| A Fluid Through A Tube Essay, Research Paper
Hypothesis This investigation starts by investigating the effect of
the length of glass tube on the rate of flow of water out of it.? The volume per second of the water flowing
out of the tube (rate), is determined by the forces acting.? The pressure force pushes the fluid through
the pipe against the resistance of the viscous force.? Therefore I would expect a longer glass tube to create more force
opposing the movement of water and therefore produce a slower rate.? In this experiment I expect to find that the
rate of flow is proportional to the inverse of the length of the tube.Variables and precautions In this investigation there were many variables that might
have affected the rate of flow of a fluid through a tube.? These were considered and the appropriate
precautions taken so that their influence could be observed and analysed and
the correct conclusions drawn. ·
Viscosity ? the fluid was kept constant as water
of constant temperature. Viscosity is the coefficient of proportionality between
the frictional force and the product of surface area × velocity gradient (at
the heart of a liquid). It is generally represented by the greek letter eta,
h, and has the S.I.
units [N s m-2].? A high
viscosity would therefore be expected to affect the rate of flow of a fluid by
increasing frictional force and therefore and slowing the rate of flow compared
to less viscous fluids. ·
Height ? The height of the volume of water in
the constant head apparatus above the glass tube, provides a pressure that
forces water out of the glass tube.?
When other variables were being investigated, the height was kept
constant by clamping the constant head apparatus using a retort stand, boss and
clamp. The height was not altered until all of the readings were taken for each
variable.? The height of the constant
head apparatus was selected to ensure that sufficient pressure was generated to
create a steady flow out of the glass tubes. ·
Length of tube ? A longer tube will create greater
frictional force, slowing the rate of flow out of the glass tube.? When other variables were being investigated
the length of tubes were kept constant, measured using a ruler. ·
Radius of tube ? the radius of tube was measured
accurately using a travelling microscope.?
When other variables were being investigated the radius of tubes were
kept constant.? For example when
measuring the effect of length of tube, glass tubing was cut from the same rod
so that the radii were kept identical. ·
Temperature ? The temperature of the water
affects the viscosity and therefore the rate of flow.? The temperature of the water was monitored using a thermometer
and readings were taken as quickly as possible so that there was little
variation in the temperature of the water between readings. ·
Tube connections ? The equipment necessary for
this investigation was connected using rubber tubing to create a water tight
set of interconnecting tubes linking the tap, constant head apparatus, outflow
pipe and glass tube. ·
Material of tube ? The material of the tubing
was kept constant as this might have affected the frictional force opposing the
flow of water. ·
Observations and measurements ? In order to
create accurate results, all measurements were taken at eye level, with the
highest order of accuracy possible.? All
equipment was selected to give accurate results for example a 200ml measuring
cylinder is only accurate to the nearest 1ml whereas a 5ml measuring cylinder
is accurate to the nearest 0.05ml. ·
Volume of water ? A constant head apparatus was
used to maintain a constant volume of water, and therefore pressure, to create
a steady, constant flow of water out of the glass tubes. ·
Repeat readings ? At least three readings were
taken for each measurement.? This meant
that averages could be taken which reduce the extent of any anomalies and allow
more accurate conclusions to be drawn.Apparatus The apparatus used in this investigation was carefully
selected in order to obtain accurate and reliable results. 1. Constant
head apparatus 2. Retort
stand 3. Rubber
tubing 4. Boss
and clamp 5. Capillary
tubes 6. Stop
watch 7. Glass
tubes 8. Measuring
cylinder 9. Travelling
microscope 10. Petroleum jelly Diagram General proceduresIn this investigation the effect of different variables on
the rate of flow of water was measured.?
This was achieved by setting up the apparatus as shown in the diagram
above.? Rubber tubing was used to
connect the tap, constant head apparatus, an outflow pipe and a glass tube
together so that there were no leaks.?
The constant head apparatus ensured a constant volume of water was
present to provide a constant pressure forcing water out of the glass
tube.? By keeping the pressure constant
the effect of different variables could be measured accurately so that accurate
conclusions could be drawn.Measurements were taken by recording the volume of water
flowing out of the glass tube into a measuring cylinder, and recording the time
using a stop-watch.? Once the water was
collected, the measuring cylinder was placed on a flat surface and a reading
taken at eye level to ensure precise results.?
Each measurement was repeated three times.In order to measure the effect of different variables, the
rate of flow of the water out of a glass tube had to be measured accurately.? The rate was determined by the volume of
water collected in a measured time.?
