Fluid Mechanics Applications/B-34: Sudden contraction
Introduction
[edit | edit source]The sudden contraction in the area/diameter of a fluid jet after it emerges from a circular aperture in a pressurized reservoir is called as vena contracta. Coefficient of contraction is the ratio of the cross sectional area of the jet at the vena contracta to the area of the orifice. The typical value may be taken as .64
Checking the vena contracta
[edit | edit source]Assumption P1A1v1 be the Pressure , Area and velocity of fluid in the tank,
P2A2v2 be the Pressure , Area and velocity of fluid in the orifice,
P3A3v3 be the Pressure , Area and velocity of fluid at the vena contracta,
Vena contracta plays a very important role in the minor losses in pipes.
The diameter of the vena contracta is nearly equal to .64 times the diameter of the original hole . Assume fluid is incompressible
using continuity equation
v1A1 =v2A2 (1) and v1 << v2 (2)
Ignoring energy losses due to viscosity,Bernoullis' holds for points along streamlines
P1 +1/2 d v12 = P2 +1/2dv22 (3)
where d is the(constant) density of fluid and P is the pressure using equation (2) and (3)
v22 =2( P1 - P2 )/d (4)
Considering momentum in the system, the mass flux =dvA so, momentum flux = dv2A The net flux bounded by area A1 and A2
dp/dt = d(v22A2-v12A1)=dv22A2 (5)
F ≈ P1A1 - [P1(A1 - A2)] = (P1 - P2)A2 (6)
equating to equation (5)
v22= (P1-P2)/d (7)
This is the contradiction with equation (4) based on conservation of energy. Then according to Torricelli this contradiction is resolved in nature by a contraction of fluid to area A3 after it passes through A2 The momentum flux is actually
dp/dt = d(v32A3-v12A1) ≈ dv32A3 ≈ 2P1A3 (8)
According to Bernoulli's equation in the limit that P3 << P2 The force that causes this change is now
F ≈ P1A1 - [P1(A1 - A2) + P3A3] = (P1A2 - P3A3) ≈ P1A2 (9)
we estimate vena contracta to be A3= A2/2
[2] http://www.physics.princeton.edu/~mcdonald/examples/vena_contracta.pdf