The effect of nonlinear concentration profiles on the breakup of jets undergoing mass transfer

Shorter Communications

1465

obtained by the present method of tracer inJection and analysis, but for the bulk region of packing only These measurements agreed with the expenmental estimates of holdup reported by Shulman. Ulnch and Wells[4], who determined column hold up by dmzzct welghmg so that theu estimates Included both the wall and bulk regions fkpartmenf of Chemrcal Umuersrty College Smgleton Park, Swansea Wales I-

I

O’i 0

Fig 2 Dependence

G 02

Rotlo. dir0 of the permeability ratio KJK different values of GIG,

upon d/r0 at

pach3, 3&=,

a

ubeb

K

Thus if the ratios &,iib and KJK ax similar it follows that the liquid holdup in the wall region IS s1m11arto the liquid holdup in the bulk regmn The experimental estimates of interstitial velocity shown in Table I and the measured permeabdities agree within+596 for the same hydrodynamic conditions so that the wall and bulk holdup agree within the limits of experimental precision There IS other evidence to show that liquid holdups in the wall and bulk regions of packing are sumlar The experimental estimates of hquid holdup reported by Farid and Gunn[Z] were

The effect of nonlinear

concentration

(Receroed

d eb e, G G, K dw 6 t u Ub uw I

concentration of output pulse concentration of input pulse diameter of packing liquid hold up m the bulk of the packmg liquid holdup in the wall region of packing Gas flow rate for unit cross-section Gas flow rate at flooding liquid permeability in the bulk region average permeability in the wall region radius of column time interstitial velocity interstitial velocity in the bulk interstitial velocity in the wall region axial co-ordinate

REFERENCE.5

[I] Gunn D 121 Farid M 131 Farid M [4] Shulman 217

Engng SCI 1978 33 1221 M and Gunn D J . Chem Emmy Scl 1978 33 1221 M and Gunn D J , Chem En&ii Scr 1979 34 579 H L , Ulnch C F and Wells N , A ICh EJ 1955 1

J . Chem

profiles on the breakup transfer

for pubkatton

The effect of mass transfer on the breakup of a lammar liquid Jet in a gas has been examined by Burkholder and Berg[l] usmg linear hydrodynamic stability analysis of a simphtied model in which the undisturbed concentration profile varied linearly with radial position within the Jet The most sigmficant predictions of the analysis are that (1) transfer of a surface-tension-lowermg solute out of the Jet stab&es the system, producing an increase in the unbroken Jet length, while transfer of the same solute mto the Jet produces lengths shorter than those corresponding to no mass transfer, (2) adsorption of a surfactant strongly reduces or elunmates the effect of mass transfer on the stabrllzatlon or destabduatlon of the Jet. while without mass transfer, the presence of surfactant has little effect on Jet stability, and (3) the wave length of the preferentially-amplfied axisymmetnc disturbance wave length (directly proportional to the resulting drop size) increases for mass transfer of a surface-tension-lowering solute out of the let and conversely for transfer into the Jet Expemental results corroborate qualitatively the first two resuks of Burkholder’s anaIysls but not the third Measurements of the drop slzc[2] under various mass transfer conditions show a great deal of scatter but suggest that the preferred disturbance wave length IS not influenced appreciably by mass transfer even

M M FARID D J GUNN

NOTATION

c

01

Engmeenng

14 September

of Jets undergoing

mass

1979)

when the Jet length IS increased or decreased more than twofold It has been speculated121 that this discrepancy between prediction and experiment may be traced to the use in the analytical model of a linear undisturbed radial concentration protile in the let In reality, concentration varies across only the relatively thin diffusmn boundary layer that develops prior to let breakup We observe that the Marangoni number (a measure of the longitudinal surface tension variation responsible for the mass transfer effect on Jet stability) apparently required to affect significantly the wave tength IS approximately a factor of three larger than that required to alfect the Jet length (see Figs 3 and 7. Ref (11) It IS anticipated that this predicted difference may broaden sign&antly if a more realtstic concentration profile IS employed The object of the present study IS to refine the Burkholder model by replacing the linear imtial concentration profile of his analysis with the “broken” profile pictured in Fig I The use of such a profile to represent concentration or temperature variation has been successful in modellutg other systems for hydrodynamic stability analysis[3,4] The predictions of the refined analysis could then be compared with the Jet length and drop size predictions of Burkholder to determine, m particular, if there are

Shorter Communfcatroons BURKblOLDER PROF I LE

L=slnfO,and

(2)

29rW V=k,=7ra2A*

(3)

The group characterrzing the surface tension forces which lead to the mass transfer effect IS the Marangom number ACTUAL

PROFILE

AC IS the drfference between the solute concentration m the Jet at the tnterface and at the Jet axrs The shape of the undisturbed concentration profib IS described by the parameter

