Heat Pump
1
Preparing a Lab Report for ENG590
A scientific report usually consists of the following sections:
1. Title 2. Introduction 3. Method 4. Results 5. Discussion 6. Conclusion 7. Bibliography
Title
The title should be less than ten words and should reflect the factual content of the lab report
and can be the same as the title on the lab sheet.
Introduction
The introduction defines the subject of the report. It must outline the aims and objectives for
the experiment performed and give the reader sufficient background to understand the rest of
the report. Care should be taken to limit the background to whatever is pertinent to the
experiment. A good introduction will answer several questions, including the following: Why
was this study performed?, What knowledge already exists about this subject? , What is the
specific purpose of the study?
Method
As the name implies methods and equipment used in the experiments should be reported in
this section. The difficulty in writing this section is to provide enough detail for the reader to
understand the experiment without overwhelming him or her. Generally, this section attempts
to answer the following questions: What materials were used?, How were they used?
Results
The results section should present the data from the experiments. The data should be
organized into tables, figures or graphs. All figures and tables should have descriptive titles
and should include a legend explaining any symbols, abbreviations, or special methods used.
Figures and tables should be numbered separately and should be referred to in the text by
number, for example: Figure 1 shows that the acceleration at different times.
Discussion
This section should not just be a restatement of the results but should emphasize interpretation
of the data. What do the results show? Do they agree with theory? How significant are the
errors in your measurements?
Conclusion
In the conclusion you should summarise what has been done and the results which have been
obtained. How do they meet your original aims?
Bibliography
This section lists all articles or books cited in your report.
The following pages contain a sample lab sheet followed by an example of a lab report.
2
LAB SHEET - VISCOSITY OF GLYCERINE
Aim:
To measure the viscosity of glycerine using Stokes' method in which steel balls are allowed to
fall through glycerine.
Theory:
(i) If a body of mass m falls through a viscous fluid, it will accelerate until
the combination of the viscous force (or drag) FD, and the buoyancy
force FB balance the gravitational force Fg (= mg)
FD + FB = Fg (1)
When this equilibrium is reached, the body continues to fall, but at a
constant velocity, called the terminal velocity.
(ii) Archimedes' Principle states that the buoyancy force acting on a body
immersed in a fluid is equal to the weight of the fluid displaced. If the
body immersed is a sphere of volume V and radius r, the volume of fluid
displaced is also V. Thus if the density of the fluid is L,
FB = VLg
= 4
3 r3
L g (2)
(iii) Stokes showed that for a sphere of radius r moving through a fluid of viscosity , the
viscous drag is
FD = 6vr (3)
where v is the steady velocity.
(iv) If the density of the sphere is S , then the gravitational force is
Fg = 4
3 r3
S g (4)
(v) Substituting (2), (3) and (4) into (1)
6vr + 4
3 r3
L g =
4
3 r3
S g
4
3 r3(S – L)g = 6vr
r2 = v g)(2
9
LS
(5)
The terminal velocity v can be determined by measuring the time t for steel balls to fall through a fixed distance s
v = t
s (6)
Substituting this expression into (5) gives
t
s
g r
)(2
9
LS
2
3
t
kr 1
So 2 (7)
Now we can see that, if the time t for steel balls of varying radius r to fall at terminal velocity through the fixed distance s can be measured, a plot of r2 against 1/t should yield a straight line of slope k.
Rearranging (8) s
gk LS
9
)(2
(9)
To be able to calculate the viscosity of glycerine, , experimental data are required for the
gradient k, the fixed distance s, the density of the sphere S and the density of the liquid L
Procedure:
1. Drop a medium sized ball into the column of glycerine and make a starting mark
close to the top of the column, but at a position at which the ball has achieved
terminal velocity (ie constant velocity), and a finishing mark close to the bottom. The
distance between the marks is the fixed distance s.
2. Now it is required to time balls of varying r when falling through that distance s
between the two marks.
Measure the diameter D of each ball with a micrometer before dropping it into the
glycerine and measuring t. Take about 8 readings of t for a wide range of r.
3. Plot a graph of r2 versus t
1 and determine a value for the slope k (without
uncertainties).
4. To determine the density of the steel balls, and because all balls were manufactured
from the same melt, it is necessary only to measure mass and volume of one large
steel ball ( = m/V).
