power requirements of gas-liquid

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Power Requirements of Gas-LiquidAgitated Systems

B. J . MICHEL and S. A. MILLERU n i v e rs i t y of Roches ter , Roches ter , New York

Power data for dispers ion of ai r in l iquids by means of a s ix-bladed f lat blade turb ine arepresented in the form o f a logari thmic plot of actu al power consumed against a funct ion of speed,impeller diameter, gas f low rote, and impeller power characterist ics. The data are those ofM i c he l (7), Bimbenet (2 ) , Sachs (72), and Oyama and Endoh (10) .Turbines of f rom 3- to 8-in. diameter were run in tanks o f from 6.5- to 18-in. diameter, withth e D / J never exceeding 0.47. The f luids tested covered a density range of 0.8 to 1.65 g./ml.and a viscosity range of 0.9 t o 100 centipoises.

Data are also presented on dispers ion of ai r in a 50% by volume batch of carbon tet rachlor idein water and dispers ion of ai r in a suspension of Alundum part ic les in water, and compared wi thdata on water in a s imi lar system.

The qual i tat ive ef fect of a surface act ive agent i s demonstrated in a comparison of data fora 0.1% by weight mixture of Pluronic L-62 in water wi th data for pure water wi th the sameapparatus.

In the field of agitation much workhas been done to show that the powerrequired by an agitator varies syste-matically with density of a homogene-ous liquid (11 or an immiscible liq-uid pair (8) that is being agitated.The introduction of a gas into a liquidin which a mechanical agitator is ro-tating, on the other hand, brings abouta mixture for which one is unable tocalculate an average density signifi-cantly related to the power consump-tion of the impeller. The theory ofpower requirement of an impeller ro-tating in a homogeneous fluid is im-perfectly developed, consisting mainlyof the relatively unrigorous theory ofmodels; that of an impeller handlinga gas-liquid mixture has never beentreated.

Gas-liquid contacting in mixing ves-sels has been investigated with regardto chemical reactions and mass trans-fer coefficients by Oldshue ( 9 ) , Bar-tholomew, Karow, Sfat and Wilhelm(1 ) and others with no special atten-tion being given correlation of powerdata. Cooper, Fernstrom, and Miller( 4 ) reported that the agitation powerwith a high rate of gas feed to theimpeller could be as little as 25% ofthe impeller's no-gas power but de-clared that the dependence of impellerpower on gas rate is erratic and un-predictable. Foust, Mack, and Rush-ton ( 5 ) presented power data for anarrow-head disperser in the form ofa plot of K (horsepower in water with-. J. Michel is with the Mixing Equipment Com-pany, Incorporated, Rochester, New York.Page 262

air divided by horsepower in water ata constant speed) vs. the superficialair velocity in feet per second basedon volume flow rate of gas and cross-sectional area of the tank. Data takenin air-liquid contacting with a stand-ard flat-blade turbine (six blades) inwater and viscous oils were presentedby Sachs ( 1 2 ) .A more recent investi-gation of gas-liquid agitation by Bim-benet ( 2 ) produced data on air-liquidcontacting by use of the same kind ofturbine in water and corn syrup solu-tions.

In 1955 Oyama and Endoh (10)attempted to correlate power in gas-liquid agitated systems by the K fac-tor of Foust, Mack, and Rushton ( 5 )and the dimensionless group N , =Q/ND'. They succeeded in a limited cor-relation, that of the effect of gas rate onthe power of a particular impelleroperating at constant speed. LaterCalderbank (3) offered the same cor-relation to data obtained with a flat-blade turbine dispersing air in a num-ber of liquids (density ranges 0.74 to1.6 g./ml., viscosity range 0.5 to 28centipoises). His plot of P , / P , againstN , at constant impeller speed resultedin tw o intersecting straight line sec-tors that only approximated the shapeof the curve of Oyama and Endoh.