BROKEN PROFILE APPROXiMATlON

F-a-b a (JET

AXIS)

(JET

SURFACE]

whtch IS seen to be the fraction of the undrst~~ Jet radrus by the mass transfer boundary layer (cf Ftg 1) When b = 0 (F = I), the profile IS that of Burkholder and Berg[l], when b = a (F = 0). the case of no mass transfer IS recovered Computatlons are camed out yleldrng both the maxfmum dlmenstonless growth constant, fi* =&xz*/~, and the correspondmg dimenstonless wave number, k’= &*a, as functions of the Marangont number for varrous values of F Results are shown m Ftg 2 for cases in which all the other drmensronless groups ansing m the anaiysrs [ I ] are Bven vaIues correspondmg to the ftutd propertses of water Further results and det;uls of the computatrons are presented elsewhere[7] The most important result IS that the computed values of s* and i* cbffer from those of Burkholder by less than 1% for values of F from 10 down to approxlmateiy 0 01. 1 e down to the pomt where the d&uslon boundary layer occupres approxrmateiy 1% of the jet ra&us Typically, m Burkholder’s experiments, mvolvmg a Jet radius of 0 076cm. the estrmated average dlffuston boundary layer ttuckness was approxtmately twze thus or more One must thus conclude that the hnear profile assumption should not affect the valrdlty of Rurkbolder’s predrctions for the condltlons of hrs experiments On the other hand, the results reported here show slgntficant devlatlon from those of Burkholder for F values of 0 005 or less Such condltrons may prevad for larger diameter Jets or for srtuatrons m whrch the Jet length IS much shorter, correspondmg to shorter contact time and hence to thmner boundary layers It may be noted that the effect of boundary layer thmness IS Indeed to broaden somewhat the occupd

Fig 1 Shape of radial concentratmn profiles m the Jet durmg mass transfer comparison of Ihe approxlmatlons of Burkholdet and Bergf I] and the present study v&h the actual profile condltlons for which the disturbance growth constant IF influenced to a greater extent by mays transfer than the wave length of the preferentlatly ampltfied dtsturbance, and m a broader \ense, to assess the sultabdtty of usrng hnear concentratlon profile5 m modeflmg masq transfer effects on the stabdlty Of Jet5 ANALYSISAND RESULTS With the exceptlon of the specification of the undisturbed solute concen(ratlon profile, the model of the hqutd Jet system IS the same as that of Burkholder and Berg[l], VIZ an mfirute, cylmdrlcal column of nonvolatde, mcompresslble. Newtoman hqurd m steady, lammar plug flow through an mviscrd me&urn m the absence of external force fields The system IS Isothermal, and a solute IS present m small amounts m either or both phases The surface IS assumed to be m local adsorptron equdlbrium with both the adJacent bulk phases In the undrsturbed state, the radral concentration profile of the solute m the Jet IS that of the broken profile shown in l+g 1, I e concentration varies lmearly across the annular region representmg the difiusron boundary layer and IS constant m the core of the Jet All bulk fluid properties are constant, and surface tension varres lmearly with solute concentratron at the surface The stabdity analysis also parallels that of Burkholder and earher workers[5,6] A small axlsymmetrlc disturbance of the form 17= Real part (v,, e@‘**z)

’

(1)

IS Imposed Q IS a typrcal term m a Fourier serves representation of the drsturbance amplitude Accorclmg to the postulates of Rayle~h[~], drshrrbances of a part~ular wave number wdl be amphfied more raprdly than all others (I e correspond to the largest growth constant, ,9) and dommate the breakup of the Jet hsturbances m the other relevant system variables m the format of (I) are also rmposed and made to satrsfy the lmearzed Navter-Stokes and dtffuslon equattons sub]ect to the appropriate boundary condrtrons (cf Ref [I]) The result IS the “charactertstlc equation” relatmg the system parameters and fluId propertIes to the wavelength and the disturbance growth constant For a gwen set of proper&s and parameters, the wavelength IS found for which the growth constant ESmaxrmum Thus both the charactenstlc disturbance wave length and correspondmg growth constant are obtamed as functions of the system propertles and par&meters These are related directly to the Jet length and the drop volume m the usual way, VIZ,

08 i;' 06

0-

IO’

IO’ *MO

(al

IO0

IO9

01

lo*

IO’

loe

tog

bM0

(bl

Fig 2 The effect of concentration profile shape on thedependence of (a) the dimensionless growth constant, p, and (b) the dunenstonless wave number, k”. on the Marangom number

