Problem 4.153 - Problem 4.153 [Difficulty: 4] Given: Data...

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Problem 4.153 [Difficulty: 4] Given: Data on vane/slider Find: Formula for acceleration, speed, and position; plot Solution: Apply x momentum Assumptions: 1) All changes wrt CV 2) Incompressible flow 3) Atmospheric pressure in jet 4) Uniform flow 5) Constant jet area The given data is ρ 999 kg m 3 = M3 0 k g = A 0.005 m 2 = V2 0 m s = k 7.5 Ns m = Then k UM a rf u 1 ρ VU () A [] u 2 m 2 + u 3 m 3 + = where a rf dU dt = u 1 = u 2 0 = u 3 0 = Hence k U M dU dt ρ 2 A = or dU dt ρ 2 A M kU M = The acceleration is thus a ρ 2 A M M = The differential equation for U can be solved analytically, but is quite messy. Instead we use a simple numerical method - Euler's method Un 1 + U n ρ n 2 A M kUn M t + = where t is the time step For the position x dx dt U = so xn 1 + x n () Un t + = The final set of equations is 1 + U n ρ n 2 A M M t + = an ρ n 2 A M M = + x n t + =
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The results can be plotted in Excel t (s ) x (m) U (m/s) a (m/s 2 ) 0.0 66.6 0.1 6.7 28.0 0.2 0.7 9.5 16.1 0.3 1.6 11.1 10.5 0.4 2.7 12.1 7.30 0.5 3.9 12.9 5.29 0.6 5.2 13.4 3.95 6.6 13.8 3.01 0.8 7.9 14.1 2.32 0.9 9.3 14.3
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Problem 4.153 - Problem 4.153 [Difficulty: 4] Given: Data...

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