Previous to this, in 1955, Kalinske(6) discussed the power required bya flat-blade turbine operating inaerated sewage. He presented the datain the form of a plot of a term di-rectly proportional to K vs. (1- , )

A.1.Ch.E. Journal

and obtained a good correlation. Goodagreement was reported on scale-upof the data from the 30-in. tank inwhich the experimentation was per-formed to a full scale sewage installa-tion, but no quantitative comparisonwas made.The investigation of power responsethat is reported here was undertakenin a search for understanding of thegas-dispersion phenomenon and in-creased reliability in the prediction ofthe power drawn by a gas-dispersingimpeller. Mixco standard flat-blade tur-bines 3 and 4 in. in diameter wereoperated in flat-bottom cylindrical tanks,6.5 and 11.4 in. in inside diameter, asindicated in Table 1. Both tanks werefully baffled. Metered air was intro-duced beneath the turbine by a spargerring in the larger tank and by anopen-end tube in the smaller. Theauthors' experience, supported byOyama and Endoh, is that no differ-ence in agitation power results fromthese different methods of gas supply.The impellers were driven by a varia-ble-speed motor resting on a cradletype of dynamometer of establishedaccuracy.For additional details of apparatusand procedure and for the record ofall the data taken the dissertation ofMichel (7) should be consulted.*D I S C U S S I O N OF RESULTS

The data were first plotted as Kagainst N,. Unlike the data of Oyamaand Endoh (10) and of Calderbank( 3 ) hey did not correlate as a straightline or as a smooth curve on eitherarithmetic or logarithmic coordinates.Instead they produced a family ofcurves, one for each gas rate (Figure1).

0 Tabular materia1 has been deposited as docu-ment 7077 with the American Documentation In-stitute, Photoduplication Service, Library of Con-gress Washington 25 D. C and may be obtainedfor $3.75 for photobrints 'br $2.00 for 35-mm.microfilm.May, 1962


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Liquid

TABLE. SUMMARYF SYSTEMS TUDIEDViscosity atDensity, operat. temp., Tank Impeller(g./ml.) (centipoise) diam., (in.) diam, (in.)

Water 1.00 0.8 12 4ccl4 1.60 1.0 12 4Glycerol [weak]Glycerol [strong]

1.12 3.6 12 41.19 13.5 12 4

Mineral oil 0.87 28.0 12 4CCIJwaterSO% by volume 1.30 - 12 4Alundum/water - - 12 4Water 1.00 0.8 12 3Water 1.00 0.8 6.5 30.1w t. % Ph on ic L-62 1.00 - 12 3inwaterRange of variables6.5-in. diameter tankGassed power-0.006 to 0.018 hp.Gas rate-0.02 to 0.47 cu. ft./min.*Liquid volume-0.92 gal.Static liquid d e p t h 4 n.Turbine position-4 in. off bottom+11.4-in. diameter tankGassed power-O.002 to 0.10 hp.Gas rate-maximum 1.0 cu. ft./min.*Liquid volume-4.4 gal.Static liquid depth-9 in.Turbine pos it io n3 in. off bottom

* At impeller condition.t Clearance measured from tank bottom to center plane of turbine, turbine coaxial with tank.

The failure of the data to yield tothe simple correlation of Oyama andEndoh may be the result of the rela-tively broad extent of experimentalconditions investigated. The range ofpower values measured in this studywas fifteenfold, a consequence of widevariation in both gas rate and impel-ler speed. Oyama and Endoh on theother hand held their impeller speedconstant and varied their gas ratesuch that a twofold span of powervalues resulted. Thus their curve isanalogous to any that might be drawnthrough points of constant speed inFigure 1.In the absence of fruitful theory asearch was begun for an empiricalcorrelation that would successfully re-late impeller power to speed, gas rate,fluid properties, and scale size over anextensive range of these variables.Logarithmic crossplots of N . againstP , at constant K and of N , against Kat constant P , were linear, indicatingthat N , a PP." K aor, by rearrangement

P , =[P:/NJ~4aFigure 2 shows a logarithmic plotof P , against P."/N. fo r a typical set

of data , those for the system carbontetrachloride/air being agitated by a4-in. turbine in a 12-in. tank at threedifferent air rates. The data for eachair rate form a separate line, the linesbeing substantially parallel with slopesbetween 0.40 and 0.42.

