Experimental study of grease flow in pipelines: wall slip and air entrainment effects

Chemical Engineering and Processing 44 (2005) 805–817

Experimental study of grease flow in pipelines: wall slipand air entrainment effects

M.A. Delgado, J.M. Franco∗, P. Partal, C. GallegosDepartamento de Ingenier´ıa Quımica, Facultad de Ciencias Experimentales,

Universidad de Huelva, Campus Universitario del Carmen, 21071 Huelva, Spain

Received 21 July 2004; received in revised form 18 August 2004; accepted 20 September 2004

Abstract

Prediction of pressure drop gradient and evaluation of wall slip and air bubbles entrainment effects observed during the piping flow oflubricating greases were investigated. With this aim, viscous flow tests in rotational rheometers and pressure drop measurements in pipelineswere carried out using different geometries with both smooth and rough surfaces. The Sisko model was applied in the experimental rangeo ly viscousm g a modifiede n based ont finition oft©

K

1

pdittiurds

di

iloteratebest

inesof-

etryer, aiour

ductmongtionis isThemea-,citythelding

0d

f flow rates for predicting pressure drop gradient. Air entrainment occurring when the pumping system was primed with a highaterial as lubricating greases significantly decreases pressure drop gradient. This air entrainment effect can be corrected usin

xpression to evaluate the drag ratio defined for non-Newtonian liquid/air intermittent flows. On the other hand, a new expressiohe internal relative roughness of pipelines was proposed to correct wall slip effect. Eliminating these two effects, the classical dehe friction factor for a non-Newtonian fluid,f= 16/Re′, can be applied to predict pressure drop of grease flow in pipelines.

2004 Elsevier B.V. All rights reserved.

eywords:Laminar flow; Lubricating greases; Multiphase flow; Non-Newtonian fluids; Rheology; Wall slip

. Introduction

Lubricating greases are generally highly structured sus-ensions consisting of a thickener, usually a metal soap,ispersed in mineral or synthetic oil[1–3]. The thickener

s added to prevent loss of lubricant under operating condi-ions but, evidently, this implies a considerably resistance tohe flow of these materials. The widespread use of greasesn centralized lubricating systems makes necessary a betternderstanding of the flow behaviour in pipelines. Thus, a cor-ect prediction of pressure drop is a prerequisite for optimumesign of pumping systems and minimizing the power con-umption.

In the case of highly viscous materials like greases, slightiscrepancies in the prediction of the pressure drop may imply

mportant differences in the power consumption of the pump

∗ Corresponding author. Tel.: +34 959 019995; fax: +34 959 019983.E-mail address:[email protected] (J.M. Franco).

or even a malfunction in grease distribution. Using a pscale pumping system or a capillary viscometer that opwithin the shear rate range of product concerned is theway for predicting the pressure drop gradient in pipelbut it demands rigorous works and longer times. Moreten, empirical models developed from rotational viscomare used for predicting pressure drop gradient. Howevcorrect quantification of the non-Newtonian flow behavfaces several difficulties in some structural fluids[4,5] be-cause the flow parameters are not only specific of the probut also depend on the range of shear rate analysed. Athese, the most common difficulty found is the modificaof the flow curves when using different geometries. Thusually the result of fracture and wall slip phenomena.appearance of wall slip leads to an apparent decrease insured viscosity at a given shear stress[6]. On the other handfracture phenomenon induces a discontinuity in the velofield, which may lead to the ejection of the sample fromgeometry (i.e. plate–plate or cone–plate geometries), yie

255-2701/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
oi:10.1016/j.cep.2004.09.003

806 M.A. Delgado et al. / Chemical Engineering and Processing 44 (2005) 805–817

a sudden reduction of apparent viscosity. Lubricating greasesshow a complex rheological behaviour in which both men-tioned phenomena may be observed[7]. In this sense, thedifficulty to predict grease pumpability due to discrepanciesobserved in the flow curves with the gap width was alreadypointed out some decades ago[8].

Many previous studies had reported wall slip phenomenain disperse or colloidal systems being the most common ex-planation the formation of a depleted layer at the boundariesas, for instance, the wall of the sensor systems in a rheome-ter or the wall of pipes or tubes, with lower viscosity thanthe bulk, which induces a lubrication effect[6]. This low-viscosity layer may have a thickness of the order of microns oreven less and, consequently, it is very difficult to distinguishbetween true slip and wall depletion effects using conven-tional visualisation methods. The visualisation of grease flowin simple geometries put in evidence the non-homogeneity ofthe deformation field in the gap of cone and plate geometryand the fracture of the sample[9]. Previously, Magnin andPiau[10] reported some photographs in which apparent wallslip and fracture were observed during the flow of siliconegreases and others authors[7,11] found some rheological ev-idences of the wall slip phenomena in greases manufacturedusing metallic soap as thickeners.

The Rabinowitsch–Mooney equation[12] may be usedt e ap unders timei is nos and,c ont wasi yd[ lyaa

hlyv ub-b ult ina hasefl bee cts.A thec onse-q pres-e en thew rages nsid-e f air,w ld ac byu s-t flowoa olina ghly

viscous materials like lubricating greases in which extremelylow flow rates may be achieved.

