Tiêu đề 7 Rotating Vessels.pdf ✅

HGE 7: Rotating Vessels 1. Open Rotating Tanks. When a liquid mass is rotated about a vertical axis at a constant angular speed ω (in rad/s), every particle experiences a normal acceleration of an which is equal to v 2⁄r = ω 2 r, where r is the particle’s distance from the axis of rotation. This acceleration causes a centrifugal force equal to CF = man = mv 2 r tan θ = CF W = mv 2 r mg tan θ = v 2 gr = (ωx) 2 gx tan θ = ω 2x g From calculus, slope = dy dx = tan θ tan θ = dy dx dy dx = ω 2x g dy = ω 2x g dx Integrating both sides, ∫ dy = ∫ ω 2x g dx y = ω2x 2 2g Where: y = vertical distance of the particle being analyzed measured from the vertex. x = horizontal distance of the particle being analyzed measured from the vertex. At the edge of the tank, y = h and x = r, h = ω2r 2 2g Since the final water surface forms a paraboloid, it is important to recall its volume (see MSTC 77: Volumes using Integration). V = 1 2 πr 2h