Parameters of Transport of Non Newtonian Fluids Through the Pipes

8/2/2019 Parameters of Transport of Non Newtonian Fluids Through the Pipes

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PARAMETERS OF TRANSPORT OF NON-NEWTONIAN FLUIDSTHROUGH THE PIPES

Rudarsko-geoloSko-naftnibornik

Emil EBERL and Uro3 EBER LId lu te for Mining Geotechnologyand Environment (IRGO),SbvenLeva 93, Ljubljana, Slovenia.

UDC 62275 32 13:539.501 Izvomi manrtveniELanakVol. 7

Key-words: Pulp viscosity, Tub e viscometer, Rheology, T ranspo rtthrough the pipesThe Institute for Mining,Geotechnology and Environment hasdeveloped a tube viscometer which i s enabling estblishment of therheological properties. This paper is describing how it is possible,by the help of the experimental results, to calculate the pressuredro p of the concentrated suspension by transport through thepipes, being th e basis fo r dimensioning the pipes.

SV. 65-69

IntroductionIn t he m echanical process engineering as well as inthe mineral processing the flow of suspensionthrough equipm ent, tubes etc. is playing an essentialrole. In mining the transport of tailings has becomeof special importance in connection with minebackfill and lately als o with disposal of problematicwaste (slag, ash, flotation tailings, gypsum fromprocess of scrubbing sulfur-laden flue gasses etc.)wich has proved of great importance inenvironmental protection.The theoretical principles, wich are serving us as

basis for our calculation of transport through thetubes, are gained in our laboratory throughrheological examination (viscometry) since thefeasibility of tran sporting a m ixture of coarse and fineparticles in water seems almost solely dependent onthe prop erties of th e suspensio n of fine particles only( B o t h e & M e i e r , 1 9 8 7 ) .At IRG O (Institute for Mining,Geotechnology andEnvironment) we have developed our own tube- orextrusion rheom eter which is enabling us to establishthe rheological prope rties of suspensions o n the basisof which we can dimension the tubes fortransportation of concentrated suspensions in the

indrustrial mining. (E b e r 1 & E b e r 1,1995).

Zagreb, 1995.

Non - Newtonian fluidsThe properties of non-Newtonian fluids cannot bedescribed with th e Newton's law of viscosity as theviscosity of these fluids proves to be dependent onthe rate of shear. Moreover, the viscosity of thesefluids can increase o r decrease due to th e changes ofth e rate of she ar which, again, is subject to th e natu reof th e fluid.The non-Newtonian fluids are according to theirproperties, established in the steady flow, dividedinto the fluids the properties of which aretime-dependent and into the fluids the properties of

which are time-independent.

IUjuZne Ndl: Viskozitet suspenzija, Cijevni viskozimeter,Relogija, Transp ort cjevovodomU Institutu za rudarstvo, geotehnologiju i okolinu (IR GO ) razvilismo vlastiti cijevni viskozimetar pomoCu kojega je mogute odrediti

reoldka svojstva suspenzija. Prikazano je na koji je naBn mogutep o m d u eksperimentalnih rezultata izra hn ati pad tlaka pritransportu gustih suspenzija kroz cjevovod Sto sluZi za o sn w udimenzioniranja cjevovodova.

In our case we take interest in the time-independentfluids only in which the shear stress proves to beconstant at any shear rate all the time. The properties ofsuch fluids depend only on the magnitude of theimposed shear stress and not on the duration of thestress.The group of non-Newtonian time-independentfluids includes Fig. 1.Pseudoplastic (shear thinning) fluids encompassingmost non-Newtonian fluids. In these fluids the shearstress is increasing proportionally and the viscositydecreasing by growth of the shear rate (gradient ofspeed). This group practically includes all dispersionsystems (e.g. suspensions of particles in water).In dilatant (shear thickening) fluids, however, th eshear stress is increasing superproportionally and sodoes the viscosity with growth of th e shear rate.Plastic fluids start to yield only at a certaincharacteristic minimum yield stress to. If the shearstress z>zo linearly increases in dependence on theshear rate we ar e speaking of th e Bingham fluid.While the dilute and more concentrated suspensionsof mineral particles in water behave like pseudoplastic(shear thinning) non-Newtonian fluids, the highlyconcentrated suspensions (paste) exhibit nearly suchproperties as t he Bingham fluids do.