The method proposed by Rein and McGahey[24] to esti-mate the pressure drop during the piping flow of lubricatinggreases has been traditionally followed in the industry. Thisis a practical method from the viewpoint of the man in theplant, which deals with a simple substitution of the Newto-nian viscosity,µ, for an apparent non-Newtonian viscosity,η, at each flow rate, in the Hagen–Poiseuille equation forlaminar flow. This procedure takes into account the appar-ent viscosity defined in the ASTM D 1092, but as Cho etal. [19] have pointed out, its use can overestimate the cor-rect apparent viscosity by approximately 100% for a NLGIgrade no. 2 greases whose power-law index are of the or-der of 0.1–0.3. Other simple predictions of the pressure dropgradient during the piping flow of lubricating greases havebeen described. Thus, Cho et al.[19] proposed a calculationprocedure based on the extension that Metzner and Reed[25]made on the Rabinowitsch–Mooney equation to estimate thepressure drop of lubricating grease flow in a large pipe using ashear rate at the wall corrected by using the power-law model.More recently, Turian et al.[5] compiled more accurate cal-culation procedures to estimate the pressure drop in othercomplex systems by using more elaborated non-Newtonianempirical models. However, neither the correction of possi-b thep stud-i

alls g thep thep ntali linesa ootha

2

up-p GIg den-s e-ha lc

llowi er-c ntin-u gres-s odelM mc ureo eca-1 r toa

o quantify the flow of any time independent fluid insidipe based on three basic assumptions: (a) the fluid isteady-state laminar flow, (b) the fluid properties arendependent under prevailing conditions and (c) therelip effect between the fluid and the wall. On the other hlassical analysis for the correction of wall slip, basedhe modification of the Rabinowitsch–Mooney analysis,ntroduced by Mooney[13] and further developed by Oldro14] and Jastrzebski[15]. More recent procedures widepplied were presented by Yoshimura and Prud’home[16]nd Yilmazer and Kalyon[17].

Another problem concerning the pipeline flow of higiscous non-Newtonian fluids is the entrainment of air bles when the pumping system is primed, which may resreduction of the pressure drop, depending on the two-pow regime[18]. The drag reduction phenomenon mayxplained qualitatively as a result of two opposite effet a given flow rate, the introduction of air decreasesross-sectional area in the pipe for grease flow and, cuently, grease velocity rises. On the other hand, thence of air bubbles decreases the interfacial area betweall and the liquid, which leads to a reduction in the avehear stress at the wall. Lubricating greases may be cored non-compressible fluids but a substantial amount ohich could be entrapped in the grease matrix, may yieertain degree of compressibility, for instance up to 15%sing a piston pump[19]. Although a great number of inve

igations have been reported concerning the multiphasef non-Newtonian liquid/air systems[20–23], most of themre referred to relatively low viscous liquids such as kand clays slurries or CMC dispersions and not to so hi

le wall slip nor the air bubbles entrainment effects onrediction of the pressure drop were considered in these

es.The overall objective of this work is to evaluate the w

lip and air bubbles entrainment effects observed duriniping flow of greases, which allows to correctly predictressure drop along pipelines. With this aim, experime

nvestigations on the pressure drop in horizontal pipend rotational rheometry were carried out by using smnd grooved surfaces and different gaps or diameters.

. Experimental

A commercial lithium complex soap grease kindly slied by Verkol S.A. (Navarra, Spain), classified as NLrade no. 2[26], was tested. This grease presents aity of 914 kg/m3 at 20◦C and a marked shear-thinning baviour with apparent viscosities ranging from 6× 104 Pa st 0.01 s−1 to 3 Pa s at 1000 s−1 at 25◦C. Other technologicaharacteristics are shown inTable 1.

A pumping system was designed and constructed to anvestigations of the piping flow behaviour of this commial grease. The grease from the hopper tank was coously pumped through stainless steel tubes by a proive cavity pump equipped with a force feed screw, mV2.6IVA10 (PCM Moineau, France) having a maximu

apacity of 0.190 m3 h−1 and maximum discharge pressf 10 bar. A controlled-speed electric motor, model Varm0 (Leroy-Sommer, France) was incorporated in ordepply 18 different flow rates, ranging from 0.04 m3 h−1 to

M.A. Delgado et al. / Chemical Engineering and Processing 44 (2005) 805–817 807

Table 1Some technological properties of the commercial lithium complex greasestudied

Physical properties

Worked penetration, dmm (deci-millimetre) at 25◦C 265–295Operating temperature (◦C) −30 to 180Oil separation (% (w)) <5Dropping point (◦C) 260Texture SmoothColor Dark amber–tanBase oil Mineral oil

Table 2Dimensions of the pipeline test sections

Pipe Internaldiameter,D(mm)

Relativeroughness

L/D Entrancelength(mm)

Exit length(mm)

(1/2)-S 17.12 0.0029≈ 0 60 1773 200(1/2)-R 15.84 0.036 60 1773 200(3/4)-S 22.45 0.0022≈ 0 60 1453 200(3/4)-R 20.85 0.037 60 1453 2001-S 27.86 0.0018≈ 0 60 1128 2001-R 25.46 0.043 60 1128 2001(1/4)-S 36.62 0.0014≈ 0 60 603 2001(1/4)-R 34.22 0.033 60 603 200

0.26 m3 h−1. Flow rates were directly measured by collect-ing the fluid in a weighting tank and the mass of the sampleobtained in a given time was derived. In all cases, the flowregime was clearly laminar.

Four horizontal stainless steel tubes of different diameterswere used (seeTable 2). Stainless steel meshes fixed to theinternal wall of the pipeline were used to modify the rough-ness; the dimensions of the different meshes used are shownin Table 3. Relative roughness lower than 3× 10−3 mm wereassumed negligible. Entrance and exit lengths higher than40D and 30D, respectively, were selected in order to avoidany end effects. Pressure drop gradient measurements werealways made over the sameL/D ratio in the pipeline test sec-tion. Geometrical characteristics of these sections are sum-marized inTable 2. The cycle was completed by returningthe lubricating grease to the hopper tank from the pipelinetest section. The pressure drop across the pipeline test section was measured by using Bourdon-type manometers witha stainless steel membrane separator. This device has a measurement range of 0–10 bar and 1% error. The piping flow ex-periments were carried at room temperature (22± 2◦C). Themanometers were always calibrated previously to each mea-surement with a new pipeline test section. The reproducibilityof the measurements on the same pipeline test section was

Table 3M

M)

123

evaluated by replicating the pressure drop measurements atleast three times. The maximum coefficient of variation forreplicates was 4% with an average value at 1.2%.

