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Capillary viscometers

Instruments used to measure the viscosity of liquids can be broadly classified into seven categories:

y Orifice viscometers High temperature high shear rate viscometers Rotational viscometers Falling ball viscometers Vibrational viscometers Ult i i t Ultrasonic viscometers

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A number of viscometers are also available that combine features of two or three types of viscometers noted above, such as:

Friction tubeNorcrossBrookfieldViscosity sensitive rotameterContinuous consistency viscometers

A b f i t t l t t d f tiA number of instruments are also automated for continuousmeasurement of viscosity and for process control.

Common rheological instruments

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CAPILLARY VISCOMETERS

Capillary viscometers are most widely used for measuring viscosity of Newtonian liquids.

They are simple in operation; require a small volume of sample liquid, temperature control is simple, and inexpensive.

Capillary viscometers are capable of providing direct calculation of viscosity from the rate of flow, pressure and various dimensions of the instruments.

Most of the capillary viscometers must be first calibrated with one or more liquids of known viscosity to obtain “constants” for that particular viscometer.

The essential components of a capillary viscometer are:

1. A liquid reservoir2. A capillary of known dimension,3. A provision for measuring and controlling the

applied pressure4. A means of measuring the flow rate5 A thermostat to maintain the required temperature5. A thermostat to maintain the required temperature.

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Classification of commercially available capillary viscometers based on their design:

1. Modified Ostwald viscometers

2. Suspended-level viscometers

3. Reverse-flow viscometers

Glass capillary viscometers a) UBBELOHDE b) OSTWALD

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CANNON-FENSKEReverse-Flow Viscometer

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Common rheological instruments

Operates on the principle of measuring the rate of rotation of a solid shape in a viscous medium upon application of a known force or torque required to rotate the solid shape at a definite angular velocity.

Th h l d t th t k th tt ti ti l l t

Rotational viscometer

They have several advantages that make them attractive particularly to study the flow properties of non Newtonian materials.

Some of the advantages are: 1. Measurements under steady state conditions2. Multiple measurements with the same sample at different shear

ratesrates3. Continuous measurement on materials whose properties may be

function of temperature4. Small or no variation in the rate of shear within the sample

during a measurement.

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Partial section of a concentric cylinder viscometer

The concentric cylinder geometry is most suited for fluids of low viscosity (<10 Pa s).

At very high values, loading problems appear and entrapment of air b bbl i diffi lt t li i t

Rotational viscometer

bubbles is difficult to eliminate.

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The determination of the shear stress and shear rate within the shearing gap is valid only for very narrow gaps wherein k , the ratio of inner to outer cylinder radii, is > 0.99.

S l d i h b d ib d hi h d ff t d

concentric cylinder viscometer

Several designs have been described which overcome end effects due to the shear flow at the bottom of the concentric cylinder geometry.

These include the recessed bottom system which usually entails trapping a bubble of air (or a low viscosity liquid such as mercury) beneath the inner cylinder of the geometry.

Alternatively the ‘ Mooney–Ewart ’ design, which features a conical bottom may, with suitable choice of cone angle, cause the shear rate in the bottom to match that in the narrow gap between the sides of the cylinders.

The Mooney–Ewartgeometrygeometry

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For k = 0.99, the shear rate may be calculated from:

(1)

where R2 and R1 are the outer and inner cylinder radii respectively, and Ω is the angular velocity.

The shear rate for non-Newtonian fluids depends upon the viscosity model itself.

For k > 0.5 and if the value of (d lnT /d lnΩ) is constant over th f i t t ( t ) th f ll i

For the commonly used power-law fluid model, the shear rate is a function of the power-law index.

the range of interest (τR1 to τR2 ), one can use the following expressions for evaluating the shear rates at r =R1 and r =R2

respectively:

(1a)

(1b)

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Many commercial instruments employ k > 0.9 and it is not uncommon to calculate the shear rate by assuming the fluid to be Newtonian.

It is therefore useful to ascertain the extent of uncertainty in i hi i iusing this approximation.

The ratio of the shear rates at rR1 , for a power-law fluid (γPL) and for a Newtonian fluid (γN) is given as:

(1 )

Evidently for a Newtonian fluid, n = 1, this ratio is unity

(1c)

For typical shear-thinning substances encountered inindustrial practice, the flow behaviour index ranges from~0.2 to 1. Over this range and for k > 0.99, the error in usingequation (1) is at most 3%.It rises to 10% for k = 0.98 and n = 0.2.

