Minimum fluidization velocities for gas—liquid—solid three-phase systems

POWDER TECHNOLOGY ELSEVIER

Powder Technology 100 (1998) 113-118

Minimum fluidization velocities for gas-liquid-solid three-phase systems J.-P. Zhang, N. Epstein *, J.R. Grace Department of Chemical and Bio-Resource Engineering, The University of British Columbia, 2216 Main Mall, Vancouver, V6T 1Z4, Canada Received 17 November 1997; received in rev:tsed form 25 March 1998

Abstract Cocurrent upward gas-liquid fluidization of coarse solids is actuated primartly by the motion of the liquid at relatively low gas velocities and by the momentum of the gas at zero or low liquid velocities. Our gas-perturbed liquid model, which has previously been shown to give good predictions of the minimum liquid fluidization velocity, U~,,f,at a fixed low gas velocity, is shown here also to give reasonable agreement with U~mfmeasurements for inverse three-phase fluidization at a given upward gas velocity, using the coefficient in the gas hold-up equation of Yang et al. [X.L. Yang, G. Wild, J.P. Euzen, Int. Chem. Eng. 33 (1993) 72] as an adjustable parameter. It is further shown that a liquidbuoyed solids/liquid-perturbed gas model can predict with moderate success the minimum gas flnidization velocity, Ugmf,for three-phase cocurrent upward fluidization of coarse solids at zero or low liquid velocities. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Gas-liquid-solid; Fluidization velocities; Coarse solids

1. Introduction Gas-liquid-solid fluidized beds are of major importance in hydrocarbon and biochemical processing operations. It is essential in considering such three-phase systems to be able to predict the minimum gas and liquid flows needed to initiate and sustain full fluidization conditions. Cocurrent upward gas-liquid fluidization of coarse solids (dp> 1 mm) is actuated primarily by the motion of the liquid at low gas velocities (Ug < 0 . 2 m / s for air and water as the gas and liquid). For such systems, we have shown [ 1 ] that the minimum liquid fluidization velocity, Ulmf, for a given modest gas superficial velocity is predicted quite well by a gas-perturbed liquid model, in which it is assumed that the solid particles are fully supported by the liquid, the role of the gas being simply to occupy space, thereby increasing the effective velocity of the liquid. A total of 264 data points, 72 from our laboratory and the remainder from the literature, were thus encompassed, with independent variables covering the following ranges: dp= 1.0 to 6.1 mm, Ps = 1170 to 7510 k g / m 3, th=0.68 to 1.0, th = 995 to 1160 k g / m 3,/z~ = 0.0009 to 0.0112 Pa s, column diameter=0.076 to 0.152 m, and Ug = 0.01 to 0.18 m/s. Presumably, bubble-induced flow can be neglected under these conditions. Although the liquids involved in all cases were Newtonian, a recent study [2] has * Corresponding author. Tel.: + 1-604-822-2815; Fax: + 1-604-822-6003

shown that the same model can be successfully adapted to non-Newtonian liquids, while we have also demonstrated elsewhere [3] that the same approach can be used to predict the minimum liquid velocity for particle transport from threephase fluidized beds.

2. Tile gas-perturbed liquid model This model, applied to three-phase fluidization involving Newtonian liquids, equates the liquid-buoyed weight of solids per unit bed volume to the frictional pressure gradient given by the Ergun [4] packed bed equation applied to the liquid-solids part of the incipiently fluidized bed. The result is Re,mr =

~/[ 1 5 0 ( 1 -

~ m f ) / 3 . 5 ( ~ ] 2 + ~naf ( 1 -- Ofmf)3 m r l / 1 . 7 5

- 150( 1 - amf)/3.5~b

( 1)

where O~mfis the gas hold-up divided by the total fluid (gas+liquid) hold-up at minimum fluidization. By means of the two approximations by Wen and Yu [5] relating the minimum fluidization voidage, ~r,f, to the sphericity, th, i.e., (~b~,f) - l = 14 and ( 1 - Emf) / t ~ 2 ~ "3

=

11

Eq. ( 1 ) may be simplified to R e l m = X/33.72 + 0.0406 Arl( 1 -

0032-5910/98/$ - see front matter © 1998 Elsevier Science S.A. All rights reserved. PIIS0032-5910(98)00131-4

