Direct Yield Stress Measurement With the Vane Method

Direct Yield Stress Measurement with the Vane MethodNguyen Q. Dzuy and D. V. Boger

Citation: Journal of Rheology (1978-present) 29, 335 (1985); doi: 10.1122/1.549794 View online: http://dx.doi.org/10.1122/1.549794 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/29/3?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in A slotted plate device for measuring static yield stress J. Rheol. 45, 1105 (2001); 10.1122/1.1392299 Uniaxial Compression of Bonded and Lubricated Gels J. Rheol. 29, 671 (1985); 10.1122/1.549821 Couette Viscometer Data Reduction for Materials with a Yield Stress J. Rheol. 29, 369 (1985); 10.1122/1.549818 Yield Stress Measurement for Concentrated Suspensions J. Rheol. 27, 321 (1983); 10.1122/1.549709 Relaxation of Shear and Normal Stresses in DoubleStep Shear Deformations for a Polystyrene Solution. A Testof the DoiEdwards Theory for Polymer Rheology J. Rheol. 25, 549 (1981); 10.1122/1.549650

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Direct Yield Stress Measurementwith the Vane Method

NGUYEN Q. DZUY* and D. V. BOGER,Department of Chemical Engineering, University ofMelbourne,

Parkville, Victoria, 3052, Australia

Synopsis

In the vane method for measuring the yield stress, the conventional analysisassumes that the stress is uniformly distributed on a cylindrical sheared surfaceto calculate the yield stress from the maximum torque and vane dimensions. Byusing two simple procedures, the present work shows that this assumption isjustified at the moment of yielding. The yield stress calculated using the proposedmethods compares favorably with that obtained with the conventional proce-dure. A comparison with the yield stress independently determined by othermethods again confirms the usefulness of the vane technique as a simple butaccurate method for direct yield stress measurement.

INTRODUCTION

The yield stress of concentrated suspensions can be determinedor measured by a large number of techniques.l "" Unfortunately,many of the existing methods are either tedious to perform orlimited in their applicability. Also, it is not uncommon to findthat, even for a given material, the yield stress values obtainedmay vary with the experimental conditions employed." With in-creasing interest in fluids with a yield stress in numerous indus-trial applications.Y' ' it is not surprising that considerable efforthas been made to develop better and more reliable techniques foryield stress rneasurement.l T':" Recently, in a critical review onthe subject;' we have demonstrated that the yield stress of floc-culated bauxite residue slurries (red mud) could be directly as-sessed using a simple vane method adapted from soil mechanics.The yield stress measured with the vane was shown to be in good

*Present address: Department of Physical Chemistry, University of Melbourne,Parkville, Victoria, 3052, Australia.

1985 by The Society of Rheology, Inc. Published by John Wiley & Sons, Inc.Journal of Rheology, 29(3), 335-347 (1985) CCC 0148-60551851030335-13$04.00


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336 DZUY AND BOGER

, ) N (rpm)

-,--,--

"" 0 .'

f-------i---DT ----------

~i , - - . :, ,

i - - I

H

--.-z

Fig. 1. Schematic diagram of a four-bladed vane in operation.

agreement with the results obtained by the more conventionalrheological methods. The applicability of the vane method hasalso been extended to other systems, e.g., concentrated suspen-sions of titanium dioxide, uranium oxide and brown coal.!" Fol-lowing this development, similar successes with the vane on sev-eral commercial greases have also been reported."

The principle of the vane method has been fully described else-where.l!'' Basically in this method, a four-bladed vane (see Fig-ure 1) immersed in a sample is rotated slowly at a constant rate todetect the yielding moment when the torque exerted on the vaneshaft reaches a maximum value. The presence of such a max-imum in the torque response is a characteristic of yield stressmaterials which can be explained by the concept of structuraldeformation and breaking of bonds in flocculated systems. I Calcu-lation of the yield stress from the measured maximum torquerequires knowledge of the geometry of the yield surface and theshear stress distribution on this surface,

