Thomas Non-newtonian suspensions part II.pdf

8/13/2019 Thomas Non-newtonian suspensions part II.pdf

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-D A V I D G . T H O M A S

Part ZZ. TURBULENT TRANSPORT CHARACTERISTICShe object of these studies was to relate the turbulentT ransport characteristics of non-Newtonian suspen-

sions prepared from symmetrically shaped particles ofmicron-sized metal oxides to the easily measured laminar-flow and hindered-settling characteristics of the suspen-sions. In the first par t of this two-part series Indusfrialand Engincning Chernrsfry,November 1963), the Binghamplastic model was chosen to represent the laminar-flowdata, largely on the basis of engineering convenience.The principal factors affecting the magnitude of theBingham plastic coefficients of a given suspension werefound to be the concentration and particle sue. Con-sideration ofthe physicochemical forces between particlesindicated that the extrinsic physical property relation-ships (that is, those which are affected by suspensionflmculation) could be expected to be quite general,applying to aqueous suspensions of metal oxides an dsolids which ac t as reversible gas electrodes. Experi-mental da ta for laminar flow and hindered settlingof floc-culated suspensions of such metallic oxides as titania,urania, kaolin, thoria, magnesia, and alumina were inaccord with this prediction provided the particles wereroughly symmetrically shaped.

This part of the series summarizes the turbulent trans-port characteristics of the same suspensions used in thelaminar-flow studies. The turbulent transport charac-teristics were shown to be unique functions of the intrin-sic and extrinsic physical properties of the suspensionindependent of the composition of the solid phase.Although not proven conclusively, there is some evidencethat particle shape affects turbulent flow data in a waywhich cannot be predicted from simple laminar-flowmeasurements. However, the deviations are sufficientlysmall that they are relatively unimportant for engineer-ing purposes.

A major question that must be resolved during thecourse of any study of the turbulent transport character-istics of nowNewtonian fluids is the appropriate viscosity

AUTHOR David G. Thomas i s a DeueloprnenfEngineer in fhbReactor Div ision of Oak Ridge Notional Laboratory, OakRidge, Tenn. This nstallafion is operated by Union Carbide

term for use in calculating parameters, such as thReynolds number or the dimensionless velocity profilcoordinates, of importance in analyzing the turbulenflow data. It is an axiom of rheology that even thsimplest proposition must be verified experimentallyFor instance, it was not until 1911 that sufficient information on friction loss of Newtonian fluids had beeaccumulated to permit Blasius 26A) to correlate thdata empirically and prove that the turbulent friction factor curve for different liquids was a unique function othe Reynolds number when the Reynolds number wacalculated using the laminar viscosity. Then in 1913 thNational Physical Laboratory 29A) showed that thsame empirical curve was valid for both Newtonialiquids and gases.

I t is not entirely clear that the question of the mosmeaningful viscosity term for use in turbulent nonNewtonian flow correlation has been resolved (24 354In all probability the question can he resolved expermentally without resorting to ex cafhcdradecisions. Onof the simplest criteria is that the Reynolds number bdefined in such a way that the turbulent non-Newtoniafriction factors would agree with the turbulent Newtoniafriction factors 7 4 . This would probably be misleading since it implies that the scale and intensity of turbulence in a non-Newtonian system is substantially thsame as in Newtonian systems provided there isReynoldsnumber similarity. The general consensus seems to bthat this is not so,and that the elements responsible fonon-Newtonian behavior must damp the turbulenccharacteristics in some way. Another viscosity that haheen used ( 3 4 70A) s the apparent viscosity evaluateat the wall shear stress. This is the natural viscosity focorrelating laminar-flow data. However, this form othe viscosity has not been tested over a sufficient rangeof tube diameters to insure that the tube diameter &ectsare absent in turbulent flow. Re-examination 4 A) opreviously published data 70A) howed that there was slight diameter effect when using the apparent viscosityto correlate the data even though there was only a fourfold variation in tube diameter. Recently, the priomethods (34 7A, IOA, 27A) of calculating the apparenviscosity fir turbulent flow correlations has been ques


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-The friction factor is a unique function of RGnalds number,independent of tube diameter for any par tim lar suspmsion.-The turbdent non-Newtonian friction factors are alwaysless than the values fw Newtonian flu ids , presumably due tosuppression of turbulewe by dissolved or suspended non-Newtonianelements.-The turbulent non-Newtonian friction factors are alwaysgreater than the laminar ualues for Neut onian f[uid s.

however, there are insufficient data available to checkthis proposal.

A set of phenomenological criteria for a visksity thatwould be particularly useful in engineering is:

Undoubtedly, other criteria can be developed basedon velocity profile measurements and ultiniately onfundamental turbulence measurements on the non-New-tonian materials themselves. In the absence of thesemore fundamental criteria, the desirability of havingan engineering correlation that fulfilled the first itemabove was given precedence in these studies.