Viscous flow measurements were carried out using threedifferent controlled-shear rate or controlled-stress rheome-ters, which complement each other. Controlled-stress mea-surements were performed in both a Bohlin CS rheometer(Bohlin, Sweden), in a range of shear stress of 4–1400 Pa,using a plate–plate geometry (25 mm diameter, 1 mm gap),and a Rheoscope rheometer (Thermohaake, Germany), ina range of shear stress of 300–1200 Pa s, using plate–plate(35 mm diameter, 1 mm gap) and cone–plate (35 mm diam-eter, 1◦, 0.052 mm gap) geometries. These measurementswere supplemented with those obtained in a RV20–CV20Ncontrolled-shear rate rheometer (Haake, Germany), in a rangeof shear rate of 10–3000 s−1, using a coaxial cylinder geome-try (15 mm external diameter, 12 mm length, 0.545 mm gap).Different plate–plate geometries with both smooth and roughsurfaces (roughness: 0.4 mm) were used. All measurementswere done at 25◦C following the same protocol (30 min rest-ing time at measurement temperature) and replicated at leastthree times using new unsheared sample.

3. Results and discussion

3

od toe rials[ nesso up ton

ro-t and

F ootha for thel

esh dimensions used to modify the internal roughness of pipelines

esh number Screenopening (mm)

Threaddiameter (mm

0 2.180 0.600 0.990 0.400 0.606 0.32

-

-

.1. Rheological measurements

The use of rough surfaces is a widely accepted methliminate wall slip phenomena in a great variety of mate

11,27,28], although the mechanism through the roughf the surfaces influences the wall slip phenomenon isow not fully understood.

Fig. 1 shows the viscous flow curves obtained inational rheometers using different geometries, gaps

ig. 1. Viscous flow curves obtained from rotational rheometry using smnd rough plate–plate and cone–plate geometries and different gaps

ubricating grease studied.


surfaces (smooth and rough) in a relatively wide range ofshear stress. As it can be noticed, a marked influence of theroughness of the tool used and gap effect when using differentgeometries exist. These effects were widely accepted as ex-perimental evidence of wall slip phenomena. However, thesedifferences are mainly found at intermediate shear stressesaround a critical value for which a sudden drop in viscositytakes place. This drop in viscosity occurs at significant lowervalues of shear stress when using smooth surfaces in compar-ison with the measurements performed with rough surfaces.Only slight differences were observed at very low or veryhigh values of shear stress, which is according to the gen-eral evidence that the slip phenomenon is only detected in acertain range of shear stresses[6]. This result supports thegeneralized idea that the wall slip phenomenon is confined toa certain range of shear rates associated with almost constantvalues of stress, yielding a jump or plateau region, in whichflow instabilities are predominant[6,7]. This jump is usuallypromoted by a dramatic change in the microstructure of thesample under shear.

Significant differences between measurements obtainedusing smooth cone–plate and plate–plate geometries werefound, especially at shear stress higher than 400 Pa, whichcorresponds to the above mentioned stress jump. However,a slight influence of smooth plate–plate gap on the flowc vis-c therh sig-n ughs ab-s

cy toc e ap-p plef s ath sheart ratel re-f ncy,fi en-t

τ

ey’sm nge.F ngeo ft ntalc ap tot tiono

(

Fig. 2. Fitting of viscous flow curves to the Sisko model in the same experi-mental range of shear rate than obtained in piping flow measurements, usingboth smooth (a) and rough (b) plate–plate with 1 mm gap.

Re′ = ρuD

η∞G(n,X)

(1 + X)(3)

where

G(n,X) =[1 + 4

[(n+2n+3

)X +

(2n+12n+n

)X2 +

(n

3n+1

)X3

]](1 + X)3

(4)

X = m

η∞γn−1 (5)

In addition to this, a higher flow index can be observed forthe fitting made on the flow curve obtained with smooth ge-ometries, which indicates a more dampened drop in viscositythan the observed by using roughened geometries (Table 4).

3.2. Piping flow measurements

Pressure drop measurements in the fully developed lami-nar flow conducted in horizontal pipelines with four differentdiameters and both smooth and rough internal surfaces areshown inFig. 3. As may be seen, the pressure drop gradi-ent, (−�P/L), increases with the volumetric flow rate ap-proximately following a power law dependency in all cases,showing flow indexes ranging from 0.111 for the pipe 1(1/4)-St sured andr thatt ssure

TV roughp te thano

SR

urve may be detected, yielding asymptotic values ofosity when the gap is increased over 1 mm. On the oand, as expected, the gap width does not exert anyificant influence on the flow measurements when rourfaces are used, which is indicative of the wall slipence.

As above described, all these curves show a tendenoincide at relatively high shear rates. Nevertheless, thearance of fracture followed by the ejection of the sam

rom the gap does not allow to observe the flow curveigher values of shear stress. However, in all cases, a

hinning behaviour with a slight tendency to a high shearimiting viscosity,η∞, can be observed in all cases. Theore, the Sisko model, which takes into account this tendets fairly well the flow behaviour obtained in the experimal shear rate range observed in the piping flow tests:

= η∞γ + mγn (1)

However, both Ostwald-de Waele and Herschel-Bulklodels could be applied within a shorter shear rate raig. 2shows the quality of these fittings in the restricted raf application in the pipeline flow.Table 4shows the values o

he parameters obtained from the fitting of the experimeurves measured in plate–plate geometries with 1 mm ghe Sisko model, which have been applied for the calculaf both pressure drop gradient and Reynolds number[5]:

−�P) =[

8η3∞γ4wL3

uD2(1 + X)3G(n,X)

]1/3

(2)

to 0.276 for the pipe (1/2)-R (seeTable 5). According tohe rheological results previously shown, different presrop values were found in tests carried out with smoothough pipes, for the same diameter, in spite of the facthere should not be influence of the roughness on the pre

able 4alues of Sisko’s parameters obtained using both smooth andlate–plate (1 mm gap) in the same experimental range of shear rabtained in the piping flow measurements

n m(Pa sn) η∞ (Pa s)

mooth surfaces 0.125 625.0 0.89ough surfaces 0.098 842.0 2.54


Fig. 3. Experimental pressure drop measurements in the fully developedlaminar flow conducted in pipelines with different diameter and both smoothand rough internal surfaces.

drop gradient in the laminar regime. This discrepancy foundin the laminar flow regime indicates once again the existenceof wall slip phenomena in the pipelines.

