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Problem 4.190 - Problem 4.190[Difficulty 3 NOTE ERROR...

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Problem 4.190 [Difficulty: 3] NOTE ERROR: Retarding torque is 0.05 N.m! Given: Data on rotating spray system Find: Differential equation for motion; steady speed; troque to stop Solution: Basic equation: Rotating CV Assumptions: 1) No surface force; 2) Body torques cancel; 3) Steady flow; 5) Uniform flow; 6) L<<r The given data is Q 15 L min = R 225 mm = d 5 mm = ρ 999 kg m 3 = T 0.05 N m = For each branch V 1 2 Q π 4 d 2 = V 6.37 m s = A π 4 d 2 = A 19.6 mm 2 = The basic equation reduces to a single scalar equation (FOR EACH BRANCH) T 2 V r 2 ω V × r × α r × + ( ) × ρ d A r V xyz ⎯⎯ × ρ V xyz ⎯⎯ d = where T is the retarding torque α is the angular acceleration But r 2 ω V × r × α r × + ( ) × 2 ω r V α r 2 + = (r and α perpendicular) The volume integral is then V r 2 ω V × r × α r × + ( ) × ρ d ω R 2 V α R 3 3 + ρ A = For the surface integral (FOR EACH BRANCH)
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