Transcript
Page 1: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371DOI 10.1007/s00397-014-0759-1

ORIGINAL CONTRIBUTION

Optimization of the vane geometryApplications to complex fluids

Aminallah Rabia · Samir Yahiaoui ·Madeleine Djabourov · Francois Feuillebois ·Thierry Lasuye

Received: 18 June 2013 / Revised: 14 January 2014 / Accepted: 15 January 2014 / Published online: 8 March 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract The use of nonstandard geometries like the vaneis essential to measure the rheological characteristics ofcomplex fluids such as non-Newtonian fluids or particle dis-persions. For this geometry which is of Couette type, there isno analytical simple model defining the relation between theshear stress and the torque or relating the angular velocity tothe shear rate. This study consists on calibrating a nonstan-dard vane geometry using a finite volume method with theAnsys Fluent software. The influence of geometrical param-eters and rheological characteristics of the complex fluidsare considered. First, the Newtonian fluid flow in a rota-tive vane geometry was simulated and a parametric modelis derived therefrom. The results show an excellent agree-ment between the calculated torque and the measured one.They provide the possibility to define equivalent dimen-sions by reference to a standard geometry with concentriccylinders where the relationships between shear stress (resp.shear rate) and the torque (resp. the angular rotation) areclassical. Non-Newtonian fluid flows obeying a power lawrheology with different indices were then simulated. Theresults of these numerical simulations are in very good

A. Rabia (�) · S. Yahiaoui · M. DjabourovESPCI ParisTech - Laboratoire de Physique Thermique, 10 RueVauquelin, 75231 Paris Cedex 5, Francee-mail: [email protected]

S. Yahiaouie-mail: [email protected]

F. FeuilleboisLIMSI, UPR 3251 - CNRS, BP 133, Bat 508, 91403 Orsay Cedex,France

T. LasuyeINEOS ChlorVinyls, Chemin des Soldats, 62670 Mazingarbe,France

agreement with the preceding Newtonian-based model insome ranges of indices. The absolute difference still under5 % provided the index is below 0.45. Finally, this study pro-vides a calibration protocol in order to use nonstandard vanegeometries with various heights, gaps, and distance to thecup bottom for measuring the rheology of complex fluidslike shear thinning fluids and concentrated suspensions.

Keywords Vane geometry · Complex fluids ·Non-Newtonian · Shear thinning fluids

Introduction

The vane in cylinder geometry has been used for threedecades in view of characterizing the rheology of complexfluids. The interest in this nonconventional geometry firstlycame from the possibility that the vane would avoid wallslip effects which are currently observed with some flu-ids in standard Couette geometry. For example, a wall slipmay disturb the characterization of the elastic properties andyield stress of concentrated dispersions. Earlier works, see,e.g., Nguyen and Boger (1983), Nguyen and Boger (1985),and Keentok (1982), were concerned with the procedure tocharacterize the yield stress using a vane geometry with acontrolled rotational speed and recording the torque versustime until a maximum torque was obtained. It was assumedthat the region of the suspension close to the edges of thevane blades deforms elastically (that is, the torque varies lin-early with time) due to the stretching of the bonds (Nguyenand Boger 1983; Sherwood and Meeten 1991). The vanegeometry gained an increasing interest in food research(Roos et al. 2006; Krulis and Rohm 2004) and concentratedparticle suspensions (Rabia et al. 2010), and it is generally afamiliar tool in soil mechanics. Apart from the reduction of

Page 2: Optimization of the vane geometry

358 Rheol Acta (2014) 53:357–371

wall slippage, the vane tool has other practical advantagesas follows: the insertion of the geometry inside a concen-trated suspension minimizes the disturbance of the internalstructure of the suspension which recovers its cohesion afterthe vane has been immersed in the medium (thixotropicfluids). A standard use of the vane geometry requires a pre-cise description of the stress and strain factors (Barnes andCarnali 1990; Derakhshandeh et al. 2010; Estelle and Lanos2012; Fisher et al. 2007; Boger 2013), like for the classicalgeometries (Couette, cone-plate . . . ) (Ovarlez et al. 2011).

The analogy between the vane geometry and a Couettedevice has been examined in several papers. The numberof blades, arranged at equal angles, is not the same in alldevices: Savarmand et al. (2007) used a six-blades vane,while Nguyen and Boger (1983) and Ait-Kadi et al. (2002)used four blades and Sherwood and Meeten (1991) usedtwo. The importance of the number of blades was usuallynot examined. The blades were considered to be equiva-lent to an inner cylindrical surface, and some authors simplyassumed that the diameter of this inner cylinder is equal tothe vane diameter. This is in contradiction with the experi-mental determination made by Keentok (1982) who foundcorrections as large as 10 % of the radius, depending onthe material properties. The shear rates of the liquid atthe bottom and the top were taken into account by addinga corrected length to the height, assuming a power lawdependence of the stress with the radial position r (Nguyenand Boger 1985) for the yield stress measurement. Their

Fig. 1 A 3D view of the vane in cup geometry with six blades. H vaneheight, z vane immersion, S1 wall surface of the cup, S2 bottom surfaceof the cup, d distance to the cup bottom, R1 vane radius, R2 cup radius,and e gap

Fig. 2 Isometric view of the used mesh for “Geo20-30-5”

corrections are only valid for their specific vane and cupdimensions. In the recent publication by Savarmand et al.(2007), a numerical method clearly showed the end effectsof the finite length of the vane and results were testedagainst experimental data. The correction to the vane lengthderived from the simulations and experiments depends onmaterial properties, like the index for a power law fluid orthe yield stress for a Bingham fluid (Potanin 2010).

Savarmand et al. (2007) performed numerical simula-tions on a vane geometry in a cylindrical cup, with anarrow gap. This choice of the gap does not always corre-spond to the experimental conditions used for characterizingcomplex fluids such as particulate suspensions. They simu-lated experiments on various types of fluids (silica particlesuspensions, Xanthan solutions, or a Newtonian oil) and

Fig. 3 (Color online) Middle hight cutting plane (z = 15 mm) show-ing isovalues of the velocity magnitude for e = 5 mm with ω =1rad/s

Page 3: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371 359

Fig. 4 (Color online) Top and side cutting planes showing isovaluesof the velocity magnitude for a gap e = 5 mm and a distance to thebottom of geometry d = 5 mm and ω = 1rad/s

introduced end-effect corrections of the effective height ofthe vane which depend on the fluidity index of the solutionsto be characterized.

