Charles Hannigan: Finding the Viscosity of Energol ® 320 Oil and How it Varies with Temperature 1 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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Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
1
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.
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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)
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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.
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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)
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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)
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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)
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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.
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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).
Charles Hannigan: Finding the Viscosity of Energol® 320 Oil and How it Varies with Temperature
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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