Viscosity 080808

8/7/2019 Viscosity 080808

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Viscosity

Motor oil A non-Newtonian fluid Non-drip paint

Industrially, understanding the viscosity properties of liquids is extremely important and

relevant to the transport of fluids as well as to the development and performance of paints, lubricants and food-stuffs.

In a Newtonian fluid, the relation between the shear stress and the strain rate is linear with the constant of proportionality defined as the viscosity. In the case of a non-

Newtonian fluid, the flow properties cannot be described by a single constant viscosity.

Some non-Newtonian fluids thicken when a shear stress is applied, e.g. cornflower

suspensions, and some can become runnier under shear stress, e.g. non-drip paint.

Objectives

1. Learn to use a falling-ball viscosimeter 2. Determine the relationship between concentration and viscosity of a sugar

solution

3. Determine the relationship between temperature and viscosity of a Newtonianfluid

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Key safety

The experiment involves use of a water bath to control the temperature of the viscometer.

At temperatures > 50 °C, care should be taken when touching the viscometer, water bathand tubing to avoid the risk of burns. The liquid from the viscometer should not be

removed when it is hot and the temperature controller of the water bath should not be setabove 70 °C.

Apparatus

Thermo falling ball viscometer

Haake P5 circulating water bath

6 viscometer ballsCleaning plunger

Cleaning brush

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Principles

A body moving in a fluid is acted on by a frictional force in the opposite direction to its

direction of travel. The magnitude of this force depends on the geometry of the body, its

velocity, and the internal friction of the fluid. A measure for the internal friction is given by the dynamic viscosity η. For a sphere of radius r moving at velocity v in an infinitely

extended fluid of dynamic viscosity η, G.G. Stokes derived an expression for the

frictional force:

r vF ⋅⋅⋅= η π 61 (1)

If the sphere falls vertically in the fluid, after a time, it will move at a constant velocity v,and all the forces acting on the sphere will be in equilibrium: the frictional force F 1 which

acts upwards, the buoyancy force F 2 which also acts upwards and the downward actinggravitational force F 3. The latter two forces are given by:

gr F ⋅⋅⋅= 13

23

4 ρ π (2)

gr F ⋅⋅⋅= 2

3

33

4 ρ

π (3)

ρ1 = density of the fluidρ2 = density of the sphere

g = gravitational acceleration

And the equilibrium between these three forces can be described by:

321F F F =+ (4)

The viscosity can, therefore, be determined by measuring the rate of fall v:

ν

ρ ρ η

gr

⋅−⋅⋅=

)(

9

2 122 (5)

where v can be determined by measuring the fall time t over a given distance s. The

viscosity then becomes:

st gr ⋅⋅−⋅⋅= )(

92 122 ρ ρ

η (6)

In practice, equation 1 has to be corrected since the assumption that the fluid extends

infinitely in all directions is unrealistic and the velocity distribution of the fluid particles

relative to the surface of the sphere is affected by the finite dimensions of the fluid. For asphere moving along the axis of a cylinder of fluid of radius R, as in a falling-ball

viscometer, the frictional force is:

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⎟ ⎠

⎞⎜⎝

⎛ ⋅+⋅⋅⋅⋅= R

r r vF 4.2161 η π (7)

And equation 6 becomes:

)(4.21

1)(

9

2 122

Rr s

t gr

⋅+⋅

⋅⋅−⋅⋅=

ρ ρ η (8)

If the finite length L of the fluid cylinder is taken into account, a further correction of the

order r/L is necessary.

This experiment utilizes a falling-ball viscometer which determines the viscosity of

fluids. The substance under investigation fills the vertical tube of the viscosimeter and the ball falls down this tube over a calibrated distance of 100 mm. The resulting falling time t

is a measure of the dynamic viscosity η of the liquid according to the equation:

t K ⋅−⋅= )( 12 ρ ρ η (9)

where ρ1 is the density of the fluid under investigation. It can be seen that equation 9 isderived from equation 8 by combining all the constant multiplying terms into one

calibration constant K . This calibration constant and the density of the ball ρ1 may be read

from the test certificate of the viscosimeter.

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The purpose of this experiment is to investigate the concentration and temperature

dependence of the viscosity of a sugar solution. Aqueous sugar solutions can beconsidered as Newtonian fluids. A Newtonian fluid has a shear stress that is linearly

proportional to the shear rate; the constant of proportionality is the viscosity:

dxduη τ = (10)

where τ is the shear stress exerted by the fluid (“drag”), η is the viscosity and du/dx the

velocity gradient perpendicular to the direction of shear.

