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 reading 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 200 ml measuring cylinder is only accurate to the nearest 1 ml whereas a 5 ml measuring cylinder is accurate to the nearest 0.
05 ml... 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 procedures In 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.
4 mm 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 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 250 mm. 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 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 l 2. The effect of radius of tube Once 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 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 r 2 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 below Table 4 Height Volume of water (ml) Time (seconds) Rate (ml / s ) Average rate (ml / s 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 Poiseuilles equation for pipe flow dV dt = p hr g r 4 8 h l Where t is time, V is volume, h is height, r is radius, r is density, l is length and his 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 deprivation of the formula is shown below Therefore, the volume of water flowing per second should be: 1. proportional to the height 2. proportional to the radius 4 3. inversely proportional to the length of tube 4.
inversely proportional to the viscosity The results of my investigation reveal the following clear relationships Rate of water flow is: 1. inversely proportional to the length of tube 2. proportional to the radius squared However, 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 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 poiseuilles 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 poiseuilles equation. To be sure of this fact, I have plotted a graph of the results with error bars, 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 poiseuilles 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 r 2, 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 +/- 1 mm because each reading is accurate to +/- 0. 5 mm. This would probably only have contributed a small error in the investigation... Measurement of radius The measurement of radius is accurate to +/- 0.
1 mm because each reading is accurate to +/- 0. 05 mm. 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. 2 mm. 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. 01 seconds. But human error makes readouts accurate to only around +/-0. 1 s... Measurement of volume The measurement of volume was accurate to +/- 1 ml.
This meant that for example a volume reading of 200 ml had an error of 0. 5%. However, volume readings such as that of 30 ml 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, 2 nd edition, 1993, P 235 2. A laboratory manual of physics, Tyler F, 2 nd edition, 1964, P 63.