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The purpose of this project was to study the behaviour of fluid flow in a pipe. The fluid being investigated in this case was oil and its viscosity was found to be approximately 3.2 Ã- 10-2 Pa.s. Since the type of oil was not known it was not possible to compare this value with a literature value. The Hagen-Poiseuille (HP) equation used to find this value was considered to be valid for the Reynolds number (Re) values obtained ranging from 195 to a maximum of 2626. A value greater than 2626 was not feasible because the pressure transducers used had a maximum operating gauge pressure of 100 kPa, which led to limitations in the accuracy of the experiment. As a result the project was restricted to the study of laminar and transitional flow only.
Fluid flow in pipes is a vital constituent in industrial applications. The transfer of fluids such as petroleum using widespread pipeline networks around the world would not be possible before fully understanding and controlling the physics of the flow. Furthermore, other areas use fluid dynamics as a basis, such as in weather forecasting for instance, where changes in the atmosphere are modelled and evaluated using computer software in order to make accurate predictions about weather.
The objectives of this investigation were to first of all accurately measure the viscosity of the oil sample supplied and determine the specific range of Re values for which the Hagen-Poisseulle (HP) equation was applicable. Subsequently, the friction factor for laminar and turbulent flow had to be calculated, as well as the pipe entry length. Finally, the last aim was to produce a qualitative description of the fluid motion at different volumetric flow rates.
In order to simplify the calculations performed using the HP equation, underlying assumptions were used; a steady fully developed laminar flow, stationary conditions for the fluid at the pipe walls (i.e. no slip condition) and that the fluid was incompressible .
The study of fluid dynamics in this experiment involved different types of flow; therefore it is essential to define them before going any further. These modes of flow were distinguished in the experiment of Osborne Reynolds. The experiment involved injecting a dye into a transparent pipe to detect the changes in flow behaviour as the volumetric flow rate of the fluid was increased . At low flow rates, the dye streak appeared straight and the flow was defined as being laminar since all flow particles were moving in streamlines (i.e. straight lines parallel to the direction of flow). In going from low flow rates to high flow rates a transition occurred as the streak appeared wavy, i.e. transitional flow. At high flow rates, the distinct streak line was no longer visible as the dye spread throughout the pipe, corresponding to turbulent flow, i.e. the flow particles motion was irregular and random.
From this Reynolds investigation came the introduction of the Reynolds number which is the ratio of inertial forces to damping forces and is defined as:
Where Re is the Reynolds number, U is the velocity of the fluid (m.s-1), D is the diameter of the pipe (m), and Ï is the density of the fluid (kg.m-3).
Experimental results from experiments similar to Reynolds' have shown that laminar flow in pipes occurs for Re less than 2000, for turbulent flow the Re value is greater than 4000, and for transitional flow, the Re value is found in between 2000 and 4000 .
Pressure drop in the horizontal pipe of this experimental investigation on fluid flow was easily determined using the Hagen-Poisseuille equation, defined as the following:
Where Î”P is the pressure drop (Pa) across the length of a pipe, L (m), of radius r (m).
Q (m3.s-1) refers to the volumetric flow rate of the fluid inside the pipe, with viscosity Î¼ (Pa.s)
3.1 Description of apparatus:
As can be seen from Fig. 1 below, the oil provided was circulating within a closed system using a gear pump. The upper horizontal brass pipe used for this circuit is 6.1 m long and has a diameter of 0.019m. At the start of the experiment, the fluid was released from the reservoir and pumped into the settling chamber. A parabollic bell mouth (not present on diagram) inside this chamber allowed transfer of the fluid into the pipe. The pressure drop inside the pipe was monitored by taking pressure readings by means of 19 digital transducers along this channel. Transducer 18 was different from the other pressure sensors since it measured the pressure across the cross section of the pipe . As the oil discharged from the pipe outlet into the'perspex' deflector, the fluid jet could be observed, allowing the nature of the flow to be determined. Following this, the volumetric flow rate could be calculated by allowing the oil to accumulate in the weighing tank using a ball valve (BV1) and taking reading from the electronic balance, before releasing the oil back into the reservoir. This flow rate was controlled by changing the extent to which the by-pass valve V2 was open (i.e. maximum flow rate meant minimum opening).
BV1 (ball valve)
V2 (By-pass valve)
V1 (Supply valve)
Figure 1: Simplified schematic representation of the experiment (not to scale)
Since the oil being used was an irritant substance, potential contact with the skin and the eyes was established as a potential hazard; therefore it had to be avoided. In the case of an incident occurring, water had to be used thoroughly to rinse off any remains of the oil on the skin. In addition to this, other precautions had to be taken regarding the apparatus being used. For instance, the gear pump had to be switched on after both turning the isolator handle to horizontal position and opening valve V1. This prevented the pump from running dry which would have caused it to overheat and eventually it would have stopped working. Moreover, the accumulation of oil within the weighing tank was never to go beyond 15 kg to prevent overflow . For this reason valve BV1 which controls the flow of oil in the weighing tank could only be closed when readings to determine the flow rates had to be taken.
