On the Viscosity of Energol 320

Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature

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On the Viscosity of Energol® 320 Oil and How it Varies with Temperature

Charles S. Hannigan 30/04/2012

1. Abstract

Using a rotational viscometer, the relationship between the viscosity of Energol® 320 oil

and temperature was found to be μ=(7.48±1.13)e-(0.059±0.007)T. The investigation found

systematic discrepancies between the relationship from this experiment and that of the

manufacturer’s data. Though this would likely be too small to affect any practicable

application of the oil, the reasons behind the error are fully explored in the discussion

section. Secondarily, the viscometer used in the investigation was calibrated, which has

uses for any future experiments carried out using the apparatus.


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2. Introduction

2.1. Aims of the Experiment

This experiment aims to measure the viscosity of Energol® 320 Oil over as large a

temperature range as possible using a rotating cylinder method. The final aim being

to find the relationship between viscosity and temperature for the oil - the collected

data will then be compared with the manufacturer’s own data.

2.2. Viscosity and Fluids

Viscosity is to fluidity as resistance is to conductance – they are essentially the

same thing but defined in inverse ways. Viscosity can be simply put as the

‘thickness’ of a fluid but is more technically the resistance to flow through a fluid due

to shear and tensile stresses [1]. So the lower a substance’s viscosity, the more fluid

it is, which leads to some people suggesting solids are simply fluids with

exceptionally high viscosities. Of course, in this report, ‘fluid’ will be used in the

context of liquids and gases.

2.3. Applications

Finding the viscosity of a substance can be vital for knowing how a system works or

ensuring a device will operate correctly within its environment. Knowledge of

viscosity can also be vital to other areas of physics just as Robert Millikan found out

when he attempted to find the charge of the electron with his [now] famous oil drop

experiment [2] – his value for the viscosity of air was slightly too low, making his value

for e wrong by several times the error he quoted in his experiment.

Viscosity also plays a part in public safety in volcanically active regions – the

viscosity of magma can be used to approximate the range of a flow, thus being able

to determine risk to any populated areas. In addition to this, the fluidity of molten or

semi-molten material plays a part in many aspects of geology [3].

These examples and others from the worlds of cooking, manufacturing and

chemistry, show knowing how fluid a substance is and how this property changes

with factors such as temperature and pressure has real world practical applications.

2.4. Energol® 320 Motor Oil

A major one of these real world applications will be touched upon in this

investigation – the use of lubricants in engines, gearboxes and many other moving

components. For example engine oil has to have viscosity of the appropriate


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magnitude throughout the operating range of the engine – this can be from a cold

winter start to hundreds of degrees in the combustion chambers. Without a good

lubricant an engine can be subject to excessive wear or even to seizing up [4].

Therefore, the manufacturers of these oils test their viscosities exhaustively to

ensure they are up to the job.

2.5. Previous Methods of Finding Viscosity

Many methods have been proposed and used to measure the viscosity of a fluid,

with most following one of two principles: firstly there are gravitational methods

whereby an object is dropped down a tube full of a particular fluid (e.g. a falling

sphere viscometer) or the fluid flow rate down a tube is measured (U-tube

viscometers [5]). There is also the rotational viscometer, which relies on measuring

the torque required to rotate an object in a fluid, which is related to the viscosity. The

set up in this investigation will very closely mirror that of Searle’s rotational

Viscometer, which he set out in his 1912 treatise on the viscosity of treacle [6].

3. Theory

3.1. Dynamic and Kinematic Viscosity

There two different kinds of viscosity, though they are closely related to one another

– dynamic viscosity (μ) is a measure of a fluid’s resistance to shear flow, while

kinematic viscosity (ν) is the dynamic velocity divided by the density of the

substance [7]. As dynamic viscosity and density will both change with temperature,

kinematic viscosity has two variables dictating its value.

3.2. Accompanying Formulae

This experiment has been carried out previously by Dr Searle, under almost identical

experimental conditions – in his 1912 paper he laid out the derivation required to find

viscosity [6]:

Where μ is dynamic viscosity and g is acceleration due to gravity, while the other

constituents of the formula are best left to be defined once the design of the

viscometer has been described. The derivation makes the assumption that the

cylinder used in the experiment is of infinite length, l, which in reality is an impossible

notion and so requires a correction factor outlined in section 4.3.

(1)


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Figure 1: Diagram showing laminar flow around a cylinder rotating slowly (left) and flow around a cylinder rotating above the turbulent threshold, note the non-laminar flow lines with eddy currents and vortices.

