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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