Sugar solutions are characterized by a sharp decrease of viscosity at high temperature

which can be characterized by an Arrhenius type equation. Concentration has a strong

positive effect on viscosity, and the combined temperature/concentration effect can be

described by the generalized exponential equation:

[ BC RT E K a += )(exp0 ]η (11)

where K 0 and B are constants, Ea is the activation energy for viscous flow and

R=8.314Jmol-1

K -1

is the gas constant.

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Experimental Setup

Temperature control

The Thermo Falling Ball Viscometer may be controlled in the temperature range 10 to

80ºC using the HAAKE P5 liquid circulator provided.

The sample should rest for at least 15 minutes in the measuring tube at the testtemperature before the measurement is started.

Loading the sample

All parts of the viscometer in direct contact with the sample

must be kept clean and dry.

A sample volume of approximately 45 cm3

is poured into

the measuring tube 1 up to 20 mm below the tube. The ball2 is then introduced into the tube and the hollow stopper 13

placed on top. The liquid should reach a level just beyond

the top of the capillary 15 in the stopper; top up if necessary. The sample in the tube must be free of air

bubbles.

Before measurements are recorded, the ball should run up

and down the tube at least once to improve thehomogeneity of the samples and its temperature uniformity.

Selection of the balls

Ball selection is made on the basis of the expected viscosity of the liquid under investigation according to the specifications given in the table below; test measurements

can be made if necessary.

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Ball

No.

Material Density

ρ

(g/cm3)

Diameter

(mm)

K

(mPa⋅s⋅cm3/g⋅s)

Recommended

measuring range

(mPa⋅s)

1 boron silicaglass

2.2 15.81 ± 0.01 0.0088 0.6 - 10

2 boron silicaglass

2.2 15.6 ± 0.05 0.09 7 - 130

3 nickel iron

alloy

8.1 15.6 ± 0.05 0.09 30 - 700

4 nickel iron

alloy

8.1 15.2 ± 0.1 0.7 200 - 4800

5 7.9 14.0 ± 0.5 4.5 800 - 10000

6 7.9 11.0 ± 1 33 6000 - 75000

More accurate ball parameters will be available in the lab.

Sometimes it may be necessary to use two different balls in order to cover a wider measuring range, i.e. when the viscosity is measured over a wide temperature range; in

this case, you insert two different balls into the measuring tube at the same time with the

smaller ball inserted first.

Measuring the falling times

The measuring tube assembly locates into a fixed position of 10º to the vertical. By

turning the assembly over through 180º, the ball is set to the measuring position. Use astopwatch to measure the time taken for the ball to fall between the lines on the

measuring tube. The measuring period starts and ends when the lower edge of the ball

crosses the upper and lower lines, respectively. Turning the assembly through 180º againreturns the ball to its starting position.

For falling times of > 5 min, the measuring time can be reduced by timing the ball over

the distance between the centre line of the tube and either the upper or lower line (i.e.

over a distance of 50 mm instead of the usual 100 mm). In this case, remember to adjust

the ball constant accordingly.

Record the falling times for the 7 sugar solutions provided. Also record the falling times

for the 40 wt% solution as a function of temperature over the temperature range 20-70 ºC.

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Evaluation of the results

Use equation 9 to calculate the viscosities of the liquids investigated. Take the densities

of the sugar solutions from the table below.

Concentration (wt%) Density ρ (g/cm3)

0 1.00

10 1.04

20 1.08

30 1.12

40 1.17

50 1.23

60 1.29

Plot the appropriate graphs to determine whether the viscosity of the sugar solution

behaves as expected as a function of concentration and temperature. Use your results toestimate the activation energy for viscous flow (Ea) of the 40 wt% sugar solution.

Unknown sugar solution

You will be given an additional sugar solution with an unknown concentration. Use whatyou have learned about the dependance of the viscosity on concentration and temperature

to determine experimentally the unknown concentration of the solution.

A non Newtonian fluid

You will also be given a non Newtonian fluid. Measure how the viscosity of this fluid

changes with temperature. Discuss the qualitative differences with your measurements of

the sugar solutions.

References

Fluid Mechanics, L. D. Landau, Pergamon (Oxford), 1959 (QA 901)

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Viscosity 080808

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