The influence of pipe diameter on the wall slip phenom-ena may be easily deduced from the typical plots of natu-ral log of wall shear stress (−�PD/4L) versus the naturallog of the flow characteristic (8u/D) which coincides withthe actual wall shear rate in the case of Newtonian fluids[4]. Fig. 4shows the experimental results obtained from thepipeline flow experiments for individual straight pipes in thisform. As above mentioned, theτW–8u/D relationship fol-lows a power law model for each pipeline within its ownmeasured flow characteristic range, with the power law flowparameters (k′, n′) shown inTable 5. However, if all exper-iments are considered together, two separate curves can beobserved, one corresponding to pipes with smooth surfacesand another one corresponding to rough surfaces (Fig. 4),with a clear tendency to a constant viscosity at high values ofthe flow characteristic as detected in rotational viscometry.In addition to this, no significant influence of the diameteron the pressure drop gradient can be detected in spite of dif-

Table 5Values of the Metzner–Reed power-law parameters (k′, n′) found in thepipeline flow for each tube

P

((11((11

Fig. 4. (−�PD/4L) vs. (8u/D) plot obtained from pipeline flow tests con-ducted with both smooth and rough internal surfaces. Solid line shows thewall slip correction based on the relative roughness (Eq.(23)).

ferences found by modifying the internal roughness, whichindicates that not detecting neither diameter influence nor gapinfluence is not necessarily indicative of non-slip conditions.Moreover, the effect of roughness is almost independent ofthe diameter. Thus, an important difference between mea-surements performed with rough and smooth surfaces wasobtained in these cases in spite of the same qualitative rheo-logical behaviour.

In order to correlate the flow behaviour obtained from rota-tional rheometry with those found in pipelines, pressure dropwas obtained from Eq.(2)and the flow characteristic was cal-culated from shear rate values as shown inAppendix A. Thisprediction ofτW–8u/D relationship is presented inFig. 5forselected smooth and rough pipes. As it can be observed, thepredicted curves significantly differ from the experimentaldata. However, approximately the same deviation, 22± 2%,was found with all pipes, independently of the roughness and

Fl ur-f

ipes n′ k′ (Pa sn)

1/2)-S 0.225 418.93/4)-S 0.183 474.6-S 0.132 568.0(1/4)-S 0.111 607.21/2)-R 0.276 455.23/4)-R 0.191 610.7-R 0.165 720.0(1/4)-R 0.121 711.6

ig. 5. Comparison of experimental and predicted log(−�PD/4L) vs.og(8u/D) curves obtained for 1′′ pipes with (a) smooth and (b) rough saces.


diameter, calculated as follows:

Average absolute error

= 1

n

n∑i=1

∣∣∣∣experimentali − predictediexperimentali

∣∣∣∣ × 100 (6)

As it can be seen, general differences between experimen-tal and calculated pressure drop gradient for the same flowcharacteristic range in smooth cases were similar to thosefound in rough cases, which indicates that the magnitude ofwall slip found in experimental flow measurements made inboth pumping system and rotational rheometer was approxi-mately the same.

3.3. Correction of pressure drop for the two-phaseair/grease pipeline flow

The experimental evidence of similar differences foundbetween experimental and calculated pressure drop gradi-ent with both smooth and rough surfaces also indicates thatthere is no influence of slip on such deviation, which mustbe attributed to another phenomena independent of the rel-ative roughness. Taking into account that priming the pumpin highly viscous materials may lead to a significant air en-t op n int wt rela-t asefl ichm plug(

en-t urredd on-ta ens ghlyv .

andRt ase

ses.

plug flow model may be used, in principle, to correlate thepressure drop in the grease/air piping flow:

φ2L = 1

λL

Re′L

Re′TP

(7)

whereφ2L is defined as the ratio of the two-phase pressure drop

gradient, (−�PTP/L), to the pressure drop gradient for theliquid flowing at the same volumetric flow rate as in the two-phase mixture, (−�PL/L), λL is the input volume fraction oflubricating grease calculated according to Eq.(8), andRe′

TPandRe′

L are the generalised Reynolds numbers for two-phase(based on the mixture velocity,uSG+uSL) and liquid flow(based onuSL), respectively.

λL = uSL

uSL + uSG(8)

beinguSG anduSL the superficial liquid and gas velocities,respectively.

Although the idealised plug flow model is likely to un-derestimate the magnitude of two-phase pressure drop, someauthors[21,32]have concluded that it gives reasonable esti-mations of drag ratios foruSG< 1 m s−1 andRe′

TP < 500. Forother conditions, the slip between the two phases becomessignificant and the experimental values of drag ratio are sig-nificantly greater than those predicted from the model[33].A lon-g e-t filmb thes gasb wp b-b tess

hev al-c no n ofp oughl na-l hee engthc test.T /4)-Sp th oft wass fiedb thet gtho tiona ng ana es oxi-m voidf m-e hase

rainment in the pipeline[19], the consideration of the twhase air/grease pipeline flow could explain the deviatio

he predictions of pressure loss.Fig. 6shows the grease flohrough a transparent tube in which the entrainment ofively large air bubbles is well demonstrated. This two-phow is clearly non-homogeneous and intermittent, whay be considered a not well-developed slug flow or a

elongated bubble)[29].A large number of investigations have reported experim

al measurements of the two-phase pressure drop incuring the flow of gas and shear-thinning liquids in horiz

al pipes. These have been summarised by Mahalingam[30]nd Govier and Aziz[31]. However, not all authors have givufficient details in theirs conclusions to be applied on hiiscous pseudoplastic materials like lubricating greases

The empirical approach proposed by Heywoodichardson[32] to estimate the drag ratio,φ2

L, consideringhe shear-thinning behaviour of the liquid for a two-ph

Fig. 6. Visualisation of air bubbles during the flow of lubricating grea

lso, this correlation might be used for slug and plug (eated bubble) flow patterns[29]. The main differences b

ween slug and plug (elongated bubble) flows are thatelow the large gas bubble is considerably thinner inlug flow and there is significant entrainment of smallubbles in the liquid slug[29]. Thus, the two-phase floattern shown inFig. 6 fits better to a plug (elongated bule) flow model, which is found in all range of flow ratudied.

