MASS TRANSFER IN GAS-LIQUID CONTACTING SYSTEMS A critical review with suggested generalized correlations
Gas-liquid : ' . contacting with atid without mechanical agitation is used i n a variety of industrial applications and has bccn studicd from niaiiy points of view in vavious ty-pes of equipincnt. Most of thc work reported is spccific to the apparatus used and is applicable only within its particular esperiiiiental rangc. This inforination u s d l y docs not allow a clear insight into the individual factors which affect the inass transfer taking place in these systeins. The present work represents an attcinpt to gain insight into this iniportant phenoinenon of inass transfer i n gasliquid dispersions and to obtain a general framework within which all available wbrk and seine of the contradicting resrilts ,nay be better understood. Factors Affecting Gas-liquid Mars Transfer Rater
There.are niany factors which affect inass transfer in gas-liquid dispersions. They include : --physical properties of gas and liquid -type of gas distributor, orifice diameter, spacing, and position --diinensions of column or tank, baffles (number, position, size) -type of mechanical agitator, size aiid relative dimensions -velocity of rotating impeller and energy input -gas flow rate -continuous phase flow rate in countercurrent flow system -presence of chemical reaction, concentration of electrolytes 32
INDUSTRIAL AND E N G I N E E R I N G C H E M I S T R Y
-positioii of downcoiners in indtiplate countercurrent syste1ns -presence of solid catalysts For a givcn system, the dependent parameters include: -bubble size holdup -bubble velocity of rise or relative slip velocity -actual power input
-gas
The inter-relations of all these factors make a sing1 general correlation iinpossible to achieve at this time. I t is useful, however, to combine the separate reports in the literatiire which individually attempt to correlate solile of these factors for various types of apparatus within a certain range of physical properties and experimental variables. The paraiueters most coininonly used in correlations are the volumetric holdup ratio, @, iiieaii bubble diameter, d, siiperficial gas velocity, V,, impeller Reynolds number, &, specific power input, ( P / V ) or (VD), aiid sparger configuration. These will be treated in detail here. The quantitative effects of the various systein geometry variables nientioned above on the rate of inas$ transfer is considered outside the scope of this work. For these latter variables, one should refel to Westerterp's excellent studies (50, 57) 011the effects of tank aiid iiiipeller diameters (especially the ratio of these) on the interfacial area and on the volurnetric transfer coefficient at high agitation rates. Baric Relationships
The rate of mass transfer in practically all gas-liquid contacting systems is contmlled by the liorrid nhase re-
ships between the latter and the impeller velocity. E ation 3 may be employed only with seometrically sirm r agitation equipment, whereas the relationship including the specific power input is independent of the geometry (7). The use of the latter form, however, is more di&ult since actual power input is ysually unknown, being in itself a complicated function of the type of impeller, its diameter and tip velocity, and the gas flow rate. By analogy to Equation 3, a similar relationship may be obtained for a bubbling gas-liquid system with no mechanical agitation :
% ’
B depends on the ratio ofdispersed and continuous phase
SAMUAL SIDEMAN JAMES W. FULTON ONER HORTACSU sistance, to transfer. Various definitions of the volumetric m a s transfer coefficient are commonly used depending on the type of experiment conducted. For example, for oxygen absorption in water or sulfite solution conducted under unsteady state conditions:
kLa =
1
dCL
(C,- C,)
t
A somewhat different equation is applicable to steady state:
