An incompressible fluid is flowing through a vertical pipe with a constriction.

Question

An incompressible fluid is flowing through a vertical pipe with a constriction. The wide section is 2.00cm in diameter and is at the top of the pipe. The pressure of the fluid in the wide section at the top is 200 kPa. The velocity of the fluid in the wide section is 4.00m/s. the narrow section is located 4.00m below the wide section. What is the diameter of the narrow section for the pressure of the fluid in the narrow section to equal the pressure in the wide section? ( density of the fluid is 1000 kg/m^3)

Explanation / Answer

The pressure P1 = 200 kPa The diameter of the wide section d1 = 2.0 cm The speed in the wide section is v = 4.0 m/s The heigth variation h = 4.0 m when the pressure are equal by applying Bernoulis thorem        P1 + (1/2) v1^2 + gh1 = P2 + (1/2)v2^2 + gh2                0.5 v1^2 + g(h) = 0.5 v2^2                  0.5(4m/s)^2 + (9.8 m/s^2)(4) = 0.5 v2^2                         39.2 + 8 = 0.5 v^2 then   v = 9.72 m/s From continuity equation           A1 v1 = A2 v2           (d1/2)^2 v1 = (d2/2)^2 v2              d1^2 v1 = d2^2 v2 Therefore the diameter of the narrow section              d2 = [ (v1/v2) d1^2 ]                   = [ 4/9.72)] (2cm)                     = 1.28 m              

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