At the beginning of the experiment, the pressure transducers were all switched on and reset to 0 kPa. A verification, regarding both the supply and by-pass valve (V1 and V2 respectively), had to made in order to check that these valves were fully opened (as mentioned in the safety section). The isolator was activated by setting its handle to horizontal position.
Afterwards, the balance supply had to be switched on and the ball valve BV1 was opened. At the exit of the pipe, the oil jet could be observed from the 'Perspex' deflector. In order to calculate the volumetric flow rate of the fluid in the pipe, BV1 had to be closed and simultaneously the balance was set to zero and the stop clock was started. As the balance reading approached 15 kg, the stop clock was stop and BV1 was opened allowing the oil back into the reservoir. In order to measure the pressure drop along the pipe, the pressure readings of the 18 transducers were recorded.
The above procedure was then repeated for 9 additional different flow rates by reducing the opening of V2 to a certain amount (i.e. altering the fluid flow rate). Finally, the pump, the transducers, the balance supply were switched off, the isolator handle was set to its original position (i.e. vertical position), but all valves remained open.
4. Results and Discussion:
Shortly after the gear pump was switched on, the rising level of fluid could be observed from the 'Perspex' settling chamber. At first, the oil level appeared to be stationary, however soon after it started to rise at a faster pace before slowing down again as the fluid reached the pipe inlet. The transducers readings then started to increase from zero up to a relatively constant value (i.e. the readings were fluctuating), as oil was flowing through the pipe, enabling observations of the pressure drop at different locations along the pipe. The oil jet coming out of the horizontal pipe could be described as having a parabolic shape. The flow regime observed was laminar due to the smooth and undisturbed appearance of the flow. Once all data had been recorded, the plunger near the pipe entrance was pushed inwards causing the jet of fluid in the deflector to become turbulent.
The volumetric flow rate for each of the 10 attempts was calculated using the following equation:
In Equation (3), Q is the volumetric flow rate (m3. s-1), m is the accumulated mass of the oil (kg) in the weighing tank, t is the time (s) and Ï is the density of the oil (kg.m-3). The results obtained for each reading were then tabulated in Table 1 below.
Table 1: Volumetric flow rates calculated from their respective mass of oil and time taken
Q (m3 s-1)
Preventing the mass of the oil in the weighing tank from reaching 15 kg was a very challenging task as can be seen by the mass recorded data in Table 1. Although the mass did go beyond 15 kg, no overflowing took place probably due to the fact that the maximum load capacity of the tank was much greater than this value. The volumetric flow rate obtained for the 10th reading corresponds to the maximum flow rate, i.e. at minimum opening of valve V2, and in comparison the minimum flow rate was recorded at reading number 1.
For each flow rate, pressure readings recorded from each transducer, spread out along the whole pipe length, were recorded and used to obtain a graphical representation of the pressure at different locations from the pipe inlet (Fig. 2).
From Fig. 2 below, it can be seen that from the pipe inlet to the pipe outlet, the pressure of the fluid was decreasing at a relatively constant rate since each line was relatively straight. This rate of change of pressure with respect to distance, i.e. gradient of each line, is defined as the pressure gradient. The pressure values at the exact location of both the inlet and the outlet were not recorded since there was no transducer at these positions. Although all lines tend towards zero gauge pressure near the pipe outlet, at higher flow rates the pressure gradient was much greater in magnitude than at lower flow rates. This was because the initial pressure at high flow rates was much higher than at lower ones.
Figure 2: A graph of pressure against distance from the pipe inlet at different volumetric flow rates, Q (m3 s-1). 
The pressure gradient for each value of Q was then plotted in an additional graph to allow the measuring of the viscosity of the oil using the HP equation, as shown in Figure 3.
Figure 3: Graph showing the pressure gradient across pipe for each volumetric flow rate calculated. 
At maximum flow rate, the pressure near the inlet was beyond 100 kPa therefore the first two transducers were no longer able to measure the pressure. This meant that the pressure drop for the maximum flow rate (1.48Ã-10-3 m3 .s-1) was not measured across the same distance than for all the other flow rates (i.e. length from transducer 3 to transducer 19); the pressure gradient corresponding to this flow rate corresponds to the blue cross on the graph in Fig. 3. The linear relationship between the pressure gradient in the pipe and the volumetric flow rate implies that the HP equation is being obeyed, since according to this equation the pressure gradient is directly proportional to Q.
The slope of this graph was found to be 107 Pa.s.m-4 and from this value the viscosity of the oil could then be calculated by rearranging Equation (2) as follows:
L is the length of the pipe (m) from transducers 1 to transducer 19.
This value was then used to calculate the Reynolds number for each flow rate using Equation (1), allowing the flow regime to be determined. The volumetric flow rate of a fluid is defined as:
With Q being the volumetric flow rate inside the pipe (m3 s-1), U being the velocity of the fluid (m.s-1) and A = (Ï€d2)/4, as the cross sectional area of the pipe (m2).
Since the velocity of the fluid was unknown, Equation (5) had to be rearranged and substituted into Equation (1) the following way:
A summary of the results obtained can be seen in Table 2 below.