Olmf) 3 --

33.7

(la)

J.-P. Zhang et al./ Powder Technology 100 (1998) 113-118

114

In the absence of gas fow, i.e., for O~mf=0 , Eq. ( l a ) becomes equivalent to the Wen and Yu [5] equation for minimum liquid-solid fluidization. For three-phase (gasliquid-solid) fluidization, however, an estimate of O~mfis required in order to solve Eq. (1) or (la). A good estimate at minimum fluidization [ 1] is provided by the empirical equation of Yang et al. [6] for cocurrent upward flow of gas and non-foaming liquid through fixed beds of solids, which can be written as 0.16U~

(2)

O~mf~---

~mf ( Ug + Ulmf)

forx--- Ug/ ( Ug + U~) _<0.93. Eq. ( 1) or Eq. ( 1a), together with Eq. (2), must be solved simultaneously or iteratively for a,nf and U~mf.The left sides of Figs. 1-3 show experimental data [ 1,7] and corresponding predictions for minimum air-water fluidization of 1.5 mm glass beads, 4.5 mm glass beads and 1.2 mm steel shot, respectively. In each case there is seen to be quite reasonable agreement.

'4T ,2

•

10-

0

Ugmf VS. UI

8"0

6-

~o0

4-

eme#

...~,

~

eqn.(7) withe~. (8)

2-

16o

0

2;0 3bo 460 s60 o",60O Ug (or Ugmf), mm/s

700

Fig. 1. Minimum air-water fluidization velocities for 1.5 mm glass beads in an 82.6 mm column. Points are experimental; lines are predictions. 40-

35!

•

v mlvs, vg

3oi

0

UgmfVS. Ul

25-

~

20-

:/eqn.

(la) with eqn. (2) ' eqn . (1) with eqn . (2)

'"....

•

sl

"" "'.

/

"'"'"'260 460 660 s60 ~000 12'00 14'00 Isoo 0

0

'

0

eqn. (7) with eqn. (8)

'

0 0

'

I

O,

,4

.

Ug (or Ugmf ), mm/s Fig. 2. Minimum air-water fluidization velocities for 4.5 mm glass beads in an 82.6 mm column. Points are experimental; lines are predictions.

30

25 t

20

•

u, msvs. ug

0

UgmfVS. U,

f-.

3"

15.

~,

/oqn. (I)with~. (2)

10- "_ ~_

....

/ / e q n . (la)with eqn. (2)

O 5-

...... O

eqn. (7) with eqn. (8) -- 4 O

0

260

460

660

"'"

800

1000

1200

1400

Ug (or Ugh/), rnm/s Fig. 3. Minimum air-water fluidization velocities for 1.2 mm steel shot in an 82.6 mm column. Points are experimental; lines are predictions.

3. Inverse fluidization

Liquid anti liquid-gas fluidization of solids whose density is less than that of the liquid has received some attention during the past decade, e.g., for wastewater treatment [8]. For such systems, the liquid flows downward, while the gas travels countercurrently upward. Only two studies [9,10] have addressed the minimum fuidization problem for such inverse systems. Both report that as the upward gas velocity is increased, which might intuitively lead one to expect that downward fluidization by the liquid would be thereby impeded, the minimum liquid fluidization velocity actually decreases. This observation is, however, consistent with what should occur if the sole role of the gas is to occupy space. In other words, the same qualitative trend was observed as for upward cocurrent gas-liquid fluidization and as predicted by our gas-perturbed liquid model. We have therefore tested our model quantitatively against these literature data. The properties of the particles used in the above two studies are summarized in Table 1. Legile et al. [9] assigned particle diameters of 3.5 mm to both their cork granules and their polyethylene cylinders. In Eq. ( 1) or Eq. ( l a ) , however, the required dianteter in the Reynolds number is the equivolume sphere diameter, dp; the value of dp recorded in Table 1 for the cork 'granules' is, at best, a crude estimate, while, for the polyethylene cylinders where the diameter and length are given by the authors to only one significant figure, our calculated value of d o = 4.2 mm may also be significantly in error. The low value assigned by the authors [9] to the cork density ( 106 kg/m 3) may also be inaccurate in the absence of any stated means for preventing the cork from being permeated by the liquid. The averaged value of emf=0.35 reported [ 10] for the 4 mm hollow propylene spheres is also unusually low. The gas in both studies was air, while the liquid in one study [9] was water and in the other [ 10] was a 5% aqueous sodium chloride solution at 28°C. Solution of Eq. ( 1 ) or ( 1a) for inverse fluidization requires an equation for Ogmfvalid for counter-current gas-liquid flow