Friction Loss CharacteristicsAnalysis of data for the turbulent friction loss of sus -

pensions of titania, ho lm, and thorium oxide showed thatthe limiting viscosity a t high rates of shear, 9, ulfilled thethree criteria given above for a suitable viscosity 30A.32.4 . This analysis also showed that the effectiveviscosity (Equation 25 in the November article) was notsatisfactory since its use resulted in a definite diametereffect for tubes having a ninefold range in diameters.Typical data for two different suspension concentrationsare shown in Figure 12. Th e coordinates of the diagramon the left permit the correlation of the laminar-flow databy a single line with the turbulent flow data for differenttube diameters branching off n different lines. Th egood agreement of the laminar-flow data , independent ofthe tube diameter, ensured that the wall effects werenegligible for the systems studied and, therefore, thecoefficient of rigidity and yield stress were unique prop-erties of the suspension. On the right side of Figure 12,the same data are plotted as f versus N E , = D V p / pcoordinates which, with the proper selection of the vis-cosity, permit the correlation of the turbulent flow datafor any given suspension by a single line. The goodagreement of the turbulent flow data demonstrates thatthere was no detectable wall effect for tubes from to1 inch in diameter and that friction factor similaritywith high yield stress suspensions was obtained using thelimiting viscosity at high rates of shear, 9, n the calcula-tion of the Reynolds number.Data Correlation. Turbulent friction facton fornon-Newtonian suspensions were always below thosefor Newtonian fluids; however, two entirely differenttrends with increasing Reynolds number were observeddepending on the value of the non-Newtonian proper-ties 32A). For yield values less than 0.5 Ib.,/q. ft. thesuspension friction facton tended to approach those forNewtonian fluids as the Reynolds number was increased.

W I L L M I 1 S1KBSr.LB.dn.2Figure 72. Pseudo-shcm dingram and Faming friction-factor plot forconcentrotcd suspmrionr showing og8emmf of laminar and ttububntdata, respcc6ively,as ubs dzamatrr IUOS wried. AN ruspenrionr WETC fthoria. For t h c j r s t three rc63 of points listed in the key, surpcnrionprofimfics wcm as follow^: 0 = 0.70, = 0.69 lb.,/ft.z, 7 = 5.7cp For the sccond three, 0 = 0.76, . = 1.25 1b.,/fLP, and 1 =?O. l cp. Thc CpUnliOn of fhc two lines dcsignoted A ) ir ?/dy4.0 lag N E J d f i- 0.40

i

t. .

VIELO SIKESS, LO.i/SP. RFigure 13. E&t of yield dress m BIosiur cm mts rdati.; ihr. .eyM[drnlrmbnan d th Fmulingjricrimr actor .:

.., . . . .

OY P

Figure 9 from Part I . Fricrionfuctof-Reynoldr numbn d i ~ p mmlamimrpour of Binghamplustiz motnidx in roundpipes. A largmchar t 4ppCms on P q c 27 of tht Novmbn mticle. Thc light diagonallincr hnve thc equmkm:

lElwDLDS awua r -


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I n contrast to this, t he friction factors for suspensionswith yield values greater than 0.5 lb.,/sq. ft. tended todiverge from the Newtonian values as the Reynoldsnumber was increased. Th e extent of the change inbehavior is illustrated in Figure 13, which shows data forthe Blasius coefficient, B, and exponent, b , plotted,\as afunct ion of yield stress. (The use of the yield stress in thisplot is only a convenient way of indicating the degree ofnon-Newtonian behavior and does not imply that itis the only variable or importance.) Th e values of B andb were determined by fitting the friction factor line foreach suspension with the expression :

f = BlvRe-bDimensional analysis shows that , aside from the Reynoldsnumber, two different combinations of suspensionphysical properties are possible, V M and gCpr,x2/pz.These two parameters must be combined additively inorder to represent the change of slope shown in Figure13. Th e expression for the coefficient and exponent ofthe Blasius equation (Equation 32) for the entire range ofyield values becomes :

B = 0.079 (0. + W (33)b = 0.25 (& + *) (34)

where 0 = p / v and @ = gcprvx2/pL2.When the yieldstress is zero and 7 and /L are equal, Equations 33 and 34reduce to the commonly accepted values for Newtonianfluids. Th e constants and exponents of these equationswere determined using a nonlinear least squares pro-cedure ; the results were 3 2 A )

a = 0.48 c = 0.15d = e = 2 x = 6 x lO-+ft.

Friction loss da ta a re also available for non-Newtoniansuspensions of flexible 5 4 9A) and rigid (704 13A)needle-like particles. In general, the behavior of sus-pensions of such particles is more complicated than t ha t ofsuspensions of equiaxial particles. For instance, laminar-flow studies show tha t suspensions of rigid needle-likeparticles display time-dependent shear-thinning or shear-thickening behavior 13A). With flexible fibers, thereis the possibility of entanglement resulting in viscoelastic-ity and/or shear thinning behavior with the added com-plication of flow with an annular region of suspendingmedium next to the wall 5A). Turbulent friction lossdata for rigid needle-like particles are difficult to com-pare with data for equiaxial particles because in bothcases there are no fundamen tal turbulence data and evenon a macroscopic scale there is the problem of defininga unique viscosity. One way of making this compari-son 35A) is on the basis of the magnitude of the frictionfactors predicted for a particular set of data by the pro-posed correlations for equiaxial 3 2 A ) and for needle-shaped particles (1O A) . In two particular cases, ex-tensive data were available for a suspension of equiaxialparticles. Th e physical property da ta required for usein the two proposed correlations were evaluated objec-tively from the laminar-flow shear-diagrams takingespecial care that there were good laminar-flow data from

