Construction and Building Materials 25 (2011) 3443–3449

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Construction and Building Materials

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Torque–speed relationship in a concrete rheometer with vane geometry

Aminul Islam Laskar ⇑, Rajan BhattacharjeeCivil Engineering Department, National Institute of Technology, Silchar 788 010, Assam, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 June 2010Received in revised form 15 January 2011Accepted 1 March 2011Available online 1 April 2011

Keywords:RheologyVane rheometerYield stressPlastic viscosity

0950-0618/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2011.03.035

⇑ Corresponding author. Mobile: +91 9957601712.E-mail address: aminul.nits@gmail.com (A.I. Laska

Derivation of mathematical relationship between torque and speed during shearing in concrete rheom-eter with vane geometry has been presented in this paper. Resistance offered by concrete below andabove the vane as well as effect of concrete resistance from the annulus were taken into considerationto represent actual flow condition of concrete during shearing. An expression for total shear stress hasbeen derived from where shear stress versus torque and shear strain rate versus rotational frequencyrelationships have been established for the vane geometry of a concrete rheometer.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

‘‘Rheology’’ is the scientific study of the deformation and flow ofmatter. The devices which use principle of fluid rheology to mea-sure the shear stresses of fluid at varying shear rates are called rhe-ometers. Concrete as a fluid is most often assumed to behave like aBingham fluid with good accuracy [1,2]. In Bingham model, flow isdefined by two parameters: yield stress and plastic viscosity. Yieldstress gives the quantitative measure of initial resistance of con-crete to flow and plastic viscosity governs the flow after it is initi-ated. To determine the Bingham parameters with a rheometer,fresh concrete is sheared at increasing steps to overcome thixot-ropy before the rheological test. And then, shear rate is decreasedgradually and stress is measured. The relationship between shearstress and shear rate is plotted as flow curve (Fig. 1). The interceptat zero shear strain rate is yield stress, so while the slope of theflow curve is plastic viscosity l.

There are several concrete rheometers such as Two-point[3,4], BML [5], CEMAGREF-IMG [6], IBB [7], BTRHEOM [8,9] andICAR [10]. An important advantage of vane over conventionalgeometries in the measurement of yield stress of suspensionsis that yielding occurs approximately along a localized cylindri-cal surface circumscribed by the vane. This means that yieldingof material takes place within the material itself and the prob-lems and errors associated with the slip flow on the boundaryare absent [11]. Nguyen and Boger [12,13], and Keentok [14]adopted the vane for use in the direct yield stress measurementof some concentrated suspensions. Since then, vane has been

ll rights reserved.

r).

successfully used by several workers in rate controlled modefor direct yield stress measurement of suspensions of red mud,titanium dioxide, laterite, alumina and lubricating greases [11].Keating and Hannant [15] studied the effect of rotational speedand measuring system stiffness on yield stress of cement slurrieswith the vane. More recently, Koehler and Fowler [10] developeda vane rheometer for rheological measurement of concrete at theInternational Centre for Aggregate Research (ICAR) at the Univer-sity of Texas at Austin. ICAR vane rheometer is now commer-cially available. The rheometer is capable of measuringconcrete with slumps greater than 50–75 mm up to self com-pacting concrete. However, the mathematical relationship be-tween torque and speed during shearing of concrete in a vanerheometer was not attempted. The objective of the present studyis to derive torque–speed relationship in a vane rheometer forconcrete taking into account the resistance offered by concretein the annulus, top and bottom of the vane.

2. Derivation of torque–speed relationship

In existing vane rheometer such as ICAR, arrangement for no-slip condition at the top of concrete specimen is not provided.Therefore, the top surface of concrete may sometimes rotate dur-ing shearing (partial slip). It is also equally likely that the veloc-ity of material at the top is exactly zero (no-slip) depending onthe shear rate for a given geometry of the rheometer. The otherpossibility is that velocity is zero at some depth measured fromthe top, particularly when angular velocity is small. The possiblevelocity profiles are shown in Fig. 2. In the present study, it isassumed that ribs are provided on vertical sides, top as well asat bottom of cylindrical container to ensure no-slip condition.

Fig. 1. Graphical representation of Bingham’s equation.