Because the rate is calculated by dividing the volume by the time
measured on a stop-watch, the specific time for each measurement did not have
to be kept constant.? However, the time
in which a volume of water was collected was kept sufficiently long to prevent
inaccuracies associated with reaction speeds and small time values.? Although not essential, the time used to
collect the water was kept relatively constant so results could be compared and
trends in the results could be observed immediately.? In this way, any anomalous results could be identified quickly
and experiments repeated.A preliminary experiment was carried out in order to
determine the approximate rate of water flow out of a glass tube.? This allowed the selection of the correct
apparatus, to minimise inaccuracies.? It
also allowed observations to be made on any flaws in the investigation that
could be corrected to ensure more accurate results.In the preliminary investigation a capillary tube, and
larger glass tube of radii 0.6 and 3.4mm respectively were used.? The results are shown in the table below. Radius (mm) Volume (ml) Time (s) Rate (ml/s) 0.60 2.6 71.49 0.036 3.40 198 4.98 39.8 In the experiment using the capillary tube, I encountered
several difficulties; the capillary tube (being very thin) made it difficult to
create a water tight seal with the rubber tubing connecting the tube and the
constant head apparatus.? Water also had
a tendency to run back along the capillary tube rather than being collected in
the measuring cylinder.? To solve this
problem, petroleum jelly was positioned on the underside of the capillary tube
where the water flows out.? This enabled
a proper flow of water directly out of the tube to be maintained.? However, the time taken to collect an
adequately large volume of water was very long, which meant that all
measurements (especially repeat readings) would take a significant amount of
time, and would limit the number of measurements that could be carried
out.? From the results in the table, it
is clear that a glass tube of larger radius produces a much faster flow of
water.? Using a larger tube meant that
there were no problem with water flowing back along the tube, and readings
could be taken quickly and efficiently.?
From the results of this preliminary experiment, it was decided that
capillary tubes would not be used in the main experiments in the investigation.? 1. The
effect of length of tube Initially the effect of the length of tube on the rate of
flow of water out of the tube was measured.?
The height of the glass tube (and therefore pressure) and the radius of
the glass tubes were kept constant.?
Five different lengths of glass tubing were used and three readings were
taken for each length.? By repeating the
readings, the results could be averaged which reduces the extent of any
anomalies and allows correct conclusions being drawn.? The results are shown in Table 1 below.Table 1 Length of tube Volume of water (ml) Time (seconds) Rate (ml/s) Average rate (ml/s) 1 2 3 1 2 3 1 2 3 335 70 72 71 10.31 10.27 9.95 6.79 7.01 7.14 6.98 250 98 140 130 10.14 10.26 10.30 9.66 13.65 12.62 11.98 188 118 118 116 10.19 10.18 9.92 11.58 11.59 11.69 11.62 95 218 212 210 10.27 10.17 10.22 21.23 20.85 20.55 20.87 52 210 215 212 5.16 5.26 5.30 40.70 40.87 40.00 40.52 The results from this experiment have been plotted in the
graph ?A graph to show the variation of the rate of flow with length of tube?.This graph shows that the rate of flow of water out of the
glass tube decreased with increasing length to form a curved graph.? However, observation of the trend line
reveals a possible anomalous result for the glass tube of length 250mm.? In order to check this possibility, the measurements
were repeated and the graph re-plotted ?A corrected graph to show the variation
of rate of flow with length?.Table 2 Length of tube Volume of water (ml) Time (seconds) Rate (ml/s) Average rate (ml/s) 1 2 3 1 2 3 1 2 3 335 70 72 71 10.31 10.27 9.95 6.79 7.01 7.14 6.98 250 98 99 103 10.14 10.26 10.3 9.66 9.65 10.00 9.77 188 118 118 116 10.19 10.18 9.92 11.58 11.59 11.69 11.62 95 218 212 210 10.27 10.17 10.22 21.23 20.85 20.55 20.87 52 210 215 212 5.16 5.26 5.30 40.70 40.87 40.00 40.52 The graph plotted using the new measurements shows results
that more clearly follow the trend line.?