The basic physical principle behind the theory is that when applying a couple (a

system of forces resulting in no acceleration of the centre of mass, meaning it can

still induce rotation [8]) to a cylinder in a fluid it experiences an opposing couple. This

resistance is dependent on the viscosity of the fluid and the dimensions of the

viscometer.

3.3. Molecular Level

Viscosity’s origins lay at the molecular scale – in the case of liquids the strength of

the intermolecular bonds dictate the viscosity [9]. If the bonds between molecules are

weak – either due to the electronic structure of the compound or because the kinetic

energy of the molecule is high compared to the bond potential (i.e. at higher

temperatures) – then the ‘attachment’ between molecules within different flow layers

will be lower and shear stress between the layers will be lower thus giving a smaller

value for viscosity. In the case of gases the intermolecular bonds do not play a role

in the viscosity, instead the diffusivity of the gas is the main factor.

3.4. Turbulence

Viscosity as has been described so far only holds for low flow velocities – above a

certain velocity turbulent flow ensues. Turbulent flow means that flow layers are not

well defined and parallel as in the case of the left hand side of figure 1 but are

disordered with eddy currents as shown on the right of figure 1. Turbulent flow leads

to a phenomenon called eddy viscosity – an additional viscous force that increases

the apparent viscosity of a substance.


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This means throughout the experiment laminar flow must be preserved –

fortunately there are two easily calculable dimensionless numbers that dictate

whether a flow is in the laminar or turbulent regime. The first is the Taylor number,

which gives a comparison of the relative importance of the viscous and centrifugal

forces in a rotating fluid. It is defined by:

Where ρ is the fluid density; b is the radius of the rotating inner cylinder [see

section 4.2]; a is the stationary outer radius of the viscometer; T is the time period

of rotation and μ is the dynamic viscosity [10]. Above a Taylor number of ~1700

turbulent flow arises.

Secondly there is the Reynolds number – it gives a similar comparison to the Taylor

number in that it expresses the relative importance of the viscous forces in a fluid to

the rotational, inertial forces. It is given by:

Where ρ is the density of the fluid; v is the velocity at which the fluid is moving over

the surface; l is the length of surface in contact with the fluid and as before, μ is the

dynamic viscosity [9]. Reynolds numbers over ~10000 correspond to the onset on

turbulent flow.

(3)

(2)


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4. Experimental Method

4.1. Apparatus Set-up

The viscometer (shown in figure 2) simply consists of two concentric cylinders – the

outer cylinder is fixed, while the inner cylinder is free to rotate around a firmly (but

not excessively) tightened screw at the top and another bearing at the bottom. The

cylinder is rotated by weights placed on two trays pulling thread off of a drum via

two ball-bearing pulleys. Each of these conditions focusses on minimising the

resistance to the cylinder’s rotation from friction. The period of rotation of the

cylinder was measured by a slotted optical sensor that registered every time a strut

attached to the thread drum passed through it.

The inner cylinder had a length scale engraved on it, which could be read through a

Perspex panel in the outer cylinder. The scale showed how much of the inner

cylinder was immersed in oil, which was a variable that could be changed

throughout the experiment.

Another variable that could be changed was the temperature of the oil. This was

controlled by heated or cooled water pumped from a thermostatically controlled

heating unit through the hollow jacket of the outer cylinder. The temperature of the

water was measured by a thermometer in the heater, while the oil was measured

with an electric thermometer inserted into the top of the viscometer.

Figure 2: Diagram showing the set-up of the viscometer.

Inner Cylinder

5

4

3

2

1

Length Scale

Oil

Mass Trays

Hollow-Jacketed Outer Cylinder

Ball-bearing Pulley

Top Bearing

Support

Thermometer

Heater Water Pipes

Thread Drum

Lower Bearing

Optical Sensor


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D

b

a

Figure 3: Diagram showing the important dimensions of the viscometer.

4.2. Viscometer Attributes

Figure 3 shows the three dimensions of the

viscometer necessary in the calculation of

viscosity. D is the diameter of the thread drum; b is

the radius of the rotating inner cylinder and a is the

inner radius of the outer cylinder. It was important

to take the inner radius as the hollow jacket of the

outer cylinder means there was a significant

difference between its outer and inner radii. Each

was measured using callipers for maximum

precision.

The masses of the two weight trays were also

measured with a precision balance, as they had a not-insignificant mass. The thread

that connected the mass trays to the drum was assumed to have negligible mass.