The aim of this paper is first to provide a practicalparametrization and optimization of the vane geometry,based on a rigorous numerical calibration for Newtonianfluids. We then demonstrate that the wide gap parametrizedvane geometry provides a reliable tool for the measurementof the rheology of non-Newtonian liquids and power lawfluids under certain conditions.

It is natural to start with a Newtonian fluid. We performboth a systematic set of experiments and for each case theflow field is calculated with 3D computer fluid dynamicssimulation using the finite volume method (FVM).

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

140

Fig. 5 Dimensionless torque versus immersed height for differentgaps e, and for d = 0.5

0 0.2 0.4 0.6 0.8 140

60

80

100

120

140

Fig. 6 Dimensionless torque versus gap in total immersion (H = 3)for various values of the distance d

The free surface and end effects due to the bottom ofthe blades are taken into account. The influence of the dis-tance between the blade and the bottom of the cup is studiedsystematically. By analogy with a Couette geometry, wethen deduce the equivalent height Heq and the equivalentradius Req of the vane normalized by the geometrical radiusof the vane. Recommendations for optimization of proce-dures are derived from the comparison between theory andexperiments.

We then consider non-Newtonian fluids. The analogyestablished with a Couette device first allows to obtain thestress and the strain factors for any configuration of the vaneand the cup. We then analyze the discrepancy obtained whenusing the vane with a non-Newtonian power-law fluid andspecify the range of validity of the Newtonian approxima-tion using a suitable definition of the average distance insidethe gap.

0 0.5 1 1.5 2 2.5 30

50

100

150

Fig. 7 Interpolation of the total torque versus immersion for a gape = 0.1

Page 4: Optimization of the vane geometry

360 Rheol Acta (2014) 53:357–371

Table 1 Linear interpolations of the numerical values of the torque versus immersion for various gaps and for d = 0.5, allowing �1/z to becalculated

Linear fit (Slope=�1/z) R − square SSE

e = 0.1 45 z + 4.978 1 0.2817

e = 0.2 29.51 z + 5.283 1 0.05873

e = 0.3 23.19 z + 5.046 1 0.06491

e = 0.4 19.83 z + 4.947 1 0.06979

e = 0.5 17.78 z + 4.936 1 0.0871

e = 0.6 16.42 z + 4.984 0.9999 0.1196

e = 0.7 15.41 z + 5.061 0.9999 0.1537

e = 0.8 14.74 z + 5.134 0.9999 0.1925

e = 0.9 14.16 z + 5.23 0.9999 0.2575

e = 1 13.61 z + 5.348 0.9998 0.3678

R − square and sum of squares due to error (SSE) indicate the precision of these interpolations

Model of the vane for newtonian fluids

Definition of the equivalent Couette device

The purpose of the modeling is to provide a calibration ofthe vane geometry. By analogy with a Couette device, theequivalent inner radius R1eqv and height Heqv are calcu-lated.

If we note the shear stress form factor Fγ and theshear rate factor Fσ of the geometry, they are, respectively,defined as the shear rate over the angular velocity ω and theshear stress over the torque � by the following relations:{γ = Fγ · ω : shear stressσ = Fσ · � : shear rate

As the viscosity η is given by the ratio σ/γ , the relationshipbetween the vane rotational velocity and measured torquemay be written as follows:

ω= η

(1)

This relationship is exact for a Newtonian fluid. For a coax-ial cylinders geometry with a large gap, these factors arederived as for instance, Couarraze and Grossiord (2000):1

Fγ = R22 + R2

1

R22 − R2

1

and Fσ = R22 + R2

1

4πR22R

21H

(2)

where R1 and R2 stand here for the inner and outer cylinderradius, and H is the height of the cylinders.

End effect are not considered in these relationships. Inthe case of a vane geometry (see Fig. 1 where the usualnotations are displayed), the immersed height is replaced byan equivalent immersion (effective immersion) zeqv and theradius of the mobile cylinder is replaced by the equivalent

1The shear stress factor is calculated for a mean stress between theouter and inner radius: σ = σ(R1)+σ(R2)

2

radius R1eqv , so that Eq. (1) may be written in the followingform:

ω= η

4πR22R

21eqv

R22 − R2

1eqv

zeqv (3)

The equivalent immersion may also be written as zeqv =Heqv −H + z with Heqv the equivalent height of the vane.

Equation (3) is made dimensionless by normalizing thetorque with η ωR3

ref , where the reference length Rref istaken as the geometrical radius of the blades (Rref =Rvane). Equation (3) then becomes:

� = 4πR22R

21eqv

R2ref

(R2

2 − R21eqv

)zeqvCCC (4)

where the overbar (�, z, . . . ) denotes dimensionlessquantities.

Model equations

Let us consider a Newtonian fluid with a viscosity η. Forthe fluid velocities are very slow or the viscosities arevery large, inertial forces are small compared with vis-cous forces (i.e., the Reynolds number2 is low compared tounity). Stokes equations apply with appropriate boundaryconditions for the fluid flow:

{∇p = η∇2u

∇ · u = 0(5)

2In the experiments presented in section “Experimental validation ofthe numerical model for a Newtonian fluid,” the Reynolds numberrange is 0.0025 � Re � 0.04 and the Froude number is Fr = 10−3

Page 5: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371 361

The boundary conditions for the velocity are as follows:

⎧⎪⎪⎨⎪⎪⎩

uwall = 0 : No-slip on the outer cylinderur = uz = 0; uθ = rω : No-slip on the surface of the

bladesuz = 0 : At the free surfacez = H

(6)

where uθ and uz denote the components of the fluid veloc-ity u in a cylindrical coordinates system attached to theouter cylinder and p is the pressure. A representation of theboundary conditions is displayed in Fig. 1. The torque onthe blades has two main contributions:

