy force F1 and the downward interfacial tension force F2as shown in Figure la. If only a static force balance is performed, then the force normal to the upward and downward forces has no influence on the drop formation. Experimental observations have shown, however, that the volume of the detached drops is reduced due to the drag exercised by the flow of the continuous phase. In order to accommodate the drag force of the moving ambient liquid in the static balance, we shall assume that this additional force can be simulated by adjusting the term cos 4, which represents the influence of an inclination of the nozzle. In other words, we shall assume that the formation of drops in a moving continuous phase can be simulated by drop formation from an inclined nozzle in a quiescent liquid. Thus, the effect of the velocity of the continuous phase is replaced by a virtual inclination of the nozzle (see the broken line in Figure 1). The influence of the force acting on the drop by the continuous phase flow
will be interpreted in terms of the reduction of the interfacial tension due to a virtual inclination of the nozzle. This inclination will be estimated for the cases of very low and very high continuous phase flow rates. At very low continuous phase, the virtual inclination of a nozzle is represented by the vector sum of the upward force Fl and the drag force due to the continuous phase flow F3as shown in Figure la. It is assumed in this case that the center of a drop does not move horizontally because the velocity of a continuous phase is not high and the drag force, due to it, is small. If the Stokes law is used to evaluate the drag force F3, then the virtual inclination of a nozzle is given by
4 = tan-l (F3/F1) (6) where F1 = ApgV, and F3 = 6 ~ p u ( 3 V ~ / 4 a ) ’ / ~ . The volume of the drop formed at very low velocities of the ambient (continuous) phase is determined from eq 5 and 6 using a trial-and-error method, such as that of “false position” or the “Newton-Raphson” method. At very high velocities of the ambient phase it is assumed that the drop formed at the nozzle tip is displaced horizontally by the flowing continuous phase and the virtual inclination of the nozzle is determined by this horizontal displacement (Figure lb). The center of the drop moves away from the nozzle in the horizontal direction at the speed u which is the linear velocity of the continuous phase. Therefore, the drop is displaced by a distance uVz/Q in the horizontal direction when it is detached from the nozzle tip. The virtual inclination of the nozzle in this case is
The volume of a drop formed at very high velocities of the ambient (continuous) phase is determined from eq 5 and 7 by a trial-and-error method. The actual volume V of the drop at the time of detachment will be somewhere between the two extreme cases, Vl and V,, and a simple exponential interpolation formula will be tested
v = v1(1- e-+b) + VZe*/ub
(8)
in which the parameters a and b will have to be determined from experimental data. When the continuous phase flow rate is very high, eq 8 becomes V = V,. For very low continuous phase flow rates, eq 8 reduces to V = VI. Constants a and b in eq 8 were determined as a function of the nozzle diameter using the experimental data for the formation of benzene drops in a uniform water flow published by Itoh et al. (1979a) u = 0.516/dN2.21 (9) b = 0.914/dN0.46(
(10)
638 Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981 07, systm’water-tmzene 0
0=695~10~’cc.fe~ ’
I
I
1
I
I
’/
/1
Sfstem wafer-benzene &erin sq -benzene
d M = 0.058 cm
dN = oio8
cm
Present work Kumw aK Kuloor 11970) ltoh et PI :1979!
L
-1
!
I
u
l
,
I
!
I
[cm/sec]
Figure 2. Effect of continuous phase flow rate on drop size. (Data of Itoh et al., 1979a).
.,I
08
I
l
l
I
system: w a t e r - m z e n e
a,,
’
I’
,’,‘*/ ,.
= 0.042-0 ai9 cm
/
7
i (12)
02
J
01
0 cd
‘m]
Figure 3. Comparison of experimental and calculated values of size of drops formed in a uniform flow normal to the nozzle. (Data of Itoh et al., 1979a).
