Simplification of Bubble Size Estimation in a Bubble Swarm
INTRODUCTION In a previous publication in this journal, Yianatos et aL (6) used drift flux analysis for estimating mean bubble diameter, db, in a bubble swarm. The approach involved measuring the relative bubble to liquid (or slip) velocity, Usb and comparing with a model estimation of slip velocity where bubble diameter is the search parameter. The bubble diameter which gives a match between calculated and measured slip velocities is the estimated bubble diameter. The approach was accurate within about _+15%. This procedure has been reexamined and a simpler, yet equally accurate approach has become evident. DRIFT FLUX ANALYSIS The slip, or relative, velocity of gas and liquid in countercurrent bubble-liquid flow with uniform sized bubbles is defined (with Jg positive upward and Jl positive downward to preserve common usage) by Usb = Jg +
Jl
.
The slip velocity is related to a single bubble's terminal rise velocity in an infinite pool, Uv and gas holdup, Eg. A frequently used relationship is (Shah et al., (5)) Usb = U T . ( 1 _ ~g)rn-I
[21
In the previous approach, m was taken as function of Reynolds number, the analogy with the result for settling of solid particles derived by Richardson and Zaki ( 3 ), and Usb was related to bubble diameter by analogy With the hindered settling equation used by Masliyah (2). In the present case, m is fixed at 2, following the suggestion of Wallis for fine bubbles (db ~< 2 m m ) , and the Schillar and Naumann (4) expression is used for bubble terminal velocity, UT, g Apd 2 U x = 18#1(1 + 0 . 1 5 R e °'687) ( d b ~ 1 . 5 m m ) ,
[3]
where Re is Reynolds number, given by
[11
Re = dbplUT
[4]
TABLE I Bubble Diameters, Measured and Predicted Frother" ppm
Jg (cm/s)
JL (cm/s)
eg (%)
Re Eq. [4]
ds2bb
dbe
dbd
(ram)
(ram)
(ram)
5 10 15 20 25 10 15 15 15
1.0 1.0 1.0 1.0 1.0 2.1 1.5 0.5 0.8
0.91 0.85 0.82 0.85 0.77 0.30 0.30 1.00 1.00
9.5 12.9 15.8 15.5 16.2 15.7 14.0 12.3 17.0
157 97 77 79 74 265 165 37 55
1.20 0.86 0.77 0.69 0.73 1.51 1.13 0.62 0.67
1.11 0.87 0.75 0.77 0.74 1.40 1.11 0.55 0.64
1.11 0.87 0.76 0.77 0.74 1.40 1.11 0.55 0.64
a Dowfroth 250C. b Bubble size measured using photography. c Bubble size predicted using Ref. ( 1). d Bubble size predicted using present approach.
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Journal o f Colloid and Interface Science, Vol. 140, No. 1, November 1990
NOTES The routine now is to measure Usb (by measuring Jg, J~, and %), estimate UT from Eq. [2] (with m = 2), and iteratively solve for db in Eqs. [3] and [4]. Other expressions for Ux, provided db ~< 1.5 mm, could be substituted (Dobby et al. (1)). One simplification that becomes apparent on use is that there is now only one definition for Re instead of, as before, one for the determination of m and another for the determination of U~b. Using this new routine the data of Yianatos et al. (6) were reexamined; an extract of the results, selected to cover the full range in db, is given in Table I. Essentially no difference with the previous result is found. CONCLUSION The simplified approach, based on a drift flux analysis with m = 2 and an expression for UT applicable to db ~< 1.5 mm, gives adequate estimation of db in the tested range of 0.5 to 1.5 mm, APPENDIX: NOMENCLATURE db g Jg Jl m Re Usb Ux
bubble diameter, general term, cm acceleration due to gravity, c m / s 2 superficial gas velocity, c m / s superficial liquid downward velocity, c m / s parameter, Eq. [ 1] bubble Reynolds number, Eq. [4] slip velocity between bubbles and liquid, e m / s bubble terminal velocity, c m / s
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Greek Symbols eg fractional gas holdup ol liquid density, g / c m 3 //1 liquid viscosity, g / c m . s REFERENCES 1. Dobby, G. S., Yianatos, J. B., and Finch, J. A., Canad. Metall Q. 27(2), 85 (1988). 2. Masliyah, J. H., Chem. Eng. Sci. 34, 1166 (1979). 3. Richardson, J. F., and Zaki, W. N., Trans. Inst. Chem. Eng. 32, 35 (1954). 4. Schiller L., and Naumann, A., Z. Ver. Dtsch. Ing. 77, 318 (1933). 5. Shah, Y. T., Kelkar, B. G., and Godbole, S. P., AIChE J. 28(3), 353 (1982). 6. Yianatos, J. B., Finch, J. A., Dobby, J. S., and Xu, M., J. Colloid Interface Sci. 26 ( 1 ), 37 ( 1988 ). MANQIU X u J. A. FINCH
Department of Mining & Metallurgical Engineering McGill University Montreal, Quebec Canada H3A 2A 7 Received March 5, 1990; accepted April 13, 1990
Journalof ColloidandlnterfaceScience,Vol. 140,No. 1, November 1990