I000

K0

01No

Fig. 1. K vs. N, fo r the 4-in. turbine operatingin the 12- in. tank f i l led with water .

A cross plot of Po against Q (Fig-ure 3) shows that P , cc Q".". This rela-tionship in combination with the pre-ceding one indicates

A logarithmic plot of P, against( P 2ND8/Q0.'") for the data obtainedin the 12-in. tank with the 4%. tur-bine is shown in Figure 4. Includedare data for air dispersed in water,carbon tetrachloride, mineral oil, andtwo strengths of glycerine solution(Table l ) , covering three orders ofmagnitude of the abscissa quantity.*A line of slope 0.43 represents allthese data within 217%. The rela-tively fewer data collected for the sys-tem water/air with a 3-in. turbine in12- and 6.5-in. vessels also correlateas straight lines of the same slope butwith different intercepts, Figure 5.The line for the 3-in. impeller in the12-in. tank lies almost within thespread of the data for the 4-in. tur-bine (about 20% above the 4-in.);the line for the 6.5-in. tank is about35% below the latter.Also shown for comparison in Fig-ure 5 are the data of other investi-gators: those of Oyama and Endoh

*Actually only about one-third of the morethan 400 data points measured and calculated areshown on the graph, because many of them over-lay one another. The extreme points are all plottedhowever.

16. 2 4 6 8 d' 2 4 6 8iO" 2 4 6 SlO-' 2 4 6 8d 2 4 6 810'P 1Na

Fig. 2. P# VS. P. ' /N. fo r the 4-in. turbine operating in the 12411. tankf i l led w ith carbon tetrachlor ide.

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(10) or water/air in a 10.8-in. tankwith a 3.6-in. turbine, those of Bim-benet (2) for air dispersed in waterand corn syrup (viscosity from 1 to90 centipoises) by 3- and 4-in. turbinesin a 12-in. tank and by 5- to 8-in. tur-bines in an 18-in. tank, and those ofSachs ( 1 2 ) for air dispersed in waterand 105-centipoise oil by a 6-in. tur-bine in an 18-in. tank. Inasmuch asnone of these three sources reportedno-gas power data, values for Po hadto be estimated before the abscissagroup could be computed. The methodof estimation was that of Rushton,Costich, and Everett ( 1 1 ) , who re-ported a power correlation directlyapplicable to the turbines used byBimbenet and Sachs and suggestedcorrection factors for differences inturbine proportions that permit esti-mation of the power requirements ofthe impeller used by Oyama andEndoh. Remarkable as it may seem, asingle line of slope 0.45 will representwithin tolerable engineering limits thebehavior of air-liquid systems involvingliquid densities between 0.9 and 1.6g./ml. and liquid viscosities between1 and 100 centipoises in a one andone-halffold range of tank size and arange of impeller size such that impel-ler-to-tank diameter ratios lie between0.25 and 0.44. t is likely that a widerrange of liquid viscosity could be cor-related in large equipment, that iswith the agitation in the turbulent re-gime. The equation of this line consti-tutes a relationship by which thepower required by a gas-dispersingagitator can be calculated:

!1lo 3

The value of the coefficient c prob-ably depends importantly on the ge-

' ~-fl-J -;

lW4 2 4 6 8 0 3 2 4 68102 2 4 6 8 U 1 2 4 68100 2 4 68101Po' ND'-6Fig. 4. P, vs . Po2N1)8/Q0.58or the 4-in. turbine in the 12-in. tank.