3444 A.I. Laskar, R. Bhattacharjee / Construction and Building Materials 25 (2011) 3443–3449

Nguyen and Boger [13] performed experimental measurementsfor such a case of a vane rheometer to calculate shear stress. Thetorque attributed to side (Ts) was assumed equal to shear resis-tance offered when the material just yields, as usually assumedin soil vane shear. The distribution of stress below the vane wasrepresented with an integral in terms of an unknown function of

Fig. 2. Possible shear rate field near

bottom shear stress, so that total torque, T = Ts + 2Tb where Ts isthe resistance from side and Tb is the resistance from bottom. Tosolve this equation, additional assumptions were made by Nguyenand Boger [13] for different conditions to obtain yield stress of sus-pensions. But concrete is a Bingham material, characterized byyield stress and plastic viscosity and cannot be treated as a mate-rial like pure clayey soil.

US 7,624,625 [16] disclose a vane rheometer that is used tomeasure yield stress and viscosity of a cement-based material.The greatest weakness of this rheometer lies in the fact that forsample above and below the impeller, it is assumed that stress atthe impeller-material interface is equal to yield stress when thematerial just shears. This assumption is not correct for cement-based material where one has a variation of shear rate across thegap. In fact, this assumption is applicable to soil vane shear testwhich is used in geotechnical engineering. Such soil shear vanetests are standardized by Geotechnical Codes all over the world.

In the present paper, it is assumed that concrete behaves like aBingham material that is, workability is high and the workability interms of slump is more than 75 mm as mentioned in some litera-tures. For the derivation of the total shear stress over the entire

the top wall of the rheometer.

Fig. 3. Velocity profile along horizontal and vertical directions.

Fig. 4. Designation of vane dimensions in a rheometer.

A.I. Laskar, R. Bhattacharjee / Construction and Building Materials 25 (2011) 3443–3449 3445

volume of the concrete in a cylindrical container, the total volumeis divided into several parts depending on the shear rate field. Fig. 3shows the velocity profile at the salient surfaces in horizontal and

vertical directions. The geometrical features of the vane and cylin-drical container are shown in Fig. 4 where H represents height ofthe vane, Z1 and Z2 respectively denote the height of concreteabove and below the vanes.

Total torque T required rotating the impeller at N revolution permin. (rpm) is given by:

T ¼Z

dT ð1Þ

Total torque T is the summation of the torque componentscontributed by the volumes ABDC (V1), BDPO (V2), ACKL (V3),CJFK-AMEL (V4), DIHP-BNGO (V5), and CJID-AMNB (V6). Individualtorque components have been presented in the subsequentparagraphs.

2.1. Volume ABDC (V1)

Here, velocity along the shaft axis is zero and velocity along thecircumference circumscribing ABDC is (xD/2), where x isthe angular velocity of the vane in radian per second and Dis the diameter of the vane. Shear strain rate along radial directionis equal to:

_c ¼ ðxD=2Þ=ðD=2Þ ¼ x: ð2Þ

3446 A.I. Laskar, R. Bhattacharjee / Construction and Building Materials 25 (2011) 3443–3449

Torque contribution is given by:

T1 ¼ ðso þ lxÞpDHD2

ð3Þ

2.2. Volume BDPO (V2)

Consider a circular element dr along BD at a radial distance rfrom the impeller shaft. Linear velocity at this radius = r x;x = angular velocity of the vane in radian/s.

Shear strain rate, _c ¼ xr=Z2.Torque on this elemental disc is expressed as:

dT ¼ ðs0 þ l _cÞ2pr2dr

Total torque; T2 ¼Z D=2

0dT ¼ pD3

12so þ

pD4x32Z2

l ð4Þ

2.3. Volume ACKL (V3)

Torque T3 contribution can be deduced in a similar manner as inSection 2.2 and is given by:

T3 ¼Z D=2

0dT ¼ pD3

12so þ

pD4x32Z1

l ð5Þ

2.4. Volume of hollow cylinder DIHP-BNGO (V5)

Referring to Fig. 5, consider an elemental layer of thickness dz ata height z from bottom on the cylindrical surface BDPO. The veloc-ity along the radial direction on the surface of BDPO is given by:

v r ¼z

Z2

xD2¼ vz

Z2

Therefore, at a height z from bottom, shear stress sr ¼ so þ l vzZ2g

where g = (Dt � D)/2 is the effective gap of the annulus.