Equations 9 and 10 are purely empirical formulas without any physical significance since only the nozzle diameter was varied in the set of experimental data used here. The magnitudes of the coefficients and exponents in these two formulas must necessarily depend on other geometrical parameters. Figure 2 shows that the match of our model with Itoh’s data is good, i.e., significantly better than the match of the correlation proposed by Itoh et al. (1979a) and far better than that by Kumar and Kuloor (1970). Figure 3 shows the good agreement between predided and measured drop volumes for a wide variation of nozzle diameters. It could be argued that the good fit of our model with Itoh’s data is simply due to a careful curve-fitting procedure and that this fit does not say anything about the virtues of the model or, in other words, that other models with two adjustable parameters could provide the same service. It must be recognized, however, that Itoh’s (1979a) correlation
and a = 0.94; a = 3;
0 = -0.28 (coflowing stream)
0 = -0.5 (cross-flowing stream)
In the region of low ambient phase velocities, predictions obtained by eq 12 agree relatively well with experimental results. At high continuous phase flow rates, however, their results are significantly different from the experimental data. Their model does not predict the sharp reduction of the drop volume with the increase of the velocity of the continuous phase. This sudden decrease of drop volume is well predicted by both Itoh’s (1979a) model and by our eq 8, but the average deviation of eq 11from the experimental data is 11.0%, while that of eq 8 is only 6.9% (see also Figure 3). To test further the predictive capability of our model, we have compared its predictions with another set of experimental data by Itoh et al. (1979b) obtained in stirred tanks. Itoh et al. (1979b) measured the sizes of benzene drops released from a nozzle (dN= 0.108 cm) into water and aqueous glycerin solutions in a stirred tank. We have used their data to test the utility of eq 8 and the comparison between measured and calculated drop diameters is shown in Figure 4. The average percentage error is 8.2%, which is satisfactory considering the difficulty with which accurate experimental data can be obtained. It needs to be pointed out that, in carrying out the comparison shown in Figure 4, we have used the local mean continuous phase velocity at the tip of the nozzle rather than the uniform flow velocity. Modeling in the Region of High Flow Rates of the Dispersed Phase. Although the model developed above holds very well for the formation of drops in flowing liquids, its usefulness is limited when used for bubble formation at high gas flow rates. This is best illustrated by comparing the model with the data of Ulbrecht and Ranade (1979), who measured the sizes of bubbles formed from a gas chamber rotating in a pool of water and aqueous polymer solutions. The nozzle diameter was 0.16 cm located at a radial distance 2.8 cm from the axis of rotation. The angular velocity of the chamber was varied between zero and 6.5 revolutions per second and the gas flow rate between about 1.2 and about 20 cm3/s, This is a much higher flow rate than that used by Itoh et al. (1979a,b). It will be recalled that the model derived in the preceding section of this work is based on a static force balance, and
Ind. Eng. Chem. Process Des. Dev., Vol. 20, No. 4, 1981 639 12
I
I
l
I
I
/
I
I f
/
12-
-
i
I
l , l
I
Ulbrecht and Ranade (19791 Q=tz-z15cc/sec
,’
0.8I
P
b
08 -
I
06a
a
E 04-
,(/
02-
o A
-
CC/%
1.2- 50 5.0-100 100- 21.5
-
006-
2 n
/
0
I 02
I 04
I 0.6
DCSl
I 08
I 10
, 1.2
-
[cml
Figure 5. Comparison of experimental and calculated values of size of bubbles formed in a stirred tank. (Data of Ulbrecht and Ranade, 1979).
D cal
[cm]
Figure 7. Comparison of experimental and calculated values of size of bubbles formed in a flowing liquid at high gas flow rates. (Data of Ulbrecht and Ranade, 1979; and Takahashi and Miyahara, 1978).