ometry of the impeller; for the flat-blade turbine used in this study theauthors' data, Bimbenet's, and Oyama'sindicate that c is 0.08 when the unitsare horsepower, revolutions per min-ute, feet, and cubic feet per minute.The exponents of Q and of the group[ P o 2Q"."/N,] also may be functionsof equipment geometry, but probablyless sensitively so than c.The closeness to 0 .5 of the slopes ofall the lines plotted tempts one toconclude that P J P , should be apower function of NDS/QO5' , a rela-tionship much more attractive forscale-up calculations. An effort to cor-relate the data of Bimbenet and theauthors' water/air data in this fashionwas not successful.The results of two experiments in-volving three-phase systems are in-cluded in Figure 4.Air was dispersedinto a 50-50 volume % mixture ofwater and carbon tetrachIoride andinto a slurry of fine (120 mesh)Alundum particles in water, the 4-in.turbine and 12-in. tank being used.The exact composition of the Alundumslurry is unknown, but the solids con-

10.8643

Fig. 3. Pg vs. Q a t P ," /N , =1.0 x lo-' for the 4-in. turbine operatingin the 12-in. tank fi l led with carbon-tetrachloride.Page 264 A.1,Ch.E. Journal

tent was on the order of 5 wt. % Thedata fall well within the band of theauthors' two-phase data, with no dis-cernible systematic segregation. Theimpeller speed range, hence the powerrange, used in these experiments wasconsiderably less than in the two-phase experiments; power values laywithin a fourfold spread. This nar-rower experimental span was imposedby the lowest speed that would per-mit a satisfactory liquid-liquid or liq-uid-solid suspension.T h e E f f e c t o f I n t e r f a c i a l T en s io n

The liquid-gas interfacial tension ofthe two-phase systems investigated inthis research ranged from 27 to 72dynes/cm. One might expect the ef-fect of interfacial tension to be re-flected in the Weber number N, , =D 3 N " p / u . The extremes of the valuefor N, , encountered were 1.2 X lo"for water/air at lowest impeller speedand 3. 6 x lo" for carbon tetrachloride1air at highest impeller speed, a thirty-fold range. The data for these two sys-tems are correlated by the same line inFigure 4, ndicating no effect of inter-facial tension on impeller power.Because of the industrial impor-tance of liquid-gas suspensions contain-ing a surface-active agent, in one ex-periment 0.1% of Pluronic L-62 brandsurface-active agent was added towater, and a run was made with the3-in. turbine in the 12-in. tank. Asshown in Figure 6 the resulting datalie appreciably below the correspond-ing line for water/air, the value of P ,observed being as low as 60% of thatexpected. The surface tension of thePluronic solution reported by the man-ufacturer of the agent is 37 dynes/cm., corresponding to a maximumvalue of Nw . =2.4 x lo6. Inasmuchas this is only two-thirds of the ex-treme value of N,". for carbon tetra-chloride/air, the nonconformity of thedata to the general correlation mayreflect the uncertainty of surface-ten-

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lob642

10 -86

pg 4

21 0 28

64

2

10-3

I

10 ' 2 4 6 8 lo-' 2 4 6 8 10. 2 4 6 8 lo-' 2 4 6 8 10 2 4 6 8 10Pz ND'

IS

Fig. 5. P, VL . P02NL)9/Q0.58or da ta o f Oyama and Endoh (70) and B imbene t (2). Representa-tions of t h e d a ta o f M i c h e l (7) and Sachs (72).

sion observations for solutions of sur-face-active materials. In any event theintroduction of a surface-active agentmay affect the power of a gas-dispers-ing agitator in a way not reliably pre-dictable from the general correlationreported in this paper.CONCLUSIONS

The power required by an agitatorimpeller dispersing gas into a liquid isusualIy less than the power requiredby the impeller operating at the samespeed in a gas-free liquid. For equip-ment of a single size or of a narrowrange of scale sizes, the power (P , , )can be calculated in terms of the no-gas power ( P o ) ; thus

