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Unformatted text preview: Laboratory Report Experiment 4: Radial Flow between Parallel Disks Report Prepared By: Karan Shah Naman Kandoi Luke Malapira Nitin Bansal October 19, 2010 Lab Preparation Division: 06 ME 309L – Fluid Mechanics Laboratory Abstract: The Bernoulli Equation is a simple equation in fluid mechanics, which measures pressure, velocity, and elevation in a fluid. The objective of this experiment is to compare the measured pressure distribution for radial flow between parallel disks with the theoretical pressure distribution calculated from a uniform, inviscid flow model using the Bernoulli equation. Airflow is used in the experimental setup to set the stagnation pressure to 215 mm H2O. The actual pressure distribution (gage) between the disks is measured as a function of radius by moving the slider. The slider is incremented by 1/16th of an inch to get consistent reading. After taking readings, the location where the minimum gage pressure occurs is noted. The analysis to predict the gage pressure between the disks as a function of position gives the minimum gage pressure (Pmin). A plot of theoretical dimensionless pressure distribution and measured pressure distribution is made to compare theoretical pressure distribution calculated from inviscid flow with the measured pressure distribution for a radial flow between parallel disks. Finally, the experimental uncertainty of the measured values and the calculated results was calculated to account for the systematic and random errors during the experiment. Introduction In this experiment, an adjustable pressure tap is connected to a pressure transducer. A pressure transducer is an instrument that measures the pressure acting on it and converts it to electrical, magnetic or pneumatic signal. The slider on the pressure tap is increased by 1/16th of an inch each time a reading for the pressure between the parallel disks is taken. Fig. 1: Experiment Setup and Apparatus To predict the gage pressure between the discs as a function of the radial position i.e. to calculate p(r) – patm as a function of r, the expected pressure is normalized by the minimum gage pressure (pmin ­patm) and the radial position r, is normalized by the disk radius R as follows: ȹ r r ȹ p( r) − patm = f ȹ , min ȹ ȹ R R Ⱥ pmin − patm The only two factors that contribute to the uncertainty of the measured pressure distribution are pressure and radius. The general uncertainty principle is given by: € x ∂y uy i = i * * ux i y ∂x From this formula, the uncertainty of the pressure distribution is calculated as: uρ = ( uPmin ) 2 + ( uR min ) 2 where uρ = Uncertainty in €ressure distribution p uPmin = Uncertainty in pressure distribution due to minimum gage pressure uR min = Uncertainty in pressure distribution due to minimum radius € The following paper provides a comparison between the measured value of pressure € distribution and the theoretical values of the pressure distribution using plots, € schematics and other calculations. Flow from the top hose enters the control volume fully developed and exits through all sides of the disk. Fig. 2 shows the horizontal velocity profile assuming the no slip condition. The gist of this experiment is to show that laminar planar poiseuille flow is only run by the pressure distribution (pressure gradient), which is a function of varying radial position. € Materials and Methods Fig. 2: Velocity profile 1. 2. 3. 4. 5. The equipment required for this experiment is as listed below: Pressure Sensor (Pressure transducer) Slider measure Gauge Pressure monitor Pressure Valve Two parallel disks A pressure transducer is made of a sensing element which records the magnitude of the pressure acting on the sensor. Another part of the transducer converts this pressure into an electric, magnetic or pneumatic signal, which can be recorded and seen on an indicator. First, the least count of the slider ruler is determined. Then, the airflow is turned on by opening the pressure valve and is adjusted to a stagnation pressure of 215 mmH2O. This is the pressure when the slider on the pressure tap is at zero. The slider is moved by 1/16th inch increments from 0 in to 2 in and the pressure between the disks is measured at each increment. The location at which the pressure is minimum is determined as rmin. Refer to Fig. 1 for details. The gage pressure in between disks is determined as a function of radius by using conservation of mass principle to obtain all values in dimensionless form. A table of each pressure for corresponding radial position is made and then plotted on a graph. Results: The calculations and derivations for dimensionless theoretical data and the dimensionless measured data are provided in Appendix A. As seen in Fig. 3, the trend of the dimensionless theoretical data and that of the dimensionless measured data is quite different. From the data sheet attached in Appendix B, it is clear that the minimum pressure of  ­191.6 mm H20 occurs at 3/8 in. The dimensionless radius is calculated by dividing r by rmin. The dimensionless theoretical data is calculated from equation (1) from Appendix A. The dimensionless measured pressure is calculated by dividing the measured pressure by Pmin. From the plots as seen in Fig. 3, upto 3/8 in, the graph of the dimensionless theoretical data is decreasing while that of the dimensionless measured data is increasing. But after this point i.e. after the vena contracta, the graphs look alike. (Dimensionless) Pressure vs Radius 45 40 Pressure (Dimensionless) 35 30 25 20 15 10 5 0 0.00000 1.00000 2.00000 3.00000 4.00000 5.00000 6.00000  ­5 Radius (Dimensionless) Measured Theoretical From the sketches as seen in Fig. 3 and Fig. 4, the vena contracta occurs at the sharp corner. Since the flow hits a wall and is forced to go in a different direction, it causes stagnation pressure. This in turn lowers the pressure at the corner and causes there to be a turbulent region at the corner. This phenomenon is clear from the sketches. Fig. 3: Dimensionless Pressure vs. Dimensionless Radius Fig. 4: Pressure Distribution Fig. 4: Flow field with Stagnation point and Vena Contracta Appendix A For Control Volume I Conservation of mass: d ∫ ρdV + ∫ ρurel dA = 0 dt CV CS CS ∫u rel dA = 0 from Steady State r 2Vr = R 2VR Bernoulli Equation € For Control Volume II Conservation of Mass d ∫ ρdV + ∫ ρurel dA = 0 dt CV CS CS ∫u rel dA = 0 rmin 2Vmin = R 2VR Bernoulli Equation € Combining the two Bernoullis equations: Eq. (1) This equation gives the value of Normalized Theoretical Pressure as a function of radius Pmeasure, normalized = Pmeasured / Pmin rdimensionless = r / rmin Appendix B Calculations: Normalized theoretical pressure for r=1/16 in Ⱥ 2 Ⱥ 2 1 − Ⱥ Pr − Patm Ⱥ1 /16 Ⱥ Ⱥ = = 38.3177 Pr min − Patm Ⱥ 2 Ⱥ 2 1 − Ⱥ Ⱥ 3 / 8 Ⱥ Ⱥ Normalized measured pressure for p = 199.1 mm H2O = € P 199.1 = = −1.039 Pmin −191.6 € Dimensionless radius = r rmin Also, see attached excel sheet. € = 1 /16 1 = 3 /8 6 € Appendix D Uncertainty Calculations: ...
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