J.-P. Zhan g et al. / Powder Technology 100 (1998) 113-118

115

Table 1 Inverse fluidization particle properties Reference

Particle type

Dimensions (mm)

dp (mm)

p~ ( k g / m 3)

~'rnf

Legile et al. [ 9] Legile et al. [ 9] Ibrahim et al. [ 10l Ibrahim et al. [ 101

Cork granules Polyethylene cylinders Polypropylene spheres Polypropylene spheres

d = 3 to 4 d = 4, 1= 3 d=6 d=4

3.5 4.2 6 4

106 880 862 877

0.36 0.47 0.38 0.35

through a fixed or incipiently fluidized bed. The only equations for a~nf available in the literature, other than for rings and saddles [ 11 ], are for cocurrent upward gas-liquid flow, as in the case of Eq. (2), or cocurrent downward gas-liquid flow. For this reason, an equation of the form of Eq. (2), namely O/mf ~"~-

,Umf (Ug "~-Ulmf)

35

•

--,

•

Ps=106kg/m 3,dp=3.5mm

.......

•

Ps = 880 kg/m3, dp = 4.2 mm

3025-

(3)

was applied, with C as an adjustable parameter. Some justification for this procedure is provided by the fact that Yang et al. [6] themselves found that Eq. (3) was equally valid for both non-foaming and foaming liquids, but with C = 0.16 for the former and C = 0.28 for the latter. An additional rationale for using the same form of equation for both cocurrent and countercurrent flow is that both upward and downward cocurrent flow have been successfully correlated by similar functional relationships (differing only in constants), with the velocity ratio as the key variable [12]. Eq. (la) rather than Eq. ( 1 ) was solved in conjunction with Eq. (3) so as to minimize dependence of the solution on knowledge of emf and its uncertainties. From the twelve three-phase data points of Legile et al. [9] and the seven of Ibrahim et al. [ 10], the best overall fit was for C = 0.22, midway between the two values of C proposed by Yang et al. [6]. The model predictions with C = 0.22 are compared with the data points in Figs. 4 and 5 for the data of the two studies. The model follows the trends of the experimental data for all four systems (two investigated by Legile et al. [9] and two by Ibrahim et al. [ 10] ). In the former case, there is even good quantitative agreement, with moderate agreement also in the latter case. Some of the discrepancies may be due to the uncertainties in particle properties noted above. A separately adjusted C for each experimental system would allow even better agreement, but complicating the model in this manner is not warranted in the absence of experimental information on amf for gas-liquid counter-current flow. A parity plot for all the inverse fluidization data points, including the additional four points for two-phase liquidsolid fluidization (i.e., for zero gas flow) is shown in Fig. 6. The average absolute deviation (AAD) between the predicted and experimental results is 28%, nearly as good as for predictions of cocurrent upward minimum fluidization [ 1] (where AAD = 26%), and even for widely accepted equa-

st "~i....... o

.

0

2

4

6

.

.

.

8

.

.

10

.

.

12

.

14

.

.

.

16

18

20

Ug , m m / s Fig. 4. Experimental data points of Legile et al. [9] for inverse fluidization comparexl to predictions of Eqs. ( I a) and (3) with C = 0.22.

20

- - ,

l

S

5.

,

~

"',,,

...... '

•

•

dp=6mm

•

dp = a t o m

•

0

Ug , m m / s Fig. 5. Experimental data points of Ibrahim et al. [ 10] for inverse fluidization compared to predictions of Eqs. ( 1a ) and (3) with C = 0.22.

tions figrtwo-phase fluid-solid systems (e.g., Ref. [5], where AAD = 25%). It has very recently been reported [ 13] that inverse airwater fluidization of solid particles having particle density close m the liquid density (ps=934 kg/m 3 compared to p~= 997 kg/m 3) can be induced with zero liquid flow if the gas velocity is increased sufficiently. Apparently the increa,;ed gas hold-up in the upper region of the bed eventually becomes sufficient to reduce the effective fluid (gas-

J.-P. Zhang et al. / Powder Technology 100 (1998) 113-118

116

Table 2 Predicted values of Ug,,,~at U~=O for air and water as the gas and liquid compared with experimental data [7]

10 2

Ps(kg/m3), dp (ram) •

•

,

,

[] 0 A V

862 877 106 880

4

6 4 3.5 ~ ~

Particles

10 ~

Glassbeads Glassbeads Steel shot

;e

/ 10 0

,

. . . . . . .