1467

Shorter Commumcattons difference between the Marangoru number requued to slgmlicantly alter the growth constant (hence let length) and that required to alter the wave number, as was antlclpated In the Burkholder analysis, the difference m this requved Ma IS approximately a factor of three, whereas for a value of F equal to 0 001, the difference IS approximately a factor of SIX The duecnon of the boundary layer effect IS consistent wtth the observation that Jet length IS mfluenced under condltlons for whuzh drop size IS not Finally, the comparison of these results wuh expenmentally-determined drop sure suggests that let breakup durmg mass transfer may be controlled by the concentrahon profile exlstmg dunng the very early stages of Jet life, when the dlffuslon boundary layer IS extremely thin Acknowledgemenr-This work was supported the National Science Foundation

by a grant from

LARRY E TARR t JOHN C BERG Department of Chemical Engrneenng Unruersrty of Washrngton Seattle, WA 98195, USA

BF-10

c 9 F

undisturbed let radius approximation of radial locanon depth (cf Fig 1) concentration drffuslvny (a-b)/a

tTo whom correspondence

Greek symbols p growth constant * maximum growth constant $ dimensionless maximum growth constant A* preferred wave length vlscoslty p u perturbation of Jet radms us perturbation of jet radius at orifice 0- surface tenslon REFERJZNCES

NOTATION

a b

wave number preferred wave number dlmenslonless preferred wave number Marangoru number, cf eqn (4) radial coordinate tame Jet velocity drop volume axial coordinate

of diffusion

penetration

HI Burkholder H C and Berg J C , A I Ch E J 1974 20 863 PI Burkholder H C , Ph D Thesis m Chemical Engmeermg, Umversny

of Washington

1973

r31 Luck W , J Flutd Mech I%5 21 565 [41 Vldal A and Acnvos A, Ind Engng Chem

Fundls 1968 7 53 PI Raylelgh Lord, Proc Roy Sot (London) 1879 29 71 Weber C, Z Angew Math Mech 1931 11 136 :; Tarr L E , M S Thesrs In Chemical Engmeenng, Umversny of Washmgton 1976

should be addressed

c!u?#auaI &gmcma&!Scrnn Vd 35 pp 1467-1470 PuailmonPressLtd 1980 PrInted m GreatElnialn

Interfacial area m trickle-bed reactors: comparison between ionic and organic liquids and between Raschig rings and small diameter particles (Received

30 July 1979, accepted

Studies on the determination of gas-liquid mterfaclal area m tncklmg two-phase downward flow through packed bed reactors are rather few m literature The correspondmg results have been obtamed by gas absorption accompamed with fast chemical reactlon of pseudo m, nth order They concern exclusively the carbonatauon of so&urn hydroxide [IA] and the oxnlatlon of ditiomte[3] or of sodium sulfite[S]. I e lughly ionic solutions Packmgs generally used are of large size, the smallest used by Huose et al [2] bemg glass beads of 2 59 mm dla The purpose of this study IS to present and compare results obtained for two packings, I 16 mm dla glass beads and 6 48 mm nommal dra glass Raschlg rings by using three gas-hqmd systems Two of these systems are gonlc, carbonatatlon of sodium hydroxide and oxidation of sodium sulfite The thud. carbamatlon of cyclohexylamme. IS purely orgaruc TIM will, allow for a comparison between the data obtamed with orgamc and loruc hqulds for two different shapes of packmg Expenments have been carried out in a 5 cm dra column packed to a 49 cm height The charactenstics of the packmgs and the operating condltlons are grouped on Table I It IS interesting to notice the small porosity of the glass beads packmg whuzh IS probably due both to the heterogeneous diameter drstnbutlon and to the non-perfect sphenctty of the particles The pressure drop

7 August

1979)

was measured with pressure taps located above and under the packmg and the ltqmd holdup was determmed by the welghmg methodf6, IS] The kmetlcs of sodium hydroxide carbonatation and oxidation of sodrum sulfite catalyzed by cobalt sulfate (3 65x IO-. kgmole m-“) at pH = 8 5 have been studied by Laurent[7] and the reactlon of carbon dloxlde with cyclohexylamme m organic solvent (toluene + 10% by volume of tsopropanol) has been studied by Sndharan el al [g,9]. BelhaJ[lO] and Alvarez et al [ 11,12] All these expenments have been camed out m a wetted film laboratory equipment As for the carbonatatlon and the carbamatatlon in the packed column, the expenments have been worked out m open reactor with gas chromatographrc analysrs at reactor inlet and outlet The inlet Cq mole fraction vanes between 2 and 4% Assummg that the two phases work m plug flow and the total pressure vanes linearly wrth the packmg he&t. the mass balance m gaseous phase and the solute conservatlon enable to write the two followmg equations, by neglectmg the resistance m the gaseous phase [I31 Qo dY=&a QdGa The

tions IS gven

- Ceo) = zQo( K - Y)

of the specfic m Table I

expression

dh

flux p employed

0) (2)

m these equa-