Note on drop formation at low velocity in quiescent liquids

Shorter Commumcatlons NUI’ATION

dlmenslonless tracer concentratron dlspersron coefficient residence time dlstnbution distance along vessel reactor vessel length parameter defined below eqn (8) peclet number based on fresh feed flow rate peclet number based on total vessel flow rate recycle ratio laplace transform vmable fluid velocity time time constant, L./U dimensionless distance along vessel, I/L dtmenslordess time t/r(l + R) dtrac delta function first moment of E _m*& dimensional variance of ti c? dimensionless variance of E

c

1

2

I

0

3

REBERENCES

8

Fig 4 Residence

ttme drstnbutlons

Acknowledgement-This ENG 74-13520

by NSF Grant No

VIJAY K MATHUR

Ctty Uruversrty of New York New York NY 10031. USA

Chlcal En#ma~ Pcrgunon Press Ud

TRAM-R, u* = 0 75

work was supported

Electronrc Associates Inc West Long Branch NJ 07764, USA

[l] Redly M J and Schnutz R A, A ICh E 3 1%6 22 153 [Z] McGowm C R and Perlmutter D D , A I Ch EJ 1971 17 831 [31 Perlmutter D D, Stab&y of Chemrcal Reactors PrenticeHall, Englewood Cliffs, New Jersey 1972 [41 Fu B , Wemstem H , Bcmstem B and ShafFer A B , Zna! Engng Chem Proc Des Lkv 197110 501 [51 van der Laan E T , Chem Engng SCI 1958 7 187 WI Mathur V K , Micromuung effects on reactor dynamic response Ph D Thesis, Ilho~s Institute of Technology, Clucago 1977 [71 Bellman R E, Kalaba R G and Locket J, Numerical Znuersron of fhe Laplace Transform Amencan Elsevrer, New York 1966

HERBERT

WEINSTEIN

Scwnce Vol 35 pi 1452 1454 1980 Pnntcd m Great tIntam

Note on drop

formation at low velocity in quiescent liquids (Received

10 Aprd 1979)

Drop formatlon has continuously been the subject of research III relation to mdustnal processes Numerous design problems associated wth, for example, hqmd-hqmd extraction. d&dlatlon, Aotatlon. spray combustion. spray evaporation, heat transfer eqmpment and emulslfymg systems are intimately connected wth the physics govemtng drop formation Specifically, the accurate prediction of mterfaclal area resulting from drop formation IS requved for calculatmg heat and mass transfer effects wth confidence in processes of the type mentloned above The subject of drop formation has been extensively reviewed and discussed m Refs [l-4] From these works, it may be concluded that there IS a dlshnct lack of mfonnatron concerning the fluid mechamcs of drop formatlon at low flow rates m liquni-liquid systems Although the problem has been alleviated m part by more recent studies (among others, Refs [7] and [8]), accurate final sue predlcfions of drops detachmg from nozzles are dtfficult to obtain Practlcalty all studies attemptmg to predict detached drop dtameters are based on the coupling of observed experimental results with theoretical considerations involving the forces present durmg drop formation and detachment The theoretical approaches vary in degree of detad and complexity wth no agreement emcrgmg among researchers regarding the

manner m which inertial effects in the force balance should he included The ObJectwe of this note IS to provide a simple but useful correlation for predicting detached drop diameters in hquidliquid systems at low flow rates and m which the host fluid IS quiescent Even though the comIation should only apply to drop formation at low Reynolds number based on nozzle velocity (Re G 100). it IS found that the addition of a Reynolds number dependent term allows its extension to higher flow rates (Re=MOO), while yielding droplet diameters to within an accuracy better than 7% on average DERIVATloN OF THE CORREL.ATlON Expenmental evdence From expenmental results at IS known that a typIcal plot of leadmg edge drop velocity vs ttme IS as mdlcated m Fig 1 The figure shows three stages which, from left to right, correspond to (I) Spherical growth, dtstortlon effects are small (II) Nechng. dIstortIon dlects arc large (III) Detachment and fall At low flow rates, mert~al and drag effects on the drop are small and the force balance is prunardy between Interfacial and

1453

Shorter Communications

With an expression available for h,,, It IS now requued to determme the form of the correlation which most smtably determmes detached drop diameter A little re!iection shows that a linear correlation IS consistent wrath earlier assumptions and IS also the simplest to fit Usmg the expenmental data avadable m Ref [9] yields g