a small tube viscometer corresponding to the wall shearstress of the turbulent flow points obtained with tubesof large diameter. (The existence of laminar flow for thedata taken with the small tube was proven by two facts:visual observation of the suspension as it jet ted from theviscometer tube and the good agreement of the lam inar -flow data with the Hedstrom line on a plot off versusD V p / ? ; that is, Figure 9 [see page 281. The re-sults of such a comparison 3 5 A ) showed that in gen-eral the correlation for equiaxial particles (Equations32 to 34) was in good agreement with the experimentalda ta for suspensions of equiaxial particles, whereas thecorrelation for turbulent friction loss of needle-like par -ticles predicted friction factors 22 to 36Oj, larger than theexperimental values for equiaxial particles when using t helaminar-flow properties for equiaxial particles evaluatedusing the procedure recommended by Dodge andMetzner 70A). Thus particle shape apparen tly affectsturbulen t flow dat a in a way which cannot be predictedfrom simple laminar-flow measurements. However, thedeviations are sufficiently small that they are relativelyunim por tant for engineering purposes.

For values of the yield stressless than 0.3 lb.,/sq. ft., the @ term contributes less than5% to the final value of the Blasius coefficient an d ex-ponent of Equations 33 and 34. This means a consider-able simplification in design procedure for suspensionswith a moderate yield value because a single plot of fric-tion factor versus Reynolds number can be prepared withv / p as the non-Newtonian paramete r. For larger valuesof the yield stress, the complete expression for B and bmust be used. In either case, the Hedstrom numbershould be evaluated and checked against Figure 9 toensure that flow is not laminar for the given value of theReynolds number, D Vp/v. Aqueous suspensions of sym-metrically shaped particles of titan ium oxide an d magne-sium metal have been shown to possess friction loss charac -teristics when flowing through fittings (valves, ells, etc.)which are virtually the same as those for Newtonian fluidsof the same density 38A).

Phenomenological Analysis. The value for x inEquations 33 and 34 is of the same order of magnitudeas the ratio of Y / U * for the suspending medium (forexample, for a viscosity of 0.9 cp. and a density of 1.0g./ml., the value of Y / U * is 6.0 x 10-B ft. when r w =7.5 lb.,/sq. ft. and the suspension density is 1.5 g./ml,),Since u* is experimentally proportional to u , the fluctu-ating component of the mean velocity 25A), then xalso is the order of the scale of eddies for which viscousforces are important ( 7 5 A ) that is, x is the distance overwhich the yield stress might be expected to exert aninfluence on the structure of turbulent flow.

By application of Prandtl's simplified equat ions for thefriction loss in round pipes to the suspension data 26A,3 2 A ) , it was possible to infer additional information aboutthe phenomena which occur during turbulent flow ofnon-Newtonian suspensions. The relation between thedimensionless wall thickness layer, N = uw* 8/11 and thesuspension characteristics was found to be :N = 11.4 ( V / ~ ) O . ~

Design Procedure.

(35)V OL. 5 5 NO, 1 2 D E C E M B E R 1 9 6 3 29


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This equation reduces to the commonly accepted valuefor Newtonian fluids when ?I = p. Such a thickrningof the wall layer has been indicated experimentally inseveral different studies (3A, 7A, 27A) ; however, noprior attempts have been made to evaluate the thickeningquantitatively.

One possible explanation of the gradual thickeningmay be obtained by comparison wi th the results of de-tailed studies of Newtonian fluids. These studies showedtha t the production of turbulence was at a maximum veryclose to the wall, at a distance y + -11.4 = llr-i.e., thevalue given by Equation 35. However, the energy ofturbulence was not completely dissipated where it wasproduced, but there was a displacement toward the axisdue to diffusion in the direction of decreasing turbulentintensity (25A). Now, flocculated solids in suspensionwould be expected to markedly suppress the turbulentintensity, particularly in the vicinity of the tube axiswhere the shear stress is less than the apparent yieldstress. Following the szme arguments advanced forNewtonian fluids, damping of the turbulent fluctuationson the tube axis due to the flocculated solids results in anadditional displacement of the maximum in the turbu-lence dissipation and production away from the wall, andconsequently a thicker wall layer, as given by Equation35. Thi s explanation is also consistent with the observa-tion that friction fzctors of low yield stress suspensionsconverge toward Newtonian friction factors as theReynolds number is increased. That is, the increase inReynolds number is accompanied by an increase in thewall shear stress, hence by decreasing values of rV T ~ ,husweakening the strength of the sink on the centerline a ndconsequently causing less deviation of the wall layerthickness from the values expected for Neivtonian fluids,an d in general a behavior more characteristic of New-tonian fluids.

In contrast to the behavior of low yield stress suspen-sions, the friction factors of high yield stress suspensionsdiverged from the Newtonian fluid line. This suggeststha t turbulent fluctuations were dam ped throughout thetube cross section by the suspended solids, and that theflow was becoming more laminar in nature. This issupported by the effect of yield stress on the von Kar ma ncoefficient, K , calculated from the friction factor valuesusing Prandtls equations 32A). The value of K wassubstantially constant up to a yield stress of 0.5 lb.,,sq.ft. and decreased thereafter. Since K is commonly con-sidered to be a measure of the average intensity ofturbulent fluctuation Q A ) he decrease in K means thatthe turbulent intensity was damped for suspensions withyield values greater than 0.5 lb.,/sq. ft. Similar deduc-tions have been reported by Vanoni 37A) in studies ofNewtonian suspensions; in addition, he observed that theeffect increased with increasing concentration and de-creasing particle size. By replacing rY by Equation 15and x by v /u* , the expression of @ in Equations 33 and34 becomes :where A1 is the force constant in Equation 15 whichcharacterizes the attraction between particles, D, s the