10 0

i

101

10 2

Experimental Ulmf, m m / s Fig. 6. Parity plot comparing predicted U~,~values with "allavailable minimum inverse-fluidization velocity data for three-phase systems (solid points) and corresponding two-phase points for zero gas flow (open points).

liquid emulsion) density immediately below the particles to a point where the bed expands downwards into a fluidized state. This is a case where the gas fulfils a role (fluid density reduction) beyond merely occupying space; the present model is unable to handle this situation.

4. Zero liquid velocity We next return to cocurrent upward gas-liquid fluidization, considering the other end of the spectrum where the liquid velocity is very low or non-existent. In such cases fluidization of the solids is activated by the gas. In the absence of liquid, the Wen and Yu [5] equation for gas-solid minimum fluidization can be written Regmf" = 1/33.72 + 0.0408 Arg - 33.7

(4)

Although the situation we wish to consider initially involves no liquid flow, the column remains filled with liquid, as in conventional bubble columns. Eq. (4) is therefore modified in one respect. If it is assumed that the solids remain wetted by, submerged in, and thus buoyed by the liquid, the buoyancy term (p,-pg)g in the gas-phase Archimedes number is replaced by (p~- p~)g, so that Regmf = 1/33.72 + 0.0408 Aqg - 33.7

p, (kg/m 3)

el,

1.5 4.5 1.2

2530 2490 7510

0.39 0.37 0.41

Ugmf

Ugmf"

Ugmf

(m/s) expt.

(m/s) Eq. (4)

(m/s) Eq. (5)

0.55 1.07 1.32

0.78 1.51 1.39

0.59 1.36 1.23

gas. All three types of particles were uniform in size and spherical in shape. Their densities, diameters and static bed voidages are given in Table 2. Further details of the experiments have been given by Zhang [7]. As expected, Eq. (4) always overpredicts •½mf, since it takes no account of the buoyant support provided to the solids by the liquid. Eq. (5) shows considerably improved agreement with the experimental results for both sizes of glass beads. In the case of the steel shot, for which overprediction by Eq. (4) is smallest, Eq. (5) actually underpredicts Ugmr by a similarly small amount. The agreement of the data with Eq. (5) is rather surprising in view of the assumption that ce,,f= 1 (i.e., liquid holdup = 0), which appears to be inconsistent with the fact that, even though U~= 0, the momentum of the gas is transferred to the solids largely through the medium of the liquid, the holdup of which must therefore exceed zero. There is a considerable body of literature, summarized by Abraham et al. [ 14], in which Ugmffor fluidization of solid particles in three-phase bubble columns is measured and correlated empirically. These investigations have been primarily focused on systems with particles of diameter < I mm. What we have called Ugmris usually referred to in these studies as the 'critical gas velocity' for the suspension of solids. The derived empirical equations typically show this velocity to be dependent not only on the variables incorporated in Eq. (5), but also on the column diameter and solids loading. In our work, we did not vary the column diameter, but we did vm3, the solids loading. Fig. 7, which is typical of our results (all of which correspond to d p > 1 mm), indicates that Ugmf at U~= 0 is independent of static bed height, Ho, a measure of solids loading.

(5)

where Ar~gis the gas-phase Archimedes number with liquidbuoyed solids given by Arlg = p~, ( p ~/-~p ~ ) g d ~

dp (mm)

(6)

Table 2 compares experimental data at zero liquid flow for two sizes of glass beads and one size of steel shot with the predictions of Eqs. (4) and (5). The experimental measurements were obtained in a column of diameter 82.6 mm using water at 8°C as the liquid and air at atmospheric pressure as

5. Low liquid velocities Turning now to the closely related situation where there is a small liquid flow, far less than that required to fluidize the solids, we revert to Eq. (5), inserting O~mf(which can now be appreciably less than 1). By analogy with Eq. ( l a ) , we write Re, mr= ~ . 7 2 + 0.0408 Arlg oL3f-- 33.7