The conventional approach employed in soil mechanics" as-sumes that the material yields along a cylindrical surface havingan area of -rrDH + 2(-rrD2/4), where D and H are diameter andlength of the vane, respectively, Also, the shear stress is assumedto be uniformly distributed everywhere on the cylinder, and equal


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YIELD STRESS MEASUREMENT 337

(1)

to the yield stress (T) when the torque is at maximum (Tm)' Withthese assumptions, a simple relationship between Ty , Tm' and thevane dimensions is obtained: 1,1l

TrD 3 (H 1)Tm = -2- ,D + 3 Ty Although the assumption of the cylindrical yield surface has beenconfirmed both experimentally3,1l- 14 and numerically.l" severalquestions have been raised concerning the validity of Eq. (1).Firstly, there is experimental evidence3,12 - 14 to suggest that thematerial may yield along a diameter CDs) that is larger than theactual diameter of the vane. Earlier studies by Wilson 12 and Ar-man et al.!" have shown that for soils the observed increase in theyield area was small and hence introduced insignificant errors tothe yield stress calculated using Eq. (1). Recently, Keentok eta1.3,14 have found that the ratio Ds/D can be as large as 1.05 forsome greases, and is apparently dependent on the plastic, thixo-tropic and elastic properties of the material. However, for inelas-tic and plastic substances the data published by these workershave suggested that Ds/D is very close to 1.0. Thus, until theindividual effects due to the fluid rheological behavior can beseparated, and the mechanisms involved are understood, the ac-tual yield area remains a matter for speculation. Furthermore, itis noted that all the published results I 2 - 14 were based on the ob-servations long after yielding had occurred.

The other reason for suspecting the validity of Eq. (1) is thatthe shear stress is not necessarily uniform everywhere on thecylindrical sheared surface. As briefly mentioned in the previousarticle,' the exact nature of the stress distribution has not beenfully understood. A recent attempt using finite elements for anideal plastic fluid has merely shown that the stress is highestnear the edge of the vane blades.l" It appears intuitively correctthat the stress is uniform along the side (or wall) of the shearedcylinder in slow flows, but may vary with radial position on eachof the two circular end surfaces. Theoretical determination of theend shear stress distribution, however, is a complex three-dimensional problem which should take into account the elasticbehavior of the material prior to yielding.

In the present investigation, no attempt is made to solve theproblem analytically. Instead, two simple analyses are proposed


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338 DZUY AND BOGER

(2)

to calculate the yield stress from the vane data and indirectlyexamine the uniform shear stress assumption at the moment ofyielding. In the first analysis, an approximate distribution for theend shear stress is assumed, whereas in the second method onlythe contribution of the wall shear stress to the total stress isconsidered. In both analyses the diameter of the cylindrical yieldsurface is assumed equal to the vane diameter. The results ob-tained are compared with the yield stress calculated using Eq. (1)as well as with the yield stress independently assessed by conven-tional techniques.

YIELD STRESS CALCULATIONS

Figure 1 is a diagram of a vane fully immersed in a suspensioncontained in a large beaker. The depth of immersion of the vane isdescribed by the distances Zl and Z2' It is assumed that the inter-ference caused by the solid boundaries (wall and bottom of thecontainer) is absent, and shearing due to the immersed section ofthe vane shaft is negligible.

When the vane is set in motion at a constant speed, the totaltorque (T) experienced by the shaft is the sum oiT; and 2T e due toshearing on the side and two ends of the cylindrical shear surface,respectively. Thus,

T = T; + 2Te

In terms of the shear stresses, it can be shown that 1

T = (; D2H )Ts + 2(2'TT r/2 Te r2dr ) ,where Ts is the constant shear stress along the side of the shearedcylinder, and Te is the end shear stress of unknown distribution. IfTe is assumed constant and equal to Ts> the solution for Eq. (2) isEq. (1) at the yielding moment, when T = Tm, and Ts = Te = Ty-

Method I

As an approximation, Te may be arbitrarily assumed to varywith radial position (r) according to a power relationship:"

Te(r) = (~')mD Ts> a ~ r~ D2' (3)


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(4)

where m is a constant describing the radial distribution functionof Te With Eq. (3), the solution for Eq. (2) is

T= TrD3(H + _l_)T2 D m + 3 s,

or in terms of Tm and Ty at yielding

TrD3 [H 1]Tm = 2" D + m + 3 Ty'

To determine both Ty and m simultaneously from experimentaldata, Eq. (4) may be rearranged in the following form:

2Tm _ H TyTrD3 - TYD + m + 3' (5)

(6)

Thus, if this method is valid, a plot of 2TmlTrD3 versus HID forvanes of different dimensions will be linear with a slope of Ty andan intercept of Tyl(m + 3).

