MEE 390 sp11 Laboratory Experiment #3: Experimental Verification of Bernoulli Equation and Pitot Probe Measurements in the Wind Tunnel Instructor: Milivoje Kostic TA: Chris Edwards February 22,2011 www.kostic.niu.edu/390/390HW‐assignments.html Part 1: Experimental Verification of the Bernoulli Equation Apparatus: Bernoulli (Venturi‐type) flow apparatus, Local atmospheric pressure column‐
manometers, Pitot tube impact probe, Graduated Cylinder, Stopwatch Useful Links: Bernoulli Equation Specifications: Bernoulli (Venturi‐type) flow apparatus: cross‐sectional flow area varies from 199 to 380 mm2 Local atmospheric pressure column‐manometers: 5 mm graduations Pitot tube impact probe: Moveable, Graduated Cylinder: 1L capacity, 20mL graduations Stopwatch: students provide their own Objective: Experimental verification of Bernoulli equation for fluid flow in a flow channel of varying cross‐
sectional area. Theory: In order to solve problems in fluid flow, it is often necessary to determine the variation of pressure with velocity from point to point throughout the flow field. As a rule, such a relation between pressure and velocity may be extremely complex, involving changes in pressure and velocity in the three space coordinates, as well as other factors such as friction. However, by making suitable simplification and approximations, it is possible to obtain a relatively simple expression relating fluid pressure and velocity. This expression is called the Bernoulli equation or Bernoulli's law. Bernoulli's law relates the pressure changes to velocity and elevation changes. For example,it indicates that, if an inviscid fluid is flowing along a pipe of varying cross section, then the pressure is lower at constrictions where the velocity is higher, and pressure is higher where the pipe opens out and the fluid slows down. The well‐known Bernoulli equation is subject to the following assumptions: 1. Fluid is incompressible (ρis constant) 2. Flow is steady: 0 3. Flow is frictionless: 0 4. Flow is along a streamline 5. Pressure and gravity are the only forces When all the above assumptions are met, then the Bernoulli equation may be expressed mathematically as ∗
(1) where (in SI units) o P is fluid static pressure at the cross section [N/m2] o ρ is density of the fluid [kg/m3] o g is acceleration due to gravity [m/s2] o V is mean velocity of fluid flow at the cross section [m/s] o Z is elevation head of the center of the cross section with respect to a datum Z = 0 [m] o h*is total (stagnation) head [m] 2
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Figure 1: Bernoulli verification setup Legend: 1 – Static pressure ports 4 – Quarter‐turn flow control valve 2 – Local atmospheric pressure column‐
5 – Graduated Cylinder manometers 6 – Pitot tube impact probe 3 – Venturi flow section The terms on the left‐hand‐side of the Equation 1 represent the static pressure head (h), velocity head or dynamic pressure (hv), and elevation head (Z), respectively. The sum of these terms is known as the total head (h*). According to Bernoulli's theorem of fluid flow through a pipe, the total head at any cross section is constant. In a real flow, due to friction and other imperfections as well as measurement uncertainties, the results will deviate from the theoretical ones. In this experimental setup, the centerline of all the cross sections lies on the same horizontal plane (which may be chosen as the datum, Z = 0), and thus, all the Z values are zero so that Equation 1 reduces to: ∗
(2) Also, for this experiment, the static pressure head is denoted as , where i represents the cross section along the axial length of the apparatus, and the total head is h*. Procedure: 1. Check if the drain gate valve is open and keep it wide open and check whether the outlet pipe goes to the drain. Initiate flow through the Venturi test section by opening inlet valve(s). 2. Check that all manometer tubes are properly connected to the corresponding pressure taps and are air bubble free. If needed, flush the air bubbles by slowly closing the exit valve and drawing the water (and the air bubbles) through the manometer tubing. 3. Adjust both (inlet and outlet) valves so that you get the maximum difference in levels between tapping point #7 and #8. 4. Wait for some time for the level in manometer tube #8 (h*) to stabilize (if it never fully stabilizes, take the average of the maximum and minimum of the fluctuating reading). 5. After the steady state is achieved, redirect the water outlet hose into a container whose capacity is known (1L, for example) and record the time taken for the water to fill it up (this is the primary technique for volume flow measurement). Take at least 5 measurements and record the elapsed time to fill the container in order to calculate the (average) volume flow rate. 6. Gently push (slide) the Pitot (total head measuring) tube, connected to manometer #8, so that its end reaches the cross section of Venturi tube at static pressure port #2. Wait for some time and record the readings from manometer #8. This value is the sum of the pressure and velocity heads, i.e. the total/stagnation head (h*), because the pitot tube is held against the flow of fluid forcing it to a stop (zero velocity). The reading in manometer #2 measures just the static pressure head (hi for i = 2) because the pressure tap does not obstruct the flow when connected to the Venturi tube. Thus, the static pressure is measured perpendicular to the flow streamline. 7. Repeat step 6 for other static pressure head manometer locations (#3 – #7). Observations: Table 1: Volume Flow Rate Observation No. Volume, ΔV [mL] Time, Δt [sec] Flow Rate, ∆ ⁄∆ [mL/sec]
