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Pitot Stagnation Tubes Under Laminar Conditions *P7.24 Jace Benoit February 15, 2007

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Problem Statement Air at 20°C and 1 atm flows past the flat plate in Fig. P7.24 under laminar conditions. There are two equally spaced pitot stagnation tubes, each placed 2 mm from the wall. The manometer fluid is water at 20°C. If U=15m/s and L=50cm, determine the values of the monometer readings h 1 and h 2. Determine these values if the manometer fluid is replaced with mercury. Fig. P7.24 Fig. P7.24

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Sketch Fig P7.24 u1u1 u2u2

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Assumptions Laminar Flat-Plate Flow Incompressible Flow Constant Temperature No Mixing at Liquid-Air Interface Frictionless Flow Steady Flow

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Solution The following gives a summary of the densities and kinematic velocity from tables A.1, A.2, and A.3. AirWaterMercury Density (ρ) 1.20 kg/m 3 998 kg/m 3 13,550 kg/m 3 Kinematic Velocity (v) 1.50 E-5 m2/s Not Required

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Solution Since this is laminar flow along a boundary layer, the Blasius equation (7.21) can be used to determine η.

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Solution Now that η is known, Table 7.1 on page 462 can be used to approximate u/U. For η = 2.8, u/U = 0.81152 and for η = 3.0, u/U = 0.84605, so for η 1 = 2.828: Fortunately, the u/U value corresponding to η = 2.0 can be read directly from Table 7.1 to be 0.62997.

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Solution Plugging U = 15 m/s into the u/U expression yields the velocities at the entrances of each manometer.

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Solution With the inlet velocities known, Bernoulli’s equation (3.77) can be used to solve for the inlet pressure assuming atmospheric pressure at the outlet of the manometer.

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Solution Rearranging this equations with the appropriate cancellations yields the following:

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Solution Now that the inlet and outlet pressures are known, this problem can now be solved as a manometer problem.

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Solution Finally, the manometer heights for water are as follows:

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Solution All that needs to be done to determine the height if mercury is the manometer fluid is changed the density in the previous equation to 13,550 kg/m 3.

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Application to BME The manometer with one end being submerged in water and the other in another medium can mimic a catheter which is used to measure various biological effects such as cardiac output and blood pressure. The catheter is filled with a saline solution similar to how this manometer is filled with water and mercury, and pressure is exerted at each ends of the catheter.

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Reference White, Frank M. Fluid Mechanics. 5th Ed. McGraw-Hill Companies, Inc.: New York. 2003.

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