(7)

and by analogy with the gas-perturbed liquid model at relatively low gas velocities, Eq. (7) can be considered a liquid-

J.-P. Zhang et al. / Powder Technology 100 (1998) 113-118

1000-

©

900. 800-

• /I,,

///

-m-- H : 770 rnm - - o - - H~ = 440 mm I1~

I

700"

t

¢' O"

5001

40011 o.o

~'-0 , u .z

o.5

1.I0

115

£.o

2.~

117

ing from the experimental points (open and closed circles). The ahrupt intersection occurs because the middle region, in which both the liquid and the gas play a role in providing direct ,or indirect support to the solids, has not yet been modelled..An attempt by us to do so by applying the Ergun [4] equation additively with respect to the gas and liquid phases was urtsuccessful. Of the empirical equations for the critical gas velocity to suspend fine particles in a liquid tabulated by Abraham et al. [ 14], only that of Pandit and Joshi [ 15,16] takes account of liquid flow, and this equation overpredicts Ugmffor the present conditions to a much greater extent than Eqs. (7) and (8), except in the single case of Ugmffor 1.5 mm glass beads at U~=0 [7].

Superficial Gas Velocity, Ug, m/s Fig. 7. Effect of particle inventory on Ugmrat Uj = 0 in an 82.6 nun diameter column for air and water as the gas and liquid with dp= 1.5 ram, p, = 2530 kg/m-~; Ap was measured over intervals of 300 and 600 m m inside dense bed.

buoyed solids/liquid-perturbed gas model for low liquid velocities. The values of x = U g / ( U g + UO for those cases in which relevant experiments were performed fall between 0.98 and unity. Unfortunately, while Eq. (2) above, which is required to solve Eq. (7), provides a reasonable fit to the experimental data of Yang et al. [6] for 8rnfO~mfVS. X up to X= 0.93, it fits poorly at higher values ofx. To rectify this situation, the nonfoaming data plotted in Fig. 16 of Yang et al. [6], with particular weight to those for which x > 0.93, were fitted by the empirical equation -- /tan/~°6nf= Emf L ~.1.076

1.4 + t a n ( l . 4 )

(8)

6. Conclusions

( 1 ) The gas-perturbed liquid model, which has previously been shown to give reasonably good predictions both of flm f for cocurrent upward three-phase fluidization of coarse (d v > 1ram) solids at relatively low gas velocities [ 1,2] and of minimum liquid velocity for particle transport from threephase fluidized beds [3], is shown here to be capable of predicting Ujmr for inverse three-phase fluidization, using the coefficient in the gas hold-up correlation of Yang et al. [6] as a fitt:tng parameter. (2) For zero or low liquid velocities, a liquid-buoyed solids/liquid-perturbed gas model is shown to give reasonable predictions of Ug,,f for particles larger than about 1 mm in a bubble column. (3) The intermediate region where both liquid and gas play significant roles in fluidizing coarse solids remains to be modelled, as does the transition from coarse (dp> 1 mm) to fine (dp< 1 mm) solids systems.

(all angles in radians) Note that for x = 1 (i.e., U~=0), Eq. (8) yields ~,,famr =0.406, implying that since C~mr= l for this condition, er,r = 0.406, a reasonable value for the minimum fluidization voidage of impermeable spheres. Eq. (8), with x = Ugmf/(Ug.~r+ Ut), was solved simultaneously with Eq. (7) for C~mfand Ugmfat lOW values of U~, again for air-water (8°C) fluidization of 1.5 mm glass beads, 4.5 mm glass beads and 1.2 mm steel shot. The computed results are shown towards the right of Figs. ( 1 ) - ( 3 ), respectively, as dashed lines together with corresponding experimental results obtained in our 82.6 mm diameter column. The data for the case where U~= 0 are also shown. It is seen that Eqs. (7) and (8) are in reasonably good agreement with the experimental data for the 1.5 mm glass beads (Fig. 1), while they moderately overpredict most of the data for the other two types of particles (Figs. 2 and 3). The predicted lines intersect the Gas-Perturbed Liquid Model lines abruptly, rather than merging with them smoothly, as they shouldjudg-

