MECHANICS of FLUIDS LABORATORY - Mechanical Engineering

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m · 2 s ρ 2 2 A 2 2 ⎝ ⎛ 1 - ρ 2 2 A 2 2 ρ ⎠ ⎞ 2 1 A 2 = 2 R γ 1 γ - 1 T 1 ⎢ ⎡ p 1 - ⎛ 2 ⎞ ⎣ ⎝ ⎠ γ - 1 γ p 1 For an ideal gas, we write ρ = p/RT. Substituting for the RT 1 term in the preceding equation yields · m s 2 A 2 2 = 2ρ 2 2 γ γ - 1 ⎝ ⎛ p 1 ρ 1 (γ - 1)/γ ⎠ ⎞ 1 - (p 2 /p 1 ) 1 - (ρ 2 2 A 2 2 /ρ 2 1 A 2 1 ) For an isentropic process, we can also write or p 1 ρ 1 γ = p 2 ρ 2 γ p ρ 2 = ⎛ 2 ⎞ ⎝ p 1 ⎠ 1/γ ρ1 from which we obtain ρ 2 p 2 = ⎛ 2 ⎞ ⎝ ⎠ p 1 2/γ ρ1 2 Substituting into the mass flow equation, we get after considerable manipulation Equation 16.1 of Table 16.1, which summarizes the results. Thus for compressible flow through a venturi meter, the measurements needed are p 1 , p 2 , T 1 , the venturi dimensions, and the fluid properties. By introducing the venturi discharge coefficient C v , the actual flow rate through the meter is determined to be · m ac = C v ·ms Combining this result with Equation 16.1 gives Equation 16.2 of Table 16.1. It would be convenient if we could re-write Equation 16.2 in such a way that the compressibility effects could be consolidated into one term. We attempt this by using the flow rate equation for the incompressible case multiplied by another coefficient called the compressibility factor Y; we therefore write m · 2(p ac = C v Yρ 1 A 2 √⎺⎺⎺⎺⎺⎺ 1 - p 2 ) ρ 1 (1 - D 4 2 /D 4 1 ) We now set the preceding equation equal to Equation 16.2 and solve for Y. We obtain Equation 16.3 of the table. The ratio of specific heats γ will be known for a given compressible fluid, and so Equation 16.3 ⎦ ⎥⎤ could be plotted as compressibility factor Y versus pressure ratio p 2 /p 1 for various values of D 2 /D 1 . The advantage of using this approach is that a pressure drop term appears just as with the incompressible case, which is convenient if a manometer is used to measure pressure. Moreover, the compressibility effect has been isolated into one factor Y. Orifice Meter The equations and formulation of an analysis for an orifice meter is the same as that for the venturi meter. The difference is in the evaluation of the compressibility factor. For an orifice meter the compressibility factor is much lower than that for a venturi meter. The compressibility factor for an orifice meter cannot be derived, but instead must be measured. Results of such tests have yielded the Buckingham equation, Equation 16.4 of Table 16.1, which is valid for most manometer connection systems. Calibration of a Meter Figures 16.4 and 16.5 show how the apparatus is set up. An axial flow fan is attached to the shaft of a DC motor. The rotational speed of the motor, and hence the volume flow rate of air, is controllable. The fan moves air through a duct into which a pitot-static tube is attached. The pitot static tube is movable so that the velocity at any radial location can be measured. An orifice or a venturi meter can be placed in the duct system. The pitot static tube has pressure taps which are to be connected to a manometer. Likewise each meter also has pressure taps, and these will be connected to a separate manometer. A meter for calibration will be assigned by the instructor. For the experiment, make measurements of velocity using the pitot-static tube to obtain a velocity profile. Draw the velocity profile to scale. Obtain data from the velocity profile and determine a volume flow rate. For one velocity profile, measure the pressure drop associated with the meter. Graph volume flow rate as a function of head loss ∆h obtained from the meter, with ∆h on the horizontal axis. Determine the value of the compressibility factor experimentally and again using the appropriate equation (Equation 16.3 or 16.4) for each data point. A minimum of 9 data points should be obtained. Compare the results of both calculations for Y. 42
TABLE 16.1. Summary of equations for compressible flow through a venturi or an orifice meter. m · s = A 2 ⎨ ⎧ ⎩ 2p 1 ρ 1 (p 2 /p 1 ) 2/γ [γ/(γ - 1)] [1 - (p 2 /p 1 ) (γ - 1)/γ 1/2 ] 1 - (p ⎭ ⎬⎫ 2 /p 1 ) 2/γ (D 4 2 /D 4 1 ) (16.1) m · ac = C v A 2 ⎨ ⎧ ⎩ 2p 1 ρ 1 (p 2 /p 1 ) 2/γ [γ/(γ - 1)] [1 - (p 2 /p 1 ) (γ - 1)/γ 1/2 ] 1 - (p ⎭ ⎬⎫ 2 /p 1 ) 2/γ (D 4 2 /D 4 1 ) (16.2) Y = √⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺⎺ γ [(p 2 /p 1 ) 2/γ - (p 2 /p 1 ) (γ + 1)/γ ](1 - D 4 2 /D 4 1 ) γ - 1 [1 - (D 4 2 /D 4 1 )(p 2 /p 1 ) 2/γ ](1 - p 2 /p 1 ) Y = 1 - (0.41 + 0.35β 4 ) (1 - p 2/p 1 ) γ (venturi meter) (16.3) (orifice meter) (16.4) rounded inlet manometer connections motor pitot-static tube axial flow outlet duct fan FIGURE 16.4. Experimental setup for calibrating a venturi meter. venturi meter manometer connections rounded inlet motor axial flow fan outlet duct pitot-static tube orifice plate FIGURE 16.5. Experimental setup for calibrating an orifice meter. 43
Page 1 and 2: A Manual for the MECHANICS of FLUID
Page 3 and 4: TABLE OF CONTENTS Item Page Report
Page 5 and 6: as its strong points. Suggest exten
Page 7 and 8: EXPERIMENT 1 FLUID PROPERTIES: DENS
Page 9 and 10: EXPERIMENT 2 FLUID PROPERTIES: VISC
Page 11 and 12: L level R i y zeroing weight torus
Page 13 and 14: 5. Is a low value of the standard d
Page 15 and 16: pivot balancing weight lever arm wi
Page 17 and 18: Questions 1. Can a similar procedur
Page 19 and 20: actual flow rate (measured in the l
Page 21 and 22: manometer orifice meter venturi met
Page 23 and 24: otameter tank valve motor static pr
Page 25 and 26: Pressure Measurement Equipment A Wi
Page 27 and 28: EXPERIMENT 10 DRAG FORCE DETERMINAT
Page 29 and 30: Experiment II For a number of wings
Page 31 and 32: axis) versus (Q 2 /b 2 h 0 3 g). De
Page 33 and 34: Therefore, Q = ∫ 0 Integration gi
Page 35 and 36: Develop a hydraulic jump in the cha
Page 37 and 38: At any upstream (of the hump) locat
Page 39 and 40: EXPERIMENT 16 MEASUREMENT OF VELOCI
Page 41: Incompressible Flow Through a Meter
Page 45 and 46: placed on the torque arm to reposit
Page 47 and 48: ∆H = H 2 - H 1 = p 2 ρg + V 2 2
Page 49 and 50: Specific Speed A dimensionless grou
Page 51 and 52: 1 large orifice volume flow rate in

MECHANICS of FLUIDS LABORATORY - Mechanical Engineering

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