viscosities which affects the internal circulation. This circulation, in turn, affects the external transfer coefficient (rigid or circulating sphere). Its value also depends on whether the Sherwood or the modified Sherw w d number is correlated, The additional geometric variables here are ( H / T ) and ( D / T ) or other similar combinations. Equations 3 and 4, together with Equation 2, which defines the interfacial area, a, conveniently display the important parameters to be discussed. It must be realized that the choice of the dimensionless groups is somewhat arbitrary. The gas flow group can conveniently be exchanged with a modified Weber number representing the ratio of inertia to surface tension forces. Thus the gas flow group can be substituted by (DpV$/v) in Equation 3 and (dpV?/c) in Equation 4. A number of other dimensionless correlations, mostly concerned with the maximum bubble diameter associated with the breakup of the dispersed phase, have been suggested (3, rg, 47). For a given fluid system and mechanically agitated equipment, Equation 3 reduces to the familiar form:
kLa = A’V,@N‘ = A”V,B(P/V)’”
The specific area of the gas bubbles, a, is given by: a = -
6d
d For mechanically agitated gas-liquid dispersions, dimensional analysis gives:
where the term on the left side is a modified Sherwood number and the terms on the right side are the Schmidt number, a gas flow group (24), and the impeller Reynolds number. The viscosity ratio has been included in accordance with theory (47) and experiment (4, 48). Additional dimensionless groups may be incorporated into Equation 3 to account for geometric variables, such as ( H / T ) , ( D / T ) , ( L / H ) , etc., where H a n d L are the height of the liquid and impeller in the tank, respectively. It should bc noted that Equation 3 can be transformed to include the specific power input using known relation-
(5a) where, for completely developed turbulence, > lo’, 7‘ = The simplified form of Equation 4 is : k IBtVrVT’dT’-2 (5b) L r i b The limitations inherent in Equation 5a explain some of the discrepancies in the experimental constants and exponents obtained under various system geometries and operating conditions. The various experimental results are summarized in Table 111. It is evident from the equations above that one can gain a better insight into the complicated problem of mass transfer in gas-liquid dispersions if the m a s transfer coefficient, k, (as distinguished from the volumereic mass transfer coefficient, kLa), may be assumed to be AUTHORS At the time this paper was written Samuel Sideman was Visiting Rofesmr, Oncr Horla$nr was a Gaduak Studcnt, and James W. Fulta was Assistant Professor in the School of Chemical Engineering at Oklahoma Stat8 University. Samuel Sideman is now on the staff of Technion-Imael Institute of Tech-
nology, K ~ Vrmmi. ~, VOL 58
NO. 7
JULY 1 9 6 6
33
constant. The justification and limitations of this assumption will, therefore, be discussed shortly. First, it is necessary to introduce the power input term, P, which will appear frequently in subsequent considerations. An appreciable reduction in power input is noted when gas is introduced into a mechanically agitated liquid. Correlations of the type
have been suggested (4, 8, 22, 25, 32) and the correlation constants tabulated (44) for various impeller types. More recent work (26, 30, 50) indicates that the above type correlation is suitable for constant V , only, since different curves are obtained for each value of V,. Furthermore, it was noticed by Michel and Miller (30), that the correlation given above failed for large Po ranges and predicted higher P values than were measured (30, 34). Michel and Miller’s data were taken in a 12-inch diameter tank with a 6-blade turbine impeller (3- and 4-in. diameter), for 0.8 < Vs < 1.4ft./min. and for different liquids-Le., different densities (1-1.6 gm./ml.) and different viscosities (1-29 cp.). The data were correlated to give : (7)
with an accuracy of 17y0 (34)’where Q is in cu. ft./min. This correlation, as well as Equation 6, has not yet been tested for high superficial gas velocities. The successful application of Equation 7 to data obtained with a 16blade vaned disk was reported recently (16). I t is important to note that Equation 7 yields P a ;1’3.1b for gas-liquid systems as compared with P a AT3, which is commonly used but was originally suggested for liquid mixing at high Reynolds numbers. The surprisingly small difference seems to justify the common use of the latter relationship. For moderate increases in gas rate a t a constant impeller speed, the power input to the agitator is essentially linear (34). Furthermore, for each power level of a mechanical agitator, there exists a limiting gas feed rate above which the tank will froth o\7er, but this limiting gas rate has not yet been defined (34).