1 2 3 4 5 3
Average volume flow rate: Qave[cm /sec] = Qave[mm3/sec] = Note: 1 mL = 1 cm3 = 1000 mm3 Table 2: Manometer Readings and Calculations Cross Using Bernoulli Equation Using Continuity Equation *
Section No. h [mm] hi[mm] Vib[mm] Ai[mm] Vi[mm] V
V /V [mm] 2 380 3 302 4 263 5 238 6 214 7 199 Note: ⁄4 (Di = 22.0, 19.6, 18.3, 17.4, 16.5, 15.9 mm for i = 2, 3, 4, 5, 6 and 7, respectively. ∗
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(3) ⁄ (4) For the Laboratory Report: 1. Plot the variation of both the static pressure and the calculated velocity as a function of the local diameter of the channel. How is the relationship consistent or inconsistent with expected results from fluid mechanics? 2. Calculate the uncertainty in the resultant velocity due to height and pressure measurands. 3. Describe why the speed to fill each vertical manometer tube was different. Part 2: Pitot Probe Measurements in a Wind Tunnel Apparatus: Static‐stagnation pitot probe, Wind tunnel, Water manometer, Useful Links: Pitot Tubes Specifications: Static‐stagnation pitot probe: Attaches to water manometer, 360° rotation Wind tunnel: Adjustable frequency blower, 11.4 to 28 Hz blower frequency range Water manometer: 0 – 12 mm reading, 0.1 mm graduations Objective: Generate streamwise velocity as function of input motor speed and flow orientation within the wind tunnel. Theory: The wind tunnel is a testing tool for fluid mechanics similitude. Rather than fabricate a complete airplane to test its performance, non‐dimensional values of the Reynolds number and other parameters are maintained as constants while the experimental model is reduced in size. For this reason, wind tunnels operate at speeds that are sometimes much faster or much slower than flow experienced in true operation. 2
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1 Figure 2: Test section and manometer of wind tunnel Legend: 1 – Water manometer 3 – Test section 2 – Pitot tube The flow within the wind tunnel is directed from the nozzle end to the output fan that draws the air into the test section. Given the orientation of the flow, velocity should be maximum along the length of the test section with minimal transverse flow. This experiment attempts to quantify the change in measurements if the devices within the wind tunnel are not oriented properly. Procedure: Locate the wind tunnel in EB 254. The test section is the rectangular region constructed of transparent plexiglass. To the left of the test section is the nozzle that accelerates the ambient air through the test section. To the right is a large blower that draws the air through the wind tunnel. The rotational speed of the blower is controlled by a frequency meter in the electrical box with blower power. The frequency is selected by turning the dial (“frequency control knob”) adjacent to the test section. 1. Mount the combination static‐stagnation pitot probe at the center of the test section. 2. Point the probe toward the nozzle end of the wind tunnel attempting to keep it parallel with the side walls. 3. Attach the tubes from the pitot probe static pressure and stagnation pressure ports to the provided water manometer. 4. Before turning on the blower, turn the frequency control knob to the left such that the indicator is at approximately 9 o’clock on the dial. Confirm that no SAE Aero Team members are downwind of the blower. It may be best to shout, “Is there anyone down there?” Turn the large power switch on the electrical control box. 5. The blower slowly accelerates to the frequency setting of the control knob if the system is set to “Manual” and not “Auto”. The counter on the electrical panel eventually reaches steady state at the current frequency control knob setting. 6. Take pressure measurements from the manometer as the frequency setting is between 15 Hz and 20 Hz. Be sure that the system has reached steady state before recording values. 7. Select one frequency setting in the provided range (or have one specified by the TA). Upon the wind tunnel reaching steady state, take manometer measurements for the variety of orientations of the pitot probe in the wind tunnel. Choose several angles between 0° and 90°. 8. Record data in an appropriate table for calculation of the wind tunnel velocity. 9. Repeat the measurements of pressure as a function of orientation for the pitot probe at two additional height settings: A) one‐third of the test section height and B) the probe placed 1.5 – 2 cm from the bottom surface. 10. Again, record the manometer readings for the velocity calculation as a function of pitot probe orientation. 11. Now place the Pitot tube in line with the airflow and at the center height of the test section. Take pressure measurements at several different frequency settings and record in a table. Probe height Center 1/3 height ~2cm Table 3: Manometer Readings for Select Angles Pitot Probe Orientation [deg] Manometer Readings Table 4: Manometer Readings from Frequency [Hz] Manometer Reading [mm]
For the Laboratory Result: 1. Plot velocity (with indicated uncertainty) as a function of input wind tunnel blower frequency. 2. Develop a calibration equation for the velocity as a function of blower speed. 3. Plot the velocity as a function of pitot probe orientation for all three positions within the wind tunnel. 4. Describe why the pressures change between the static and stagnation pressure ports and the reason for scaling differences (i.e. is the flow truly “uniform”?) 5. Hypothesize the influence of cross‐winds on aircraft pitot probes.