Method II

The second proposed analysis makes use of the general torquebalance in Eq. (2) at yielding:

(Tr ) (Vl2

T m = 2 D 2Ty H + 4Tr Jo Ter2dr.

Thus, for vanes of the same diameter but different lengths, thesecond term in Eq. (6) should be constant, and an experimentalplot of T m versus H would be linear for a given suspension. Fromthe slope of this line, the yield stress can be readily calculated,regardless of the nature of the end shear distribution.

EXPERIMENTAL

Rheological Studies

The suspensions used were red mud which is a residue in theprocessing of bauxite to extract alumina. Typical physical, chemi-cal and rheological properties of red mud suspensions have beendescribed previously.l,10,15 The red mud, provided by Alcoa ofAustralia Limited, was tested over a range of solids concentration


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340 DZUY AND BOGER

3000

67.6~

% wt Solids

100

400

~t:a; 200

YIELD STRESS MEASUREMENT

103iii Iii i I

341

N

'Ez

~ 102

"">=

.)--/~

Trueyieldstre:/s-I Inferred) ~

/- ..

o ......58 60 62 64 66 68

% wt Solids

Fig. 3. Comparison of the yield stress values determined using differentrheological methods: (0) direct extrapolation, extrapolation using (e) Binghammodel, (0) Herschel and Bulkley, (~) Casson.

Vane Method

Description of the vane apparatus employed and the method ofoperation have been given in an earlier paper.' It was establishedthat the vane should be operated at rotational speeds below 10rpm to avoid the influence of viscous resistance and instrumentinertia on the measured maximum torque. The standard vanespeed employed throughout the present work was 0.1 rpm. Basedon a detailed study!" of the possible effects of vane dimensionsand system boundaries, the following criteria have been estab-lished for satisfactory measurements with the vane method: HID< 3.5, DTID > 2.0, ZllD > 1.0, and Z21D > 0.5. (See Figure 1 fordefinition of the symbols.)

The typical vane test curves are shown in Figure 4 for two redmud samples at 67.6% and 68.2% solids. These curves, presentedin terms of torque (T) versus angle of rotation (8), were generatedwith a single vane (D = 26.15 mm, HID = 1.92). Three distinctregions of behavior may be observed from the shape of the T-8curves. At angles of rotation less than about 10 degrees, T islinearly proportional to 8. In this region, the red mud behaves asan elastic solid with the degree of elasticity being described by


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342 DZUY AND BOGER

20 r----r---..,..----,,....--.,.......-......-----r--~

- 15fi'z

M

b~ 10a;::>tr

isI- 5

o 20 40 60 80 100 120 140Angle of rotation, 0 (deg.)

Fig. 4. Typical vane test curves (vane dimensions: D = 26.15 mm, HID =1.92; vane speed: 0.1 rpm).

the slope of the linear response. In the second region, an increasein shear leads to a continuing rise in torque but at a progressivelydecreasing rate. Yielding of the sample occurs at e of 20 to 22degrees, where T reaches a maximum value. Finally with furtherincreasing shear, the torque declines slowly toward a constantlevel.