mation time, tc2,at very high continuous phase velocities is given by
/!I! Figure 6. Model of bubble formation in a flowing liquid at high gas
Ut& = dN 2
2
1/3Q1/3tc21/3 (15)
2
Usually the first term of the right-hand side of eq 15 is negligible compared with the second term. Therefore
flow rates (high continuous phase velocity). V2
it does not take into account the transport of momentum from the stream of the dispersed (gas) phase. Nevertheless, it is instructive to cany out the comparison in order to see what is the range of the dispersed phase flow rate in which eq 8 can be safely used. From the comparison shown in Figure 5 , it can be concluded that eq 8 will hold well up to rates of about 5 cm3/s from a single nozzle. Since inductrial spargers will have somewhere between 150 and 300 of individual nozzles, the upper limit of the “low gas flow rate’’ model will be between 0.05 to 0.1 m3/min of gas throughput. The magnitude of the upper limit depends, of course, also on the diameter of the nozzle, as it is shown in Figure 12 of the paper by Kumar and Kuloor (1970). At high gas rates through the nozzle, interfacial tension no longer controls the size of the growing bubble and the buoyancy and the inertia forces become dominant. When neglecting all terms in eq 4, except terms due to inertia and buoyancy, we have
+-=dN + &)
= Qtc2 = 0.489Q3/2~-3J2
(16)
An exponential interpolation formula, which is applicable for the entire continuous phase flow rate range, is
v = v1(1 - e-a‘/u”) + vze*’/u”
(17)
Integration of eq 13 with appropriate boundary conditions leads to
where a’ = 1.00 and b’ = 0.242. Constants a’and b’ were determined from the experimental data obtained by Ulbrecht and h a d e (19791, who used a sparger rotating in water and polymer solutions. Equation 17 correlates the data with an average deviation of 13% as shown in Figure 7. In the same figure, we have also compared data of Takahashi and Miyahara (1978), who measured bubble sizes in flowing water. The average deviation in this case is less than 6%. The larger scatter of Ulbrecht’s and Ranade’s (1979) data around the prediction of eq 17 is probably due to the influence of the viscosity of the continuous phase. Although Acharya et al. (1978) reported that the viscosity of the continuous phase has very little influence on the mechanism of bubble formation when the gas flow rate through the nozzle is in the range from 0.5 to 60 cm3/s, this conclusion applies only to situations when the surrounding liquid does not flow past the sparger.
This is the Davidson and Schuler (1960) model, which agrees reasonably well with experimental data for 4 cm3/s < Q < loo00 cm3/s as shown in Figure 9 of Kumar and Kuloor’s (1970) paper. The bubble volume VI or the bubble formation time, tel,calculated by eq 14, will hold, however, only at very low velocities of the continuous phase past the nozzle. We shall assume now that the formation time at very high velocities of the continuous phase is controlled by the following mechanism. The center of the bubble is displaced horizontally by an amount u in unit time and the detachment takes place when the center of the bubble is displaced by a distance equal to the s u m of the radius of the bubble and the radius of the nozzle (Figure 6). For-
Conclusions A simple model has been derived which predicts the diameter of drops and bubbles detached from nozzles when a uniform flow is superimposed over the nozzle. The average percentage error of 6.9% in drop diameter predictions was found. This model was also tested with data on drop formation in a stirred tank and the average percentage deviation was only 8.2%. Two versions of the model were proposed for the formation of bubbles from spargers operating in flowing liquids: one version applies to low flow rates and the other to high flow rates of the dispersed phase. The model was tested with experimental data in the range up to 20 cm3/s of gas flow through a single nozzle, i.e. up to 0.4 m3/min for a typical industrial sparger. Unfortunately, no other published data are available to test the model at higher flow rates.
640
Ind. Eng. Chem. Process Des. Dev. 1981, 20, 640-646
Nomenclature a, b = constants in eq 8 a', b' = constants in eq 17 Bo = (ApgdN2/y),Bond number D = drop diameter dN = nozzle diameter Fl = buoyancy force F2 = interfacial tension force F3 = drag force g = acceleration due to gravity n = flow index in power-law model Q = volumetric dispersed phase flow rate t = time t , = detachment time u = linear velocity of continuous phase V = drop volume V, = volume of force-balance drop u, = velocity of expanding drop u, = dispersed phase velocity through nozzle W e = (u2pDo/y)(Do/dN),Weber number x = distance measured from nozzle tip y = (V/V,) Greek Letters 0 = constants in eq 11 y = interfacial tension Ap = difference in the densities of two phases cy,
p p
= viscosity
4 = angle between the center of the nozzle and the vertical
+ = Harkins-Brown correction factor
Subscripts 0 = quiescent continuous phase 1 = very low continuous phase flow rate 2 = very high continuous phase flow rate c = continuous phase
d = dispersed phase Literature Cited Acharya, A.; Mashelkar, R. A.; Ulbrecht, J. J. I d . Eng. Chem. Fundem. 1978, 17, 230. Chuang, S. C.; Goldschmlt, V. W. J . Besic Eng. 1970, 92, 705. Davklson. J. F.; Schuler, B. 0. 0. Trans. Inst. Chem. Eng. 1980, 38, 335. Harklns, W. D.; Brown, F. E. J. Am. Chem. Soc. 1019. 41, 499. Heart@, P. M.; de Nle,I. H.; de Vrles. H. J. Chem. Eng. 1971, 26,441. Itoh, T.; Hlrata, U.; Inoue. K.; Kitagawa, Y. Chem. Eng. Jpn. 1979a, 5, 288. Itoh, T.; Hirata, U.; Inoue, K.; Yamamoto, T. Chem. Eng. Jpn. 1979b, 5 ,
a/.