Force on this elemental area; dF ¼ so þ l vzZ2g

� �pDdz

Total force ¼Z Z2

0dF ¼ pD so þ

lv2g

� �Z2

Torque; T5 ¼pD2

2Z2 so þ

l2g

xD2

� �ð6Þ

2.5. Volume of hollow cylinder CJKF-AMEL (V4)

As in Section 2.4 above, torque contribution T4 can be deducedand is given by the following expression:

Fig. 5. Velocity profile in the annulus between vane and container.

T4 ¼pD2

2Z1 so þ

l2g

xD2

� �ð7Þ

2.6. Concrete in the annulus CJID-BNMA (V6)

Velocity and shear rate along CD (or AB) are given by v = xD/2and v/g respectively.

Torque; T6 ¼ so þ lvg

� �pDH

D2

ð8Þ

If the vane is at the center of the concrete specimen, Z1 = Z2 = Z(say) and therefore T2 = T3 and T4 = T5.

Since the volume V1 does not shear during the experiment i.e.vane-in-cup rheometer, [13,17,18], T1 = 0.

It may be mentioned here that during shearing, deformationsand hence the velocity at the common boundaries (shown by dot-ted lines in Fig. 3c) are equal for adjacent parts due to compatibilitycondition, and have been taken into account while deriving expres-sions for torque components. Moreover, torques T2–T 6 are all di-rected towards the longitudinal axis of the shaft. The magnitudeof resultant torque is, therefore, an algebraic sum of magnitudeof component torques. That is, total torque (T) is given by:

T ¼ 2T2 þ 2T4 þ T6

Or; T ¼ pD2 H2þ Z þ D

6

� �so þ

p2D3

120N

D4Zþ H þ Z

g

� �l ð9Þ

From Eq. (9), it may be observed that total torque T is a linear func-tion of rotational frequency N.

3. Fabrication of a vane rheometer

To validate the torque–speed relationship, a vane rheometer forconcrete was fabricated in the laboratory for testing fresh concrete.Subsequent sections present description of the fabricatedrheometer.

3.1. Geometrical requirements of rheometers

The general rule for concrete rheometers is that gap size shouldbe in the range 3–10 times the maximum size of aggregate [19,20].This is important to minimize the effect of change in particle pack-ing near walls. Ratio of outer to inner diameters in case of coaxialrheometers has been suggested as 1.2 [2] or 1.1 [19] to ensuresmall variation in shear rate across the gap and to minimize thespeed range at which plug flow occurs. The recommended valueof height to radius ratio of coaxial cylinder rheometer is less than1.0 to minimize the contribution of the bottom of the cylinder.

In ICAR rheometer, the vane is immersed in concrete in a cylin-drical container with 400 mm diameter and 375 mm height. Thevane has a diameter 125 mm and a height equal to 125 mm. Thedistance between the top of the concrete and the top of the vaneis 125 mm. Similarly, the distance between the bottom of the vaneand the bottom of the cylindrical container is also 125 mm. Asmentioned earlier, the top of the concrete surface is open and isfree to rotate during shearing.

3.2. Fabrication

A schematic diagram of the fabricated rheometer is shown inFig. 6 to describe its components and working principle. As men-tioned, it consists of 125 mm diameter vanes driven by an induc-tion motor through a gear box. The height of the vane is 125 mmand it is mounted coaxially with a cylindrical container of effective

Fig. 6. Schematic diagram of the present rheometer.

Fig. 7. Calibration chart for torque.

A.I. Laskar, R. Bhattacharjee / Construction and Building Materials 25 (2011) 3443–3449 3447

diameter 275 mm (total diameter being 315 mm) with sleeve andbearing arrangement to ensure accurate alignment. The torque andspeed of rotation of the vane is controlled manually by varying in-put voltage with a 10 A alternating current (AC) variac. The cylin-drical container is provided with vertical ribs of 20 mmprojection at a pitch of 60 mm along the circumference to preventslippage near wall. Ribs are also welded to the bottom of the cylin-drical container to achieve no-slip condition during shearing. Theeffective gap between the bottom and the shearing surface is100 mm. The effective concrete height above the vane plate is also100 mm. No-slip condition of flow at top of the cylinder (which isotherwise missing in ICAR rheometer) is achieved by providing20 mm high mesh of blades. The ‘no-slip’ condition is an essentialcondition to be achieved during shearing so that mathematicalexpression for shear stress in fluids can be applied for the deriva-tion of torque–speed relationship. The mesh can be detached afterthe experiment is over. The number of revolution of the vane ismeasured automatically with a non-contact infrared digitaltachometer, by focusing at the retro-reflective tape glued to thespindle or shaft.