@ = [ ( A 1 / O , * ) j ( P U * Z / g e ) ] (36)

10 2

10-3

D I A IN HT TR APm 3 3 l S 378 635 1 6 x 34

10-2 VI ;33 175 196 48 x I O 5f

04 I 05 IC6R E Y N O L O S NUMBER

vFzgure 74 . Hea t transfer a n d j u i d j 7 o u . characteristics o f thorza Jlur-ries using the lamttang vascosity at high rates o f shmr t o calcvlute theIieynolds and Prand tl numbers. For these data, 7 = 8.2cp. = 0 0055Ib./ft.-spc. ry= 0.46 lb. /sq. ft . . and p = 7 3 3 b./cu. f tparticle size, + is the volume fraction solids, and u* is thefriction velocity. O n this basis, is proportional to thevolume fraction solids cubed and the ratio of the at trac -tive forces between particles to the disruptive forces dueto turbulent fluctuations. Hence, the relative impor-tance of the turbulent fluctuations at any given flow rateis diminished by the addition of more solids or by theincrease of the attractive force term by a reduction in par-ticle size. This is in agreement with the observationsof Vanoni, cited above.H eat T r ans fer to F loccu la fed Suspens ions

Turbulent heat-transfer measurements were made onnon-Newtonian suspensions in the same equipment usedfor the friction loss tests ( 3 0 A ) . A comparison of theresults of the two different kinds of measurement allowedthe general features of non-Nebvtonian suspension heattransfer to be readily identified. A s with the friction lossdata. the limiting viscosity at high rates of shear wasshown to be a suitable viscosity for correlating the heat-transfer data.

Laminar and turbulent data for a suspension having ayield stress of 0.5 lb.,/sq. ft . taken with tubes of 0.318 and1.030 inches in diameter are shown in Figure 14. Thesedata show the characteristic displacement of the laminardata with tube diameter correspondin5 to the differentvalues of the Hedstrom number, gcpr,D2/q2. The heat-transfer and fluid-flow data having the same Hedstromnumber show that the departure from laminar flow oc-curs at identical Reynolds number for both the heat-transfer and flow dat a. Th e heat-transfer data for fullydeveloped turbulent flow for both tube diameters arecorrelated by the conventional Newtonian line :

- 0.027 N E e - 0 , 2 N p 7 - 2 3 ( ~ , / q ~ ) . ~ (37)h- -CPGwhen lVpr= c p v / k , provided that the Reynolds numberis 3 to 5 times the critical value for the onset of non-

30 I N D U S T R I A L A N D E N G I N E E R I N G C H EM I ST R Y


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laminar flow. This range of Reynolds numbers for thetransition region corresponds very closely to the heat-transfer transition range observed with Newtonian fluids,and in fact, except for the displacement in critical Reyn-olds numbers due to the non-Newtonian laminar charac-teristics of the suspension, the non-Newtonian heat-transfer data are very similar in appearance to New-tonian heat-transfer data.

Design Procedure. The suspension heat-transferda ta shown in Figure 14 show a dip region at Reynoldsnumbers extending from the critical value for thetransition to about 4 N R J C . A similar dip regionwasfirst identified for Newtonian fluids by Colburn 8A).Since the value of the critical velocity for the transition(3 to 8 ft./sec. for slurries with values of the yield stressfrom 0.075 to 0.5 lb./sq. ft.) is already approaching therange of velocities commonly used in heat exchanger de-sign, a graphical procedure similar to the one originallyproposed by Colburn (BA) appears to be the most suit-able method for avoiding the ambiguities associatedwith design for this dip region. Th e design procedurerecommended is :

1. Calculate the value of the Hedstrom numberand identify its location on a Fanning friction factor-Reynolds number ( D V p / v )plot containing the Hedstromnumber grid as a parameter (Figure 9 of the Novemberarticle, repeated on page 28).

Locate the turbulent-flow friction-factor line onthe same plot by using Equations 32, 33, and 34.

Calculate the laminar-flow j-factor from Equation31 by using the value of the critical Reynolds number ,

A r R e) c ,etermined from the intersection of the laminar-and turbulent-flow lines obtained in steps 1 and 2 above.Plot this value on a Newtonianj-factor-Reynolds numberplot, and locate the laminar heat-transfer line from thispoint with a slope of 2 / 3 .

Connect the laminarj-factor point at N R J c o theturbulent-flow Newtonian j-factor curve at N,, = 4

with a smooth curve characteristic of the j-factorcurve in the dip region. O r if desired, an analytical

2 .3.