'The theoretical principles of rheological examinationsare serving us a basis for calculations of transports ofconcentrated and highly concentrated suspension (paste)within the laminar flow. For this purpose we need amodel conveying the ratio between the shear stress andshear rate. Pastes, consisting of mineral particles andwater, are from rheological point of view rated amongthe pseudoplastic (shear thinning) fluids for which theBingham's law is applied enabling us to defineproperties of th e concentrated suspensions with a goodapproximation, viz (Ba r n e s et al., 1989):

where:


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66 Rud.-geo1.-naf.zb.,Vol.7,Zagreb,19952z = shear stress in N/m , 2z = yield limit o r minim um yield stress in N/m ,

vp = plastic viscosity in P as ,dvldr = 3 - shear rat e (gradient of speed) in sIn the Bingham model the plastic viscosity isrepresented by the ratio between the shear stressabove minimum yield stress and shear rate. Theapp aren t viscosity, however, is the rat io between t heshear stress and shear rat e if it depends on t he rate ofshear; it is also called sh ear viscosity. Sinc e th e highlyconcentrated suspensions, when flowing, do notbehave exactly according to the Bingham's law, wewould define the viscosity of these fluids as anapparent one.Buckinham-Reiner equation of volume flow rateof concentrated suspensions through t he tubes is asfollows (B o t h e & M e i e r, 1987):

where:Q = volum e flow ra te in m3/sD = tube diameter in m,Ap = required pressure in Pa,L = tube length in m.Both authors have for the required pressureformed th e following equation :

where:v = average flow rat e im m/s.In the equations (2) and (3) the first term standsfor the New tonian fluids.Since the flow behaviour of the concentratedsuspensions is not or is only partially conforming tothe Bingham model we will use another method tocalculate parameters for transport of thesesuspensions through th e tubes in laminar conditions.The maximum shear stress tsn the tube is (S u 1 a n,1988,G e r t h, 1981):

n' = non-Newtonian rheological constant in theflow or shear diagram, which has for ordinate thevalueTs= DApI4L and for abscisse p,= 8 ; ~ .

d . nt ,n'=-In?\The data needed to plot a flow diagram areobtained by means of the tube viscometer (E b e r 1& E b e r l , 1995).If the vallues t&displayed in the diagram, aredivided by j we get the Poiseuille's equatio n:DAp , 85v=7'- 8)

= is apparen t viscosity (in t he non-N ewtonianfluids) in Pa.s.By means of t he flow diagram we can ascertain th etube diameter at a determ ined pressure d rop or viceversa.Th e pressure drop is calculated from:

Rheological properties of suspensions andcalculation of tran spor t of concentrated suspensionsKnowing the rheological properties ofconcentrated suspension we can dimension thepumping facility in terms of the required pressure,diameter and length of tubes as well as transporting

rate or we can adapt the rheologic properties ofmaterials to som e requested transport concept.It is obvious that besides the laboratoryexaminations of rheometry it is also necessary todetermine the size distribution of material, itsdensity, mineral properties, sedimentation insuspension as well as other features of material likehardening and binding strength, elutriation, and pHvalue.From the indu strial point of view it is necessary tocheck the flow properties of suspension as the data,obtained through lab-examinations, are valid forsuspensions with fine granules ( ~ 2 5 0m ), while wealso have to determine the influence, exerted by thecoarse particles in suspension.This relation is valid for all fluids, the Newtonian In this paper we ar e focusing solely onas well as the non-Newtonian ones. dimensioning the tubes on the basis of hydrologicalThe shear rat e for th e Newtonian fluids at the wall researches of concentrated suspension of fineis equal to: granules.