MASS TRANSFER COEFFICIENT Effect of Gas Flow Rate on Mass Transfer Coefficient
For a given gas-liquid system without inechanical agitation, no effect of gas (and liquid) flow rates on the value of the liquid phase mass transfer coefficient has been detected by Calderbank (6). Similar conclusions have also been reached by Calderbank (5) for gasliquid agitated systems. This important conclusion should be considered with some reservations since the transfer coefficient depends on the bubble diameter which, as will be seen later, sometimes depends on the gas flow rate. I t would thus be more correct to state 34
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
that the gas flow rate does not affect the mass transfer coefficient, kL, provided the bubble diameter does not pass the transition zone, as will be discussed next. Effect of Bubble Size on Mass Transfer Coefficient
The effect of bubble (and drop) diameter on the transfer coefficient has been discussed in some detail elsewhere (45). I n general, one may accept Calderbank’s correlations (6, 7), whereby the transfer coefficient does not change with bubble size within the range of “small” (d < 2 mm.) and “large” bubbles (d > 2 mm.). The transfer coefficient varies rather sharply in the transition zone between these two size ranges. Unfortunately, this transition zone is not uniquely defined, being highly dependent on very small changes in surface tension. Accepting for simplicity the above classification for “large” circulating and ‘%mall” rigid bubbles, the ratio of the transfer coefficients for any one gas-liquid system is given by
____ large = 1.35 (a4rso)i/c (kL)Sdl
(8)
I t is, however, important to note that this ratio is only valid for single bubbles or at low gas holdup (approximately up to 10% holdup). As gas holdup increases and the mean free path between bubbles decreases, the ratio reduces to 1.00, as shown by Calderbank’s sieve plate studies (6). The rigid sphere characteristics exhibited by the bubbles in multibubble systems might indicate that the alternating velocity fields produced by local turbulence prevent internal circulation from becoming fully developed. Effect of Agitation on Mass Transfer Coefficient
I n view of Calderbank’s data (6, 7), it will presently be assumed that the liquid phase mass transfer coefficient is practically independent of agitation intensity. This is substantiated by findings reported by Hyiiian (23) that the mass transfer coefficient is relatively constant, even at high rates of shear. I t is, however, noteworthy that theoretical consideration (6, 7) for high mixing rates yields k, (p,/r’)@.?2 a since under these conditions,
J-0.73
(9)
( P / V ) oc -1-30’ (10) Empirical considerations based on mass transfer between a liquid and suspended solids led Verineulen ($8)to suggest that k , a >]
[email protected] a (plv)O.? (11) An aliiiost identical relationship w a s obtained by Ua\ries, Kilner, and Ratcliff ( 9 ) who studied gas absorption rates in exceedingly clean water surfaces in a stirred cell. This is also in fair agreement with the enipirical correlation suggested by Yoshida (SJ), which reduces to k , a A70.42 K (P/T’)n.14 (ttirbine) (12) k , a A\’0.54 a (P/T’)OJ8 (vaned disk) (13 ) Similar theoretical and experimental relationships have been found for the heat transfer coefficient (6,13, JJ). I t should be noted that the scatter of data in all the
empirical correlations is rather large. Since the exponent on ( P / V ) is quite small, the independency on power input may be accepted, at least for low power inputs, with &15% accuracy. I n this connection it is relevant to note that effects of impurities in ,the liquid on kL vary markedly with the intensity of turbulence in the liquid (9). The effects are small at low levels of agitation and strongest a t higher agitation levels, causing k, to reduce by as much as 700/,. Thus, the effect of power input is actually counterbalanced by the surface stresses caused by the presence of relatively small surface-active impurities. This would tend to explain the observed independency of k, on P / V power input, which should be expected if the experiments were conducted without particular care for removal of the various impurities. I t should be noted, too, that kL was found (35, 40) to decrease with increased stirring rates above 100 r.p.m. when oxygen was absorbed in a sulfite solution. Mass Transfer Coefficient Summary
A summation of the effects on k, shows that the liquid phase mass transfer coefficient may be assumed to be practically independent of the operating conditionsi.e., column height (52), gas flow rate, bubble diameter, and mixing intensity. This simplification allows one to assume that for all practical purposes the volumetric transfer coefficient, kLa,depends only on the variation of the specific area with the operating conditions.