313. Kumar, R.; Kuloor, N. R. Adv. Chem. Eng. 1970. 8 , 255. Sada, E.; Yasunishi, A.; Katoh, S . ; Nlshioka, M. Can. J . Chem. Eng. 1978, 56, 669. Sulllvan, S. L.; Hardy, B. W.; Holland, C. D. A I C E J. 1984, 70, 848. Takahashi, T.; Mlyahara, T. Proceedings of the Meetlng of Society of ChemC cal Engineers, Japan, 1978. Ulbrecht, J. J.; Ranade, V. R. Paper No. 1161%presented at the 72nd Annual AIChE Meeting, San Franclsco, 1979.
Received for review August 11, 1980 Revised manuscript received April 13, 1981 Accepted April 20, 1981 The work was supported in part by NSF Grant No. CPE 7916866.
= density
Thermal Decomposition of Inorganic Sulfates and Their Hydrates Jacob Mu and Daniel D. Perimutter" Depaflment of Chemical Engineering, Universiiy of Pennsylvania, Philadelphia, Pennsylvania 19 104
A study is reported of the controlled decompositions of a series of inorganic sulfates and their common hydrates, carried out in a thermogravimetric analyzer (TGA),a differential scanning calorimeter (DSC), and a differential thennal analyzer (DTA). Various sample sizes, heating rates, and ambient atmospheres were used to demonstrate their influence on the results. The purposes of this study were (1) to reveal intermediate compounds, (2) to determine the stable temperature range of each compound, and (3) to measure reaction kinetics. In addition, several solid additives: carbon, metal oxides, and sodium chloride, were demonstrated to have catalytic effects to varying degrees for the different salts.
The thermal decomposition of common inorganic sulfates has long been an important class of reactions in the chemical industry. Applications may be found, for example, in such diverse areas as ore beneficiation (McWilliams and Hixson, 1976), in metallurgical dead-roasting (Pechkovskii and Ketov, 1957), in the preparation of catalysts and molecular sieves (Wagner, 1963), and in proposed thermochemical cycles for water-splitting (Cox, 1977). Kinetic studies of such decompositions are rarely truly isothermal, for it is very difficult to establish an isothermal condition before a substantial degree of reaction has occurred in the solid. When this is the case, dynamic techniques are preferable since they monitor the change of a selected parameter in a sufficiently large temperature interval continuously. For efficient use of experimental time, dynamic kinetic studies are commonly run at relatively high heating rates 0196-4305/81/1120-0640$01.25/0
(10 OC/min or 20 OC/min). The solid reactants have been assumed to follow closely the programmed temperature increase; however, this may be a rather crude approximation when thermal decomposition is a strong endothermic reaction. Under such conditions temperature inhomogeneities may develop in the solid, as well as temperature differences between the heating phase and the solid reactant. Inconsistencies in reported values for initial decomposition temperatures of some metal sulfates were summarized by Kolta and Askar (1975). The differences in the values are as large as 150 OC. Another disadvantage to the use of high heating rates is the possible by-passing of intermediate compounds, particularly among salt hydrates. In the study reported here a series of ten common sulfates were decomposed in a thermogravimetric analyzer (TGA) with three objectives: (1) to identify intermediate 0 1981 American Chemical Society