This means that the new set of results are likely to be more
accurate.? Further analysis of the shape
of the graph indicates that the rate of flow may be proportional to the inverse
of the length of the tube.? To test this
relationship, another graph has been plotted using the rate of flow against
1/length of tube? (see ?A graph to show
the variation of rate of flow with the inverse of the length of tube?).? This graph illustrates a clear trend where
by the points are positioned in a straight line through the origin.? This demonstrates that the rate of flow of
water out of the glass tube is proportional to the inverse of the length of the
tube.? V ??????1 ?t??????? l2. The
effect of radius of tubeOnce I had found the relationship between the rate of flow
and the length of glass tube, I decided to investigate another dimension, the
radius of the glass tube.? Glass tubes
of varying radius were cut to the same length using a glass cutter.? The edges of the tubes were smoothed using
sand paper for reasons of safety but also to create a smooth edge for the water
flow.??? In order to measure the radius
of each glass tube, a travelling microscope was employed. ?Each glass rod, in turn, was clamped in to a
secure position using a boss, clamp and retort stand.? The travelling microscope was positioned in front of the opening
of the glass tube and focussed so that the cross hair was aligned with the
middle of the inside edges.? A reading
on the scale was taken and the microscope realigned so that the cross hair lay
on the opposite inside edge.? A second
reading of the scale was taken and the difference between the measurements gave
the diameter.? The measurements for the
diameter were halved to obtain the radius values.? The equipment was used to measure the rate of flow of
water from the glass tube three times for each radius.? The results are shown in Table 3 below.Table 3 Radius (ml) Volume of water (ml) Time (seconds) Rate (ml/s) Average rate (ml/s) 1 2 3 1 2 3 1 2 3 0.40 12 12 11 20.11 20.36 20.27 0.60 0.59 0.54 0.58 1.2 66 65 66 10.16 10.15 10.17 6.50 6.40 6.49 6.46 2.2 230 224 230 10.14 9.75 10.12 22.68 22.97 22.73 22.79 3.4 200 198 199 4.98 5.06 5.04 40.16 39.13 39.48 39.59 4.0 198 200 190 3.35 3.33 3.00 59.10 60.06 63.33 60.83 Using these results, the graph ?A graph to show the
variation of rate of flow with radius? was plotted.? This graph shows a clear trend whereby the rate of flow of water
increases with the radius of the glass tube.?
The graph is curved upwards indicating a possible power law.? To test this relationship, a graph of rate
of flow against radius squared and radius cubed was drawn.? The graph ?A graph to show the variation of
rate of flow with the radius squared? displays a trendline that is a straight
line through the origin on the graph.?
The graph ?A graph to show the variation of rate of flow with radius
cube? has a curved trendline.? This
comparison of the graphs has shown that the rate of flow of water is
proportional to the radius squared.V???? r2 t 3. The
effect of height Another possible variable that should vary the rate of
flow of water out of a glass tube was the height at which the tube was
positioned.? I decided to alter the
height of the constant head apparatus rather than the tube itself.? This was because, altering the position of
the glass tube may have introduced inaccuracies associated with keeping the
glass tube exactly horizontal.? By
keeping the glass tube lying flat on the work surface leading to a sink, the
height of the constant head apparatus was varied by altering the position of
its clamp on the retort stand.? The
variation of height should vary the pressure which is forcing the water out of
the tube. ?Measurements were repeated
three times for five different heights.?
The results are shown in table 4 shown belowTable 4 Height Volume of water (ml) Time (seconds) Rate (ml/s) Average rate (ml/s) 1 2 3 1 2 3 1 2 3 540 124 124 117 5.25 5.24 4.99 23.62 23.66 23.45 23.58 340 100 110 110 4.82 5.13 5.19 20.75 21.44 21.19 21.14 240 91 96 96 5.17 5.21 5.21 17.60 18.43 18.43 18.15 140 74 76 72 5.25 5.26 5.10 14.10 14.45 14.12 14.22 40 46 40 42 5.37 5.37 5.36 8.57 7.45 7.84 7.95 The graph ?A graph to show the variation of rate of flow
with height? was plotted using the results.?
The graph shows that the rate of flow of water increases with height
difference between the capillary tube and the constant head apparatus.? Small initial increases in height have a
large influence on the rate of flow (indicated by the steep part on the graph),
but further increases in height become less significant (the graph levels
off).? Conclusion Poiseuille?s
equation for pipe flow dV dt = p hr g r4 8 h l Where t is
time, V is volume, h is height, r is radius, r is
density, l is length and h?is visocity. The French physician Poiseuille discovered
the above law in 1844 while examining the flow of blood in blood vessels.
