SHORT INTRO

An experiment was conducted to find the viscosity of oil by
using the relationship between diameter and terminal velocity of a sphere. This
was done by dropping balls of different materials and masses and recording the
time taken to fall a certain distance. The results I obtained were INSERT
RESULT HERE. The oil that was used is called SAE30 and has a real value of
INSERT RESULT HERE, this is quite different to the values that were calculated
with a percentage error of INSERT RESULT HERE.

INTRODUCTION

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The main application for this experiment is to find the
viscosity of an oil with unknown viscosity, this can give applications such as
being able to predict how the oil will behave, and can also help in
transportation and production of the oil as well. However, the viscosity
calculated only applies to one temperature as viscosity varies greatly with
temperature.

The oil used was SAE 30 which
according to the manufacturers website has a dynamic viscosity of 0.2393 at a temperature of 20oC, the website that I
am using did not have any errors associated with it. According to the website
they use a rotational viscometer. This involves rotating a probe in a sample of
the liquid and the force required is used to calculate the viscosity. Another
value for SAE from a different source is 0.310 and
again does not come with errors. This is quite different to the other supposed
value, this could be due to the fact the website states three different methods
used; capillary tube viscometer, Saybolt viscometer, and a rotational viscometer.
A saybolt viscometer calculates viscosity by heating the oil until it fills a
container of known volume, and a capillary tube viscometer measures the time
taken for a known value of oil to flow through a capillary with known diameter
and

length.

The
method that I used involved dropping a spherical ball into a cylinder of the
oil and calculating the terminal velocity which I then, using stokes law and
other equations allowed me to calculate the viscosity of the oil. To have an
accurate answer two different types of materials were used, this way the
viscosity of the oil should be the same in both cases and it would give further
credibility to our answer. However, the values I obtained varied by nearly
double, which then led me to believe that a mistake was made during the
experiment most likely when dropping the WHICHEVER HAD THE MOST DIFFERENCE as
this was nearly double the reported value of the viscosity for the oil.

THEORY

The experiment that I did finds the value for the dynamic
viscosity of an unknown oil. This is mainly done by using stokes law where  is viscous drag force and  is the viscosity of the fluid:

In the case of our experiment the total acceleration needs
to be zero as the sphere will be travelling at terminal velocity. This can be
done by formulating an equation using newtons second law and making the
resultant force equal to its weight minus the buoyancy force .

In order for the acceleration to be zero the resultant force
must equal  as this means that the ball will be travelling
at terminal velocity. Now with  equation one and two can be substituted
together.

The buoyancy force in the case of the sphere is equal to  with  being the buoyancy of the liquid. The buoyancy
force  can now be substituted into the equation and  can be changed into  as this will allow for easier simplifying
later with  being the density of the sphere and  as the volume of the ball. Now substituting
both of these into the equation and simplifying you get the equation:

Elimanting some variables and swapping  for  the equation can be simplified to make it
equal to the terminal velocity in terms of the diameter of the sphere dropped
in the liquid.

This now means that the visocosity of the oil can be
calculated after plotting a graph of terminal velocity against diameter
squared, this then means, if the other variables are known, the viscosity of
the oil can be found.

For this model there are several assumptions, one of the
main ones being that the sphere is actually travelling at terminal velocity
when it passes the two markings as this is the mian theory for the experiment,
another being the temperature doesn’t change during the experiment as this
would cause the value for viscosity to change because it is very temperature
dependent. This also applies to measuring the density of the oil because if
this is measured at a different time to the experiment the temperature could
have a difference that affects the final value.

METHOD

As shown in the theory section of this report the
measurements required are terminal velocity, the density of the sphere and the
oil, and the diameter of the spheres that have been dropped. The main set up
for the experiment is to find the terminal velocity of the spheres and the
values can be obtained with little equipment such as a scale and a micrometer.

Figure 1: Basic diagram of the experiment.
A cylinder filled with the oil and with three lines at equal distance apart
from each other and a stopwatch to record the time for the sphere to travel
these distances

To collect all the data several other measurements are
required which involve equipment not shown in Figure 1. These include using a
density bottle to find the density of the oil and using a micrometer to find
the diameters of the spheres, then a scale to find the density of each sphere.
Once these measurements had been collected I could use the setup in Figure 1.
This involved dropping the ball and recording the time taken to drop from the
first line to the second, then separately record the time taken to drop from
the second line to the third line. These two times are then averaged to get the
value for the time taken to drop a known distance. This was then repeated for
spheres of different diameters made of steel and nylon so that a graph could be
plotted.

PARAGRAPH ON ERRORS

ANALYSIS AND RESULTS

In this section I will be looking at my results and
comparing them to the real values as provided by the manufacturers and other third-party
companies. I will also be commenting on the validity of the experiment and the
analysis methods used to obtain values for the viscosity. As said in a previous
method the raw data was measured using a stopwatch and this was then used to
find a value for terminal velocity, as this allowed the viscosity to be found
more easily. Figures 1 and 2 are both shown below with them representing
terminal velocity against diameter squared with Figure 1 showing the results
for steel balls and figure 2 for nylon balls. For both Figure 2 and 3 have data
that very closely resemble their respective linear fit which also means the
error bars are easily in the best fit line.

Figure 2: Graph
showing terminal velocity against diameter squared with a least squares fit
shown. This graph is showing the results for when steel was the material of
the spheres.

Figure 3: Graph
showing terminal velocity against diameter squared with a least squares fit
shown. This graph is showing the results for when nylon was the material of
the spheres.

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