1. Normal stress (essentially the pressure on blades) on themain part of the blade surface

2. Tangential viscous stresses on the tip of the blades

For our configuration, the torques on the set of blades andon the cup have opposite values. It is then more convenientto obtain the torque by surface integration on the cup S sincethe viscous stresses there are more regular, so that a standardmesh is sufficient:

� =∫Sr ∧ σv · ndS (7)

where r = rer is the position vector from the axis, n is aunit vector normal to the surface and pointing here into thefluid, and σ v is the deviatoric (viz. the viscous) part of thestress tensor for a Newtonian fluid:

σ = 2ηD (8)

with D = 12

(∇u+∇T u)

the rate of strain tensor. The sur-face S on which the viscous stresses apply is composed oftwo parts: the lateral surface S1 and the one at the bottomof the cup, S2. At its top, the fluid is in contact with air.The interface is assumed to be horizontal since the capillarynumber (ratio of viscous to surface tension forces) is usu-ally low. From the zero tangential stress condition, there isthen no contribution to the torque.

The total torque is evaluated by its axial component (zcomponent) and can be written as follows:

� =∫S1

rσvdS +∫S2

rσvdS (9)

It will be seen later (from the data in Fig. 5) that thetorque varies linearly with the immersion height z. In thelimit z = 0, the only contribution to the torque arises fromthe end-effect due to the bottom (surface S2), which is thesecond term in Eq. (9): �2 = ∫

S2rσvdS.

The first term is the lateral contribution (surface S1) dueto the shearing inside the gap. It can be written as follows:

�1 =∫S1

rσvdS = z

∫s1R2σvdl = z�1/z

�1/z is the derivative of the torque versus immersed height,or torque per unit of immersed height. Here, dl is a linearelement of the circumference delimiting the flow field.

The normalized total torque can thus be written as fol-lows:

� = z�1/z + �2 (10)

Numerical results and discussion for newtonian fluids

The Stokes equations and associated boundary conditionsfor the steady flow field were solved by the finite vol-ume method (FVM) using a commercial software ANSYS-Fluent. The used spacial discretization scheme is theUPWIND scheme, whereas the computation was performedusing the iterative scheme: “semi-implicit method for pres-sure linked equations.”

0 0.2 0.4 0.6 0.8 10.92

0.925

0.93

0.935

0.94

0.945

0.95

0.955

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

a

b

Fig. 8 Equivalent dimensions of vane geometries compared to cylin-drical Couette. a Shows the equivalent radius. b [Heqv −H ], where His the dimensionless geometrical height of the vane. Tables 10 and 11in Appendix A shows the data points coordinates

Page 6: Optimization of the vane geometry

362 Rheol Acta (2014) 53:357–371

Table 2 Values of equivalent dimensions for typical dimensions forwhich Rref = Rvane = 10mm

Gap Bottom R1eqv Heqv

e(mm) distance d(mm) (mm) (mm)

Geo20-22-1 1 1 9.51 32.18

Geo20-30-1 5 1 9.32 37.25

Geo20-36-1 8 1 9.26 39.21

Geo20-22-5 1 5 9.51 30.57

Geo20-30-5 5 5 9.32 32.34

Geo20-36-5 8 5 9.26 33.01

All the geometries and meshes mentioned in this articlewere designed using ANSYS Workbench (ANSYS Design-Modeler to design geometries and ANSYS meshing togenerate the mesh). A mesh sensitivity study has been per-formed before starting computations to check the meshindependency and to choose the best element type.

The solution convergence was obtained with residuals of10−6 and the integral in Eq. (7) was calculated in Fluentwith an accuracy of 10−6 for various values of the normal-ized immersion height z, gap width e, and distance d to thebottom of the cup.

Results with equivalent dimensions are presented for arange of parameters appropriate to experiments. We arepresented then propose interpolation equations for handyoptimization of the vane geometry.

Numerical results: influence of geometrical parameters

A 3D FVM calculation was performed for each triplet ofvariables (e, d, z). Calculations were performed for a sixblade geometry rotating at ω = 1rad/s.

The mesh used in the case of geometry “Geo 20-30-5”3(triplet (0.5, 0.5, 3)) is shown in Fig. 2. It contains600,620 nodes and 2,332,202 elements. In the vicinity ofthe cup surface, a structured and progressive mesh is usedto better adjust the stress integration steps.

Figures 3 and 4 show the associated simulation results ofthe velocity field. It is noticed that a high-velocity gradientappears in the two figures at the tip of the blades and at thebottom of the vane.

Figure 5 show that the torque varies linearly with theimmersion z for a fixed distance from the bottom of the cupd = 0.5 and various values of the gap e. The values of the

3The notation ”Geo 20-30-5” here denotes a vane geometry with diam-eter d1 = 20mm, external cylinder diameter d2 = 30mm and distanceto the bottom of the cup d = 5mm.

Table 3 Summary of the experimental calibration of the vane rheome-ter for various geometries

Gap Distance R1eqv Heqv

e(mm) d(mm) (mm) (mm)

Geo20-22-1 1 1 9.43 ± 0.21 33.56 ± 3.56

Geo20-30-1 5 1 9.3 ± 0.48 37.06 ± 3.44

Geo20-36-1 8 1 9.63 ± 0.73 39.34 ± 6.49

Geo20-22-5 1 5 9.36 ± 0.2 32.62 ± 3.1

Geo20-30-5 5 5 9.39 ± 0.39 32.65 ± 1.7

Geo20-36-5 8 5 8.96 ± 0.58 33.95 ± 3.86

torque versus immersion are shown in Table 8 in AppendixA. According to Eq. (10), the value of �1/z is provided bythe slope of the straight line in Fig. 5.

In addition, the total torque depends on the distance tothe bottom of the cup, d . Values of the total torque ver-sus the gaps for different values of d are shown in Fig. 6for a totally immersed vane. It is clear that the total torquestrongly increases when the gap decreases (e → 0). Italso appears that d has a greater influence for larger gaps.Indeed, increasing the outer radius increases the bottom sur-face of the cup. All results of the torque versus gap areshown in Table 9 in Appendix A.