4

procedure given by Petersen and Christiansen 2 4 Amay be used to determine the transition region curve.Heat- and Momentum-Transfer Analogies

Newtonian heat- and momentum-transfer analogies76A) developed by Colburn, Prandtl, von Karman

Martinelli, an d Metzner an d Friend (ZOA) were in goodagreement with the experimental data for non-Newton-ian suspensions. This was to be expected since thesuspension Pra ndt l number was about 10. Althoughthe best fit was observed with the Martinelli analogy1 6 4 ) (which was developed following von K arm an' s as-

sumption of a buffer layer between the laminar sublayerand the turbulent core), the scatter of the dat a probablydoes not warrant the use of such a complex expressionInstead, the von Karma n analogy (Equation 38) may berecommended.

f2)NP:'a= 1 + 54j7i{ 1vP, 1) + In [ I + 5 / a N ~ , 1 > 1 j

(38)If the Martinelli analogy is used, a considerable simpli-fication is afforded by using the tabulated values ofparameters given by Knudsen and Katz 16A). Withvalues from these tables. the Martinelli equation reducesto 4fT 2.120 log N,,Z/f + 17.48 (39)for a Prandtl number of 8 and todfx 1.715 log N R e 4 f + 19.07 (40)for a Prandtl number of 11. The experimental friction-factor and the j-factor da ta 30A)for Reynolds numbersgreater than four times the critical number are plottedin Figure 15 as4f3ersus N ~ ~ d j .he Martinelli linesfor the two different Prandtl numbers, Equations 39 and40, are included, together with appropriat e lines for theother analogies. All the turbulent-flow suspension heat -transfer and pressure-drop data are included on a singleplot and show reasonably good agreement with both theMartinelli and von Karman analogies.

Figure 75. For thej r s t j v e s ets of poin ts listed on the jgu re, physical properties of the slurries were: r y = 0.075 b . /sq . f t . , 7 = 2.9 cp =0.0079 lb./ft.-sec., Npr = 8.2. For the bottom two sets, these were: r y = 0.46 lb. /sq. f t . , 7 = 8.2 cp = 0.0055lb./ft.-sec., N p T= 7 7 . 2

Comparison of heat- and momentum-transfer analogies with data f o r non-Newtonian suspensions.

*0318 '378 635A 0 318 252 635A 0318 I 26 635

1030 175 196E 1030 a7 196( 0 318 378 6358 1030 I75 196

I

---- R I E N D A N D M E T Z N E RI

V O L . 5 5 NO. 1 2 D E C E M B E R 1 9 6 3 3 1


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Suspension Nucleate Boi l ingNucleate boiling heat-transfer measurements have

been made with aqueous T h o z uspensions containing upto 0.105 volume fraction solids (344). Boiling tookplace from the surface of 1/16- and l/g-inch diameterplatinum tubes submerged in slurry. Th e results showedthat the heat flux was proportional to the nth power oftemperature difference between the tube surface and thebulk fluid :

q / A = (AT) (411For the slurries studied, the heat flux at a A T of I O F. wasabout l o 4B.t.u./hr. s q . ft. regardless of slurry concentra-tion. However, the value of the exponent, n, decreasedas the volume fraction solids was increased. Th e valueof n was 3.3 with no thorium oxide present and ap-proached unity at a volume fraction solids of 0.10.Typical results are shown in Figure 16.

The maximum heat flux attainable under nucleate-boiling conditions (often called the critical heat flux orburnout heat flux) at slurry concentrations of 200 g. ofthorium per kg. of water was about the same as for water .However, at a concentration of 1000 g. of thorium perkg. of water, the burnout heat flux was 210,000 B.t.u./hr.sq. ft. compared with a value of 490,000 B.t .u. /hr . s q . ft.for water under corresponding conditions.

At constant heat flux, the temperature differencebetween the heated tube surface and the fluid saturationtemperature increased 5 to 6 F. per hour. This resultmight be explained by a soft film tha t surrounded the

i 6 I V O L U M E F R A C T I O NO S O L I D S T U B E O I A ,I N C H E S

1 / 1 6i aI8

1 A86547

i

I

1 ^?- 2 3 r 5 C 9 , 3 I c::A T F,

Fzgure 76.f rom a platinum tube.tank containing about 3gallo ns of slurry

Effect o thoria suspenszon concentration on nucleate boilingForced conuectionjow rate about one g.p.m. zn

32 I N D U S TR I A L A N D E N G I N E E R I N G C H E M I S T R Y

heated metal surface. This film was apparent ly less than1/32 inch thick and was never distinguishable as an adhering film after the tube was removed from the slurry system. N o hard cakes were observed on the surface fromwhich boiling took place during any of the tests.

The nucleate-boiling tests were made with aqueouthorium oxide slurries which had non-Newtonian laminarflow characteristics and which were almost Newtonianunder turbulent-flow conditions. No phenomena wereobserved which could be at tributed to the effect of thesolid particles on the gross physical properties of the slurry. For example, the non-Newtonian laminar-flow characteristics of the slurry had no discernible effect on the nucleate-boiling heat transfer ; instead , both the decrease inburnout heat flux and the effects shown in Figure 16 wereatt ributed to the deposition of a film of solids onthesurfaccaused by vaporization of liquid a t the surface ( 2 0 A ) .

In studies of subcooled boiling burnout with nonKewtonian suspensions, a decrease in burnout heat fluxfor the suspensions was observed when compared withwater flowing at the same velocity and degree of subcooling 72A). However, it was postulated that thisdecrease was due to a decrease in the effectiveness of thebubble convection loop caused by the suspension yieldstress, rather than a film of solids on the surface. As yetthere is insufficient data to prove whether the observedeffect is due to one mechanism or the other or to a combination of both.Flocculated Suspension Transporl

The minimum transport velocity is defined a s themean stream velocity required to prevent the accumulation of a layer of sta tionary or sliding particles on thebottom of a horizontal conduit. In studies with flocculated suspensions, two flow regimes were observed depending on the concentration of the suspension ( 3 1 A ) . Inthe first, the suspension was sufficiently concentrated tobe in the compaction zone and hence had an extremelylow settling rate. Th e second regime was observed withmore dilute suspensions which were in thc hinderedsettling zone and settled ten to one hundred times faqtathan suspensions which were in compaction.

