Fluid Mechanics Applications/B-34: Sudden contraction

Introduction

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

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 v_1² = P₂ +1/2dv_2² (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

http://www.physics.princeton.edu/~mcdonald/examples/vena_contracta.pdf