Interpolation of the torque versus the immersion

The previous numerical results of the torque versus z

are now fitted with linear interpolation using least square

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

60

70

Fig. 9 Validation of the numerical results for a specific Vane geometrywith e = 0.5 and d = 0.5. Experimental errors are shown: verti-cal error bars are related to temperature effects on glycerol viscosity,whereas the horizontal error bars are related to the position of theminuscus

Page 7: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371 363

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

β=1.051.11.31.52.233.84.65.46.277.88

± 5%

Fig. 10 Effect of the fluid index on the Newtonian approximation forvarious values of β = R2/R1 in a Couette device. The two dashedlines specify the range Fγ /FγN = 1 with a deviation of ±5 %

method. The slope gives the value of the lateral torque �1/z.Figure 7 shows the example for e = 0.1, and Table 1summarizes the whole interpolation results.

Again, we notice that the zero-intercept which representsthe bottom end-effect does not depend on the lateral gap e;the value remains in the limit of 4 % such as intercept =5.142 ± 0.206.

To avoid lengthy calculations, we did not simulate the fullimmersion of the vane geometry, although experimentallythis is often used. We did not include either in our simula-tions an upper fluid because we consider that the angularvelocity of the vane is small enough to neglect such surfaceeffects. We consider that our numerical results are valid forthee full immersion of the vane geometry.

Equivalent dimensions with reference to Couette device

By identifying Eq. (10) with the experimental expressionfor the torque (Eq. (4)), we may define equivalent dimen-sions of a given vane geometry for a Newtonian fluid. Theslope of the straight line representing � versus z determines

Table 4 Gap selection for non-Newtonian fluids. β = R2/R1

Fluid index β

0.45 < n � 1 For all β

0.35 < n < 0.45 β � 2

0.3 < n < 0.35 β � 1.4

0.25 < n < 0.3 β � 1.25

0.1 < n < 0.25 small gap

Table 5 Power law fluids with different values of the index n andconsistency M

Fluid number index n Consistency M[Pa.sn]

(1) 0.4 2.51

(2) 0.15 1.413

the equivalent radius and the intercept gives the equivalentheight.

Hence:

�1z =4πR2

2R21eqv

R2ref

(R2

2 − R21eqv

)

and

�2 = 4πR22R

21eqv

R2ref

(R2

2 − R21eqv

) (Heqv −H

)

Then one can deduce the equivalent dimensions:

R1eqv(e) =(

�1/z(e)+ 1

R22

)−1/2

(11)

Heqv(e, d) = H +(�(e, d)

�1/z(e)− z

)(12)

It is important to notice that the equivalent value of theradius depends only on gap. When the gap e increases, thetorque �1/z decreases and therefore the equivalent radius

Fig. 11 Isometric view of the geometrical mesh

Page 8: Optimization of the vane geometry

364 Rheol Acta (2014) 53:357–371

decreases. Figure 8a shows that the equivalent radius isslightly smaller than the geometric radius. This is due to therecirculation of the fluid around the singularities (limits ofthe blades) as observed in Fig. 3.

The equivalent height varies both with e and d (seeFig. 8b), so that for a fixed distance d = cst , the equiv-alent height increases with the gap. When the gap is fixedand the distance to the bottom of the cup (d) decreases, theequivalent height (Heqv) increases.

According to Eq. (12), we calculate and plot in Fig. 8bthe difference (Heqv − H) versus the dimensionless gapfor various distances to the bottom of the cup in the range(0.1, 1).

One can notice that the equivalent height is much moresensitive to changes in variables (e, d) than the equivalent

10-2

10-1

100

101

10-1

100

101

102

10-2

10-1

100

101

0

5

10

15

20

25

30

35

40

a

b

Fig. 12 Numerical validation of end correction for a power-law fluid:n = 0.4. a Log-Log scale, b semi-log scale

radius. Corrections for the equivalent radius are less than0.1, while corrections for the equivalent height vary between0.09 and 1.1.

As an example, Table 2 gives values of the equiva-lent dimensions for typical geometries for which Rvane =10 mm and height H = 30 mm.

In Table 2, we notice that the equivalent radius R1eqv issmaller than the geometric radius Rvane. This difference isexplained by the fact that the fluid between blades is notcompletely driven and can be considered as a rigid body.Figure 3 shows that the velocity stream lines are not concen-tric up to the tip of the vane so that the equivalent radius tocoaxial cylinders is smaller that the geometric vane radius.This was already observed and explained in previous studies(see for instance Ovarlez et al. (2006)).

10-2

10-1

100

101

102

10-2

10-1

100

101

102

10-2

10-1

100

101

102

0

10

20

30

40

50

60

70

80

a

b

Fig. 13 Numerical validation for a power-law fluid: n = 0.15. a Log-Log scale, b semi-log scale

Page 9: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371 365

Experimental validation of the numerical modelfor a Newtonian fluid

Experiment

In this section, the numerical predictions are compared withexperimental results on Newtonian fluids using AR 2000rheometer (TA Instruments) with various geometries.

The fluid is glycerol (with 5 % water in volume) witha mass density of d = 1.25g/cm3. Since glycerol ishygroscopic, we avoided exposing it to air for a longtime.

The viscosity of glycerol is very sensitive to temperaturechanges, viz. dη/dT for T in the range (20, 25)◦C is about−0.1 Pa s ◦C. A thermostated bath is used to maintain anuniform temperature in the sample. The water temperaturein the bath, controlled by a platinum resistance, was T =26±0.5 ◦C close to the ambient, and circulates in the jacketsurrounding the outer cylinder of the device. In these con-ditions, glycerol viscosity is η = 0.525 ± 0.05Pa.s. Thenthe torque relative error �

�= η

ηrelated to temperature

fluctuations is about 1 %.Measurements were carried out for ten immer-

sion values (z), for a combination of three gap widthse = (1, 5, 8) mm, and three distances to the bottom ofthe cup d = (1, 3, 5) mm. The vane was rotated veryslowly, thus allowing to apply the above numerical modelbased on quasi-stationary Stokes flow. The dimensionlesstorque was calculated using Eq. (4) and the viscosity η wasmeasured independently with a Couette device. Figure 15(Appendix B) shows an example of measured torque forω = 1rad/s using a gap of e = 1 mm and at a distanceto the bottom of the cup d = 5 mm. It is observed thatthe torque varies linearly with the immersion geometry,within the limits of experimental error. The straight lineshown in Fig. 15 (Appendix B) was fitted in the senseof least squares. The linear fitting formulae for variousvane geometries are displayed in Table 12 (see AppendixB). For example in Fig. 15: R1eqv = 9.36 ± 0.2 mm and

Heqv = 32.62 ± 3.1 mm. The errors on the values of theequivalent dimensions are related to deriving temperatureand to the measurement of the fluid immersions. Table 3summaries the results of the experimental calibration.