Concentrated Suspension Transp ort. The difference between the dilute and concentrated flow regimesare clearly evident when the experimental data shown inFigure 1 7 are examined. For dilute suspensions (characterized by a low yield stress in accord with Equation 15)the minimum transport velocity was substantially aconstant for any given suspension; this velocity waappreciably greater than the velocity required foturbulent flow. As the concentration was increased, theminimum transport velocity also increased, but not asrapidly as tha transition velocity, so that eventuallythe two curves coincided. Thus for sufficientlylarge concentrations, the suspensions were in compactionand the minimum transport velocity \vas essentiallythe velocity at transition to turbulent flow. ID realitythe minimum transport Reynolds number was 10% to60y0 greater than that calculated from the intersectionof the appropriate Hedstrom line (Figure 9) and the


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........................KAOLIN-HI 0? jt-

.........................I U B I OIAMEIER [INCHES)

1.045A 2.056

3.97.l I - l . . . l I . . . l i .

I -4 2 3 4 6 810-3Y d L O S I R I S S 3 B i . / S O . F1.

r i g w e 77.bulctlrflow ipuolr minimum honrport velocity at Imge vulues of y i d d stressEffect of yield stress on minimum trnnspmt uel0cily. Velocity for transition from lmninm to fur

turbulent flow l i e calculated from Equations 32. 33,and 34. In all probability the true minimum transportvelocity corresponds to the end of the transition regionrather than the beginning. To a first approximation hisvelocity is given by Equation 26.

The critical concentration dividing the dilute and con-centrated flow regions was determined experimentallyto be the same as that observed in hindered-settlingmeasurements as the floc structure bridged across thetube. In the absence of experimental data similar totha t shown in Figure 8, the critical concentration divid-ing the two regimes may be estimated from Equation 22by assuming that depth, H , equals diameter, D.In classifying dilutesuspension transport phenomena, it is useful to 'considerthe case of infini te dilution. Under these circumstances,the minimum transport problem for dilute suspensionscan be divided into two major flow regimes 36A)(Figure 18). For flow regime I, a particle in contactwith the channel wall would be immersed in an essen-tially laminar sublayer when D,u.*lv)5. Inaddition, the diagonal limes on Figure 18 for wnstantvalues of the product D,uo*/v) U,/u,*) = D,Li,/v givesthe Reynolds number (and hence the drag coefficient)for particles settling in a quiescent fluid.

For dilute suspensions in the hindered-settling range,theflocs are not in contact. Hence, energy must be sup-plied in excess of that required to initiate turbulence inorder to overcome the tendency of the floes to settle.Thus the criterion given for concentrated suspensionswould no longer be sufficient; nstead the minimum trans-port condition for dilute suspensions was found to bea function of both the floc settling velocity and theintensity of turbulent fluctuations 374. The settling

Dilute uspensionTransport.

rate provides a measure of the tendency of the: particlto settle out, whereas the turbulent fluctuations provida driving force to maintain the particles in suspensioThe measurement of the turbulent fluctuations flocculated non-Newtonian suspensions was beyond tscope of the suspension transport studies. HoweveLaufer 774 as shown that the turbulent fluctuatioof Newtonian fluids at any given radial position are prportional to the friction velocity, u* = dgx. osequently, the friction velocity was used as a measuof the turbulent fluctuations. In fact many investig

V =I I03

IO-

Figurc 18.t im Flour regime closrifd ion for minimum transport corre


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tions have shown that the ratio of particle terminal set-tling velocity to friction velocity is the principal factoraffecting the distribution of solids in a flowing fluid ( 6 47 8 A ) .

The experimental data for dilute suspension transportshowed tha t the friction velocity at the min imum trans-por t condition was substantially independent of pipesize in the range 1- to 4-inch diameter and also inde-pendent of concent rationup o the limit given by Equation22. In addition, all of the data (37A) for dilute floccu-lated suspensions of thoria a nd kaolin were in fl owregime I as defined in Figure 18. Additional regime Idata were available for nonflocculated suspensions ofglass beads in water (21A, 33A) and fine coal dust in air22A). These data are shown in Figure 1 9 using the

coordinates suggested by Figure 18. The particlediameter for flocculated suspensions was taken to be thefloc diameter, calculated in the manner described abovein the section on hindered settling of flocculated suspen-sions. A least-squares analysis gave (33,4)

The coefficient and exponent are in rather good agree-ment with the predictions of a phenomenologicalanalysis (0.0104 and 3, respectively) which assumedthat particle transport occurred when lift, due toBernoulli forces resulting from the velocity gradientacross a small particle resting on the pipe wall, wassufficiently large to overcome the gravitational forces onthe particle. (Parenthetically, it may be noted tha t thetransport process for large size, flow regime 11, particlesis quite complicated as might be guessed by examinationof Figure 19 . Since these suspensions ar e not flocculated,they are outside the scope of the present paper but addi-tional information may be found in reference 33A.)Epilogue