4.3. Length Correction

As previously mentioned, a correction needed to be carried out to take into account

the finite length of the viscometer. The fact that the inner cylinder had a bottom plate

meant there was a higher surface area for the oil to drag on than could be

accounted for by the length of the cylinder alone, effectively making the cylinder

‘longer’. Exactly how much longer was found by placing a constant arbitrary mass of

20 grams on the weight trays. The period of rotation of the inner cylinder was

measured; this was repeated for several immersion lengths of the cylinder.

4.4. Measuring Viscosity

There were several possible independent variables to work with to find viscosity but

in this case the length of the immersed cylinder was kept constant, while the applied

couple (mass) was changed. The period of rotation was measured for each applied

mass with the range of mass applied being from 10-110g, as above 110g the strut

passed through the optical sensor too fast for it to register.

The temperature of the oil was measured at each run, with the aim to complete each

set of measurements fast enough that the temperature stays constant. The

temperature of the oil was then changed using the thermostatic heater, or by adding


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Figure 4: Plot of MT against length immersed – used to calculate the length correction factor.

The gradient of the graph is (6.47±0.10) g s cm-1

, with the y-intercept at (4.00±0.41) cm.

ice into the water supply that was pumped into the hollow jacket of the viscometer. A

new set of readings were then taken at that oil temperature.

5. Results

5.1. Correction Factor

The periods of rotation that were measured in part 4.3 were then multiplied by the

applied mass. It is clear to see from equation 1 that if the length of cylinder were

infinite then the x-intercept of a plot of MT against l would be 0. In reality this isn’t the

case and the intercept is actually offset by an amount equal to the correction factor

needed for the length. The plot is shown in figure 4 below.

The equation of the graph’s best fit line is therefore:

Setting MT to zero and rearranging to give the x-intercept gives a value for the

correction factor (to be denoted k) of (-0.62±0.02) cm.

5.2. Dimensions of the Viscometer

The proportions of the viscometer that needed to be known in order to carry out the

experiment were as follows: the outer drum radius, a=(0.02486±0.0002)m; the inner

cylinder radius, b=(0.01876±0.0002)m; the thread drum diameter,

(4)


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D=(0.0192±0.001)m and the length immersed in oil, l was originally 0.04m but with

the correction factor became:

Thus making l=(0.0462±0.0005)m.

5.3. The Viscosity of Energol®

Equation one was utilised in finding the viscosity of the oil – it is clear to see that

plotting 1/T, the dependent variable in the investigation, against M, the independent

variable, yielded graphs with gradients of:

Rearranging the equation to make μ the subject and inserting all the relevant

constants yielded a value for the dynamic viscosity. Plots of this variety were made

for the full temperature range of the investigation and the corresponding viscosities

found. A plot showing the how the viscosity changes with temperature is shown

below.

5.4. Turbulent Flow

It was also necessary to check any results wouldn’t have been affected by turbulent

flow – this meant finding out the maximum Reynolds and Taylor’s number values

that would have occurred during the experiment.

0.1

1

10

15 20 25 30 35 40 45

Vis

co

sity,

μ (

Pa

s-1

)

Temperature (°C)

Figure 5: Logarithmic plot showing how the viscosity of Energol® 320 oil changes with temperature. The

black line represents the experimental data, with μ=7.48e-0.059T

; the error on the exponent is ±0.007, and

the intercept ±1.13. The red line represents the manufacturer’s reference data with μ=4.35e-0.058T

.

(6)

(5)


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For the Taylor number this meant using the lowest viscosity measured and the

smallest time period of rotation measured, as the rest of the factors in the equation 2

are constant. The maximal value Taylor number that would have occurred was 40.4,

significantly lower than the critical value of ~1700. With the Reynolds number, the

maximum would again occur at the smallest time period (giving a large velocity) and

lowest viscosity. The calculated value was 37.1, again significantly lower than the

critical value of ~10000.

6. Discussion

The final result of the investigation (shown in figure 5) shows that viscosity changes

exponentially with temperature, with fluids getting more viscous as temperature is

lowered. This relationship doesn’t necessarily hold for temperatures far outside this

range, particularly once one takes into account phase changes, which cause sudden

changes in viscosity. For example, the oil will most likely become a semi-solid, almost

jelly like substance and then solidify at lower temperatures and will become gaseous at

much higher temperatures.

The experimental data presented in figure 5 has a clear systematic difference from the

reference data provided by the oil’s manufacturer. While the intercept may be erroneous,

the gradient given by this experimental data is almost identical to that of the

manufacturer’s data – it may even be that the error in that case was overestimated.