In this geometry, the shear stress is evaluated from torque data.

(1d)

Thus, the shear stress varies as (1/r2 ) from τR1 at r =R1 to τR2

at r =R2 .

(1e)

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For k > 0.99, R1≈R2 and therefore, the two values are veryclose and shear stress is given as:

(1f)

To minimize end effects, the lower end of the inner cylinderis a truncated cone. The shear rate in this region is equal tothat between the cylinders if the cone angle, α, is related tothe cylinder radii by:

(1g)

The main sources of error in the concentriccylinder type measuring geometry:

1. End effects1. End effects2. Wall slip3. Inertia and secondary flows4. Viscous heating effects5. Eccentricities due to misalignment of the geometry

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Secondary flows are of particular concern in thecontrolled stress instruments which usually employ arotating inner cylinder, in which case inertial forcescause a small axisymmetric cellular secondarymotion(‘Taylor’ vortices).

The dissipation of energy by these vorticesleads to overestimation of the torque.

Th t bilit it i f N t i fl id iThe stability criterion for a Newtonian fluid ina narrow gap is:

(1h)

where Ta is the ‘ Taylor ’ number.

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In the case of non-Newtonian polymersolutions (and narrow gaps), the stability limitincreases.

When the outer cylinder is rotating, stableCouette flow may be maintained until the onsetof turbulence at a Reynolds number, Re, of ca.5000050000where Re=ρ Ω R2(R2-R1)/ μ(Van Wazer et al., 1963).

An important restriction is the requirement fora narrow shearing gap between the cylinders.

Di t t f h t l bDirect measurements of shear rates can only bemade if the shear rate is constant (or verynearly so) throughout the shearing gap.

But many coaxial measuring systems do notBut many coaxial measuring systems do notfulfil this requirement.

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Many (if not most) non-Newtonian fluidsystems, particularly those of industrial orcommercial interest such as pastes, suspensions

f d t i l ti l l ti lor foods, may contain relatively large particlesor aggregates of particles.

Gap size to ensure that adequate bulkmeasurements are made i e a gap sizemeasurements are made, i.e. a gap sizeapproximately 10-100 times the size of thelargest ‘ particle’ size.

The starting point lies in considering the basicequation for the coaxial rotational viscometer, whichhas been solved for various sets of boundaryconditions (Krieger and Maron, 1952):

(2)

where Ω is the angular velocity of the spindle withrespect to the cupτ is the shear stress in the fluid at any point in theτ is the shear stress in the fluid at any point in thesystemf(τ) = is the rate of shear at the same pointthe subscripts b and c refer to the bob and the cup,respectively.

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Assuming the infinite cup boundary condition, τc

(shear stress on the cup) in equation (2) becomesequal to zero and the expression may bedifferentiated with respect to τb giving:

The rate of shear may be obtained by evaluating

(3)

(graphically) either of the derivatives on the righthand side of equation (3).

In a system which displays yield stress behaviour, theintegral in the general expression for the rate of shearneed not be evaluated from the bob all the way to thecupcup.

This is due to the fact that, for such a system, noshearing takes place where τ is less than the yieldvalue, τ0 . Thus the integral need only be evaluatedf th b b t th iti l di R th di tfrom the bob to the critical radius, Rcrit, the radius atwhich τ=τ0.

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This gives:

(4)

where the ‘critical’ radius, Rcrit , is given as:where the critical radius, Rcrit , is given as:

(5)

For systems which may be described in terms of aconstant value of yield stress, equation (4) may beconstant value of yield stress, equation (4) may bedifferentiated, giving:

(6)

The following steady shear data for a salad dressing has beenobtained at 295K using a concentric cylinder viscometer(R1=20.04mm; R2=73mm; h=60mm). Obtain the true shear stressdata for this fluid. (Data taken from Steffe, 1996 .)

Example

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Since R1/R2=20.04/73=0.275 (<< 0.99), one cannot use theclose gap approximation.The shear stress at the surface of the rotating bob, τb , isgiven by equation (1f) as:

Solution

g y q ( )

For the first data point,

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SolutionThe calculation of the corresponding shear rate usingequation(3) requires a knowledge of the slope, dln(Ω)/dln(τb).

b

The given data is plotted in terms of ln(Ω) versus ln(τ ) inThe given data is plotted in terms of ln(Ω) versus ln(τb) inFigure 1.