BUBBLE S I Z E The large amount of research and the numerous correlations p-ublished on bubble size indicate the complexity of the problem at hand. A detailed review of these particular studies is outside the scope of this paper. Only the most general conclusions will be drawn, and selected correlations will be quoted to demonstrate variety as well as agreement.
case, the ultimate bubble size will be determined by the level of turbulence in the continuous phase as caused by the injection of the dispersed phase alone or by the additional energy dissipated by mechanical agitation. I t is noteworthy that considerations based on forces affecting the bubble size (3) show the bubble diameter to be independent of both orifice diameter and velocity of rise and to depend only on the physical properties-i.e., to depend on viscosities and densities when viscous forces control bubble breakup, and to depend on densities when surface tension forces control the breakup. T h e apparent effect of the higher gas flow rate on bubble diameter can probably be attributed to the secondary effect of the holdup ratio-Le., the holdup ratio increases with increase of gas flow rate, affecting average physical properties, flow characteristics, coalescence, and turbulence which, in turn, affect the bubble size. Although the above conclusions are generally independent of the distributor type, it nevertheless seems desirable to distinguish between the correlations for the nozzle-type and porous plate-type gas distributors. This distinction is desirable because the orifice diameters of these two types of distributors differ by orders of magnitude. Single- and Multiorifke Distributors
For low gas velocities (laminar flow) where bubbles form in series and where
Vo < (20 c~5D,~/p,3g~Ap~)"~ Van Krevelen (27) suggests from his work with tubes (1-9 mm. diameter i.d.) that
108 Q p
d =
(14)
whereas Leibson's (28) studies with submersed orifices give
Effect of Gas Distributor and Flow Rates on Bubble Size
I n general, for small orifices and very low gas flow rates, the bubble diameter of the slowly forming bubble depends on the orifice diameter, Do, and is only slightly dependent on V,, the gas velocity in the orifice. As the gas flow rate in the orifice increases, the bubble diameter depends more on V, and less on the orifice diameter. At high gas velocities, the dependency of bubble diameter on gas flow rate decreases with the increasing flow rates (2) as well as orifice diameter, evident from the results of Yoshida (52) and others. As orifice Reynolds numbers exceed 2100 (in the turbulent regime), "double bubbles" are formed as the smaller second bubble (formed rapidly after the larger bubble leaves the orifice) is forced into the preceding larger one. At higher gas velocities, there is an increasing formation of toroidal bubbles. Both the double bubble and the toroidal ones vibrate violently and eventually break into small bubbles. I n the particular case of low dispersed phase holdup, bubble size may be determined by the cosditions a t the orifice where the bubbles enter a hydrodynamic field in which the turbulence is low. I n the general
lI4
(,a,>
( V0?0Y3
(15 )
d = 4.57 Doliz ___
For turbulent flow encountered with high gas velocities where Vo > (20 ~ - 5 D , ~ / p , 3 g ~ A p ~ ) ~ ~ ~ and bubbles form in series, van Krevelen suggests
indicating that bubble diameter is independent of orifice diameter as well as viscosity and interfacial tension, but is rather strongly dependent on the gas velocity in the orifice. Quigley, Johnson, and Harris' (38) subsequent study with single orifices showed a weak dependency on the viscosity but a strong dependency on the orifice diameter, particularly at low flow rates. Their correlation (in British units) is : d = 0.22 D00.33 ( V0 A 0 ) 0 . 1 2 5
y0.02
+
3.02 X VOL. 5 8
NO. 7
(VoAo)1.09(17) JULY
1966
35
where D,is the orifice diameter and A , is the orifice cross section area. I n this equation, thc second terin is essentially a correction factor for high air flow ratcs, and iiiay be neglected for air rates u p to those normally encountered in sievc plates and for orifices up to ‘Ixinch in diaiiictcr. The agreeiiicnt between these last two correlations is quitc good for air rates above 10 C L I . ft./hr. per orificc and orificc diaiiictcrs larger than l / x inch. A correlation applicable for high turbulent flow in thc orifice (-YlLro> 10,000) was proposed by Leibson (28) for single orifices and was confirmed by Calclcrbank’s stiidies on sieve plates :