Substituting D = 0.125 m, H = 0.125 m, Z = 0.100 m, g = 0.075 min the above Eq. (9), we have the linear torque–speed relation asfollows:

T ¼ 0:0089947so þ 0:0001692Nl ð10Þ

where N is the rotational frequency in rpm. Expressing the aboveequation in Bingham’s form, we have

111:18T ¼ so þ 0:01881Nl ð11Þ

Thus, we have shear stress, s = 111.1T Pa and shear strain rate,_c = 0.01881 N s�1. Both the quantities _c and s can be observed dur-ing the experiment. By plotting the values of ( _c, s), one has the flowcurve from which so and l can be obtained.

3.3. Calibration of torque

The calibration of torque in the present rheometer was done byrotor blocking method. This type of electro-mechanical methodwas adopted by Banfill [21] and recently by Laskar and Talukdar[22]. The circuit for the purpose of calibration consists of voltme-ter, ammeter and a 10 A AC variac. A spring balance anchored toa fixed object is fitted to the pulley that is again welded to the

shaft. When the motor is switched on, the spring balance blocksits rotor and the spring balance reading is noted. This arrangementgives the braking torques at different voltages. Thus for a set ofvoltages, braking forces or torques can be obtained. In the presentcase, two spring balances were used: 20 kg up to 65 V and 100 kgfor above. The spring balances were again calibrated using an accu-rate digital balance. Finally, braking torque was plotted against voltand calibration chart was obtained using regression analysis(Fig. 7).

4. Materials and mix proportions

4.1. Cement

Cement used throughout the experiment was Ordinary Portland Cement (OPC).The physical properties of cement determined as per Indian Standard Code PracticeIS: 12269-1987 are as follows:

Specific gravity = 3.10; Standard consistency = 29%; Initial setting time = 65 minand final setting time = 8 h; 28 day compressive strength = 50 N/mm2.

4.2. Sand

Locally available river sand (specific gravity = 2.6) was used in the presentstudy. The particle size distribution is shown in Table 1. Sand was stored insidethe laboratory throughout the experimental investigation.

4.3. Coarse aggregate

Crushed stone aggregate (specific gravity = 2.6; aggregate crushing value = 20%)of maximum size 16 mm was stored in the laboratory. The particle size distributionis presented in Table 2.

4.4. Chemical admixtures

Poly-carboxylic ether polymer (PC) was used as High range water reducingadmixture (HRWRA). Ordinary tap water was used to prepare fresh concrete. Themix proportion of concrete prepared in the laboratory is shown in Table 3.

5. Methods

Concrete was mixed in a tilting mixer (laboratory type). Mixingsequence was as follows:

� Mix coarse aggregate, fine aggregate and cement for 1 min.� Add water during mixing and mix for 2 min.� Stop mixing for 1 min.� Add admixture to the mix and mix for 3 min.� Pour the concrete mix.

The prepared concrete was transferred to the cylindrical con-tainer from the same height every time. The rheological test was

Table 1Sieve analysis of sand.

Sieve size (mm) % Passing

4.75 98.22.36 96.51.70 94.61.18 91.20.60 66.30.30 20.30.15 1.6

Table 2Sieve analysis of coarse aggregate.

Sieve size (mm) % Passing

16 10012.5 42.2010 31.806.3 25.04.75 0.9

Table 3Mixture proportion in kg/m3 and mix designation.

Mixdesignation

Cement Sand Coarseaggregate

Water HRWRA Slump(mm)

Mix 1 559 444 1085 186.3 7.2 110Mix 2 470 516 1035 196 7.4 225Mix 3 450 400 1207 194 7.7 220Mix 4 532 516 1033 195.8 7.5 190

Table 4Observed values of rheological parameters.