This discussion of the transport characteristics of non-Newtonian suspensions must be considered as more inthe nature of a progress report than a final report for solittle has been done and so much remains to be done.Certainly, these studies represent the first extensive in-vestigation of laminar and turbulent heat, momentum,and mass transfer characteristics of the same suspensionsunder comparable conditions and all the data displayedremarkable internal consistency. But only integralmeasurements (for example, over-all pressure drop, meanvelocity, or over-all heat-transfer coefficients) weremade. hecessarily, the analysis of these results wasbased more on plausibility arguments than on the factsone could obtain from differential measurements such asvelocity and temperature profiles or concentration gra-dients. Despite these limitations, the investigations ful-filled one engineering requirement-that of producinggeneral relations suitable for design purposes. Applica-tion of the principles of colloid chemistry showed thatthe laminar-flow physical property relations could beexpected to be of widespread usefulness, applying toall aqueous suspensions of metal oxides or solids which actas a reversible gas electrode provided the particles

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Ffgure 19 xferimental results f o r minimum trunsfort oelocityclassiJied byjoiow regimeare symmetrically shaped. SVith the same restrictions onparticle shape, the turbulent flow correlations were alsoshown to be general in that they were functions only ofthe intrinsic properties of the two phases and the laminar-flow properties of the suspensions. In addition to pro-viding design relations, the results furnish some insightinto processes that may be occurring as well as establishlimits into which more rigorous treatments must fit.

The question remains, what can be done to improveour fundamental understanding of the transpor t charac-teristics of suspensions? Under laminar flow conditions,analysis shows that the properties of dilute suspensions ofsolid spherical particles are simple functions of the fluidproperties, the volume fraction solids, and in some casesthe particle density and diameter. Yet this representsalmost the limit of our ability to derive rigorous theories.Th ere has been some success in treat ing dilute suspensionsof nonspherical solid particles or dilute suspensions inwhich the particles are liquid drops. But attempts to ex-tend theory to concentrated suspensions, where particle-particle collisions leading to doublet and triplet forma-tion must be accounted for, leave something to be desired.Th e further complications accompanying considerationof irregularly shaped particles or the reduction of particlesize to near colloidal dimensions has hardly been touchedtheoretically. Factors tha t must be considered in thelatter case include the nature of the attractive forcebetween particles, the effect of electrolytes on the magni-tude of this force, surface roughness of the particles, thegeometrical ar rangement of particles in a floc under staticconditions, the effect of shear on the ra te of formation anddestruction of flocs, and on the arrangement of particleswithin the floc, and so forth.

O n the exper imental side, work of Mason a nd Bartok79A) on the behavior of suspended particles in

3 4 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y


9/9

laminar shear is an outstanding example of the kind ofresearch that is required to provide a rigorous test oftheory. Mention might also be made of the recentwork by Giesekus 14A) on the motion of particles sus-pended in constant velocity gradient flow field.As for turbulence studies with flocculated suspensions,the outlook is not favorable. The primary reason forthis, of course, is the great difficulties that have been en-countered in the development of turbulent flow theoryfor Newtonian fluids. On e area of turbulent floccu-lated suspension research that is important and alsowhich is within the capability of existing experimentaltechniques is the determination of accurate velocityprofiles for non-Newtonian suspensions in pipes andchannels 3 A ) . Some progress has been achieved ontheoretical and experimental studies of the statisticalproperties of flowing gas-solid suspensions 23A, 2 8 A ) butno results a re available for non-Newtonian suspensions.NOMENCLATUREA = area, sq. ft.A1 = force constant, lb.,a 6 , c, d, e = exponents, dimensionlessB = coefficient, dimensionlessc,D = diameter, ft.d u / d r = velocity grad ient, sec.fg = conversion factor, (lb.,/lb.,)(ft./sec.2)gL = gravitational constant, ft./ secZG = mass flow rate , lb.,/hr. sq. ft.h = heat-transfer coefficient, B.t.u ./hr. sq. ft. O F.j = h/~ ,pV) Np, )~ ~ ,imensionlessk = thermal conductivity, B.t.u./hr. sq. ft. F./ft.L = length, ft.NN x e = Hedstrom number, g,pruD2/+Np = Prandtl number, c , v /k eN R ~q = heat flux, B.t.u./hr.T = temperature, O F.Utu*uoVV, = transition velocity, ft./sec.x = length, O Y / U , ft.y +Greek LettersCY6K76 = p / ~ , imensionlessppLevp = density, lb.,/cu. ft.rrwrur = g,pryx2/p2,dimensionless@Subscriptsb = bulkf = floc

= suspending mediump = particles = suspensionw = wall

= specific heat, B.t.u./lb. O F.

= Fanning friction factor, DAj /4L) ( p V 2 / 2 g , ) dimensionless

= wall layer thickness, dimensionless

= Reynolds number, Dvp/q or DVp/pe

= particle settling rate, ft./se_c.= friction velocity, d g c r w / p , t./sec.= minimum transport friction velocity at infinite dilution,= mean stream velocity, ft./sec.ft. /sec.

