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The objective of this lab is to become familiarized with the operation of a digital data acquisition system, including calibration and connections to transducers. Three separate experiments were conducted. The first experiment was to investigate the mean velocity profile at the exit of a pipe, and determine the flow characteristics. The second experiment was to investigate the pressure drop along a pipe at different flow rates and calculate friction. The last one was to investigate the yaw characteristics of the pitotstatic probe, the Kiel probe, and the cobra probe [1]. 2.1 Velocity ProfileThe Reynolds number for this pipe flow can be determined using the following equation:
Since the air flow has a Reynolds’s number of 66415.7, the flow is considered partially turbulent; referring to the Moody diagram, with a roughness factor of 0 or the ‘smooth line’ for the copper pipe. Even if the pipe is much rougher, the flow is still considered partially turbulent, not fully turbulent for this Reynolds Number. 2.2 Experimental and Power Law Velocity ProfileThe experimental exit velocity profile and the estimate provided by the oneseventh power law are almost identical as illustrated in Figure 1. However, a few notable discrepancies distinguish the two curves. The power law velocity profile is not valid near the wall. According to the power law equation,
The velocity gradient is extremely high in this region. In addition, this equation does not accurately approximate the centerline velocity because it does not yield a flat slope [2]. In fact, according to the power equation a flat slope is impossible at the center; the slope at the center must be:
However, the slope at the center can be adjusted towards zero by decreasing the exponent 1/7, which increases the denominator upon differentiation. This will yield a flatter centerline velocity profile. However, this modification may propagate discrepancies elsewhere on the curve. 2.3 Adjusting the OneSeventh Power LawThe power law was determined to best fit the experimental data with an exponential value of n = 3/14, which is greater than n = 1/7. Upon inspection of the results it was apparent that the experimental profile was flatter, thus increasing the exponent n to 3/14 modified the power law towards experimental data. However, if the exponent was increased even further, the powerlaw profile nearer to the wall significantly diverged from experimental data. The exponential form of this equation limits the fluctuation of the profile to areas near the pipe wall. This unfortunately limits the attempt to flatten the powerlaw profile because most of the flattening occurs outside the centerline, not around the centerline as needed to match experimental data. This is illustrated in Figure 1, where the powerlaw evaluated with n=1/9 The symbolic relationship between n and ucl/u was calculated as shown in Appendix B. The relationship was determined to be:
The value of n necessary to get the ucl/u ratio is n = 7.548 or approximately 3/14. With this value, a closer approximation is achieved than with n=7. The improvement can be attributed to the fact that n is a function of the Reynolds Number [2]. The oneseventh power law was modeled on a particular Reynolds Number, which may be different than the Reynolds Number of the experimental flow. This generalization of the powerlaw leads to simplification of use, but more error. Also, the oneseventh power law is assuming a level of flow turbulence that may differ from the experimental data. This new value of n=7.548 is based on the ucl and average velocity of this experimental flow, thus it models the velocity profile more accurately. Figure 1 Velocity Profile at Pipe Exit Equations used for this section:
Calculations using equations 2 and 3 for friction factor from the laboratory question sheet are computed in Appendix A. The result of the friction factor using equation 2 is 0.01222 and using equation 3 is 0.019451. Equation 3 yields a result that is on the Moody chart representing “smooth” pipes. In order to compute the ratio of the friction factor from equation [3] was substituted into equation [2] and the value was determined to be 1.195. This value was similar to the experimental value of 1.208. The percent difference being 1.1 %. This small difference can be attributed to the fact that (given by the computer program) was calculated using numerical integration with only a limited number of data points. Moreover, this flow is turbulent and only simple tools (i.e. basic Pitot tube) were used to model the flow creating some error. Finally, error could also be attributed to procedural inaccuracies such as slight misalignment of the Pitot tube. All computations can be found in the hand calculations of Appendix B. Pressure drops along the pipe at the various tap locations were determined experimentally and tabulated in Table 1 shown below. Also, Bernoulli’s equation with major head losses was also used to compute the pressure drop along the pipe and compared to the actual results (Table 1). The first column indicates pressure drops using Bernoulli’s equation with ’s given by the computer program in the lab and the other column indicates pressure drops using Bernoulli’s equation with a from question 3 equation [2] of the lab. See Appendix B for sample calculations.