The presence of a peak in the T -6 curve indicates that the redmud has a yield stress. The peak becomes more pronounced withhigher solids concentration, or with suspensions exhibiting agreater degree of structure. 10

RESULTS AND DISCUSSION

In Figure 5, the quantity 2Tml-rrD3 is plotted as a function ofvane length to diameter ratio according to Eq. (5) in the firstproposed analysis. For the 64.8%, 66.5%, 67.6% and 68.2% sus-pensions, the data were obtained from measurements with threevanes with HID varying from 0.95 to 2.0. For the remaining sam-ples, six vanes with HID of 1.0 to 3.3 were used. The linearity ofthese plots for all concentrations tested confirms the validity ofEq. (5) derived from the assumed power law distribution for thestress Te in Eq. (3). Values of the yield stress and the power lawindex m, calculated from the slope and intercept (at zero HID) of


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600

'"o-.'"

Ef-'" 200

200 r--...,---r--,......--,

:/100 / 60.1

~~o 2.0 3.0 0 1.0 2.0 3.0

HID HID

Fig. 5. Plots of 2Tml-rrD3 versus HID to test Equation (5).

the lines, are summarized in Table I. Also shown in this table arethe yield stress values calculated using the second proposedmethod which requires an experimental plot ofTm versus H basedon Eq. (6). The results presented in Figure 6 for all samples testedusing vanes of the same D but different H also validates thesecond approach which does not rely on any specific assumptionsfor the end shear stress distribution.

The results obtained from both methods are compared with the

TABLE IComparison of the Yield Stress Values Calculated from Vane Data

Using Three Methods

Method I Method II Conventional Method

Solids Ty m Tv Ty (mean) Std. Dev.wt% (Nm- 2) (-) (N~-2) (Nm- 2j (Nm- 2)

57.8 15.5 -0.063 18.8 15.9 0.759.0 19.6 -0.024 22.5 20.1 0.960.1 27.7 +0.414 30.7 26.1 1.461.6 34.1 +0.332 39.7 32.3 1.463.5 39.8 +0.148 41.7 41.3 0.264.8 50.9 -0.610 52.2 53.6 1.065.8 77.9 +0.019 77.6 76.9 2.666.5 109 +0.088 111 109 167.6 169 +0.047 173 168 368.2 287 + 0.017 296 287 2


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344 DZUY AND BOGER

20 8

% wt Solids 65.8

15 68.2 6

~ 64.8z

M 10 -: 4

~63.5

~ 67.6 61.6E 601f-

5 2 59.057.8

20 30 40 50 0 20 40 60 80H(mml H(mml

Fig. 6. Plots of Tm versus H to test Eq. (6) (vane diameter = 25.23 mrn),

yield stress values calculated using Eq. (1). The latter are pre-sented in Table I as average values obtained with more thanthree vanes. The small standard deviations also shown in thetable demonstrate no significant effects of absolute vane dimen-sions on the calculated yield stress.

An analysis of the tabulated data indicates for any given sus-pension, that the yield stress calculated using Eq. (1) is essen-tially the same as the yield stress obtained from each of the twomethods employed. Also, the empirical parameter m, which de-notes an arbitrary but possible mode of distribution for Te [Eq.(3)], varies little about zero for all but three samples tested. How-ever, the extreme values of + 0.414, + 0.332 and - 0.610 recordedfor the 60.1%, 61.6% and 64.8% samples respectively, do not con-stitute a consistent trend in a total of ten observations. It shouldbe noted from Eq. (4) that a zero-value for m suggests a uniformdistribution of the stress on the two end surfaces and the side ofthe yield cylinder. Thus if the present results are an indication,the usual assumption of a uniform stress distribution everywhereon the sheared surface may be considered reasonable, at least atthe moment of yielding and in so far as yield stress calculation isconcerned. These results, however, are not sufficient to reach afirm conclusion about the exact shear stress distribution at anyother stages, i.e., prior to and after yielding.

Of the three methods employed, method II appears to be thecorrect way for calculating the yield stress from vane data since


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103Iii , I iii I I i I

Vane method

-- Rheological method

(from Fig. 31

N

'Ez

~ 102

-a0;;,::

62 64 66 68% wt Solids

6058101 1 , , , , , ,! ,! !

