IMPACT OF JET AND ITS MATHEMATICAL EQUATIONS

 

 

IMPACT OF JET AND ITS Mathematical EQUATIONS

OBJECTIVE

This blog's purpose is to explore the reaction forces caused by a fluid flow's change in momentum when a water jet contacts a flat plate or a curved surface, and to compare the results by taking into account diverse instances.

INTRODUCTION

The liquid comes out in the form of a jet from the outlet of a nozzle, which is fitted to a pipe through which the liquid is flowing under pressure. If some plate, which may be fixed or moving is placed in the path of the jet, a force is exerted by the jet on the plate. This force is obtained from Newton’s second law of motion or from impulse-momentum equation. Thus impact of jet means the force exerted by the jet on a plate which may be stationary or moving. In this blog we’ll see the Impact of jet i.e the force exerted by the jet on a plate.

JET PROPULSION

 Jet propulsion means the movement of bodies, such as ships, aircrafts, rocket etc. with the help of jet of water discharging from rear side (backside of ship).

 When a jet of fluid coming out from a nozzle strikes a plate, the jet exerts a force on the plate. The magnitude of force exerted can be determined depending upon (1) the shape of plate, whether it is flat, inclined or curved and (II) the state of plate, whether it is stationary or moving. This force exerted by jet on the plate is called as "action of jet".

As per Newton's third law of motion, every action is accompanied by an equal and opposite

reaction.

 Hence, the jet while coming from nozzle, exerts a force on the nozzle itself in opposite direction of jet. This force acting on the nozzle in opposite direction is called as 'reaction of jet'.

 If nozzle is fitted in a body/housing like flexible hose pipe (free to move), then the body will

start moving in opposite direction of jet.

 Various Cases of Impact of Jet

 IMPACT OF JET ON A STATIONARY FLAT PLATE

Consider a jet of water coming out from the nozzle, strike a flat vertical plate

V= Velocity of Jet

d=diameter of the jet

a=area of the cross-section of jet = d²

 The jet after striking the plate, will move along the plate. but the plate is at right angle to the jet. Hence the jet after striking, will get defected through 90⁰c. Hence the Component of the velocity of the jet in the direction of jet, after striking will be zero.

 Force exerted on the jet on the plate

                                  =Rate Of Change Of Movement

                                  =(Initial Momentum - Final Momentum)/Time

                                  =(Mass *Initial Velocity-Mass*Final Velocity)/Time

                                  =Mass/Time[Initial Velocity-Final Velocity]

                                  = paV(V - 0)

                                  =paV2

IMPACT OF JET STRIKING ON A FLAT VERTICAL MOVING PLATE

Let

V = Velocity of jet, and

u = Velocity of plate

Velocity of jet relative to the plate = (V - u)                                                  in 'm/sec'

ass of water striking the plate per second = pa (V - u)                                  in 'kg/sec'

The jet is moving with an initial velocity (V - u) relative to the plate.

            Initial velocity of jet = (V - u)In 'm/sec'

Also, final velocity in the direction of jet is zero.

Force exerted by jet on the plate = [Mass of water striking plate per second] *[Change in Velocity]

                                                        = pa (V - u) *(Initial Final velocity velocity - Final Velocity)

                                                        = pa (V - u) x [(V - u) - 0]

                                                    F = pa(V - u)2                                             in 'N'

Also, work done by jet on the plate per second

                                                      = F*u

                         W.D. per second = pa(V - u)² *u                                         In 'J/sec'.

IMPACT OF JET ON A INCLINED FIXED PLATE/OBLIQUE IMPACT OF JET ON STATIONARY FLAT PLATE. 

 V= Velocity of jet in the direction of x

= Angle between the jet and Plate,

A=Area of cross-section of the jet.

 Then mass of water per sec striking the plate = p x aV.

If the plate is smooth and if it is assumed that there is no loss of energy due to impact of the jet. then jet will move over the plate after striking with a velocity equal to initial velocity i.e., with a velocity V. Let us find the force exerted by the jet on the plate in the direction normal to the plate. Let this force is represented by F

 Then

 F= = mass of jet striking per second

XI Initial velocity of jet before striking in the direction of n

Final velocity of jet after striking in the direction of n]

= paV [V sin θ-0] = pav² sin

This force can be resolved into two components, one in the direction of the jet and other perpendicular to the direction of flow. Then we have,

F= component of F, in the direction of flow

= F, cos (90° - θ) = F,, sin θ= PA V² sin ex sin θ (.. F,, = pav² sin 9)

= PAV² sin²θ

And,

F= component of F,, perpendicular to flow

= F,sin(90°-θ)=F,cosθ=PAV² sin cosθ.

