Question

20 applications of conservation of linear momentum

Answer 1

Example: Application of linear momentum equation


A straight, 100 metre long, rigid, horizontal pipe of 0.25 m inner diameter discharges water into atmospheric air (P2 = 1 atm = 101 kPa). What should be the pressure at the inlet to the pipe if the flow rate at an instance is 0.01 m3/s and it increases at a rate of 0.1 m3/s2, Assume water to be incompressible and the frictional effects to be negligible. The density of water may be taken as 1000 kg/m3.



Solution:



Since the pipe is rigid, uniform in cross-section and water is incompressible, from conservation of mass, at any instance the rate at which water enters the CV must be same as the rate at which it leaves and the velocity V is constant throughout, i.e.,



A1V1 = A2V2 = AV (1)



Now from conservation of linear momentum:



(2)



since frictional and body forces are negligible, the net force acting on the C.V in x-direction is given by:



(3)



and the net momentum flux, out of the C.V is zero mass flow rate and velocity at inlet and outlet to the C.V are same at any instance, i.e.,



(4)

(5)



From the given data r is 1000 kg/m3, L is 100 m, (dQ/dt) is 0.1 m3/s2 , P2 is 101 kPa and D is 0.25 m. Substituting these values and in equations 2 to 5, we obtain:



P1 = 304.7 kPa (Ans.)