Figure 1: Evaluation of the value of dln(Ω)/dln(τb)

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SolutionThe calculation of the corresponding shear rate usingequation(3) requires a knowledge of the slope, dln(Ω)/dln(τb).

b

The given data is plotted in terms of ln(Ω) versus ln(τ ) inThe given data is plotted in terms of ln(Ω) versus ln(τb) inFigure 1.

The dependence is seen to be linear and the slope is 2.73.

Therefore the shear rate is calculated as:

The given data is plotted in terms of ln(Ω) versus ln(τb) inFigure 1.

Finally, Figure 2 shows the rheogram for this material and it appears that this substance has an apparent yield stress of about ~3–4 Pa.

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Figure 2: Shear stress–shear rate curve for the salad dressing

The cone-and-plategeometry

The test sample is contained between an upper rotating

d flcone and a stationary flat plate

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The cone-and-plate geometry

The small cone angle (< 4°)

The shear rate is constant throughout the shearing The shear rate is constant throughout the shearinggap

When investigating time-dependent systems, allelements of the sample experience the same shearhistorys o y

The small angle can lead to serious errors arisingfrom eccentricities and misalignment.

The cone-and-plate geometryBy considering the torque acting on an element of fluidbounded by r = r and r = r + dr (Figure 3).

Integration(7)

(8)

Figure 3: Schematics for the calculation of shear stress

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The cone-and-plate geometry

For a constant value of τ:

or the shear stress is given as:

(9)

(10)

The cone-and-plate geometryThe corresponding expression for shear rate is obtained byconsidering the angular velocity gradient (Figure 4).

The fluid particle adhering to the rotating cone has ap g gvelocity of rΩ and that adhering to the stationary plate is atrest.

Figure 4: Schematics for the calculation of shear rate

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The cone-and-plate geometry

The velocity gradient or shear rate is estimated as:

(11)

Since shear rate does not depend upon the value of r , the fluid everywhere experiences the same level of shearing.

(11)

For small values of α , it is justified to use the approximation tan α = α in equation (11).

Advantages

1. homogeneous shear field (for cone angles up to about4°)

2. The theory involved is straightforward and simpley g p3. Only a small volume of sample is needed (2.5 ml at

most)4. The mass and hence inertia of the platen held by the

torsion bar are low5. Both normal stress and oscillatory measurements are

il deasily made6. The technique can be used for a wide range of fluids7. It is easy to observe is the fluid is behaving strangely

e.g. fracturing

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Disadvantages

1. The maximum shear rate is limited2. Is not at all suitable for suspensions due to the

possibility of particle jammingp y p j g

A 25 mm radius cone–plate system ( α = 1°18ʹ45ʺ ) is used to obtainthe following steady shear data for a food product at 295 K. Obtainshear stress–shear rate data for this substance.

Example

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For a cone-and-plate geometry with α <4º, the shear rate is given by equation (11) and the corresponding shear stress is given by equation (10). Therefore for the first data point:

Solution

The value of α in radians is ~0.02 and for small values of α , tanα≈α:

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Fluids which have a significant elastic component willproduce a measurable pressure distribution in the directionperpendicular to the shear field (Fig. Below).

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Some cone and plate viscometers allow measurement of the resulting normal (axial direction) force on the cone making it possible to calculate the first normal stress difference, as:

(12)

Th l f diff i ith th hThe normal force difference increases with the shear rate for viscoelastic fluids. It is equal to zero for Newtonian fluids.

In this measuring geometry, the sample is contained between an upper rotating or oscillating flat stainless steel plate and a lower stationary platestationary plate

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In contrast to the cone-and-plate geometry, the shear strain is proportional to the gap height, h.

Advantages

1. It allows precise determination of rheologicalparameters in oscillatory flow.

2. Loading and unloading of samples are easier thang g pin the cone-and-plate or concentric cylindergeometries, particularly in the case of highly viscousliquids or ‘ soft solids ’ such as foods, gels, etc.

DisadvantageDisadvantage

1. When the fluid has a yield value difficulties arise ifshearing stresses fall below this value at any point.