In order to apply Eq.(8), it is necessary to consider toid fraction of gas–liquid flow because it allows the culation ofλL. Void fraction,α, is defined as the fractiof total pipe volume occupied by the gas, or the fractioipe cross-sectional area occupied by the gas phase. Alth

ocal values of void fraction are of greater interest for aytical purposes[34], average values of void fraction for tntire volume at average pressure in the test section lould be experimentally obtained with a relatively simplehus, the average void fraction was obtained for the 1(1ipeline, incorporating at the end of the line a short leng

he same pipe where a sample of air-lubricating greaseubmitted to flow along this. The air volume was quantiy subtracting the weight of this short length of pipe with

wo-phase mixture to the one it would have had if this lenf pipe had been full of grease. In this way, the void fract 1 bar and room temperature was determined, resultiverage value of 0.037± 0.006 for the volumetric flow rangtudied. This fact implies that pump always injects apprately the same air–grease volume ratio. However, the

raction along the pipeline will be different for each diater and volumetric flow, since the volume of the gas p


Table 6Selected values of void fraction, superficial velocities and two-phase generalized Reynolds numbers studied for different pipelines and flow conditions

Pipe QL (m3 h−1) uSL (m s−1) uSG (m s−1) α Re′L Re′

TP

(1/2)-S 0.041 0.049 5.09E−04 7.03E−03 0.016 0.0170.158 0.190 1.49E−03 5.80E−03 0.195 0.1980.261 0.315 2.18E−03 5.30E−03 0.483 0.490

(1/2)-R 0.041 0.049 4.77E−04 7.32E−03 0.013 0.0130.158 0.190 1.50E−03 5.69E−03 0.148 0.1500.261 0.316 2.17E−03 5.12E−03 0.355 0.360

(3/4)-S 0.041 0.029 2.27E−04 7.80E−03 0.006 0.0060.158 0.111 9.77E−04 6.47E−03 0.076 0.0780.261 0.184 1.45E−03 6.13E−03 0.192 0.195

(3/4)-R 0.041 0.029 3.21E−04 7.66E−03 0.049 0.0500.158 0.111 9.99E−04 6.17E−03 0.059 0.0600.261 0.184 1.46E−03 5.68E−03 0.147 0.149

1-S 0.041 0.019 1.07E−04 8.50E−03 0.003 0.0030.158 0.072 6.49E−04 7.26E−03 0.036 0.0360.261 0.119 1.06E−03 9.75E−03 0.090 0.092

1-R 0.041 0.019 1.63E−04 8.02E−03 0.002 0.0020.158 0.072 6.86E−04 6.85E−03 0.028 0.0280.261 0.119 1.04E−03 6.27E−03 0.070 0.072

1(1/4)-S 0.041 0.011 1.12E−04 7.92E−03 0.001 0.0010.158 0.042 4.88E−04 6.80E−03 0.013 0.0140.261 0.069 7.26E−04 6.30E−03 0.034 0.035

1(1/4)-R 0.041 0.011 1.22E−04 7.80E−03 0.001 0.0010.158 0.042 4.98E−04 6.70E−03 0.010 0.0110.261 0.069 7.37E−04 6.20E−03 0.027 0.027

is affected by the pressure gradient. Thus, the void fractionexperimentally obtained for the 1(1/4)-S pipe was correctedwith the average pressure in the test section to get an aver-age void fraction for each diameter and volumetric flow, asfollows:

α =((VG)1

(ρG)1(ρG)P

)((VG)1

(ρG)1(ρG)P

+ VL

) (9)

whereVG andVL are volumes of gas and liquid phase, re-spectively, injected by the pump at the operating pressureand room temperature,ρG is the air specific gravity at thecorresponding pressure and room temperature, and the sub-scripts 1 andP are relative to the pressure of 1 bar and theoperating mean pressure in the pipeline, respectively.Table 6shows values of this parameter for selected pipeline and flowconditions.

The determination of grease superficial velocity,uSL, wassimply obtained by dividing the grease volumetric flow mea-sured experimentally with the total cross-section. However,more complex is the determination of air superficial velocity,uSG, since the air volumetric flow incorporated is unknown.In order to calculateuSG, the average translational velocity ofgas phase,uG, was experimentally determined for a represen-t ges -l nalvt thea was

obtained as follows[36,37]:

uG =∑

i(vG)i(uG)i∑i(vG)i

(10)

where (uG)i is the average air translational velocity for eachbubble of volume (vG)i.

In order to extrapolate the average translational velocityof bubbles, from the 1(1/4)-S pipe to other diameters, is nec-essary a correlation with the average pressure in the test sec-tion, as for instance according to the Weymouth equation.The translational velocities of air bubbles are different foreach pipeline due to the pressure drop gradient; therefore,a relationship between average relative velocity,ur, and theaverage pressure in the test section,Pave, was established tocorrect the average translational velocity for each diameter:

ur = 0.004P2ave− 0.0644 (11)

whereur is the average relative velocity (uG −uL) andPaveis the average pressure in the test section.

Knowing the volumetric flow of grease, the average trans-lational velocity and the void fraction, the superficial air andgrease velocities can be calculated as follows:

uSG = uGα = QG

AGα (12)

u

be-i seeE quid

ative distribution of air bubbles in the volumetric flow rantudied, as Rosehart et al. proposed[35]. Average bubble veocities were determined by visually timing the translatioelocity of the nose of the air bubble between the 60L/D axialest section plane for each flow conditions. In this way,verage translational velocity for each volumetric flow

SL = uL(1 − α) = QL

AL(1 − α) (13)

Finally, Eq. (7) can be used for the laminar regime,ng Re′

TP andRe′L defined according to the Sisko model (

q. (3)). The generalised Reynolds number based on li


properties can be determined according to Eq.(14), as a func-tion of pipe diameter,D, physical properties (density andSisko’s parameters) and superficial grease velocity. The def-inition of Re′

TP by Eq.(15) is based on the assumption thatair entrained in the bulk of grease reduces the effective localvalues of density and Sisko’s parameters by approximatelythe same proportion, similarly to that considered with theReynolds number deduced from the power flow model[29].Consequently, both the ratiosρ/η∞ andk/η∞ were consid-ered unaffected by air entrainment in the bulk and conse-quently the generalised Reynolds number for the two-phaseflow is only a function of physical properties of lubricatinggrease and the mixture velocity,uM, defined asuSL +uSG.

