The objective of the pipe flow experiment is to understand the flow and to calculate various quantities associated with the flow through different pipes. Some of the data are used to determine Reynolds number ( Re ). The results obtained are used to draw a Moody diagram are correlated as a function of the Re and relative roughness e/D. In the lamina region where Re < 2300, f is a function of Re only. However, for very large Re, f becomes independent of Re and is therefore a function only of relative roughness e/D. The diagram contains the relations based on both the theoretical and experimental data. For turbulent flow in smooth-walled pipes, the theoretical f is obtained using the formula due to Prandtl. To cover the transitionally rough region, f is calculated by the formula by Colebrook; e/D which is required in the formula is obtained using trial and error. Another graph with the experimental and theoretical f as the y and x axis respectively is plotted for comparing how close the experimental and the theoretical data fit. From measuring the pressure drop over a section of pipe with constant diameter and different flow rate, head loss can be determined for that particular flow rate.
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The water in this experiment is assumed to be incompressible with a steady and fully developed flow. The pipes are considered are assumed horizontal, z1=z2, and there is no turbomachine present and thus specific work, wcv=0. Energy is also conserved. Each tube studied is of constant diameter which gives constant velocity by conservation of mass flow.
Equation (1) is obtained by applying the energy equation (First Law of Thermodynamics) for a control volume surrounding this portion of the piping system, and making the one-dimensional assumption of uniform flow and properties at the inlet and outlet,
where subscript 1 and 2 denotes end "1" and "2" of each pipe considered. P is pressure. Ï denotes the density of water which is 998 kg/m3.The symbol "Î±" represents the kinetic energy flux coefficient and is used as a correction factor when the velocity profile is approximated with uniform and constant velocities. V denotes the velocity of the flow. g is the gravitational force which is 9.81 m/s2 and z is the elevation of the predetermined end. (hl)tot denotes the total head loss between "1" and "2" and wcv is the specific work (work per mass) done by the control volume. With the assumptions made in this lab, equation (1) can be reduced to obtain part of equation (2)
where f is the frictional factor. L and D are the length and the diameter of the pipes respectively. Equation (3) and (4) are used to find Reynolds number, Re.
Î¼ denotes viscosity and T denotes room temperature which is 298 K. Equation (5) is used to obtain the theoretical f in the laminar region
Equation (6) is due to Prandtl and is used to obtain f for turbulent flow in smooth-walled pipes
Equation (7) is due to Colebrook and also used to obtain f for turbulent flow but in rough-walled pipes
where e/D is relative roughness which was later found to be 0.07 for pipe 3 used in the experiment.
The set-up is shown in on page 12 in the lab manual. The four pipes used are of varying inner diameters. The pipe samples are measured using a vernier caliper to determine the diameter. Make sure to "zero" the electronic pressure-measuring device and no bubbles in the tubes to be probed at each end of the pipe portion. Open appropriate valves and turn on the pump. Open the valves related to each pipe before taking each set of data. Flow-rate is measured using a flow tank and a stopwatch. For pipe with a smaller diameter (pipe 1), the flow-rate is set with a fine control needle valve and the portion of flow tank with finer scale is used.
Results and Discussion
Graph 1 shows a Moody Diagram plotted with the data obtained in the lab experiment. The diagram is plotted on a log-log scale. The three solid lines are plotted using f calculated from theoretical equations. They show the relationship between f, Re and e/D. The line relating f to Re in the region where 0 < Re < 2300 represents laminar flow which is a function of only Re. The Colebrook and the Prandlt line shows turbulent flow in rough-walled and smooth-walled pipe respectively. Both lines fall in the region for 0.01< f <0.1. The Colebrook line is higher than the Prandlt line i.e. f is higher for the same Re. This corresponds to the expectation that "fully rough" flow is independent of Re. Both the laminar and Prandlt line have significant negative slopes showing that f decreases with increasing Re from f =1 when Re = 0. The Prandlt line is more negatively sloped than the Colebrook line suggesting the effect of change in Re on f is more significant for flow in smooth-walled pipes and non-effective for "fully rough" flow.
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All data obtained in this lab fall in the turbulent region since all the Re calculated based on experimental data are much greater than Re = 2300. Thus, no experimental data fall on the laminar line. Pipe 3 has a rough surface and therefore, the data points obtained are scatted along the Colebrook line. There are various Colebrook lines with different e/D in the actual Moody Diagram. The theoretical line for this experiment is only accepted when it overlaps the experimental line. The corresponding e/D is thus determined to be 0.07. This result can be verified from the actual Moody Diagram. For f = 0.1, e/D is about 0.07 and the theoretical Colebrook line drawn is near that region and the slope of the line is small. The almost constant f values obtained for increasing Re prove that in the "fully rough" region, f becomes independent of Re and is a function only of e/D.