Comparison between numerical and experimental results

Figure 9 shows that the numerical and experimental resultsof the torque versus immersion height, for a vane geome-try of e = 5 mm, at distance d = 5 mm, are in very goodagreement, in the limit of experimental errors. It is empha-sized here that the numerical results contain no adjustableparameters. Similar results (not shown here) were obtainedwith the other geometries. It may be concluded thatour numerical study provides a precise modeling of thevane geometry for Newtonian fluids measurements withouthaving to perform lengthy experiments. The results ofSection “Equivalent dimensions with reference to Couettedevice” show that one can correct for gap and bottom dis-tance effects for a vane geometry by comparison with aCouette geometry with concentric cylinders using equiva-lent dimensions.

Applications of the vane geometry to non-Newtonianfluids

Optimization of the vane geometry in the case of complexfluids

In this section, the vane geometry is modeled in order to beused with non-Newtonian fluids. The choice of geometrygap is essential for non-Newtonian fluids, since the shearrate is highly dependent on this parameter. We consider anon-Newtonian power law fluid:

σ = η(γ ) γ = Mγ n (13)

η(γ ) = Mγ n−1 (14)

Table 6 Calculation of the viscosity at different shear rates for a power-law fluid with parameters n = 0.4 and M = 2.51Pa.sn using optimumparameters of the vane geometry

ω(rad/s) γ ηth �num ηappηappηth

(%) ηcorηcorηth

(%)

5.e−3 0.011 36.85 1.02e−5 30.06 18.4 35.35 4

1.e−2 0.023 24.31 1.37e−5 20.21 16.9 23.77 2.2

5.e−2 0.114 9.26 2.65e−5 7.82 15.5 9.19 0.6

5.e−1 1.38 2.32 6.67e−5 1.97 15.5 2.31 0.6

Where ω(rad/s) is the angular rotational velocity of the rotor, γ (s−1) the shear rate, �num(N.m) the calculated torque, ηcor (P a.s) the cal-culated viscosity including end-effect corrections, ηth(P a.s) the theoretical viscosity, and ηapp(P a.s) the apparent viscosity without end-effectcorrections

Page 10: Optimization of the vane geometry

366 Rheol Acta (2014) 53:357–371

Table 7 Calculation of viscosity versus shear rate for a power-law fluid with parameters n = 0.15 and M = 1.413Pa.sn

ω(rad/s) γ ηth �num ηappηappηth

(%) ηcorηcorηth

(%)

2e−3 0.014 53.08 1.58e−5 36.34 31.5 53.14 0.1

3e−3 0.021 37.61 1.7e−5 26.04 30.8 38.08 1.2

5e−3 0.035 24.36 1.84e−5 16.91 30.6 24.73 1.5

1e−2 0.07 13.51 2.04e−5 9.37 30.6 13.71 1.4

2e−2 0.14 7.5 2.25e−5 5.19 30.8 7.59 1.2

3e−2 0.21 5.31 2.39e−5 3.67 30.9 5.36 1

5e−2 0.35 3.44 2.58e−5 2.37 31.1 3.47 0.8

1 7.02 0.27 3.86e−5 0.18 34.1 0.26 3.7

Where ω(rad/s) is the angular rotational velocity of the rotor, γ (s−1) the shear rate, �num(N.m) the calculated torque, ηcor (P a.s) the cal-culated viscosity including end-effect corrections, ηth(P a.s) the theoretical viscosity, and ηapp(P a.s) the apparent viscosity without end-effectcorrections

where M is the fluid consistency and n < 1 the fluidityindex with concentric cylinders.

The shear rate at any position r in the gap of a Couettegeometry is, for a non-Newtonian fluid, as follows:

γ (r) = 2

n

[R2r

] 2n

[R2R1

] 2n − 1

ω (15)

Then the shear factor can be written as follows:

Fγ (r) = 2

n

[R2r

] 2n

[R2R1

] 2n − 1

(16)

where r between R1 and R2.Fγ depends on the fluid index n and on the inner to

outer radius ratio, meaning that the shear rate inside thegap depends both on the rheological parameters of the

10-2

10-1

100

101

102

103

0

0.2

0.4

0.6

0.8

1

Fig. 14 PEO viscosity measurements at a concentration 20 wt% instationary flow, using a vane geometry and a cone-plate standardgeometry

non-Newtonian fluid and on the radial dimensions of thecylinders. The stress factor remains unchanged, as calcu-lated for a Newtonian fluid, Eq. (1). In the case of large gapsbetween concentric cylinders, possible approximation of theshear stress is the average of the stress calculated betweenr = R1 and r = R2. This approximation then provides thedefinition of an average radius rav such as follows:

1

r2av

= 12R2

1+ 1

2R22

⇒ rav = R2R1

√2

R22+R2

1(17)

By keeping this definition for a non-Newtonian fluid withthe vane geometry, the average shear factor can be writtenas follows:

Fγav = 21− 1n

n

[1 + R2

2R2

1

]1/n

[R2R1

]2/n − 1(18)

The influence of the ratio β = R2/R1 and of the fluid indexn are evidenced by the ratio Fγav/FγN , where FγN is theshear factor for a Newtonian fluid (n = 1). Equation (18)can be written as follows:

Fγav

FγN

= 21− 1n

n(1 + β2)

1n−1

[β2 − 1

β2n − 1

](19)

Figure 10 shows how this ratio varies with n for vari-ous values of β (a similar plot was proposed by Ait-Kadiet al. (2002)). This plot highlights two different domainsaccording to the value of the index n:

– Minor changes of (Fγav/FγN ) versus n when n � 0.45– Significant changes of (Fγav/FγN ) when n < 0.45

It is important to notice that for n � 0.45, the ratioFγ /FγN is close to unity ±5 % (this interval is bounded by

Page 11: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371 367

the two dashed lines in Fig. 10). Therefore, it is reasonableto use the average stress factor for any fluid with an indexn � 0.45, whatever the gap value expressed with β. How-ever, for fluids with low index 0.1 � n � 0.4 the Newtonianapproximation is no longer valid in the ±5 % error limits.The choice of a small gap is essential in this case. Finally,to summarize this analysis, Table 4 suggests the most con-venient choice of the gap for different ranges of the fluidindex.