( 5 ) A fluid can be by means of sh ear - or flow diagramrated with respect to its rheological properties. Thedata on the slope of the straight-line in the flowan d for the nOn-Newtonian Ones according diagram with arithm etic (Fig. la ) o r log-log plot ofRabinovich coordinate axes (Fig. lc) and march of the viscosity. 3n'+l . 3n '+ l 8Vyv = -.ya= --n 4n Dwhere:

curve resp. (Fig. l b) ar e characterizing the flow type.In the fluid diagram with arithm etic plot all curves,with the exception of the curve representing theBingham fluid, have got the ir origin in the coord inate


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Eberl,E. &Eberl, U.:Transport of Fluids 67

system. While the Newtonian fluid is representing astraight line and the Bingham fluid almost a straightline the other s ar e developing curves. In the diagramwith log-log plot all fluids are represented by straightlines except for the Bingham fluid, which is displayedby a curve.The shear rate of the New to4an fluid in laminarflow at the tube wall is 8 v P , while in th enon-Newtonian fluid th e shear ra te in the laminarflow is a function of the value 8 v P , so that we maywrite down:

c The plotted values, obtained by means of theYixometer are,- in th e diagram DAp/4L to 8 V P , fo rA c the time-independent non-Newtonian fluids,A -entered into the same curve regardless of the

B diameter o r length of the viscometer tube and flowrate. In order to be able to plot this curve we have t oknow density of the fluid, pressure, bearing o n it , an dit s mass.D Fig. 1. Flow diagrams of various fluids:A-Newtonian, B-pseudopiastic, C diitant, D-Bingham,E-plastic;a), b) arithmetic plot of coordinate axes;c) log-log plot of coordinate axes,

"a" (y = shear rate, z= shear stress, rl = apparent viscosity)

Fig. 2.Flow diagram:s= DApJ4L Pa) - shear stress, j. = d v / d r = 8 v / D = 3 2 ~ / n ~ ~i') -shear rate, qa apparent viscosity (P as )


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6 8 Rud.-geo1.-naf.zb.,Vol. 7,Zagreb,1995

Practical exampleWe would like to calculate the pressure drop fortransport of concentrated suspension (57.5 ~ 0 1 % ) ffilter ashes quantity Q =3 0 m3/h (8.3 ~10 " m3/s)

through a tu be with inside diamter Di= 140 mm.We performed the measurements in our lab withtube viscometer, having tub e diameters D = 9.0 and6.0 mm a nd - lengths L = 900 and 600 mm at variousvalues of pressure over the free suspension surface.On the basis of the obtained results we firstcalculated the values DAp/4L and 8 v P and thenplotted th e diagram (Fig. 2).It can be se en from t he diagram (Fig. 2), showinglog-log plots of values y and t on the abscisse andordinate, that th e represented value t orms almost astraight-line which brings us to a conclusion that thissuspension does happen to be totaUy pseudoplasticnor Bingham, yet it comes closer to t he pseudoplasticthan to the Bingham fluid also TO scarcely occurs inthe latter.One group of data, obtained by experiment withtube viscometer, tube, dia D = 9.0 mm, and lenghtL=0.9 m, was Ap= 0.9 bar, p= 1575 kg/m3, fluiddischarge time t = 23.3 s and catched pulp m ass m =1.373 kg.

DAp - 0,009 - 9 lo 4= 225,0--4L 4.0,9 [pal

In the same way we obtained values DAp/4L and8i7 P also at othe r, lower pressures with tub e 9.0 andtube 6.0 mm. These values are displayed in th e flowdiagram (Fig. 2).AU calculated values, obtained with tubes 9.0 and6.0 mm, ar e in compliance with (correspond to) o ne

and the sam e curve which m eans that the examinednon-Newtonian fluid is time-independent.The apparent viscosity is according to the equation(8):

The apparent viscosity in the same way alsocaluculated at other shear stresses and displayed inthe d iagram (Fig. 2).We now carry out the calculation of the function8 v P for the tube dia . Di = 140 mm a t flow Q = 8.3 x1 0 - ~ m ~ / s .t this value of the function we take areading from the diagram (Fig. 2) of the apparentviscosity at th e flow Q through a given tube.

-= --'0'539 - 30.81 s and q=O.57D, 0.14Th e calculation of th e pressure drop (equation 9):

From the Fanning equation we can dcduce forpressure drop:

where A represents friction factor (non-dimensional)for which there is valid for th e lam inar flow:

Re-Reynolds' number, the practical limiting valueof which is at th e transition from th e laminar into th eturbulent flow for all Newtonian an d non-Newtonianfluids approx. 2100.

