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Capillary viscometers

Instruments used to measure the viscosity of

liquids can be broadly classified into seven

categories:

Orifice viscometers

High temperature high shear rate viscometers

Rotational viscometers Falling ball viscometers

Vibrational viscometers

rason c v scome ers

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A number of viscometers are also available that combinefeatures of two or three types of viscometers noted above,

such as:

Friction tube

Norcross

Brookfield

Viscosity sensitive rotameter

Continuous consistency viscometers

num er o ns rumen s are a so au oma e or con nuousmeasurement 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 reservoir

2. A capillary of known dimension,

3. A provision for measuring and controlling theapplied pressure

4. A means of measuring the flow rate

. .

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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

Glasscapillaryviscometers

a) UBBELOHDE

b) OSTWALD

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CANNON-FENSKE

Reverse-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.

Rotational viscometer

ey ave severa a van ages a ma e em a rac ve par cu ar y o

study the flow properties of non Newtonian materials.

Some of the advantages are:1. Measurements under steady state conditions

2. Multiple measurements with the same sample at different shear

3. Continuous measurement on materials whose properties may be

function of temperature

4. 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 (

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The determination of the shear stress and shear rate within theshearing gap is valid only for very narrow gaps wherein k , the ratio of

inner to outer cylinder radii, is > 0.99.

concentric cylinder viscometer

evera es gns ave een escr e w c overcome en e ec s ue

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 MooneyEwart 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 MooneyEwart

eometr

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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 is the angular velocity.

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

For k > 0.5 and if the value of (d lnT /d ln) is constant over

For the commonly used power-law fluid model, the shear

rate is a function of the power-law index.

e range o n eres R1 o R2 , one can use e o ow ng

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 fluidto be Newtonian.

It is therefore useful to ascertain the extent of uncertainty in

us ng t s approx mat on.

The ratio of the shear rates at rR1 , for a power-law fluid (PL)

and for a Newtonian fluid (N) is given as:

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

c

For typical shear-thinning substances encountered in

industrial practice, the flow behaviour index ranges from

~0.2 to 1. Over this range and for k > 0.99, the error in using

equation (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 R1 at r =R1 to R2at r =R2 .

(1e)

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For k > 0.99, R 1R2 and therefore, the two values are very

close and shear stress is given as:

(1f)

To minimize end effects, the lower end of the inner cylinder

is a truncated cone. The shear rate in this region is equal to

that between the cylinders if the cone angle, , is related to

the cylinder radii by:

(1g)

The main sources of error in the concentric

cylinder type measuring geometry:

.

2. Wall slip

3. Inertia and secondary flows

4. Viscous heating effects5. Eccentricities due to misalignment of the geometry

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Secondary flows are of particular concern in the

controlled stress instruments which usually employ arotating inner cylinder, in which case inertial forces

cause a small axisymmetric cellular secondary

motion(Taylor vortices).

The dissipation of energy by these vortices

leads to overestimation of the torque.

e s a y cr er on or a ew on an u n

a narrow gap is:

(1h)

where Ta is the Taylor number.

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In the case of non-Newtonian polymer solutions (and narrow gaps), the stability limit

increases.

When the outer cylinder is rotating, stable

Couette flow may be maintained until the onset

of turbulence at a Reynolds number, Re, of ca.

where Re= R2(R2-R1)/(Van Wazer et al., 1963).

An important restriction is the requirement for

a narrow shearing gap between the cylinders.

rec measuremen s o s ear ra es can on y e

made if the shear rate is constant (or very

nearly so) throughout the shearing gap.

fulfil this requirement.

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Many (if not most) non-Newtonian fluidsystems, particularly those of industrial or

commercial interest such aspastes, suspensions

or oo s, may con a n re a ve y arge par c es

or aggregates of particles.

Gap size to ensure that adequate bulk

, . .approximately 10-100 times the size of the

largest particle size.

The starting point lies in considering the basic

equation for the coaxial rotational viscometer, which

has been solved for various sets of boundary

conditions(Krieger and Maron, 1952):

(2)

where is the angular velocity of the spindle with

respect to the cup

system

f() = is the rate of shear at the same point

the subscripts b and c refer to the bob and the cup,

respectively.

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Assuming the infinite cup boundary condition, c

(shear stress on the cup) in equation (2) becomesequal to zero and the expression may be

differentiated with respect tob 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, the

integral in the general expression for the rate of shear

need not be evaluated from the bob all the way to the

.

This is due to the fact that, for such a system, noshearing takes place where is less than the yield

value, 0 . Thus the integral need only be evaluated

rom e o o e cr ca ra us, crit, e ra us a

which=0.

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This gives:

(4)

where the critical radius R is iven as:

(5)

For systems which may be described in terms of a

constant value of ield stress e uation 4 ma be

differentiated, giving:

(6)

The following steady shear data for a salad dressing has been

obtained at 295K using a concentric cylinder viscometer

(R1=20.04mm; R2=73mm; h=60mm). Obtain the true shear stress

data for this fluid. (Data taken from Steffe, 1996 .)

Example

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Since R1/R2=20.04/73=0.275 (

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Solution

The calculation of the corresponding shear rate usingequation(3) requires a knowledge of the slope, dln()/dln(b).

b

b

Figure 1.

Figure 1: Evaluation of the value of dln()/dln(b)

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Solution

The calculation of the corresponding shear rate usingequation(3) requires a knowledge of the slope, dln()/dln(b).

b

b

Figure 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() versus ln(b) in

Figure 1.

Finally, Figure 2 shows the rheogram for this material and

it appears that this substance has an apparent yield stress

of about ~34 Pa.

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Figure 2: Shear stressshear rate curve for the salad dressing

The cone-and-plate

geometry

Thetestsampleiscontained

betweenanupperrotating

coneandastationaryflat

plate

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The cone-and-plate geometry

The small cone angle (< 4)

gap

When investigating time-dependent systems, all

elements of the sample experience the same shear

histor

The small angle can lead to serious errors arising

from eccentricities and misalignment.

The cone-and-plate geometry

By considering the torque acting on an element of fluid

bounded 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:

or the shear stress is given as:

(9)

(10)

The cone-and-plate geometry

The corresponding expression for shear rate is obtained by

considering the angular velocity gradient (Figure 4).

The fluid article adherin to the rotatin cone has a

velocity of r and that adhering to the stationary plate is at

rest.

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:

Since shear rate does not depend upon the value of r , the

fluid everywhere experiences the same level of shearing.

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

Advantages

1. homogeneous shear field (for cone angles up to about

4)

2. The theory involved is straightforward and simple

3. Only a small volume of sample is needed (2.5 ml at

most)

4. The mass and hence inertia of the platen held by thetorsion bar are low

5. Both normal stress and oscillatory measurements are

eas y ma e

6. The technique can be used for a wide range of fluids

7. 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 limited

2. Is not at all suitable for suspensions due to the

possibility of particle jamming

A 25 mm radius coneplate system ( = 11845 ) is used to obtain

the following steady shear data for a food product at 295 K. Obtain

shear stressshear rate data for this substance.

Example

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For a cone-and-plate geometry with

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Fluids which have a significant elastic component will

produce a measurable pressure distribution in the direction

perpendicular 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)

e norma orce erence ncreases w e s earrate 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

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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 rheological

parameters in oscillatory flow.

2. Loading and unloading of samples are easier than

in the cone-and-plate or concentric cylinder

geometries, particularly in the case of highly viscous

liquids or soft solids such as foods, gels, etc.

1. When the fluid has a yield value difficulties arise if

shearing stresses fall below this value at any point.