Re′L = ρLuSLD

η∞G(n,X)

(1 + X)(14)

Re′TP = ρLuMD

η∞G(n,X)

(1 + X)(15)

Table 6 shows selected values of these parameters fordifferent pipelines and flow conditions models to illustratethe procedure. However, the experimental two-phase pres-sure drop was not correctly predicted by strictly applyingexpression(7) to determine−�PTP/L. In this sense, it mustbe pointed out that experimental studies from which this ex-pression was obtained, were carried out with relatively lowv wert seaD hes uidp flowi iquidv Con-s , i.e.cφ allers e ofd ase[n tionsb es inl son[

C

w lugf theat

φ

p r

Fig. 7. Comparison of experimental, (φ2L)exp, and theoretical drag ratio,

(φ2L)theor, obtained from Eq.(17) for the Sisko model.

among those calculated for each diameter, volumetric flowand roughness studies, whose value was 0.817± 0.060.

Fig. 7 shows the comparison between experimental andcalculated values of drag ratio of the two-phase mixture. Thedirect application of the correlation of Lockhart–Martinellireveals that it overpredicts the value of drag ratio up to 25%.However, the modified definition of drag ratio given by Eq.(17), describes almost all experimental data with maximumerrors of±10%.

Fig. 8 shows the experimental results of pressure dropfor the grease piping compared with the predictions obtainedfrom Eq.(2), for the different pipelines tested, once correctedwith the Lockhart–Martinelli expression:

φ2L = −�PTP/L

−�PL/L(18)

As it can be observed, a significantly reduction of the dif-ferences shown inFig. 5was achieved, with maximum errorsof ±5%, calculated from Eq.(6).

The influence of air entrainment has not been previouslyconsidered in the prediction of pressure drop during thepipeline flow of lubricating greases. The traditional estima-tion of the pressure drop during the piping flow of lubricatinggreases was based on a simple substitution of the Newtonianviscosity,µ, for the non-Newtonian viscosity,η, as defined int uillee o-n agni-t nt e-do pres-s tinga uillee es-s

iscous fluids showing values of the consistency index lohan 50 Pa sn, and volumetric flow range higher than thochieved in this work. Heywood and Richardson[32] andziubinski and Chhabra[20] remarked the influences of tuperficial liquid velocity and apparent viscosity of liqhase on the experimental drag ratio. Thus, for the same

ndex, the drag reduction increases as the superficial lelocity decreases or the consistency index increases.idering liquid velocities achieved and grease viscosityonsistency indexes of around 600–900 Pa sn, this effect on2L may result dramatic in this case. Moreover, the smuperficial liquid velocities are, the higher is the increasrag reduction with the apparent viscosity of the liquid ph

29,32,33]. As previously applied[32,38], the expression(7)eeds to be corrected to take into account such deviaetween predictions of drag ratio and experimental valu

aminar conditions. In this way, Heywood and Richard32] introduced a factorCL, defined as:

L = (−�PTP/L)exp

(−�PTP/L)pred(16)

hich is considered to be indicative of the efficiency of pormation but evidently it may also take into account allbove mentioned possible reasons of deviations. Eq.(7) was

hen modified into:

2L = CL

1

λL

Re′L

Re′TP

(17)

It should be stated that this modification ofφ2L is com-

letely empirical and, in this case,CL is an average facto

he ASTM D 1092, at each flow rate, in the Hagen–Poisequation for laminar flow[24]. This procedure leads to erreous predictions of pressure drop, for instance in a m

ude of ((3n+ 1)/4n)n when using the power law model. Ohe other hand, Cho et al.[19] proposed a calculation procure based on the extension that Metzner and Reed[25] maden the Rabinowitsch–Mooney equation to estimate theure drop of lubricating grease flow in a large pipe, calculapower-law viscosity but also using the Hagen–Poise

quation for laminar flow, what implies a much lower prure drop prediction, i.e. in a magnitude of (3n+ 1)/4n. The


Fig. 8. Comparison of experimental and predicted log(−�PD/4L) vs.log(8u/D) curves obtained for individual pipelines from piping flow andrheometric measurements with (a) smooth and (b) rough surfaces, once cor-rected air entrainment effects.

concordance of these simple procedures with some experi-mental results[24] may be an eventual consequence of wallslip and/or air entrainment effects above described, whichsignificantly lower the experimental measurements of pres-sure drop.

Fig. 9. Friction factor vs. generalised Reynolds number plots for the lubricati deusing the flow behaviour obtained from rough plate–plate geometries; (b) rou plageometries and (c) smooth pipelines, determined using the flow behaviour o

3.4. Friction factor–Reynolds number relationship

Fig. 9 shows the friction factor versus Reynolds numberplots calculated by following the strict theoretical treatment,based on the friction factor defined by Fanning’s equation.Using the suitable pressure drop expression for the lubricatinggrease, after correcting the air bubble effects, the frictionfactor for the homogeneous system can be determined.