I

III ,

j

0 Fi

1

m t-

growth at low velocity, spherical expansion I). neckmg (Stage II) and detachment (Stage III)

Drop

(Stage

gravitational effects If the tnne reqmred for neckmg IS short compared to the spherical growth period, then final drop dmmeter should correlate strongly wth the magmtude of the leadmg edge displacement corcespondmg to the mnumum velocity in F@ 1 (displacement m&cated as h m the figure) That thrs mdeed appears to be the case wdl be shown by the good agreement obtamed between measurements and predicttons of final drop dmmeter based on this supposition Cowelatmn for low flow rates In agreement with the arguments presented above, It wdl be assumed that the drop leadmg edge velocdy IS the sum of two velocity components, one due to spherical expansion and the other due to neckmg The later component reflects gravdatlonal contnbutions to dIstortton but may also Include weak inertial effects Therefore v,=

vs+

where it IS fauly strajghtforward component is given by [9]

v, to show

(1) that the spherical

v,=+ I+

(2)

2 ( 0, )

with V. the maxunum fluId velocity m the nozzle PoiselIe flow) The distortion velocity component m eqn (1) IS a compbx function of the physical and geometrical parameters govermng drop formation and defies detaded mathematical analysts However, a statistical mvesmatlon based on dnnensional analySISconsiderations of the data avadable m Ref 191 and reported m part m Refs E7j and [S], shows that the best correlation of the smrficant parameters affectmg V, ISgwen by *

= 7 91 (&>“‘(~>“(&J~

= 0 39(@)~“(&!!_)-0~+o

3l.j

This correlation has a regression coefficient of 0 99 and predicts the expenmental values of detached drop diameters in Ref [9] urlth an average error less than 23% (see comparison m Fig 2) The agreement between measured and predicted results IS gratefymg and substantiates the validity of the approximations made m order to denve the correlation It should be noted that the constant appeanng m eqn (6) IS physically meamngless, It IS merely a consequence of the best linear fit to the experimentat data A curved fit (possibly parabolic) forced to yield d = 0 when h nun= 0, would be physically more meaningful. but could not be camed out here due to the unavadatuhty of expenmental results m this range Equation 6 has been denved from data covermg the parameter ranges Investigated m the liquid-liquid systems m Ref [9] These correspond to lo+50 OOl<~
Yl

5x lo-‘<&

<6x

IO-’

dpdv”
where the subscnpt

-

refers to the drop phase

Ertcnsron of correlataonto hrgherflow rates At Reynolds numbers larger than 100. eqn (6) under-predicts detached drop diameters m hquid-liquid systems Thrs IS to be expected since distortion velocity IS only weakly dependent on Inertial effects at low Reynolds numbers and wdl lead to underestimates of the mass flow into the drop dunng neckmg However, by the simple adddlon of a Reynolds number dependent term, obtained by fitting data available m the hterature, predictions of detached drop chameter up to Reynolds < 1400 have been made with an average error less than z?7% The form

8-

(3)

The maximum en-or m V, gven by eqn (3) IS k 15%. and has been calculated from devtattons of the data from the least squares regression curve Substitution of eqns (2) and (3) m (1) yields the expression

Re (00

Humphrey o Hayworth.Treybal

l

o Scheele

I

0 for which it

IS

($),,

easy to show that a mmnnum =043fF)-Ob(&)+OP

(6)

I

I

I

I

I

Melster

I

P

123456789

m V, appears at

(g) n cdc (5)

Fig 2 Companson

between measured drop diameters

and calculated

detached

Shorter Commumcatlons

1454 of the correlation at high flow rates 1s

1

~=039~~)-“@(&)~m+0 n

12Re0”+039

m

Figure 2 shows a comparrson between predlchons from this work and representative measurements of others The agreement IS seen to be reasonably good Devlatlons m the data of Scheele and Melster are probably due to readmg errors from the graphlcal results for detached drop duuneters provided by these authors Equations (6) and (7) provide stmple but reasonably accurate means of estimating detached drop dmmeters m hqurd-hquld systems of engmeenng Interest Used for this purpose and over the ranges to which they apply, they should be helpful to the design engmeer JOSEPH A

C

HUMPHREY

Lkpartmenf of Mechanrcal Enguteenng Unrverslty of Cahfomra Berkeley, CA 94720. V S A