Table 1 Pressure drop along pipe at different tap locations
Figure 2 Pressure drop along a pipe Figure 2 plots all these pressure drops and compares the 2 different computed values to the actual results. It is shown that the most accurate pressure drop of the computer results occurs using Bernoulli’s equation and from question 3 equation [2] when compared to the actual results. Table 2 gives the percent differences. This can be attributed to the fact that values of and friction factor may deviate along the pipe as the pipe surface may not be consistent in terms of roughness. Moreover, fittings along the pipe were not accounted for in Bernoulli’s equation and thus caused some error.
Table 2 Percent difference The property values for air listed in the computer printout are for dry air only. The air in the lab is humid. The correct value for air density for a relative humidity of 70% is 1.135 kg/m3 and has a percent difference of 1.63% when compared to the original value [3]. These calculations are shown in Appendix B. To demonstrate the insignificance of the relative humidity, the Reynolds number calculated with the correct density only differs by 1.29%, which is insignificant because it does not change the relationship in the moody chart. Thus all the previous calculations are valid. 7.1 What each Probe MeasuresThe pitotstatic probe is composed of two concentric tubes that are placed into a moving flow in order to measure the difference between the stagnation pressure and the static pressure of the flow. The inner tube is open at its tip and is oriented in the same direction as the flow. This allows the stagnation pressure (gauge) to be measured. The outer tube is not open at the tip, but it has a set of holes spaced around its circumference (perpendicular to the flow) to allow for measurement of the static pressure of the flow. The difference between these to pressures can be utilized in Bernoulli’s equation to calculate the exit velocity of the flow from the pipe [4]. The Kiel probe has a small opening to measure the stagnation pressure and there is a shroud surrounding the probe that directs the flow. The probe is used to measure the stagnation pressure in a moving fluid without knowing the exact flow direction. The pressure measured is relative to the atmospheric pressure. The cobra probe has three openings at its tip. The pressure difference taken between the two outside inlets can be used to determine the yaw angle and direction of flow. It can also be configured to measure stagnation pressure if the flow direction is known. 7.2 Advantages and Disadvantages of each ProbeAn advantage of the pitotstatic probe is that it can readily and accurately measure the difference between the stagnation pressure and static pressure, and can be easily applied to Bernoulli’s equation to calculate the velocity of the flow. The other probes cannot measure static pressure, making their result less accurate, since it is compared with atmospheric pressure. However, a major disadvantage of the pitotstatic probe is that if the fluid stream is not parallel to the probe’s head, even at small angels (yaw), errors will occur in the pressure readings [4]. Because of the effect of yaw using a pitotstatic probe won’t be very effective in measuring pressure difference in very turbulent flow, where the magnitude and the direction of velocity can change with time. An advantage of the Kiel probe is that is can measures pressure where the flow direction is unknown or changing, since it is insensitive to yaw and pitch angle. Kiel probes can tolerate a variation in flow angle to around 50 degrees. By using the Kiel probe only stagnant pressure can be measured. In order to do any analysis it must be assumed that the dynamic pressure of the flow is equal to the atmospheric pressure. This assumption can cause deviation from the actual results when calculating velocity. Another disadvantage of using the Kiel probe is that it is not effective at high yaw angels, and the direction of flow cannot be determined The main advantage of the cobra probe is that it can be used to measure stagnant pressure and determine flow direction. The disadvantage of using it to determine stagnant pressure is that the flow direction must be know, since small yaw angles can cause deviations. Also similar to the Kiel probe the static pressure must be assume to be atmospheric in order to perform velocity calculations, and is also ineffective when measuring flow direction at high yaw angles. 