Fig. 9a shows the friction factor–Reynolds number corre-lation for grease flowing along smooth pipes in laminar andsteady regimen. Experimental pressure drop data comprise alldiameters and flow rates tested. The viscous flow behaviourapplied in this case was obtained from steady-state rheo-logical measurements using rough plate–plate geometries,in absence of wall slip effects. As expected, the use of thisflow behaviour to predict the friction factor–Reynolds num-ber relationship for greases flowing along smooth pipelinesleads to values of the friction factor lower thanf= 16/Re′as a consequence of wall slip. On the contrary, when thesame flow behaviour is used to predict this relationship inrough pipes the results fit much better the theoretical values(seeFig. 9b). In addition to this, very good correlation wasattained applying the flow behaviour obtained from steady-state rheological measurements using smooth plate–plate ge-ometries to determine this relationship in smooth pipes (seeF anceo rox-i easefl , byu cor-r encewt lv l slipi ore-o s not

ng grease flowing in the laminar regimen along (a) smooth pipelines,terminedgh pipelines, determined using the flow behaviour obtained from roughte–platebtained from smooth plate–plate geometries.

ig. 9c). Therefore, these plots put in evidence the importf using viscous flow models that takes into account app

mately the same extent of wall slip that shown by the growing along the pipeline. This fact occurs, in this casesing smooth plate–plate gaps higher than 1 mm, whichesponds to the critical gap from which no gap dependas detected with smooth geometries (seeFig. 1). In addi-

ion to this, the deviation fromf= 16/Re′ when using the reaiscous flow behaviour of the grease in absence of wals a measure of the extension of wall slip in pipelines. Mver,Fig. 9shows once again that pipeline diameter doe


exert any influence on wall slip effect in the diameter rangestudied.

In contrast to other results found in the literature[19],the expressionf= 16/Re′ may be used to theoretically pre-dict pressure drop gradient of lubricating greases and designcontinuous pumping system. However, its use in the deter-mination of two-phase pressure drop gradient, i.e. air/greasesystems, must be carefully considered since pressure dropobtained from friction factor needs to be modified attendingthe treatment for the two-phase flow previously shown.

3.5. Correction of wall slip effect

Previous studies[3,7,11]put in evidence the existence ofwall slip during the flow of lubricating greases. Traditionally,the analysis to correct the wall slip effect has been based onthe different viscous flow behaviour found modifying diam-eters or gap widths[16,17]. However, as previously shownin this case, diameter does no exert a significant influenceon the magnitude of wall slip (seeFig. 4) in the range stud-ied. On the other hand, significant differences were found inboth rheometer and pipelines using geometries with differentrelative roughness, susceptible to be corrected with a similarprocedure. Thus,Fig. 10shows the flow curves obtained em-ploying several pipes with the same length and diameter butd aluesi t thee

d byM on[ alll h ord canb thes ve-la locityi ss ncesf ers,t ian

F ss fora

fluids. These authors proposed a well-known expression forthe slip velocity, which is a function of wall shear stress anddiameter. Similarly, a modified expression based on the rela-tive roughness instead of diameter is proposed in this case:

uS = CDτW

ε/D(19)

whereCD is called the slip coefficient.Considering Eq.(19) in the classical Rabinowitsch–

Mooney treatment[12] and dividing byτW, the continuityequation can be rewritten in terms of the flow characteristicas follows:

8u/D

τW= 8CD

ε+ 4

τ4W

∫ τW

0τ2

rxγ dτrx (20)

The quantity on the left hand side of Eq.(20)is the recipro-cal of the viscosity for a Newtonian piping flow, often calledthe apparent fluidity[12]. The value of the integral term in thisequation is only a function of the wall shear stress. Thus, inthe absence of wall slip, the term 8u/DτW is only a function ofτW. However, if slip occurs, this term may change for differ-ent values ofε at the same value ofτW. It is evident from Eq.(20) that slippage is greater in tubes with smaller roughness,for a given value of wall shear stress. Consequently, it canbe seen from this equation that the effect of slip decreases as� ness,f

byp df eada ares 1n itha g-g df , us-i rrect

ifferent roughness. As it can be observed, the stress vncrease with the relative roughness. This indicates thaxtension of wall slip depends on this parameter.

In the following treatment, as in the classical proposeooney[13] and applied afterwards by Yilmazer and Kaly

17] and Kokini and Dervisoglu[38], was assumed that wayer thickness is very small compared to the gap widtiameter. The high velocity gradient over this thicknesse treated by a discontinuity in velocity at the wall wherelip velocity,uS, is defined as the difference between theocity of the wall and that of the fluid at the wall[16]. It is alsossumed that once steady state is achieved, the slip ve

s a function of stress. Jastrzebski[15] used the principleuggested by Mooney and Oldroyd, based on the differeound for laminar flow through pipelines of several diameto study the wall effects in the piping flow of non-Newton

ig. 10. Pipeline flow curves obtained using different relative roughneconstant diameter of 27.86 mm.

increases, and becomes negligible at very large roughor the same diameter or gap width (seeFig. 10).

The contribution of slip phenomenon was calculatedlotting (8u/D)/τW versus 1/ε at constantτW as deduce

rom Eq.(20). The experimental flow characteristic was rt constantτW for each relative roughness. These plotshown inFig. 11 for data set obtained for pipelines of′′ominal diameter. The experimental data fit fairly well wlinear trend (R2 > 0.990). The linearity of the plots su

ested that corrected slip coefficient,CD, may be estimaterom the slopes of these plots, at each wall stress valueng linear regression and therefore, may be used to co

Fig. 11. Apparent fluidity vs. 1/ε plots: estimation of slip coefficient.


the contribution of slip to the piping flow. Identically to thatfound in many other systems[17,38], CD appears to be de-pendent on wall shear stress. Log–log plot of the slip velocityversus wall stress was linear for eachε/D value, being the slipvelocities related toτW as follows:

uS = 3.810−26τ7.32W

ε/D(21)

The corresponding corrected flow characteristic, (8u/D)C,was then calculated:(

8u

D

)C

= 8u

D− 8uS

D(22)

and the corrected shear rate, (γW)C may be also calculatedthrough the standard Rabinowitsch–Mooney equation, as fol-lows:

(γW)C =(

8u

D

)C

[3

4+ 1

4

d ln(8u/D)ClnτW

](23)

This estimation procedure results in the calculation of thenon-slip flow curve from data obtained with smooth surfaceswhich is in concordance with that obtained from pipelineswith rough internal surfaces, as shown inFig. 4.