Table 2: Reynolds number values at each volumetric flow rates.
Q (m3 s-1)
From Table 2 it can be seen that as the volumetric flow rate of oil inside the pipe started to increase, the Reynolds number associated with the flow started to increase as well. This confirms the linear relationship between the Reynolds number and the volumetric flow rate, i.e. Re is directly proportional to Q.
If the use of the plunger is disregarded, one significant drawback of the experiment is the fact that the pump was not able to produce a high enough flow rate to achieve turbulent flow, therefore limiting the maximum calculated Re at 2626. At maximum flow rate (1.48Ã-10-3 kg.m-3), the jet of oil exiting the pipe into the deflector appeared to be turbulent, whereas it was in fact transitional. Essential, the flow of oil did not reach a fully developed turbulent state because the inertial forces did not overcome the viscous forces causing the flow of oil to remain laminar. This indicates that visual observations of the flow can only be deemed accurate enough when dealing with laminar flow only, since when transitional flow is present, it can be difficult to visually differentiate between transitional and turbulent flows. In that case, evaluating the flow regime using Equation (1) is necessary.
Following this, the friction factor corresponding to each volumetric flow rate was calculated using the Re values obtained and the pressure drop equation below:
Where is the friction factor.
Another equation which can be used to calculate the friction factor in laminar flow can be found below:
The friction factor values using Equation (8) were very similar to the results obtained using Equation (7). The two sets of values were plotted as log-log graphs on the same axes for comparison, as seen in Fig. 4.
Figure 4: A graph showing the expected and experimental values of the friction factor in the pipe. 
The two lines of best fit shown in Fig. 4 are very close together which suggests that the HP equation is valid for the whole range of Re values calculated (from 195 to 2626). On the other hand, the two lines do appear to diverge which implies that there will be a point where the Hagen-Poisseuille equation will no longer be valid. However, this exact point where the divergence is judged significant enough is not known. In addition to this, the fact that both lines of best fit are straight shows that the friction factor is inversely proportional to the Reynolds number, which agrees with Equation (8).
To complete the study of the flow of oil inside the apparatus, the entry length of the pipe for each flow rate had to be determined and compared with the following equation:
LE is the entry length (m), i.e. the "distance required for the flow to become fully developed" . A plot of LE/D against Re was plotted for the entrance length at each volumetric flow rate can be seen in Fig. 5 below for comparison with Equation (9).
Figure 5: Graph showing how LE/D varies with respect to Re. 
The gradient of the line of best fit is equal to . This result is relatively close to 1/16 = 0.0625, which refers to the given relationship in Equation (9). The percentage difference between the two values being approximately 2.2 %, and since the flow being studied was mostly laminar; this suggests that Equation (9) is a valid approximation of the entrance length for laminar flow.
5. Error analysis:
On hindsight, the results would have been much more accurate if the values of the volumetric flow rate, Q, could have been more spread out over the entire possible range of flow rates of the system. Instead, the vast majority of the flow rates studied was found within the first half of the entire range of flow rates, with the exception of the last three flow rates (Q8, Q9 and Q10). This limitation in the apparatus lead to fairly small values of Reynolds numbers. This in turn limited the observations and findings of the experiment, since the effects of higher Reynolds numbers could not be rigorously studied.
Moreover, a variety of systematic errors had an impact on the results obtained. For instance when using the stop clock to record the mass flow rate of oil, the human reaction time was added to the reading being taken when both starting and stopping the clock. This meant that the time taken for the mass of the fluid in the tank to reach 15 kg at each flow rate was less than the time recorded in Table 1. In addition to this, the action of closing valve BV1 was not instantaneous which signifies that there was scope for the oil to build up in the weighing tank in between the closure of BV1 and the start of the stop clock. Furthermore, the fluctuations in the readings of the pressure transducers of up to Â± 0.3 kPa meant that the pressure readings collected had to be approximated to average values. These fluctuations were due to the fact that these pressure sensors had an accuracy of Â± 0.05 kPa, therefore the actual pressure in the pipe at certain locations could not be given a set value if for example the actual value was found to be between 10.854 and 10.859. This systematic rounding error on the transducers therefore also has to be taken into account.
Besides this a random error was identified at the beginning of the experiment; the jet of oil emerging from the pipe outlet was not fixed in position since at low flow rates (Q1, Q2, Q3, and Q4) it was slightly oscillating from left to right, similar to the harmonic oscillations of a pendulum.
In this experiment, it was determined that the pressure gradient increased linearly with volumetric flow rate, thus that the Hagen-Poiseuille equation is valid for laminar flow (with Re < 2000), and also to a certain extent for transitional flow (Re > 2000). Using the HP equation, the viscosity of the oil was calculated to be 3.2 Ã- 10-2 Pa s.
Besides this, it was also determined that the experimental values of the friction factor and the entrance length were relatively similar to their respective predicted values, since plots of their values against Re showed that they were in line with the given relations of f = 16/Re and LE/D = Re/16 for laminar flow respectively. Since the limitations of the apparatus prevented the Re values obtained to go beyond 2626, it was not possible to verify the specific range of Re values for which the HP equation was valid.
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