The reasons for the disharmony between the two data sets need to be explored, starting

with the correction factor. The results shown in figure 4 are as expected and the straight

line fit has a high degree of certainty. The result is also consistent with Searle’s

experiment of 1912, which uses very similar apparatus; in it he states that he expects the

correction factor to be “something like half a centimetre.” [6] The correction factor is solely

a product of the viscometer dimensions; it also becomes part of the length dimension of

the cylinder, which was shown in equation 5 to be part of the gradient – this would make

any error in k fractional rather than systematic. Thus it seems that k was determined to a

good degree of accuracy and isn’t a major error source.

The outcome of section 5.4 suggests that turbulent flow cannot be the reason for these

inflated viscosity readings, as both of the calculations to determine turbulence show this

system was below the critical turbulent threshold. Though on smaller scales this may not

have been the case, the length engravings or any damage to the inner cylinder could

have caused extra drag and eddy currents that made the viscosity appear higher.


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Though unlikely, a new investigation with a fresh, unmarked inner cylinder would

conclusively resolve this.

Temperature was the independent variable over the experiment and the oil had to be

kept at constant temperature to get reliable data. The main problem with this was that

the oil wasn’t heated homogeneously: it was heated by the water in the hollow jacketed

outer cylinder, meaning there would have been a temperature gradient in the oil. While

this hasn’t been tested, it seems likely that the rotation of the inner cylinder would have

had a stirring effect on the oil, homogenising the temperature and thus removing any

problems from non-uniform temperature.

It’s probable that the discrepancy was due to the condition of the oil. This could be

partially due to small variation between batches but is more likely to be due to oil

degradation. The oil used in the investigation is not kept in ideal conditions – it is kept in

the viscometer, which has an open top – this allows particulate matter to get into the oil.

It also means there is a ready supply of oxygen and moisture for oxidation of the oil,

which is the leading cause of motor oil degradation – oxidation makes heavier

compounds, with higher intermolecular forces and so a higher viscosity [11]. The average

compound mass in the oil will also get higher over time as lighter compounds are free to

evaporate, having the same effect as oxidation. A test with fresh oil, in a clean container

would clarify this.

The errors discussed above all have the effect of increasing the apparent viscosity of the

oil if they are present – this is precisely the effect that is observed in the experimental

results.

7. Conclusion

The investigation yielded mixed results regarding the relationship between temperature

and the viscosity of Energol® 320 – the final result was that μ=7.48e-0.059T. This

relationship was found to have significant systematic error when compared to the

manufacturer’s data, which seems likely to have been caused by the differences

between the reference sample and the sample used in this experiment. The exponent

term of the relationship was found accurately though, which on the whole is more

important in determining the functional range of the oil (provided the systematic error is

small enough).


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Another facet of this investigation was the calibration of the viscometer to account for the

non-infinite length of the inner cylinder; this was done to a satisfactory degree of

accuracy and would be useful in any future investigations using the viscometer.

8. References

[1] Viswanath. D.S, Viscosity of Liquids: Theory, Estimation, Experiment, and Data,

1st Edition, Springer, New York (2007), p. 7.

[2] Heering. P, Constructing Scientific Understanding Through Contextual Teaching,

1st Edition, Frank & Timme GmbH, Berlin (2007), p.86.

[3] Lilboutry. L, Quantitative Geophysics and Geology, Springer, New York (2000),

p.190.

[4] Stewart, R.M, “The Relationship Between Engine Oil Viscosity and Engine

Performance”, ASTM International (1977).

[5] Viswanath. D.S, Viscosity of Liquids: Theory, Estimation, Experiment, and Data,

1st Edition, Springer, New York (2007), p. 17.

[6] Searle. G.F, “A simple viscometer for very simple liquids”, Proc. Phys. Soc,

London (1912), 27 119.

[7] Tipler. P.A. and Mosca. G, Physics for Scientists and Engineers, 6th Edition, W.H.

Freeman and Company, New York, p. 445.

[8] Kane. T.R. and Levinson. D.A, “Dynamics, Theory and Applications”, AIAA

Journal, vol. 24, issue 8, p. 1407.

[9] Tipler. P.A. and Mosca. G, Physics for Scientists and Engineers, 6th Edition, W.H.

Freeman and Company, New York, p. 446.

[10] Taylor, G.I, "Stability of a Viscous Liquid contained between Two Rotating

Cylinders". Phil. Trans. Royal Society (1923), A223: 289–343.

[11] Stadtmiller. W.H, “Aspects of Lubricant Oxidation”, ASTM Special Technical

Publication (1986), Issue 916.

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On the Viscosity of Energol 320

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