4. Conclusions

linefl h asw kesi itingv rain-m d fori ablet e tot flowr arriedo tio. Ina rela-t slip.C linefl n thel r an sured them e-h g thed ,s es-s r

A

822,s ac-k

Appendix A. Relationship between wall shear rateand the flow characteristic

The relationship between shear rate at the wall,γW, andthe flow characteristic, 8u/D, is derived from the solution ofthe motion equation for steady laminar piping flow and thedefinition of viscosity[39].

In this way, the velocity distribution,u(r), (Eq.(24)) andthe non-dimensional velocity profile (Eq.(25)) in a laminargrease flow for Sisko model can be obtained in the form ofaxial velocity and average velocity ratio:

u(r) = RγW

{1 − η∞γW

τW(ξ2 − Xξn+1)

−(

η∞γW

τW

) ∫ 1

ξ

(ξ + Xξn) dξ

}(24)

u(r)

u= 8

[(1 + n + 2nX) − (n + 1)ξ2 − 2nXξn+1

][(n + 1)(1+ X)G(n,X)]

(25)

where

ξ = γ

γW(26)

be-t gc

γ

a n

F r ratef

Prediction of pressure drop occurring during the pipeow of lubricating greases faced several problems sucall slip and air entrainment. The Sisko model, which ta

nto account the tendency to reach a high-shear rate-limiscosity, can be used to predict pressure drop. Air entent effect may be corrected using the drag ratio define

ntermittent two-phase flow. However, expressions availo calculate the drag ratio must be modified probably duhe extremely high viscosity of greases and the very lowates achieved. Nevertheless, further studies must be cut to better understand these influences on the drag raddition to this, a new expression based on the internal

ive roughness of pipelines was proposed to correct wallorrecting these two effects from the experimental pipeow measurements, in contrast to other results found iiterature, the classical definition of the friction factor foon-Newtonian fluid can be applied to predict the presrop of grease in the laminar regime or, even, quantifyagnitude of wall slip by introducing the correct flow baviour obtained in rotational rheometers and observineviations from the expressionf= 16/Re′

L. In addition to thisuch deviations (i.e.f= 12/Re′) may be used to estimate prure drop in smooth pipelines, once corrected with factoφ2

L.

cknowledgements

This work is part of the research project PPQ2001-2ponsored by CICYT (Spain). The authors gratefullynowledge its financial support.

Once obtained the velocity profile, the relationshipween γW and 8u/D is easily deduced from the followinonsideration:

˙ = −∂u(r)

∂r(27)

Assuming the non-slip condition at the wall, i.e.u(R) = 0,nd substitutingu(r) for Eq.(25), with the boundary conditio

ig. A1. Flow chart for the procedure adopted to predict the wall shearom a fixed value of the flow characteristic using the Sisko model.


γ = γW at r =R, this relationship is determined.

γW = (8u/D)

G(n,X)(29)

Some difficulties to calculateγW were found by ap-plying the Sisko for which an iterative treatment, i.e.Newton–Rhapson method, and/or available computer pro-grams, are necessary (seeFig. A1).

Appendix B. Nomenclature

A total cross-sectional area of the pipe (m2)AG cross-section of the pipe occupied by the gas phase

(m2)AL cross-section of the pipe occupied by the liquid

phase (m2)CD slip coefficient (m (Pa s)−1)CL correction factor for the drag ratioD pipeline diameter (m)f friction factor based on Fanning’s equationk′ Metzner–Reed modified power law consistency co-

efficient (Pa sn)L pipe test section length (m)m sn

nnP )−− ing

− ure

QQ f the

Q of

rRRR ase

R d on

uu sed

( les

uuuuu

u(r) local axial velocity (m s−1)8u/D flow characteristic (s−1)−dux/dr velocity gradient (s−1)VG volume of pipe occupied by the gas phase (m3)(VG)1 gas volume incorporated by the pump at room con-

ditions (m3)(vG)i volume occupied by individual bubbles of air at op-

erating conditions (m3)(VL)1 volume of grease by the pump at room conditions

(m3)VL volume of pipe occupied by the liquid phase (m3)

Greek lettersα gas void fractionγ shear rate (s−1)γW shear rate at the wall (s−1)ε roughness (m)ε/D relative roughnessη apparent viscosity (Pa s)η∝ high shear rate limiting viscosity (Pa s)λL input volume fraction of liquid phaseµ Newtonian viscosity (Pa s)ρ specific gravity (kg m−3)ρL specific gravity of liquid phase (kg m−3)(ρ ) specific gravity of air at 1 bar (kg m−3)(τ

τ

φ

(

R

ohlssex,

In-

i-

olugh

tedand24

ers: its1995)

sient53–

t-960)

consistency parameter for the Sisko model (Pa)flow index for the Sisko model

′ slope of plot ln(8u/D) versus ln(τW) at each pointave average pressure in the test section length (Pa�P pressure drop due to friction (Pa)�PL pressure drop due to friction for liquid phase flow

alone at velocityuSL (Pa)�PTP pressure drop due to friction for two-phase mixt

(Pa)volume flow rate (m3 s−1)

G volume flow rate of gas phase at the entrance otest section (m3 s−1)

L volume flow rate of liquid phase at the entrancethe test section (m3 s−1)radial distance (m)radius of pipe (m)

e′ generalised Reynolds numbere′

L generalised Reynolds number of the liquid phbased onuSL (m s−1)

e′TP two-phase generalised Reynolds number base

mixture velocity (uSG+uSL) (m s−1)average velocity (m s−1)

G average velocity of the entrained air bubbles (baonAG) (m s−1)

uG)i average velocity of individual entrained air bubb(m s−1)

M mixture velocity (m s−1)r relative velocity (uG −uL) (m s−1)S slip velocity (m s−1)SG superficial gas velocity (based onA) (m s−1)SL superficial liquid velocity (based onA) (m s−1)

G atmρG)P specific gravity of air at pressureP (kg m−3)

shear stress (Pa)W shear stress at the wall (Pa)2L drag ratioφ2

L)c corrected drag ratio

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Documents

Experimental study of grease flow in pipelines: wall slip and air entrainment effects