The dependelicy of hubblc diaineter on T,’ was recently demonstrated by Bridge, Lapidus, and Elgiii (2). At low gas flow iates (LIPto 1 ft.jinin ) bubble diameter (in water) increased from 2.3 to 3.4 mni. ,4t higher gas flow rates (1 < V , < 4.6 ft./min.) bubble diameter increased only slightly from 3.4 to 3.8 mm. They also found the dependency of bubble diameter on to be a function of surface tension as well as viscosity of the liquid media : bubble size remained unchanged (1.3 mm.)in a glycerol-water soliition ( V , < 1 ft./sec.) and increased linearly with V , (up to 3.4 ft./min.) from 1.5 to 2.4 mm.iii water containing some isoamyl alcohol. Porous Plates
Bubble diameter was measured by Pasveer (33) using different “incrusted glass” diffusers with pore diameter varying from 3 to 150 p. Large effects of pore diameter were noted, with bubble diameters varying from roughly 0.5 to 2 inm. Within the same range of low gas flow rates (0.04-0.1 6 ft./min.), the bubble size increased with increase of gas flow rate. A dependency of bubble diameter on gas flow rate was also measured by Eckenfelder (77), who used an Aloxite diffuser stone. I n the range of V , = 3.6 to 18.0 ft./min., bubble diameter varied between 0.3 to 0.6 mm., yielding the relationship d a:
v,T0.05-0.37
(19)
“for various operating conditions” (not specified). An exponent of 0.1 5 was also reported ( 7 7) for 35 ,u holes in a nylon spinneret (20 holes) with superficial velocity ranging between 3.6 to 18 ft./min., and bubble diameters varying from 1.2 to 1.6 mm. Except for very low gas flow rates, Shulman and Molstad (43) noticed no difference in bubble size when the bubble column operated with a coarse metal or a very fine carbon porous plate. Bubble diameter appeared to range between 3 to 5 itim. up to superficial velocities of about 18 ft./min., at which point coalescence was noted and larger bubbles were formed. Large slugs of air were formed at the bottom of the column when V , = 30 ft./min. One observation is quite possibly relevant to the coniment made earlier on the dependency of the ratio of the 36
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
“large” bubble and “small” bubble iiiass transfer coefficients (Equation 8) on gas flow rate. Hindered flow of bubbles was noticed (-43) at a superficial gas velocity of 9.0 ft./lnin., where hubblc velocity reached a inininium. This is consistent with Fair’s (7.3) observation that the bubbles lose individual identity at V,v > 12.0 ft./inin. I t is thus conceivable that the onset of turbulence, manifested as swirling of the bubbles and iioniiniform bubblc distribution, is associated with the dccreasc of k , . This behavior would explain the sharp transition of k,a between the “streainline” and the turbulent zone noted by Shulman (4,3). S o such transition was found by Yoshida (:Ti?), who used single large diameter orifices. Apparently larger bubbles cause higher turbulence and no well defined streamline zone actually exists tinder these conditions. This is suhstaiitiated by the apprecialily lower holdups found by Yoshida as compared with Shulman at the same superficial velocity. Effect of Holdup on Bubble Size
Coalescence is known to OCCLII’ a t large holdup ratios, thus increasing the average bubble size in the unstirred as well as niechanically agitated contactors. However, bubble size increasrs with increased distance from the impeller blades, even at low holdup ratios. Verme~ilen’sexperiments with a closed tank (L’, = 0) where gases were drawn froin the surface into the mechanically agitated liquid show that the mean bubble diameter depends on the holdup ratio (48). A practically linear relationship was found between diameter and holdup : d/do
=
1
F ( 4 ) cz 2.5 4
+ 0.7.5
(20)
where do is the meail bubble diameter a t + = 0.1. Combining gas holdup measurenients with those for interfacial area by means of Equation 2, Calderbank ( 4 ) obtained for the air-water system : d
=
4.15 C1
where C1 = &G/(P/
+ 0.09
V)0.4p,0.2
~111.