7.3 Usage of a Kiel and Cobra ProbeThe Kiel probe is useful in application where the properties such as pressure and velocity need to be determined, but the direction of the flow is not known. For example to analyze a unknown velocity profile around an object, the kiel probe can be placed in different locations to get pressure reading which can be used to calculate velocity. Using the Kiel probe accurate readings will be obtained, even with reasonable yaw angles. The cobra tube can be used to solve yaw angles for a particular flow. An application of this is to determine the flow pattern or streamlines exiting a vent. By taking readings at different positions an idea of the flow direction can be determined. 7.4 Ineffectiveness of High Yaw AnglesThere are many factors that make pressure probes ineffective at high yaw angles. Figure 3 compares the yaw response of the Kiel and Cobra probe. For a pitot tube at a slight angle, the stagnation point will change from its original location at the tip of the probe. New stagnation points may form at the holes spaced around its circumference of the probe, used to measure static pressure. This will cause an inaccurate pressure reading. Furthermore due to the mechanical design of the probe, yaw angles may disturb the flow around the probe, causing localized turbulence or eddy currents that can distort the measurements. The Kiel probe is also ineffective at high yaw angles. This is due to the design of the probe. At low angles the shroud helps direct the flow into the probe in order to measure the stagnant pressure. However at larger yaw angles the shroud obstructs the flow into the probe, and it also causes localized flow disturbance. These factors misrepresent the stagnant pressure readings. Similarly, the cobra probe is ineffective at high yaw angles because of its design and geometry. At large angles the flow through the inlet further downstream will be blocked by the obstruction of the center tip. Furthermore the yaw angle will cause localized turbulence and eddy currents. 7.5 Pressure Measurements Close to the WallThe flow was found to be turbulent, resulting in a very narrow and very steep velocity profile near the wall boundary. If the width of the velocity gradient is smaller than the probe’s tip, one reading at the edge may encompass the entire profile. This will result in inaccurate pressure readings, and will make determining the profile shape at the edge hard. When the probe is near the wall, part of the probe may actually be outsized the boundary of the inner wall, and only part of the flow is entering through the probe. Also, at the edge of the pipe there may be are different flow patterns, turbulence, and eddy currents. These phenomenons will also cause inaccurate readings. 7.6 Modeling Yaw Response of the Cobra ProbeIt is possible to model the yaw response of the cobra probe based upon the model of ideal flow over a cylinder. Since all the three inlets are in the same horizontal plane, and have a curvature, the front of the Cobra probe can be modeled as a cylinder. Once this assumption has been made the potential flow over a cylinder can be used. Since the angle between the two outer inlets is know, the pressure difference can be used to calculate the yaw angle of the probe relative to the flow. However this wont be valid for high yaw angles.
Figure 3 Yaw Response from Kiel and Cobra Probes In this lab, a comparison and analysis between experimental and theoretical results for the velocity profile, pressure changes, and yaw characteristics for various probes was conducted. The velocity profile of the partially turbulent flow was approximated by the oneseventh power law which, was further modified to yield a better approximation. A value of n = 7.5 yielded the best approximation. Friction factor for the pipe was computed using two equations (2, and 3) given in the laboratory. These equations were used to compute the ratio of and compare it to the experimental value. It was found to be similar with some error due to the fact that turbulent flow was difficult to model using the lab instruments provided. The pressure drop along the pipe was also computed and compared to the experimental values. Error existed and was due to assumptions in the computed results of Bernoulli’s equation. In experiment three the characteristics of the probes were investigated. The pitot tube is can accurately determine the velocity when the flow direction is known, while the Kiel probe can determine velocity where the flow direction is not know, and the Cobra probe can determine the flow direction. 1. Professor R. Macdonald, Measurement of Pipe Flow Characteristics by Means of Digital Data Acquisition, University of Waterloo, Waterloo, Ontario. Winter 2003. 2. Bruce R. Munson, Fundamentals of Fluid Mechanics, 4th Edition, John Wiley & Sons, 2002, Page 475477. 3. Michael A. Boyles, Thermodynamics: An Engineering Approach, 4th Edition, McGraw Hill, 2002. 4. http://www.personal.psu.edu/users/d/o/dob104/papers/405lab1.pdf; Date Accessed Feb 3 2003.

