Air of constant density flows steadily through the rectangular duct shown below.

Question

Air of constant density flows steadily through the rectangular duct shown below. The duct has a constant width W into the page. At the entrance, the velocity is constant across the area and equal to Uav. The velocity at the exit has a parabolic distribution across the duct, with a maximum value Um, and it is constant in the direction into the page. The pressure is constant everywhere. By using the continuity equation, find the ratio b/a such that Um = Uav- Then find the force F exerted by the flow on the duct, assuming that the wall friction is negligible.

Explanation / Answer

using integral form of continuity equation

AUav - 2 x integral of UdA from 0 to b = 0

4a2Uav = 2[integral Um(1- (x/b)2) bdx

assuming Uav = Um, = constant

4a2 = 2[bx - (1/b) (x3/3)] from 0 to b

4a2 = 2 [b2 - b2/3]

4a2 = 4b2/3

b/a = 3

b/a = 1.73

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