(c.g.s. units)
(21 (22)
For dispersions of air in electrolytic solutions (sodium chloride, sodium sulfate, and sodium phosphate), Calderbank obtained
d
=
2.25
[email protected](pd,
clii.
(23)
cm.
(24)
He also obtained d = 1.9
for aqueous solutions of the normal aliphatic alcohols. I n view of the above relationship for agitated systems, it is surprising that Calderbank (5) found the bubble diameter to be independent of the gas holdup from 10 to 40y0 in his sieve plate column, where the bubble diameter was about 3 mm. Above 40y0 holdup, bubble diameter increased with increased holdup. Shulinan (43) also noted that a n increase in bubble diameter occurred only at high holdups (>35y0) and V, > 15 ft./min. He also observed a range of constant holdup (-38o/c)
at 15 to 18 ft./min., which probably corresponds to Calderbank’s region of constant interfacial area. The relative independence of bubble size on V,7and on holdup ratio from 2 to 12% was recently reported by Bridge, Lapidus, and Elgin ( 2 ) . I n this range ( V s < 4.6 ft./min), bubble diameter increased only slightly from 3.4 to 3.8 mm. Effect of Electrolytes in Solution on Bubble Size
Because of the surface tension and the electrostatic potential of the resultant ions at the liquid surface, smaller bubbles are found in the presence of electrolytes in water, Initially, these effects on the k,a were noticed only at high agitator speeds and were not noticed in unagitated systems (53). Recendy, however, Yoshida (52) noted a strong effect of the electrolytes on the bubble size in unstirred colurnns, where “fine bubbles dotted with larger ones of several millimeters in diameter” were noticed in electrolyte solution, as compared with “less than 10 inin.” bubbles in water. The effect on holdup, however, is surprisingly small. Calderbank ( 4 ) suggested that hydrophilic solutes act to prevent coalescence and explained the variation of the exponent of the holdup ratio for various solutions (Equations 23, 24) as due to the ease or difficulty of particle coalescence. Effect of Agitation on Bubble Size
Under conditions of local isotropy, the local rate of energy dissipation is independent of space and directly proportional to the specific power input, giving the well known relationship :
E
=
( P / V ) = KhT3D2
(25)
Presumably these conditions exist when ( N R ~ ) ,>, ,lo4. ~~ The relationship between the bubble diameter and the power input depends upon which mixing regime is controlling (42, 44). I n the “viscous shear regime,” where particle breakup is mainly in the high shear zone, interfacial tension is very low, and particle diameter is smaller than the scale of turbulence, 7, d
~
N-312
a
E-‘i2 F ( P ~ P ~ )
(26)
I n the “kinetic energy regime” particle breakup is due mainly to fluctuations in the local turbulent velocity and to pressure variation on the surface of the particle. Unlike the former case, the particle is larger than the scale of turbulence and d a
1v-6/3
cc E-215
(27)
In the third regime, the “coalescence prevention regime,” local velocity fluctuations accelerate or decelerate coalescence, depending upon adhesion forces and the specific power input. Here d cc x-314 cc E--114 (28) Some comments regarding these relationships are of particular interest. I t is noted that Equation 26 was derived (47) based on Taylor’s equation for particle breakup due to viscous shear only, where the scale of