D. g h Re

detached drop diameter nozzle Internal dauneter gravitational constant dlsplacementof drop leadmg edge Reynolds number based on nozzle phase propertIes

y, lAp( pd

pd

time neckmg component of drop leadmg edge velocity drop leadmg edge velocity maximum velocity m developed lanunar nozzle flow spherical component of drop leadmg edge velocity InterfacIal tensron absolute density difference between hquld phases drop phase vlscosdy drop phase density

REFeRwCEs de Nle L H and de Vnes H J , Chem Engng Set 197126 441 [2] Resmck W and Gal-Or B , Advances IA Chenwal Engmeermng,Vol 7. p 2% Academic Press, New York 1968 [3] Soo L S , Fluid Dynamrcs of Muftlphase Systems Blarsdell. Waltham. Mass 1967 [4] Yoshrda K , Kunu D and Levensplel 0, Znd Eng Chem [I] HertJes

P

M

,

Fundls 1%9 8 402 [S] Tavlartdes L L , Coulaloglou C A, Zelthn M A, Khnzmg G E and Gal-Or B ,Ina! Engng Chem 1970 62 6 [6] Gal-01 B , Khnzmg G E and Tavlalrdes L L , Znd Eng Chem Fundls 1%9 6 21 [7j HumphreyJ A C.HummelR L andSnuthJ W,Cun.J Chem Engng 1974 52 449 [8] Humphrey J A C , Hummel R L and Smrth J W, Chem

NOTATION

d

V, V, V,, Vs

diameter

and

An efficiency parameter

drop

Engng SCI 1974 29 14% [9] Humphrey J A C , M SC Thesis, 1972

Umverslty

of Toronto

for batch mixing of viscous fluids

(Recewed for pubfrcafjon 11 September 1979)

Stured tanks are probably the most common mdustnal muting device Although there LSa good amount of expenmental data for these systems, theoretical knowledge IS hmlted Thrs IS probably due to two problems (a) the flow fields are extremely complex and (b) the enormous vanety of Impeller shapes make generaluatlon of usual fluld mechamcal approaches very difficult [I] Most theoretlcal work IS based on dlmensronal analysis (e g [l31) and concepts such as mlxmg time and power consumption Mtxmg time IS defined solely by experimental measurements, for which there IS no firm basic understandmg One of the clearest and most complete studres of mlxmg m stlrred tanks IS that of Hoogendoorn and den Hartog[4] Theu data should be sample to analyze smce they studled mlxmg of VISCOUSNewtonian thuds with the same density and vlscoslty They also used consistent expenmental methods for determmmg rational compansons between nuxmg tune, 13,. pernutting experiments For 8, determmatlon a thermal method was used in most expenments t A small quantity (l-2%) of heated (MC) hqtnd was added As rmxmg proceeds, mstantaneous temperature was measured at several “points” m the vessels To define nuxmg time It IS necessary to state a specified degree of homogenelty Hoogendoom and den Hartog define a axing time, 13,~. as the time beyond which at all measunng locatlons, the differences between the temperatures measured and the final tThree other methods were also compared conductwlty or pH, color addltlon and dlscoloratlon For detads about these the reader IS referred to [4]

temperature remam smaller than *25% of the total temperature step(~e (T-7+025(Tf-T0) Experunents were performed on the mixers shown m F@ 1 Dnnenslons and operatmg con&tions are shown m the same ligure All of these nuxers are used conunerclally to nux VISCOUS hqmds The basic expenmentai results of [4] are curves of hmenslonless nuxmg time. expressed as no,,, where n ts the stimng speed, as function of Reynolds number A typical curve IS pictured m Fii 2 The basic finding of the Hoogendoom and den Hartog’s paper IS that for high viscosity hqluds, for whazh normal operatmg condltlons are a low Reynolds number resme, no,,, IS a constant for a mven nuxer The same constant seems to hold for scaled-up rmxers 141 An analysis of “nuxmg efficiency” was attempted by plottmg &#,pI~~ agamst a Reynolds number defined as &/&sp [4] The approach IS probably useful but a basic problem remams There IS no fundamental understandmg of the arbltranly defined, expertmentally determined numbers or their meamng Here we attempt to rationahze these results m terms of mlxmg theory A contmuum model (Ottmo et al [5.6]) speclfymg nuxmg bounds IS used to compare performance of batch rmxers m viscous flow operation THEORETICALAPPROACH time IS defined by expenments A more fundamental measure of mlxmg states IS mtermatenal area density, a, [S, 61 which LSthe amount of mterfaclal area per umt volume separatmg two drstmct fhuds bemg nuxed If V, denotes a matenal volume enclosmg a matenal particle Mxmg