7. List of symbols

Arg Arl Arlg C d

a0

g

/4,, I AP Regmf

Gas phase Archimedes number = pg ( p~ - pg) gdp3 / t~f ( - ) Liquid phase Archimedes number = pl[ (p~- ¢~) [gdp3 / tz~2 ( - ) Gas phase Archimedes number with liquid-buoyed solids = pg(p~ - p~ )gdp3 / i,zg 2 ( - ) Constant in Eq. (3) ( - ) Granule, cylinder or sphere diameter (mm) Equivolume sphere diameter (mm or m) Acceleration of gravity (m/s 2) static bed height (ram) Cylinder length (ram) Pressure drop (Pa or mm water) Gas Reynolds number at minimum 3-phase fluidization = PgdoUgmf/ Id,g ( - )

118

Regmf" Retmf

v~ Ugmr

Usmf" U, Ulmf x

J.-P. Zhang et aL / Powder Technology 1O0 (I 998) 113-118

gas Reynolds number at minimum 2-phase gassolid fluidization = pgdpUgmft~'//,Lg ( - ) liquid Reynolds number at minimum 3-phase fluidization = pldpUlmf/]./,1 ( - ) Superficial gas velocity (mm/s or m/s) Superficial gas velocity at minimum 3-phase fluidization (mm/s or m/s) Superficial gas velocity at minimum 2-phase gassolid fluidization (mm/s or m/s) Superficial liquid velocity (mm/s or m/s) Superficial liquid velocity at minimum 3-phase fluidization (mm/s or m/s) Superficial velocity ratio = Ug/ ( Ug + U~) ( - )

Greek symbols O~mf

e~ emf so /xl & P,,

4,

Gas hold-up at minimum fluidization on a solidsfree basis = ~g/O~mf( -- ) Volume fraction occupied by gas at minimum fluidization ( - ) Total (gas + liquid) voidage at minimum fluidization ( - ) static voidage for particles in liquid ( - ) Gas viscosity (Pa s) Liquid viscosity (Pa s) Gas density (kg/m 3) Liquid density (kg / m 3) Solid particle density (kg/m 3) Particle sphericity ( - )

Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for supporting this work. References [ 1 ] J.-P. Zh~mg,N. Epstein, J.R. Grace, J. Zhu, Trans. Instn. Chem. Engrs. 73A (1995) 347. [2] H. Miura, Y. Kawase, Chem. Eng. Sci. 52 (1997) 4095. [3] J.-P. Zhang, N. Epstein, J.R. Grace, in: L.S. Fan, T.M. Knowlton (Eds.), Fluidization IX, Engineering Foundation, New York, 1998, p. 645. [4] S. Ergun, Chem. Eng. Progr. 48 (1952) 89. [5] C.Y. Wen, Y.H. Yu, Chem. Eng. Progr. Symp. Ser. 62 (62) (1966) 100. [6] X.L. Yang, G. Wild, J.P. Euzen, Int. Chem. Eng. 33 (1993) 72. [7] J.-P. Zhang, PhD thesis, Univ. of British Columbia, Vancouver, 1996. [8] L.S. Fan, Gas-Liquid-Solid Fluidization Engineering, Butterworth, Stoneham, MA, 1989. [91 P. Legile, G. Menard, C. Laurent, D. Thomas, A. Bernis, Int. Chem. Eng. 32 (1992) 41. [10l Y.A.A. Ibrahim, C.L. Briens, A. Margaritis, M.A. Bergougnou, AIChE L 42 (1996) 1889, supplemented by private communication from Y.I.. [ 111 H.L. Shulman, C.F. Ullrich, N. Wells, A.Z. Proulx, AIChE J. 1 (1955) 259. [ 121 J.L. Turpin, R.L. Huntington, AIChE J. 13 (1967) 1196. [ 13] H.P. Comte, D. Bastoul, G. Hebrard, M. Roustan, V. Lazarova, Chem. Eng. Sci= 52 (1997) 3971. [ 1411 M. Abraham, A.S. Khare, S.B. Sawant, J.B. Joshi, Ind. Eng. Chem. Res. 31 i 1997) 1136. [ 15] A.B. Pandit, J.B. Joshi, Rev. Chem. Eng. 2 (1984) 1. [ 16] A.B. Pandit, J.B. Joshi, Intern. J. Multiphase Flow 13 (1987) 415.