5. The density of glycerine L can be determined using a hydrometer.
6. Use equation (9) to calculate .
Compare the calculated value of with the tabulated data. Note the temperature dependence
and record the temperature at which this experiment was performed. Note also that the
viscosity of glycerine is very dependent on water content.
(8) )(2
9 where
LS g
s k
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EXAMPLE LAB REPORT - VISCOSITY OF GLYCERINE
Introduction Viscosity is an important physical property of all fluids which relates to the level of
friction a fluid experiences when it flows. An example of a fluid with a high viscosity
is syrup which requires more effort to stir than a low viscosity fluid such as water.
Higher viscosity fluids require more energy to overcome the higher levels of friction
when they flow. It is therefore important to know the viscosity of a fluid which is
being pumped through a pipeline. Measurement of viscosity is also important in the
petroleum industry where the viscosity of crude oil is strongly related to its
composition [1].
An instrument which is used to measure fluid viscosity is called a viscometer. There
are a number of different viscometers which are in use [2]. Here we use a falling
sphere viscometer which, as the name suggests, involves a sphere falling through the
liquid being measured. The principle behind the falling sphere viscometer is that
when the sphere is falling there are three forces acting on it. The first is its weight and
the second is the drag force and the third is the buoyancy force. This is shown in
figure 1.
Figure 1: The forces acting on a sphere falling through a liquid. Taken from the lab
sheet.
The buoyancy force is known to be equal to the weight of fluid displaced and acts in
the opposite direction to the sphere’s weight. The buoyancy force can be found using
𝐹𝐵 = 4
3 𝜋𝑟3𝜌𝐿 𝑔
where r is the radius of the sphere, L is the density of the liquid being tested, and g is the acceleration due to gravity. The weight of the sphere is simply its mass multiplied
by gravity. Although accurate balances are available to measure the mass of the
sphere, we instead express the weight in terms of the volume of the sphere and the
density of the sphere to be consistent with the approach for the buoyancy. In this way
the weight of the sphere is
𝐹𝑔 = 4
3 𝜋𝑟3𝜌𝑆 𝑔
where S is the density of the sphere. The drag force on a sphere was determined by George Gabriel Stokes in 1851 [2] and is given by
𝐹𝐷 = 6𝜋𝑟𝜇 𝑣 where is the viscosity of the fluid. This force opposes the direction of motion. When the sphere is falling at its terminal velocity the sum of the drag and the
buoyancy forces must be equal to the weight of the sphere. This gives 4
3 𝜋𝑟3𝜌𝐿 𝑔 + 6𝜋𝑟𝜂𝑣 =
4
3 𝜋𝑟3𝜌𝑆 𝑔.
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If the sphere is falling at its terminal velocity and it travels a distance s in time t, then
we can write v = s/t and re-arrange the above equation to give r2 = k/t , where 𝑘 =
9
2
𝜇𝑠
(𝜌𝑆−𝜚𝐿)𝑔 .
Method
The experiment was conducted using a large measuring cylinder containing glycerol.
Initially the density of the liquid and the balls were obtained. The density of the
glycerol was found using a hydrometer. This was placed in the cylinder and the
density read from the scale on the hydrometer at the surface of the glycerol. Secondly
the density of the spheres was found.. The spheres which were used were all bearings.
They all had different radii, but were made from the same material. The density was
found by measuring the radius and the mass of one of the balls. This was done for the
largest ball to minimise errors.
The next step was to determine how quickly the ball bearings reached their terminal
velocity. This was done by dropping a medium sized ball into the liquid and watching
its progress. Once it was judged that the ball had reached a constant velocity, a
horizontal line was drawn on the measuring cylinder using a red marker pen. A
second line was drawn near the bottom of the cylinder.
The following procedure was then followed with all the ball bearings.
1. The radius of the ball bearing was measured using a micrometer. 2. The ball bearing was dropped into the measuring cylinder of glycerine. 3. The time taken for the ball to travel between the two marked lines was
recorded.
This procedure was repeated for each of the ball bearings. Finally the distance
between the two red lines on the measuring cylinder was measured suing a ruler.