We are able to deduce from this equation for ourexample: . 1575.0.14.0.539 = 208Re= 0.57

being virtually an equal result to the previouslyobtained one.Metzner and Reed carried out an operation forcalculation of th e above relations in independence ofrheological properties of th e time-independent fluids(S u 1 t a n, 1988). This op eratio n is based on amodified friction factor and Reynolds' number forthe N ewtonian fluids.Th e modified Reynolds' num ber is:

P D"' J'"""Re,= K' *("'.I'where:K' = fluid consistency index a ndn' = non-Newtonian rheological constantwhich we can determin e from th e flow diagramDApl4L - 8 V P (Fig. 2) the abscisse and ordinate ofwhich have got a log-log plot, n' is representing th e

slope of the line and tangent resp. in th e curve of


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Eber4E. & Eber4 U Transport of Fluids 69this diagram and K' the value DApt4L on theordinate a t 8v/D=1.

We can deduce from the equations (ll), (12) and(14) the relation:

Since the representation of the ratio t / y in the flowdiagram (Fig. 2) does not constitutes a straight linewe have obtained the slope n' with a tangent to thiscurve (n'=0.863), with K at y=1,0 (K=0.914)resulting into:

0.14'". 0.539'2VYb"~575 = 207Re, = 0.914 . (n"6'-')and

ConclusionIn the mechanical process engineering as well as

mineral processing the flow of suspensions throughequipment and tubes is playing an essential role. Thetheoretical principles, serving as basis for calculationof transport, are obtained in laboratory by means ofrheological examinations (viscometry). At theInstitute of Mining, Geotechnology and Environment(IRGO) we have developed our own tube viscometerby the help of which we are able to determine therheological properties of suspensions. M e r therheological parameters of concentrated suspensionshave been established we are able to dimension thepumping facility in the terms of required pressure,diameter and length of tubes or, the other way round,we can adapt the rheological properties of a fluid tosome requested concept of transport.

A fluid may be, with respect to its rheologicalproperties, by the help of the shear-and flow diagramrap . referred to a certain rheological classification.

In the laboratory we carried out experiments withtube viscometer of two different tube diameters bywhich we determined the rheological properties ofconcentrated suspension of filter ashes in water.

The results are displayed in the diagram DAp14Lto 8 vP. Since the values have, regardles of thediameter of the viscometer tube, fallen into the samecurve, we find that the suspension is atime-independent non-Newtonian suspension.However, as it can be concluded from the form of thecurve, the suspension is not totally pseudoplastic norBingham yet it is coming closer to the pseudoplasticfluid.

We have presented a practical example of thecalculation of pressure drop in the tubes (pipes) fortransport of concentrated suspension on the basis ofthe results of experiments made with theconcentrated suspension, serving as basis fordimensioning of the tubes.Received ZXI1 1994.Accepted 20.V1.1995.

R E F E R E N C E SBa rn e s ,H .A . ,Hu t t o n , J . F . a n dWa l t e r s ,K . ( 1 989 ) : AnIntroduction to Rheology. Elservir, 19 9 pp., Amsterdam.B o t h e, H. & M e i e r , J . (1 9 8 7 ) : P u m p v er sa t z , e inForderverfaren in de r Umwelttechnik In: Thome-Kozmiensky

(editor): Behandlung von Sonderabfallen 3 , EF-Verlag furEnergie - und Umvelttechn~kGmb H, 603-625, BerlinE b e r I, E. & E b e r 1, U . (1995): Capa bilities of a tu beviscometer in measuring the viscosity of suspensions (~npreparing)G e r t h, C. (1981): Ul lmanus Encyk loped ie der Techn ischenChemie. Verlagsgeselschaft, Bd 5,22 pp., W einhelmS u 1 t a n, A. A. (1988): S izing P ipe fo r Non-Newton ian F low.Chemical Engineering, 19, 140-146.

Parameters of Transport of Non Newtonian Fluids Through the Pipes

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