turbulence is larger than the particle size. In this case viscous shear forces should be much larger than the inertial forces. However, other exponents of E may be derived if one uses other expressions for the critical inertia-to-viscous forces ratio. Thus an exponent of (- 1/4) may be obtained from Calderbank’s ( 3 ) derivation (his Equation 18) based on Hinze’s (19) expression for shear stress when turbulent fluctuations alone are responsible for the dispersion process. This result, however, seems to substantiate Equation 28 suggested for the coalescence prevention regime rather than contradict the relationship suggested for the viscous regime. Moreover, the sequence presented by Equations 26 to 28 is internally consistent (44). At low gas holdup, bubble breakup was found (48) to be in agreement with Equation 26-i.e., viscous forces controlled despite the fact that d>> 7 in the gas-liquid system. This is of particular interest since no experimental evidence exists to justify the application of Equation 26 to liquid-liquid systems. It is noteworthy that Calderbank ( 4 ) agrees with Vermuelen on the dependency of bubble size with viscosity ratio when solutes introduced into the water decrease coalescence. However, Calderbank uses the exponent 2 / 3 for the power input, consistent with the kinetic regime in Equation 27, rather than with the viscous regime in Equation 26. The problem is unresolved, and additional work is required before definite conclusions can be made. I n addition, it should be emphasized that the controlling regime may change with N (47) as well as T and H even for the same fluid system, and two regimes (for example viscous and coalescence) may exist together in nonhomogeneous mixtures. Bubble Size Summary
At low gas flow rates ( V , < 1 ft./min.), the bubble diameter is a strong function of orifice diameter D, and a weak function of gas velocity in the orifice, V,. At moderate gas flow rates (1 < V , < 18 ft./niin.) this functionality is reversed, and bubble diameter becomes a stronger function of V,. At high gas flow rates (> 18 ft./inin.), both D,and I/, have weak influences on bubble diameter. The ultimate bubble size in the gas-liquid contactor, however, depends primarily on the turbulence in the continuous phase. The average bubble size increases with gas holdup owing to coalescence of the smaller bubbles. The rate of increase is more pronounced at high holdup ratios. Bubble size decreases in the presence of electrolytes in solution and/or increased agitation. The relationship between bubble size and agitation seems to depend on which mixing regime is controlling-viscous shear regime, kinetic energy regime, or coalescence prevention regime.
HOLDUP R A T I O Effect of Orifice Diameter and Bubble Size on Holdup Ratio
Higher holdup ratios were obtained at the same gas flow rates when operating with porous plates as compared VOL. 5 8
NO. 7 J U L Y 1 9 6 6
37
For spherical bubbles forming in series from a single nozzle, but without any interaction with each other, van Krevelen (27),proposed
with orifices ( 7 ) . Within each of these two general groups of gas distributors, no effect of bubble size on gas holdup was noted (27, 38, 43). Somewhat larger holdup ratios than those obtained with porous plates (43) were obtained with fritted glass distributors (73). The effect of these extremely different types of gas distributors (hence bubble size) on holdup disappears only a t high mixing rates (above 1500 r.p.m.), where bubble size becomes independent of the type of distributor employed. Similar conclusions can be inferred from the dependency of k,a on gas distributors reported by Karwat (26).
V, = ( g d 1 2 ) l / ~cm./sec.
which, similar to Equation 30 for single spheres, indicatcs a strong dependency on bubble diameter. Quigley (38), operating with a single nozzle, obtained the following empirical correlation :
V,
Since the gas holdup is directly related to the velocity of rise of the bubbles, Vb, a study of this velocity is most instructive. For bubbles moving in a stagnant liquid, the holdup ratio is given by (29)
v,
The ascending bubble velocity V, is identical with the terminal velocity V , when the bubble rise is unhindered. For a single spherical bubble at low bubble Reynolds numbers (78) :
V,
=
V,
=
1.74
cm./sec.