Poiseuille’s Law for Fluid Flow in a Vessel assumes Steady, laminar flow Long rigid tube with non slip boundary flow Homogenous, newtonian fluid The derrivation of the formula is shown belowTherefore, the volume of water flowing per second should
be:1. proportional
to the height 2. proportional
to the radius4 3. inversely
proportional to the length of tube 4. inversely
proportional to the viscosityThe results of my investigation reveal the following
clear relationshipsRate of water flow is: 1. inversely
proportional to the length of tube 2. proportional
to the radius squaredHowever,
poiseuille’s Law for Fluid Flow in a Vessel assumes: ·
Steady, laminar flow ·
Long rigid tube with non slip boundary flow ·
Homogenous,
newtonian fluidThe most important of these assumptions is the steady,
laminar flow.? This type of flow creates
a linear graph plotted for rate of flow against pressure difference
(height).? The onset of turbulence, to
which poiseuille?s formula does not apply is shown by non-linearity (which is
clearly portrayed on the graph ?A graph to show the variation of rate of
flow with height?).? Therefore, the
results of my experiment are unlikely to follow exactly the relationships
described by poiseuille?s equation.? To
be sure of this fact, I have plotted a graph of the results with errorbars, and
a trace for the results that would be obtained if there was steady, laminar
flow (see ?A graph to compare the results of my experiment with results
following poiseuille’s equation?).? It
is clear from the graph that even with the errors in the investigation taken
into account, the results do not follow poiseuille?s formula.Steady, laminar flow that obbeys Poiseuille’s equation is
only created by liquid flow at low pressure, in relatively short tubes with
relatively narrow radii. This is because it applies to perfect flow, not
turbulent flow. At higher pressures, longer lengths or with wider bores,
turbulence sets in.Despite this, I have found clear relationships.? I found that the rate of water flow is
inversely proportional to the length of the tube.? This is because the volume per second of the water flowing
out of the tube (rate), is determined by the forces acting upon it.? The pressure force pushes the fluid through
the pipe against the resistance of the viscous force.? A longer glass tube creates more force opposing the movement of
water (the force directly proportional to the length) and therefore produces a
slower rate.During the course of
the investigation, I also discovered that the rate of flow is proportional to
the radius squared.? Since the cross
sectional area which the water flows through is given by πr2,
you would expect less resistance with a larger area of cross section of tube,
because less of the volume of water is in contact with the sides of the tube. Although limited by the
time available for this investigation, the effect of viscosity of the fluid
could also have been measured.? For
example, dilutions of a glycerol solution could have been created and the effect
on the rate of flow measured.? Errors and improvements There were many sources of error in this investigation,
that may account for any anomalous results or discrepancies in the results and
that could be improved in any future experiments. ·
Measurement of length – The measurement of length is
accurate to ± 1mm because each reading is accurate to ± 0.5mm.? This would probably only have contributed a
small error in the investigation. ·
Measurement of radius ? The measurement of radius is
accurate to ± 0.1mm because each reading is accurate to ± 0.05mm.? However, because the measurement of radius
involves reading the difference on the vernier scale between the two cross hair
positions, the errors must be added.?
This means that the radius measurement is accurate to ± 0.2mm. This
means that the smallest radius measurement had an error of 0.2/0.4 x 100 = 50%
whereas the largest radius measurement had an error of 0.2/4.0 x 100 = 5
%.? Therefore the radius is a
significant source of error in this investigation. ·
Measurement of time – Digital stopwatches can give
reading precise to within ±0.01seconds.?
But human error makes readouts accurate to only around ±0.1s.? ·
Measurement of volume ? The measurement of volume was
accurate to ± 1ml.? This meant that for
example a volume reading of 200ml had an error of 0.5%.? However, volume readings such as that of
30ml had an error of 3 1/3 %.? Therefore
readings where the rate of water flow was lowest i.e. less water was collected
had higher inaccuracies associated with them.?
This could be prevented in a future investigation by collecting a
relatively constant volume of water each time and measuring the time taken for
it to reach that level.? The rate could
then be calculated in the same way (by dividing the precise volume by the reading
on the stop watch).? This would mean
that there would be a constant low error with each measurement. ·
Flow of water out of tube – Steady,
laminar flow that obbeys Poiseuille’s equation is only created by liquid flow
at low pressure, in relatively short tubes with relatively narrow radii. In
order to create steady, laminar flow in a future investigation, capillary tubes
with a low water pressure should be used. ·
Temperature ? Temperature affects the viscosity of a
fluid and therefore the rate of flow. ?The temperature of the water, since it came directly out of a tap,
was impossible to control and did vary from day to day.? However, readings for one variable were
taken one after another and therefore significant variations in temperature
were unlikely. ??Therefore, the
temperature of the water is unlikely to be a significant source of error in
this investigation. ·
Error bars ? Error bars have been plotted on all of the
graphs of the results.? However, due to
very consistent measurements being taken, the errors are very small.? Therefore it is likely that relationships
and conclusions drawn in this investigation, are correct.Bibliography 1.
Physics, Duncan T, 2nd edition, 1993, P235 2.
A laboratory manual of physics, Tyler F, 2nd
edition, 1964, P63
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