This empirical equation involves ab-solutes rather than dimensionless ratiosand does not necessarily hold for allsizes of equipment, even if scale mod-els are considered. It obviously failsfor extreme values of Q; as Q ap-proaches zero, P, should approach P , ,and as Q becomes very large, P ,should approach the value requiredby the impeller rotating in liquid-freeair (a small but finite limit).The equation therefore must beused cautiously as a scale-up aid andwith extreme values of the independ-ent variables employed. Furthermore,although it holds in form for aeratedsolutions of surfactants, the coefficientand exponent may have values differ-ent from those applicable to systemscontaining no surface-active agent.Ungrounded in theory and unsuitedfor reliable extrapolation as the rela-

tionship is, it nevertheless provides themost general and successful relation-ship yet proposed between the powerrequired by a gas-dispersing impellerand the factors on which that powerdepends. It embraces a much widerrange of experimental conditions thanthe earlier correlations of Oyama andof Kalinske, and it avoids the prac-tical unknown of gas holdup or sus-pension density required by Foust andco-workers. Insofar as the equationapplies with a single set of coefficientand exponent values to a range of im-peller sizes in a given tank, as it ap-pears roughly to do, it defines quan-titatively an interesting qualitativeobservation long familiar to those ex-perienced with agitator dispersers; thesmaller the impeller selected to delivera particular amount of power in gas-free liquid, hence the higher the im-

peller speed, the greater the reductionin power when gas is supplied to theimpeller.To establish a firmer basis for de-sign calculations a better understand-ing of the theory underlying the be-havior of a gas-dispersing agitator andmore extensive data on the perform-ance of such agitators are required.Investigations in the turbulent rangeshould be extended to involve plant-size tanks and the range of impeller-to-tank diameter ratios that may beencountered in practice (for example0.2 to 0.7). The role of surfactantsalso needs delineation. The authorsare continuing both theoretical andexperimental study of the gas-disper-sion phenomenon.N O T A T I O Nc =constantD =impeller diameter, in. or ft.K =dimensionless, PJP, at equalN =rotational speed of impeller,N , =dimensionless ratio, Q / N D "N l V , =Weber number, ( D 3N a ) /u,dimensionlessP , = power consumed in the gassedliquid, hp.Po =power consumed in the un-gassed liquid, hp.Q =volumetric flow rate of gas,cu.ft./min at temperature of

liquid and static pressure atthe impeller

speedsrev./min.

P," ND"(7)correlating factor,c 1min.[ u. ft. min.hp." X -X CU. ft.

p =densityr = interfacial tension

1642 4 6 l a 2 4 6 1 @ 2 4 6 I V ' 2 4 6 1 0 ' 2 4 6 1 0 'p.' ND 3OO.E.6

Fig. 6. P, vs. POZND3/Q for d a ta on 0.1 % Pluronic L-62 n water com-pared wi th the g eneral l ine for a 4-in. turb ine in a 12- in. tank and w i ththe l ine represent ing the 3- in. turb ine operat ing i n the 12- in. tank f i l ledwi th water.

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L I T E R A T U R E C I T E D1. Bartholomew, W. H., E. 0. Karow,.M. R. Sfat, and R. H. Wilhelm, Ind .Eng. Chem., 42, 1801(1950).2. Bimbenet, J. ., M.S. thesis, Purdue3. Calderbank, P. H., Trans. Inst. Chem.Engrs . (London) , 36, 457 (1958).4. Cooper, C. M., G . A. Fernstrom, andS. A. Miller, lnd . Eng. Chem.,36, 504

( 1944).

University, LaIyette, Indiana (1959).

5. Foust, H. C., D. E. Mack, and J. H.Rushton, ibid., p. 517.6. Kalinske, A. A.,Sewage G Ind. Wastes,27, 572 1955).7. Michel, B. J ., M.S. thesis, Univ.Rochester, Rochester, New York(1960).8. Miller, S. A., and C. A. Mann, Trans.Am. Inst. Chem. Engrs., 40, 709(1944).9. Oldshue, J. Y.,Application of Mixersto Bio-Engineering Processes, Rose

Polytechnic Inst., Terre Haute, Indiana(1953).10. Oyama, Y., and K. Endoh, Chem.Eng.( J a p a n ) ,10,2 (1955).11. Rushton, J. H., E. W. Costich, andH. J. Everett, Chem. Eng. Progr., 46,395,467 (1950).12. Sachs, J. P., M.S. thesis, 111. Inst.Technol., Chicago, Illinois (1950).Manuscrht receiued Mav 2. 1961: reuision re-ceived Octdber 24 1961; pap& accepted Ootobw27, 1961. Paper Gesented at A.1.Ch.E. Cleaelandmeeting.