~
(30)
For a moderately distorted ellipsoidal bubble flowing in unrestricted media (.VR, 500) Harmathy (18) suggested :
>
V,
=
V,
=
1.53 (g+y'4crn./sec.
=
4.1 X l o 3 (T/,A,)oJ6 ft., hr.
(31)
TABLE I . Vari. abie Cove laic6 -
--
I
Ranee - of. Exberimenls .
v,
p/ Hp./Cu. FI.
VsI Fl./min.
0 0
4,
vs
N
1 .o 1 .o
...
1 .o
...
0-(-1)
,..
4
0-2.4 0-1 2 0-12.5 12.5-30
0 0 0 0
0
0-28 0-17 0-38 38-32
6
0.6-3.6
0
0
0-7
4 4 4
0
GAS HOLDUP,
I1
Ttpe of Correlation: E
%
1 .o
(33)
Note that Vb, given by Equations 32 and 33, is independent of viscosity as well as density. Furthermore, the independency of the terminal velocity on bubble size under turbulent flow conditions (Equations 31 and 33) is well established (78). I t is evident from the above relationships that the ascending velocity is independent of the gas flow rate as long as the bubble is unaffected by a following bubble. Hence, one would expect a constant ascending velocity, depending on bubble diameter, in multiorifice or porous plate columns where bubble formation is relatively slow, as compared to a single orifice column operating at a similar superficial gas velocity. I n the latter case a n impingement effect will take place whereby a newly formed bubble will be given a n additional velocity by the next bubble expanding from the orifice. Under these conditions the velocity of rise will depend on the gas velocity from the nozzle (or the superficial gas velocity) as indicated by Quigley's equation. I t should be further noted that Equation 33 was obtained when operating with a small undefined cocurrent flow of water. Hence, a somewhat higher dependency
Effect of Gas Flow Rate on the Bubble Velocity of Rise
+ = -V8
(32)
Vessel Volume, X 70%Cu. Ft.
... ... ... ... ...
j
-
D 7'
?
1
0
1749, 9680
Tank A. Diameter, 0.148, 0.165
0 0
14.24, 8.48"
i
2 2 . 2 , 367.2
0
Imptiler
None
h-one
None
None Sone None
4
12-120
0
8-60
4
0.4-36 0.5-4.5
0 0
0.11-5.5 11-55
0 0
0.76-3.2 0.33-1.5 0.33-1.5
78-340 60-350 60-350
4
4 4 @ 4 4 4 4
?
0.6-3.6
0.007-0.036
...
...
? I . .
0.01-0.2
?
,..
2.47, 14.84 (?)
0
0.8-20 1.5-10
0.84 0.8
,..
1 ,06-4,24 0.88-1.09
0
1.5-8 12-40
0.9 0.43
0.9-11 0.3-5 0.4-5
0.55 0.49-0.61 0.75
? 0-8
0.7-1.0 0.5
... I
.
...
0.45
0.8-0.6
...
0.8
0.5
...
4 4
85-854 135-790
Q
2.34
300-2200
38
0.007-0.02
2-10
...
. I .
7
0.40
...
...
0.53 0.6
0.5
...
,..
3-32
?
1 .O
I . .
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
1
170.2
0.4
1 214, 0 462
h-onc
0,208 0 . 0 8 , 0,155: 0 , 2 5 4
Sonc Sone
0.254
lionc Nonc
1.29
Vaned dirk Vaned disk Turbinc
i
( 4 2 . 4 , 159, 555
i
6.71 77 7 --.'.,
0.7
.'.
n
11.7 7 *",
i 1-5.0 0-3.3
~
0
.
- -~
-
0
...
0.47
2%
0.429 I
1
n I-