For more information, the reader can refer to theAppendix C where Fig. 16 shows the ratio Fγ /FγN , plot-ted versus R2/R1 for various values of the fluid index n.Figure 16b is an enlargement of the plots in Fig. 16a in theregion β < 1.4 for 0.1 < n < 0.3. This figure showsthat for small indices n < 0.3, the Newtonian approx-imation (Fγ /FγN = 1 with a deviation of ±5 %) isvalid only when R2/R1 is close to unity, i.e., the gap issmall.

For comparison, Bousmina et al. (1998) and Ait-Kadiet al. (2002) propose to calculate a value r∗ of r fornon-Newtonian fluids. The idea was to seek an optimumposition, r∗, where the shear rate is almost independentof n.

A particular case is the Newtonian approximation corre-sponding to the shear factor equal to that of a Newtonianfluid, i.e., Fγ (r

∗, n) = FγN . The particular result for r∗ isas follows:

r∗ = R2

⎡⎢⎢⎣1

n

R22

R21− 1

(R2R1

) 2n − 1

⎤⎥⎥⎦

12n− 2

(20)

r∗ determines the following shear factor:

F ∗γ (n) =

2

n

(R2r∗

) 2n

(R2R1

) 2n − 1

(21)

The shear factor FγN depends on the ratio R2/R1. Accord-ing to Ait-Kadi et al. (2002), their Couette analogy andNewtonian approach should be applied only in a certainranges of n and of the ratio R2/R1. The studies of Bousminaet al. (1998) and Ait-Kadi et al. (2002) state that a compro-mise has to be found for the radius ratio and n.: they proposethe radius r∗ = (R1 +R2)/2 in the limit of very small gapswhich is a classical result since it is the standard configura-tion encountered in commercial Couette geometries. Theyclaim also that it is possible to deal with large gap configu-rations under the condition to determine the stress and shearrate values at a particular location r∗ given by Eq. (8) of

their paper (Ait-Kadi et al. 2002). In the same paper, theygive an example of a very large gap configuration (see Fig.1 in Ait-Kadi et al. (2002)).

This choice is rather different from the average radiusthat we propose in Eq. (17) for the shear factor. Bycomparing the two approaches, we recall that in Fig. 10for a fluid index 0.45 < n � 1, our approximationis valid with a maximum error of 5 % compared to aNewtonian fluid, whereas in the study of Ait-Kadi et al.(2002), this variation is 12 %, for the same range of fluidindex.

To conclude, a gap β � 1.3 or lower may be usedto measure the viscosity of a non-Newtonian power-lawfluid with 0.2 < n � 1 with a ±10 % accuracy.If the fluid index is expected to be � 0.3, it is pos-sible to use a gap such as β � 1.4 a better accu-racy (±5 %). Therefore, we propose the following mea-surement protocol for an unknown fluid index (0.2 <

n � 1) with a Couette device or with a vane geometry:

Test protocol:– Choose the initial dimensions of the vane geometry

such as R2/R1 = 1.3 where the measurementis easy to perform.

– Derive from Fig. 8 the Couette equivalent dimensions ofthe vane in the context of a Newtonian fluid (see Section“Equivalent dimensions with reference to Couette device”above) and calculate the stress and shear factors,Eq. (2).

– Measure the rheological fluid index by varying theshear rate.

– If 0.2 < n < 0.3, the measurement of viscosity is accurateto ±10 %.

– If 0.3 < n � 1, the precision is better (±5 %) and awider gap may be used if needed using Table 4

Numerical calculation of the torque for power law fluids

In this section, we combine the method developedin Section “Experimental validation of the numerical modelfor a Newtonian fluid” and the protocol described inSection “Optimization of the vane geometry in the case ofcomplex fluids” to validate the approximations in the caseof power-law fluids. Consider for instance two fluids obey-ing the rheological model of Eq. (14) with constants shownin Table 5.

The protocol described in Section “Optimization of thevane geometry in the case of complex fluids” provides thebest vane geometry for each model:

– Fluid (1) with geometry “20-30-5”: 5 mm wide gap and5 mm from the bottom.

Page 12: Optimization of the vane geometry

368 Rheol Acta (2014) 53:357–371

– Fluid (2) with geometry “20-22-1”: 1 mm narrow gapand 1 mm from the bottom.

The fluid flow in the “Geo 20-22-1” vane geometry withsix blades is simulated in 3D with Ansys-Fluent software.Figure 11 shows the geometrical mesh which contains1,246,831 elements and 274,004 nodes.

The total torque was first calculated and the viscositywas derived from Eq. (3) using equivalent (effective) dimen-sions for the vane height and radius, which were determinedbeforehand by the Newtonian approximation (see Table 2).Figures 12 and 13 show a comparison between the rheolog-ical model and the numerically simulated values. It is clearthat when the end effects corrections are considered withthe equivalent dimensions, the results fit perfectly the the-oretical curve. However, when the geometrical dimensionsare used without end effects correction, a clear discrep-ancy appears that is greater than 15 %. One can see a goodagreement between the theoretical model and the approxi-mations used for the effective parameters of the vane. Foran index n = 0.4, the differences vary between 0.6 and 4 %with these approximations while they are larger than 15 %without corrections (see Table 6).