Fig. 7, Comparison of the yield stress measured with the vane method and therheological yield stress (from Figure 3).

the end shear stress is avoided in the analysis. The procedureinvolved however, is tedious because measurements with morethan two vanes of equal diameter are needed to construct a plot ofT m versus H. Also, the accuracy achieved is not greatly improvedwhen compared with the conventional method which requiresonly a single vane test, The latter is thus more attractive andpreferable for practical work because it represents a simple, quickand direct method for yield stress measurement.

Having established the procedure employed for yield stress cal-culationin the vane method, it is also of importance to ascertainwhether the measured quantity can be identified with therheological yield stress determined earlier. In Figure 7, the vaneresults (as data points) calculated using Eq. (1) are superimposedon the yield stress data (as a solid curve) taken from Figure 3.Over the whole range of concentrations investigated, the com-parison indicates a remarkable agreement between the yieldstress measured with the vane and the yield values determinedby means of the usual rheological methods. In line with the previ-ous studies'r' the present results again confirm the usefulness ofthe vane method as an accurate means for direct yield stress


14:55:30

346 DZUY AND BOGER

measurement. Also, by employing the vane method, the dubiousnature of the true yield stress often associated with the laboriousbut indirect rheological approach can be avoided.

CONCLUSIONS

By means of two simple analyses, it is shown for a yield stressmaterial sheared by a four-bladed vane, that the assumption of auniform stress distribution along a cylindrical yield surface isreasonable for yield stress calculation. The yield stress valuescalculated using the proposed procedures agree well with the re-sults obtained with the conventional method widely employed bysoil mechanic workers. The latter method which requires mea-surement with only one vane is recommended for practical pur-poses. A comparison between the vane results and the yield stressindependently determined supports the previous findings that thesimple vane method is capable of measuring accurately and di-rectly the true yield stress of concentrated suspensions.

The authors are grateful to Alcoa of Australia Ltd. for their support of this workand for the continuing support from the Australian Research Grants Scheme forresearch in non-Newtonian fluid mechanics.

References

1. Q. D. Nguyen and D. V. Boger, J. Rheol., 27(4), 321 (1983).2. J. J. Vocadlo and M. E. Charles, Can. J. Chern. Eng.. 49, 576 (19711.3. M. Keentok, Rheol. Acta, 21, 325 (1982).4. E. R. Lang and C. K. Rha, J. Text. Studies, 12,47 (1981).5. E. Condolios and E. E. Chapus, Chern. Eng., 93, 131, 145 (1963).6. T. P. Elson, J. Solomon, and A. W. Nienow, J. Non-Newtonian Fluid Mech.,

11, 1 (1982).7. E. B. Bagley and D. D. Christianson. Starch/Starke, 35(3), 81 (19831.8. J. Solomon, T. P. Elson, and A. W. Nienow, Chern. Eng. Commun., 11, 143

(1981).9. D. De Kee, G. Turcotte, K. Fildey, and B. Harrison, J. Text. Studies, 10,281

(1980).10. Q. D. Nguyen, Ph.D. Thesis, Monash University, Australia, 1983.11. L. Cadling and S. Odenstad, Proc. R. Swed. Geotech. Inst., No.2, 1950.12. N. E. Wilson, ASTM Special Tech. Publ., No. 361, 377 (1963).


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13. A. Annan, J. K. Poplin and N. Ahmad, Proc. Conf. In-Situ Measurements ofSoil Properties, North Carolina, 1975, Vol. 1, p. 92.

14. M. Keentok, J. F. Milthorpe and E. O'Donovan, submitted to J. Non-Newtonian Fluid Meek. (1984).

15. F. M. Want, P. M. Colombera, Q. D. Nguyen and D. V. Boger, Proc. 8th Int.Conf Hydraulic Transport ofSolids in Pipes, BHRA Fluid. Eng., Johannesburg,1982, p. 249.

16. N. Casson, in Rheology of Disperse Systems, C. C. Mill, Ed., Pergamon,London, 1959, p. 84.

17. H. Herschel and R. Bulkley, Proe. Am. Soc. Test. Mater., 26(11), 621 (1926).18. E. C. Bingham, Fluidity and Plasticity, McGraw-Hill, New York, 1922.

Accepted November 9, 1984


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Direct Yield Stress Measurement With the Vane Method

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