= dimensionless distance from wall, y u * / v

= volume immobilized water/volume solid, dimensionless= wall layer thickness, ft.= von Karman coefficient, dimensionless= coefficient of rigid ity, lb.,/ft. sec.= viscosity of suspending medium, lb.,/ft. sec.= effective viscostiy, lb.,/ft. sec.= kinematic viscosity, sq. ft./sec.= shea r stress, Ib.,/sq. ft.= wall shear stress, DAf /4 L , lb.,/sq. ft.= yield stress, lb.,/sq. ft.= volume fraction solids, dimensionless

LITERATURE CITED(1A) Alves, G. E., Boucher, D. F., Pigford, R. L., Chem. Eng. Progr. 48, 385 (1952)(2A) Bird, R. Byron, A.1.Ch.E. J . 2, 428 (1956).(3A) Bogue, D. C., Metz ner, A. B., IND .ENC. CHEM. UNDAMENTALS,143 (1963).(4A) Bowen, R. L., Jr., Chrm. Eng . 68, No. 15, 143 (July 24, 1961).(5A) Buglia rello, George, Daily, J. W., Tapfii 44, 881 (1961).(6A) Chien, Ning, The Present Status of Research on Sediment Transport,Trans. Am. Soc. Ciui lEng. 121 833 (1956).(7A) Clapp, R. M., Turbulent Hea t Transfer in Pseudoplastic Non-NewtonianFluids, p. 652 in Internatio nal Development s in Heat Transfer, Part 3,Am. SOC.Mech. Eng., 1961.(8A) Col burn , A. P., Tram. A.l.Ch.E. 29, 174 (1933).(9A) Daily, J. W., Bugliarello, George, Tappi 44, 497 (1961).(10A) Dodg e, D. W., Metzn er, A. B., A.l .Ch.E. J. , 189 (1959).(1 1A) Eissenberg, D. M., Measuremen t and Correlation of Turbulent FrictionFactors of Thoria Suspensions at Elevated Temperatures, A.1.Ch.E. J . , n press.(12A) Eissenberg, D. M., Boilin Burnou t Heat Flux Measurements in a Non-Newtonian Suspension, 51st dational Meeting, A.I.Ch.E., San Juan, PuertoRico, Sept. 29 1963.(13A) Gabr ysh, A. F., Eyring, H., Cutler, I., J . Am . Ceram.Soc. 45, 334 (1962).(14A) Giesekus, Hansw alte r, Rhcologica Acta 2, No. 2, 112 (1962).(15A) Hughes, R. R., IND.END.CHEM. 9, 947 (1957).(16A) Knudsen, J. G., Katz, D. L.,Fluid Dynamics and He at Transfer, McGraw-Hill, New York, 1958.(17A) Laufer, John, The Structu re of Turbulenc e in Fully Developed Pipe Flow,National Advisorv Committee for Aeronautics R eo. 1174 (1 954).I(18A) Ma rris , A. W., Can. J . Technol. 33, 470 (1955).(19A) Mason, S . G.,Bartok, W., The Behavior of Suspended Particles in Lami narFlow, p. 16 in Rheology of Disperse Systems, Pergamon Press, New York,1959.(20A) Metzn er, A. B., Friend , P. S., IND. NG.CHEM. 1,879 (1959).(21A) Mur hy, Glen, Young, D. F., Burian, R. J., Progress Report on FrictionLoss of &wries in Straight Tubes, Ames Laboratory Rept. ISC-474 (April 1,1954).(22A) Patt eno n. R. C.. J.Enr. Power (ASME1 81.43 (19591, ~ ~(23A) Peskin, R . L., Some Effects of Particle-Particle and Particle Fluid Interac-tions in Two-Phase Flow S stems, p. 192 in Proc. 1960 He at Transfer and FluidMechanics Inst., Stanford bniv . Press, Stanford, Calif.(24A) Petersen, A. W., Christiansen, E. B., to be published.(25A) Ro use, H., ed., Advanced Mech anics of Fluids, p. 302, Wiley, New York,1959.(26A) Schlicting, Herman, Boundary Layer Theory, 4th ed., p. 503, McGraw-Hill. New York. 1960.(27A) Shaver, R. G., Merrill, E. W., A.I.Ch.E. J . 5 , 181 (1959).(28A) Soo, S. L., TND. ENC. CHEM. UNDAMENTALS33 (1962).(29A) Stanton, T. E., Pannell, J. R., Phil. Trans. Ro ya lS o c . 214 199 (1914).(30A) Thomas, D. G., A.I.Ch.E. J . 6 , 631 (1960).(31A) Th omas, D. G., A.l .Ch.E. J . 7, 23 (1961).(32A) Thomas, D. G . , A.1.Ch.E. J . 8, 266 (1962).(33A) Thomas. D. G.. A.1.Ch.E. J . 8. 373 (1962).I(3 44 Thomas, D. G., Chem. Eng. Prog. Symp. Ser. 57, No. 32, 182 (1960).(35A) Thomas, D. G., Significant Aspects of Non-Newtonian Technology, p.669 in Progress in International Research on Thermodynamic and TransportProperties, e d. by J. F. Masi and D. H. Tsa i, Academic Press, New York, 1962(36A) Thomas, D. G., Transport Characteristics of Suspensions: Part IXRepresentation of Periodic Phenomena on a Flow Regime Diagram for DiluteSuspension Transpo rt, A.1.Ch.E. J. n press.(37A) Va noni, V. A,, Trans. Am. Soc. Civil Eng. 111, 67 (1946).(38A) Weltman, R. A., Keller, T. A,, Pressure Losses of Titania and MagnesiumSlurries in Pipes and Pipeline Transitions, Nat . Adv. Comm. for AeronauticsTN-3889, January 1957.

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