Question

Problem 5.4 A short circular pipe with diameter D, has water (p,u) flowing through it from left to right at a volumetric flow rate Q. At the exit a plug with diameter D2 that partially blocks the water flow is inserted (to provide upstream pressure pı (gage) to the pipe and/or to reduce flow rate). Both upstream pressure and flow rate can be measured. (a) Using the conservation equations (mass, momentum) in control volume formulation, derive an expression for the force required to hold the plug in place in terms of known quantities. State all the assumptions you are making. (b) For D2 = 0.1m, D2 = 0.075m, p = 1000 kg/m3, upstream p1 = 500 kPa and Q = 0.1 m²/s, calculate the magnitude of this force. 1_ _01_-.-.-.-D2]._._ plug pipe Patm Figure 3: Sketch of pipe with plug of diameter D2 at outlet on right.

Please show all the steps and calculations, thanks

Answer 1

I loi ALIVIIDKF - Pato Derivation of expression for forcelf) assumption : - m ūs pipe surface is smooth. fluid is Ideal, Incomprensible, non-viscurs. By contunity equation Q = A,V, = AzV2 - 77 2² v = (0,2-02) v Q=D2V, = (0,2-022) .- 0 By momentum equation 3 - rate of Force = "change in momentum in that direction 3 P, A, F= SQV₂-fQV, F = PI ID? - f Q(V2 -V,) - © put equation in equation ④ = F = P, T, ²-SQ 1 D2-1) D2-D2 TF DER DD- savil om Free PD te gaurile contes

6 Force F = I ligi - sav ( 22 Force DR F = ) Given; D=0.1m, D₂ = 0.075m. f = 1000 Ksim?, P = 500 *103N Q = 0.1 m )sec ol = 12.73 miss - 12.7 AL ñ *0.12 F= x 500 x103 x 0.2 1000 x 0.1% 12.73 = 2288.28 N -12-28 KN