Transport Characteristics of Suspensions:Part IV. Friction Loss of Concentrated-FlocculatedSuspensions in Turbulent Flow

DAVID G. THOMASC ambri dge University, Cambridge, England

Turbulent flow friction factors were determined for flocculated suspensions of thoria, kaolin,and titania in tubes %- to 1-in. diameter. The non-Newtonian laminar flow data were arbitrarilyfitte d with the Bingham plastic model. W it h this model the range of yield stress values was 0.018to 1.39 Ib.i/sq. it., with a maximum ratio of coefficient of rigidity to viscosity of suspending me-dium of 11.1. The volume frac tion solids were varied from 0.042 to 0.23.Two types of behavior were observed depending on the value of the yield stress. For yield valuesless than 0.5 Ib.i/sq. ft. the turbulent friction factors were always less than those for Newtonianfluids but tended to approach the Newtonian values as the Reynolds number was increased. Foryield values greater than 0.5 Ib.i/sq. ft. the friction factors were again less than those for New-tonian fluids but tended to diverge from the Newtonian values as the Reynolds number was in-creased. Both sets of data were correlated with the Blasius relation with the coefficie nt andexponent given in terms of the laminar flow properties and the volume fraction solids.

Th e mechanics of the flow of solidssuspended in liquids has attracted in-creased attention during the last dec-ade, encompassing not only basic re-search but also applied studies directedtoward large-scale operations, such asthe transport of coal in commercialpipelines ( I ) , and more unusual ap-plications such as high energy solidadditives to jet fuels ( 2 ) and thetransport of fertile material in homo-geneous nuclear reactors ( 3 ) .An im-portant aspect of the applied problemsis the friction loss characteristics ofthe suspensions in turbulent flow. Al-though the primary measurements re-quired for the determination of thefriction loss characteristics (that is thepressure drop as a function of flowrate) are relatively simple to make,the subsequent interpretation and cor-relation are complicated by several

David G . Thomas is on leave from Oak RidgeNational Laboratory, Oak Ridge, Tennessee.

factors. Not the least of these is thatfriction-loss correlations for Newtonianfluids are primarily empirical in na-ture. Although it has been establishedexperimentally for Newtonian fluidsthat Reynolds number similarity givessimilarity of friction factors, no equiv-alent solutions of the exact equationsof motion are available. This meansthat in order to compare the data onthe characteristics of suspension fric-tion loss with data for ordinary fluids,great care must be exercised in theselection of the Reynolds number inorder to insure similarity. The ambig-uity in the determination of a suitableReynolds number for suspensions arisesbecause of the effect of the solidphase on the laminar viscosity. Parti-cle size and shape, as well as solidsconcentration, are the principal factorsaffecting the viscosity. Suspensions oflarge symmetrically shaped particles(approximately 50 or larger) have

Newtonian flow characteristics withviscosity a function of volume fractionsolids ( 4 ) .Suspensions of smaller par-ticles or of assymetrically shaped par-ticles (platelike or needlelike) possessnon-Newtonian flow characteristicswith apparent viscosities a function ofthe rate of shear, as well as particlesize, shape and concentration ( 5 ) .Thus possible viscosities which mustbe considered for use in suspensionfriction loss correlations include:

1. The viscosity of the suspendingmedium.2 . The suspension viscosity, a func-tion only of solids concentration forNewtonian suspensions.3. The apparent viscosity, a func-tion of both shear rate and solids size,shape and concentration for non-New-tonian suspensions.4. The limiting viscosity at highrates of shear, a function only of solids

Page 266 A.1.Ch.E. Journal May, 1962

power requirements of gas-liquid

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