The data points obtained from the experiments using pipe 1, 2 and 4 are scattered along the Prandlt line and this observation corresponds to the fact that these pipes are smooth-walled. The diameter of each pipe increases from pipe 1 to 2 and to 4. The data points plotted are also scatted along the Prandlt line in this sequence although there is an overlapping of the data points obtained for pipe 2 and 4. This shows that Re is dependent on both the characteristic dimension (the diameter in this case) and velocity which is dependent on the pressure change between end "1" and "2" of the each pipe. This relationship between Re, D and V is displayed in equation (4). Graph 1 suggests that pipes with smaller cross-section have smaller Re and higher f and visa versa. Therefore, the data points obtained from pipe 1 has the smallest Re but highest f. This observation corresponds to expectation that more frictional force is encountered for flow in pipes with smaller cross-sections. This is because in pipes with smaller cross-sections, the unit mass of flow in contact with the pipe wall increases. However, the overlapping portion of data obtained from the experiments using pipe 2 and pipe 4 shows that having a higher velocity and thus higher Re can offset the effect due to a small cross-sectional area.
Graph 2 shows the relationship between the theoretical and experimental f. If the two sets of values are perfectly matched, the points should fit on the 45 degree line drawn. Thus, the further the point is from the line, the more discrepancy between the theoretical and the experimental f. The two distinct clusters of data points represent data obtained from smooth and rough-walled pips with the higher cluster corresponding to the rough-walled pipe. This fact is evident from the higher Colebrook line comparing to the lower Prandlt line in Graph 1. It seems that the theoretical and experimental f obtained for smooth-walled pipes are closer to the diagonal line showing less discrepancy. On the other hand, more discrepancies are shown between the theoretical and experimental f obtained for the rough-walled pipe 3. There also seem to be a relation between the discrepancy and the value of f. It seems that more points are shown along or very near the 45 degree line drawn for f with smaller values.
Head loss in each pipe is calculated using equation (2). Since the pipes are uniform with no fittings such as sudden contractions, minor head loss is negligible. Major head loss is due to frictional effects and there is an increasing trend of head loss in each pipe as f increases showing more energy is lost with more friction encountered.
There are several sources of error in the experiment. The pressure-measuring device fluctuated frequently while readings were taken. Human reaction error should also be taken into consideration especially when members in a group switched data acquisition role as reaction time varies among individuals. The scale for measuring volume is only accurate to 0.5 liter. The pipe diameter may not be uniform either.
Conclusion and Recommendations
This experiment shows the relationship between f, Re and e/D for pipe with different cross-sectional area and roughness. The Re determined in this experiment are greater than 2300 and therefore no laminar flow. The pressure differences and volumetric flow rate calculated from the date acquired and are used to calculate experimental f. The theoretical f values are calculated use different equations. Graph 2 shows how close the experimental and the theoretical values fit. It shows that, smaller f values fit better. This is probably due to the fact that the theoretical equation takes e/D into consideration.
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The f values are used to plot a Moody diagram. The three theoretical lines correspond to 3 regions namely the laminar, smooth-walled turbulent and rough-walled turbulent. As expected, no experimental data fall on the laminar line since the calculated values of Re are greater that 2300. The theoretical line shows laminar pipe friction factor decreases inversely with Re. The data points for the smooth-walled pipes 1,2 and 4 are scattered along the Prandlt line and those for the rough-walled pipe 3 are scatter along the Colebrook line. The latter has much gentler slope demonstrating f is a function of e/D only in "fully rough" regime. The other two lines show that f decrease with increasing Re, showing more friction is encountered for more turbulent flow.
Head loss in each pipe system is calculated and the small difference between the values obtained for pipe 3 corresponds to the fact the head loss is only due e/D for "fully rough" regime. For the other pipes, head loss increases with friction factor since more friction means more energy lost.
The entire pipe flow apparatus can be designed to be more stable so it would not move about when switching the two tubes from pipe to pipe. Thus, the tubes can be fitted more easily and also minimize the chances of developing bubbles. In cases where accuracy is of great importance, more than 1 set of reading of time for the same change in pressure and volume can be taken to obtain the average values. However, the trade-off is having a more tedious experiment.
Graph 1: A Moody diagram with data obtained in the experiment demonstrating laminar flow and turbulent flow in smooth and rough-walled pipes
Graph 2: Shows how theoretical and experimental frication factors fit