We also compare the results in the case of a very lowpower index n = 0.15 (fluid number (2)). Table 7 andFig. 13 give a comparison between the rheological data andthe simulated results.

Experimental validation of the vane geometry with a shearthinning polymer solution

A solution of high molecular weight (molar mass =5.106g/mol) PEO (polyethylene oxide) in water was cho-sen to validate our model of the vane geometry. The polymerwas dissolved in water at a concentration of c = 1wt%(mass percentage). The mixture was stirred for 1 week atroom temperature and then filtered with a 3 μm mesh sizefilter prior to measurements. We used the “Geo 20-30-5”vane geometry (with diameter = 20 mm, height = 30 mm

and distance d = 5 mm to the bottom of the cup). Accord-ing to Table 2 or to Fig. 8a, the equivalent Couette radius isR1eq = 9.32 mm, giving a radius ratio β = R2/R1eq = 1.6.The viscosity was measured with this vane rheometer forvarious shear rates, at a controlled temperature of 20 ◦C, andresults were compared to measurements performed with acone-plate geometry (diameter 6 cm, angle 2◦). This com-parison is shown in a semi-log scale in Fig. 14. It is seenthat the agreement is excellent between the two measure-ments for γ � 0.1, without any adjustment parameter orcorrection.

For shear rates (0.01 < γ < 0.1), the vane geometryis equivalent to the cone-plate geometry. For higher shearrates, the vane geometry is limited to γ = 100 s−1 (the

Reynolds number is close to 30 at 100s−1). It is important tonote that this comparison is made for validation, using flu-ids that are compatible with the cone-plate geometry. Thatgeometry appears to have a wider application for those flu-ids. Now, for fluids with a more complex structure, thecone-plate geometry would be inappropriate and the vanegeometry would show its superiority. Using the model ofa power law fluid provides a power index n = 0.49 when10 s−1 < γ < 500 s−1. The whole rheogram obeys theCross equation as follows:

η − η∞η0 − η∞

= 1

1 + aγ m(22)

with the following parameters:

– η0 = 0.989 Pa s: viscosity at zero shear rate– η∞ = 0.07 Pa s: viscosity at infinite shear rate– a = 0.212: consistency– m = 0.797: exponent

Conclusion

The aim of this investigation is to provide a rigorous pro-cedure for using the vane geometry in characterize non-Newtonian fluids. Because the vane geometry is neededto characterize complex fluids (particle suspensions, foodformulations . . . ), where the concentric cylinders or otherwell-known geometries (like the cone-plate geometry) areinappropriate, it is important to examine the various approx-imations which are introduced with this nonconventionaldevice.

We propose an analogy between the vane and Couettedevices, taking into account the contribution of the shearat the bottom of the vane (end effect) and the flow fieldaround the blades. The numerical 3D simulations of thevane geometry (with six blades) with Newtonian fluidsprovide an excellent validation of this analogy, confirmedby experimental tests. We propose therefore to includethese particular contributions in the “equivalent” dimen-sions of the vane geometry (equivalent radius and height) forNewtonian fluids. When non-Newtonian fluids are investi-gated, we propose to adapt the analytical formulae of theCouette for a large gap by defining the shear rate at asuitable “average” distance. We compare this approxima-tion with 3D simulations of power law fluids with the vanegeometry and find a very good agreement. This investiga-tion suggests a protocol for using the vane geometry in anysituation where nonconventional geometries are inadequateand provides the method of determining the rheograms witha very good accuracy.

Page 13: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371 369

Appendix A: Data results

Table 8 Total torque � data for different gaps e and different immersions z with distance to bottom d = 0.5

z\e 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 9.080 8.036 7.154 6.710 6.469 6.340 6.279 06.246 6.231 6.221

0.2 18.514 14.225 12.094 10.987 10.358 9.999 9.770 09.652 9.572 9.514

0.3 27.591 20.114 16.721 14.934 13.917 13.304 12.902 12.645 12.478 12.33

0.4 50.136 34.804 28.273 24.822 22.780 21.484 20.564 19.977 19.517 19.125

0.5 63.668 43.704 35.246 30.768 28.104 26.402 25.171 24.381 23.754 23.196

0.6 72.637 49.587 39.866 34.72 31.654 29.668 28.232 27.314 26.554 25.873

0.7 94.923 64.306 51.425 44.596 40.501 37.830 35.880 34.617 33.554 32.573

0.8 108.440 73.148 58.379 50.559 45.817 42.749 40.501 39.011 37.780 36.658

0.9 117.470 79.047 63.014 54.486 49.376 46.013 43.535 41.944 40.586 39.313

1 139.840 93.784 74.592 64.396 58.236 54.188 51.233 49.265 47.611 46.033

Table 9 Total torque � data for different gaps e and distances to the bottom cup d with immersion z = 3

d\e 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 147.26 102.42 83.102 73.081 67.083 63.152 60.372 58.504 57.075 55.819

0.2 142.77 97.249 77.777 67.77 61.781 58.01 55.008 53.116 51.681 50.351

0.3 141.09 95.212 75.878 65.736 59.674 55.698 52.828 50.927 49.423 48.126

0.4 140.31 94.208 74.977 64.845 58.79 54.707 51.786 49.839 48.333 47.007

0.5 139.9 93.808 74.582 64.388 58.229 54.175 51.226 49.254 47.678 46.287

0.6 139.65 93.701 74.439 64.176 57.978 53.907 50.93 48.918 47.318 45.925

0.7 139.37 93.601 74.293 64.067 57.844 53.748 50.731 48.707 47.111 45.608

0.8 139.22 93.527 74.231 63.991 57.775 53.659 50.614 48.582 46.926 45.464

0.9 139.14 93.497 74.189 63.945 57.728 53.598 50.538 48.496 46.851 45.342

1 139.11 93.508 74.174 63.926 57.69 53.558 50.497 48.424 46.709 45.172

Table 10 [Heqv −H ] data for different gaps e and distances to the bottom cup d . H is the geometric height of the vane

d\e 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1 0.2184 0.4094 0.5288 0.6364 0.7253 0.7992 0.8603 0.9213 0.9743 1.0096