Mixdesignation

Observed parameters withEq. (11)

Parameters with Laskar andTalukdar (2008)

Yieldstress (Pa)

Plasticviscosity (Pa s)

Yieldstress (Pa)

Plasticviscosity (Pa s)

Mix 1 306 62.4 289.2 56.6Mix 2 170.1 55.8 164.8 49.5Mix 3 434.4 50.2 478.4 46.9

Fig. 9. Flow curves with and without top and bottom resistance.

3448 A.I. Laskar, R. Bhattacharjee / Construction and Building Materials 25 (2011) 3443–3449

carried out after 15 min from the addition of water. Increasingshear stress sequence followed by a decreasing shear stress se-quence was used and the down curve was taken to draw the flowcurve. Concrete was sheared at each shear stress for 30 s and volt-age and rpm readings were noted. Flow curves for mixes presentedin Table 3 are shown using Eq. (11) and a typical flow curve is pre-sented in Fig. 8.

6. Results and discussion

To appreciate the difference if any, in the observed values of therheological parameters using Eq. (11), concrete mixes in Table 3were prepared and tested by the fabricated vane concrete rheom-eter. Samples were subjected to increasing and then decreasingshear rates. Flow curves were drawn for each samples using Eq.(11) and rheological parameters such as yield stress and plastic vis-cosity were estimated using regression analysis. Mixes presentedin Table 3 were also tested to determine rheological parameters

Fig. 8. Flow curve of Mix 3 with the fabricated rheometer.

Fig. 10. Comparison of Vane equation in this study and US Patent 7624625.

of concrete by a concrete rheometer developed by Laskar andTalukdar [22] at the National Institute of Technology, Silchar. Thedesign, fabrication, calibration and validation of which were pub-lished elsewhere [22]. Each time fresh mixes were prepared fortesting in the latter case. The observed values of rheological param-eters obtained by vane rheometer using Eq. (11) and rheometer de-signed by Laskar and Talukdar [22] are shown in Table 4. It may beobserved that the difference in yield stress and plastic viscosity inall the mixes presented in Table 3 is small. Thus, it may be con-cluded that Eq. (9) derived in the present paper can be used in avane rheometer to determine rheological parameters of concretein terms of yield stress and plastic viscosity.

To appreciate the difference between Eq. (9) and the equationneglecting resistances from top and bottom of concrete, Mix 4 of

A.I. Laskar, R. Bhattacharjee / Construction and

Table 3 was tested by the vane rheometer. The flow curves usingEq. (9) and equation neglecting resistance from top and bottomare shown in Fig. 9. It may be observed that when top and bottomresistances are not considered, yield stress is 1.45 times the yieldstress with Eq. (9). Similarly, the plastic viscosity neglecting topand bottom resistances is twice the plastic viscosity obtained Eq.(9). Thus, equation ignoring top and bottom resistance overesti-mates the rheological parameters of concrete.

To differentiate between the rheological parameters using theequation derived in the present paper and the parameters esti-mated by vane rheometer equation disclosed in US 7,624,625,Bingham plots are drawn for Mix 4 and is presented in Fig. 10. Itmay be observed from Fig. 10 that compared to the equation de-rived in the present paper, the value of yield stress calculated usingthe vane rheometer equation in US 7,624,625 is very small whereasthe value of plastic viscosity is very large. Thus, vane rheometerequation in US 7,624,625 underestimates yield stress and overesti-mates plastic viscosity.

7. Conclusion

Following general conclusions may be derived from the presentwork:

� Resistance offered by concrete at the top and bottom of the vaneis not equal to yield stress only. In fact, there exist shear rategradient across top and bottom of a vane.� It is possible to obtain a well defined shear rate field in concrete

lying above and below the vane by introducing ribs and thusresistance from top and bottom can be taken in account toderive torque–speed relationship.� Rheological parameters obtained using the derived torque–

speed relationship are in good agreement with values of yieldstress and plastic viscosity determined by a concrete rheometerat National Institute of Technology, Silchar.� If resistance at top and bottom of a vane is assumed equal to

yield stress, the torque–speed equation in such cases underesti-mates yield stress and overestimates plastic viscosity.� Torque–speed equation ignoring top and bottom resistance

overestimates both the rheological parameters of concrete.

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