All Problems Quiz4

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Concordia University *

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6281

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Mechanical Engineering

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Oct 30, 2023

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Your Answer Correct Answer Dirigibles attached to the ground by wires are often used for surveillance along the border. Winds up to 16 mph (7.15 m/s) are common and the drag forces are high. In order to determine the drag on a dirigible, a 1/15 scale model is built and tested in a water tunnel. For the dirigible shape, the drag force depends on the Reynolds number. Determine the water speed required to model the prototype. The drag measured on the model is 1.92 kN. Determine the corresponding drag on the prototype assuming that the model and prototype are dynamically similar. Assume standard conditions (1 atm, 20°C) for both the air and water properties. D= 523.993249516935 N

Solution For dynamic similarity, the dimensionless pi groups for the model and the prototype must have the same value. For the dragon the dirigible, the two pi groups are the drag coefficient and the Reynolds number I} = Cp and I, = Re The pi groups for the model and prototype are equal for dynamic similarity Hl,m = H1 and H2,m = H2 The drag coefficient is then a function of Reynolds number _ @ pVe Cp = ¢ (Re) or —%pvzfl = ¢<—” ) The Reynolds numbers is the same for model and prototype: P Vnln _ pVC Hom T The velocity of the water required to match the Reynolds number of the dirigible in air is Ay vV, = m P Cm B The velocity of 16 mph corresponds to 7.15 m/s. The density and viscosity of the air and water at standard conditions are from Tables B.2 and B.4. ke Ns Airp = 12045, 4 = 1.82x107° — m- m ke Ns Water: p = 9982 —5 4 = 1.003x 107 —- m- m The velocity for the model must then be kg 1204 — Vo= VoLl oggsm, w5 o P n 9982 = 182x107° = For the same Reynolds number, the drag coefficient is the same for both the model and the prototype. The relation is 2 _ D SV S0 Vit The drag on the prototype will then be kg 2 2 2 1204 = 7152 LV L —192x10° N- — < =) 152 = 524N 50 K& 7132

Your Answer Correct Answer ¥ Your answer is incorrect. A 1/12 scale model of an airfoil is tested in a wind tunnel at a velocity of 125 ft/s, a temperature of 70 F, and at 6 atmospheres pressure. The chord length of the model airfoil is 1.6 ft. Determine the Reynolds number for the tests. Determine the air speed of the prototype when flown at the same dynamic conditions in air at standard conditions. V= 142 ft/s

Solution A 1/12 scale model of an airfoil is tested in a wind tunnel at a velocity of 125 ft/s, a temperature of 70 F, and at 6 atmospheres pressure. The chord length of the model airfoil is 1.6 ft. Determine the Reynolds number for the tests. Determine the air speed of the prototype when flown at the same dynamic conditions in air at standard conditions. Solution The dynamic conditions are the same if the Reynolds numbers are the same. The model Reynolds number is given as Vi P Em Hm Re, = where ¢, is the model chord length. The density is determined from the ideal gas relation, where the temperature is in °R. 6 14696)& . 144 f2 slu; p=L =T _ 001397 2 RT 1716 —2— 529.7°R fr slig "R The Reynolds number for the model is then 1258.0.01397 2 .1 6t v, 17 S 3 Re, = —oetn = = = 7.31-10° " 3.82:107 = The air speed of the prototype at the same Reynolds number is determined from V = Re £ pt At standard conditions, the air properties are p = 0.002329 % andy = 3.82- 1077 llf)lzs The air speed of the prototype is then 382107 22 V=Re £ =731-10° ———"— =625% pt 0.002329 = 19.2 ft %

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