0.2 0.1203 0.2372 0.3027 0.3721 0.4308 0.4899 0.5173 0.5601 0.5987 0.6169

0.3 0.0836 0.1694 0.2221 0.2709 0.3138 0.3508 0.3779 0.4134 0.4415 0.4570

0.4 0.0665 0.1360 0.1838 0.2266 0.2648 0.2912 0.3113 0.3405 0.3656 0.3766

0.5 0.0575 0.1227 0.1670 0.2039 0.2336 0.2592 0.2755 0.3013 0.3200 0.3249

0.6 0.0521 0.1191 0.1610 0.1933 0.2197 0.2430 0.2565 0.2788 0.2949 0.2989

0.7 0.0460 0.1158 0.1548 0.1879 0.2122 0.2335 0.2438 0.2646 0.2805 0.2762

0.8 0.0427 0.1133 0.1521 0.1841 0.2084 0.2281 0.2363 0.2562 0.2676 0.2658

0.9 0.0409 0.1123 0.1503 0.1818 0.2058 0.2244 0.2315 0.2505 0.2624 0.2570

1 0.0403 0.1127 0.1497 0.1809 0.2037 0.2220 0.2288 0.2457 0.2525 0.2448

Table 11 The equivalent radius depends only on the gap e (see Eq. (11))

e 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Reqv 9.5100 9.4479 9.39223 9.3498 9.3201 9.3011 9.2788 9.2615 9.2469 9.2319

Page 14: Optimization of the vane geometry

370 Rheol Acta (2014) 53:357–371

0 0.5 1 1.5 2 2.5 3 3.50

20

40

60

80

100

120

140

160

Geo 20-22-5

Experimental: glycerol at T=26± 0,5°C

Linear Fit

Fig. 15 Dimensionless torque (�) versus z with distance to bottomd = 0.5 and for a gap e = 0.1

Table 12 Linear interpolations of the experimental values of thetorque for various vane geometries where e is the gap and d thedistance to bottom cup

e[mm] d[mm] interpolation �

Geo20-22-1 1 1 42.168z + 15.015

Geo20-30-1 5 1 17.63z + 12.438

Geo20-36-1 8 1 16.337z + 15.274

Geo20-22-5 1 5 39.975z + 10.456

Geo20-30-5 5 5 18.216z + 4.8283

Geo20-36-5 8 5 13.429z + 5.3024

Appendix B: Linear fit of the experimental torqueresults

The relative errors on R1eqv and Heqv are then calculatedfrom the relative error on the torque measurement.

1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

n=0.10.20.30.40.50.60.70.80.91

1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

n=0.10.150.20.250.3

a

b

Fig. 16 Variation of Fγ /FγN with R2/R1 for various values of thefluid index n. The two dashed lines specify the range Fγ /FγN = 1with a deviation of ±5 %. Figure 16b is a enlargement of 16a

Appendix C: Variation of Fγ /FγN versus R2/R1

Page 15: Optimization of the vane geometry

Rheol Acta (2014) 53:357–371 371

References

Ait-Kadi A, Marchal P, Choplin L, Chrissemant AS, Bousmina M(2002) Quantitative analysis of mixer-type rheometers using cou-ette analogy. Canad J Chem Engen 80:1166–1174

ANSYS� (2012) Fluent—Academic Research Release 14.0, ANSYS,Inc

Barnes HA, Carnali JO (1990) The vane-in-cup as a novel rheome-ter geometry for shear thinning and thixotropic materials. J Rheol34:841–866

Boger DV (2013) Rheology of slurries and environmental impacts inthe mining industry. Annu Rev Chem Biomol Eng 4:239–257

Bousmina M, Ait-Kadi A, Faisant JB (1998) Determination of shearrate and viscosity from batch mixer data. J Rheol 43(2):415–433

Couarraze G, Grossiord JL (2000) Initiation la rhologie. Tec & DocLavoisier

Derakhshandeh B, Hatzikiriakos S, Bennington C (2010) The apparentyield stress of pulp fiber suspensions. J Rheol 54:1137–1154

Estelle P, Lanos C (2012) High torque vane rheometer for concrete:principle and validation from rheological measurements. ApplRheol 22:12,881

Fisher DT, Clayton SA, Boger DV, Scales PJ (2007) The bucketrheometer for shear stress-shear rate measurement of industrialsuspensions. J Rheol 51:821–831

Keentok M (1982) The measurement of the yield stress of liquids.Rheol Acta 21:325–332

Krulis M, Rohm H (2004) Adaption of a vane tool for the viscositydetermination of flavoured yoghurt. Eur Food Technol 218:598–601

Nguyen QD, Boger DV (1983) Yield stress measurement for concen-trated suspensions. J Rheol 27(4):321–349

Nguyen QD, Boger DV (1985) Direct yield stress measurement withthe vane method. J Rheol 29(3):335–347

Ovarlez G, Bertrand F, Rodts S (2006) Local determination of theconstitutive law of a dense suspension of noncolloidal particlesthrough magnetic resonance imaging. J Rheol 50(3):259–292

Ovarlez G, Mahaut F, Bertrand F, Chateau X (2011) Flows andheterogeneities with a vane tool: magnetic resonance imagingmeasurements. J Rheol 55:197–223

Potanin A (2010) 3d simulations of the flow of thixotropic fluids, inlarge-gap couette and vane-cup geometries. J Non-Newton FluidMech 165:299–312

Rabia A, Djabourov M, Feuillebois F, Lasuye T (2010) Rheology ofwet pastes of PVC particles. Appl Rheol 20:11961 (9 pages)

Roos H, Bolmstedt U, Axelsson A (2006) Evaluation of new methodsand measuring systems for characterisation of flow behaviour ofcomplex foods. Appl Rheol 16:19–25

Savarmand S, Heniche M, Bechard V, Bertrand F, Carreau PJ (2007)Analysis of the vane rheometer using 3d finite element simulation.J Rheol 51(2):161–177

Sherwood JD, Meeten GH (1991) The use of the vane to measure theshear modulus of linear elastic solids. J Non-Newton Fluid Mech41:101–118


Recommended