Results
When the steel balls were dropped into the cylinder they appeared to reach their
terminal velocity within the first 30 cm. The time was, therefore, recorded between a
line marked on the cylinder 30 cm below the surface and a second line 50 cm below
the first. The measured diameters, d, of the steel balls and the time taken for them to
fall through 0.5 m of glycerine are shown in table 1. The calculated value of (1/r)2 is
also shown. The diameter was measured with a set of callipers and the error in the
reading was ± 0.05 mm. The time was measured with a digital stop watch with an
error of ±0.05 s.
d (mm) t(s) d(m) r(m)
r-2(m-2)
1.1 72.8 0.0011 0.00055 3310000
1.9 18.2 0.0019 0.00095 1110000 3 8.1 0.003 0.0015 444000
4.1 4.6 0.0041 0.00205 238000 4.8 3 0.0048 0.0024 174000 9.8 0.9 0.0098 0.0049 41600
15.3 0.2 0.0153 0.00765 17100 20.2 0.2 0.0202 0.0101 9800
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Table 1: The diameter and radius of the steel balls and the time taken for them to fall
through 0.5 m.
The density of the steel balls was found using the largest of the steel balls. Its mass
was measured as 31.5 ± 0.05 g. The density of the steal is given by
𝜌𝑠 = 𝑚
𝑉 =
3𝑚
4𝜋𝑟3 =
3×0.0315
4×3.14×0.01013 = 7303 kgm-3.
Figure 2 shows a graph of the time plotted against 1/r2. The best fit straight line
through the points has a gradient of k = 2x10-5 s.m2. All of the data points lie on or
close to this line suggesting that the relationship is linear. Using equation (9) this
gives the viscosity as
55.0 5.09
81.9)10007303(1022
9
)(2 5
s
gk LS
kgm-1s-1.
Figure 2: Graph of t plotted against 1/r2.
Discussion
The value found for the viscosity of glycerine was 0.55 kgm-1s-1. This is considerably
lower than the tabulated value of 1.5 kgm-1s-1 [3]. The error in measuring the diameter
of each ball was fixed at ± 0.05 mm. For the smallest ball this is approximately a 5%
error. The error reduces to 0.25% for the largest ball. The error in recording the time
from the stop watch is ±0.05 s, but this does not include the reaction time of the
person using the stop watch. This was estimated to be 0.2s and explains why the time
for the two largest balls is the same. This gives a percentage error or 100% for the
largest two balls and an error 0.3% for the smallest ball. There is a large error in
measuring the time for the larger balls and in measuring the radius for the smaller
balls. The results for the middle balls should give the most accurate reading. The
density of steal was found using the largest ball to give the most accurate value.
Most of the points in figure 1 lie on or very close to the bet-fit line which suggests
that the value for the viscosity should be fairly accurate. The viscosity of glycerol
changes rapidly with temperature and with water dissolved in the glycerol [4]. The
difference between the tabulated value and the value found here could be due to this.
If the experiment was done again, the temperature should be recorded and a new
sample of glycerine should be used to make sure there is no dissolved water. It would
be best to only use balls with a diameter between 3 and 5 mm and to make several
measurements for each ball.
y = 2E-05x - 1.0795
-10
0
10
20
30
40
50
60
70
80
0.E+00 5.E+05 1.E+06 2.E+06 2.E+06 3.E+06 3.E+06 4.E+06
t (
s)
(1/r)2 (m-2)
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It is difficult to know when the ball reaches its terminal velocity. Measurements could
be taken with a lower start line to see if this changes the value for the viscosity.
Conclusions
A value of 0.55 kgm-1s-1 was found for the viscosity of glycerol. This is about a third
of the tabulated value. This may be due to dissolved water in the glycerol, or a
difference in the temperature, which was not recorded. The result showed it is
possible to measure viscosity in this way. Some suggestions were made to improve
this experiment.
Bibliography
[1] Wernera, A, Beharb, J. de Hemptinnea, J. C. and, Behara, E. ‘Viscosity and
phase behaviour of petroleum fluids with high asphaltene contents’, Fluid Phase
Equilibria, 147, 343–356, 1998.
[2] http://en.wikipedia.org/wiki/Viscometer. Accessed 20/9/2012.
[3] Brown, J. Y. and Heath W. L. Table of Physical Constants. Paragon Press,
London 1985.
[4] http://www.binacchi.com/Utilites/useful/glycerine%20viscosity.pdf. Accessed
22/07/2010.