Force Exerted by a Jet on Stationary Curved Plate

(A)Jet strikes the curved plate at the centre: Let a jet of water strikes a fixed curved plate at the centre as shown in Fig, the jet after striking the plate, comes out with the same velocity if the plate is smooth and there is no loss of energy due to impact of the jet, in the tangential direction of the curved plate. The velocity at outlet of the plate can be resolved into two components, one in the direction of jet and other perpendicular to the direction of the jet.

Component of velocity in the direction of jet = - V cos  θ.

 

(-ve sign is taken as the velocity at outlet is in the opposite direction of the jet of water coming out from nozzle).

 Component of velocity perpendicular to the jet = V sin Force exerted by the jet in the direction of jet,

F₁ = Mass per sec x [V₁x -V2]

where

V1x= Initial velocity in the direction of jet = V

V2x = Final velocity in the direction of jet = - V cos θ

F= paVIV (V cos θ)] = paV[V + V cos θ]

= pav²[1 + cos θ]

F₁ = Mass per sec x [V₁y - V₂y ]

Similarly,

where

Viy = Initial velocity in the direction of y = 0

Vay = Final velocity in the direction of y = V sin θ

F₁ = paV[0 - V sin 0] = − pav² sin θ

-ve sign means that force is acting in the downward direction. In this case the angle of deflection = (180° - θ)

(B)Jet strikes the curved plate at one end tangentially when the plate is symmetrical. Let the jet strikes the curved fixed plate at one end tangentially as shown in fig Let the curved plate is symmetrical about x-axis. Then the angle made by the tangents at the two ends of the plate will be same.

 

 Let V = Velocity of jet of water.

θ =Angle made by jet with x-axis at inlet tip of the curved plate.

If the plate is smooth and loss of energy due to impact is zero, then the velocity of water at the outlet tip of the curved plate will be equal to V. The forces exerted by the jet of water in the directions of x and y are

F= (mass/sec) x [Vix - Val

paVIV cos θ -(-V cos θ)]

paVIV cos + V cos θ] =2pa² cos θ

Fy=paVIV₁y - V₂yl

=paV[ V sin θ-V sin θ ]=0

 (C)Jet Strikes the curved plate at one end tangentially when the plate is unsymmetrical.

When the curved plate is unsymmetrical about x-axis, then angle made by the tangents drawn at the inlet and outlet tips of the plate with x-axis will be different.

Let

θ = angle made by tangent at inlet tip with x-axis,

Φ= angle made by tangent at outlet tip with x-axis.

The two components of the velocity at inlet are

V₁ = V cos 0 and V₁ = V sin θ

The two components of the velocity at outlet are

V2=-V cosΦ and V₂ = V sinΦ

The forces exerted by the jet of water in the directions of x and y are

F= paVIVIxV2] = paV[V cos Φ - (-V cos Φ)]

= paV[V cos θ + V cos ] = pav² [cos Φ + cos ]

Fy = pаVIV₁y - V₂y] = paV[V sin θ - V sinΦ] = pav² [sin θ – sinΦ ].

Impact of Jet Striking Unsymmetrical Moving Curved Vanes, Tangentially at One End

If 'V' is the velocity of jet at inlet and the curved plate or vane is moving with a velocity 'u', then the jet will be striking the curved vane with relative velocity, V, = V-u.

α = Guide blade angle or Nozzle angle i.e. angle between direction of motion of jet at inlet and direction of motion of vane.

θ = Vane angle at inlet i.e. angle made by the tangent to the vane at inlet and direction of motion of vane.

      =Angle made by relative velocity with direction of motion of vane at inlet. Vw = Velocity      of whirl at Inlet 1.e. component of absolute velocity of water (V) in the direction of jet = V cos α

Vw= Velocity of flow at inlet or radial velocity at inlet i.e. component of absolute velocity of water (V) in the direction normal to the motion of vane = V sin  

β= Angle made by absolute velocity (V.) with direction of motion of vane at outlet.

Φ= Vane angle at outlet i.e. angle made by tangent to the vane at outlet with direction of motion of vane.

 = Angle made by relative velocity with direction of motion of vane at outlet.

Vw1 = Velocity of whirl at outlet i.e. component of absolute velocity (V1) of water at outlet in the direction of motion of vane = V1 cos β.

 Vf₁ = Velocity of flow of outlet i.e. component of absolute velocity (V1) of water at outlet in the direction normal (perpendicular) to the direction of motion of vane =V1 sin β.

Fig shows the inlet and outlet velocity triangles showing Impact of jet in this case. The procedure to draw triangles is explained in subsequent Section

Conclusion

At the end we are able to conclude that

·       We can apply impulse momentum equation to the given geometry of vanes and find equation for force and work done. We can Calculate force exerted by a jet, work done and efficiency for the given vane and data. We can draw velocity diagram for given curved vane with special reference to hydraulic machine, such as pumps and turbines.

 

 

Contributors:

 

1.Yash Tadge

2.Prajwal wani

3.Uday Valvi

4.Veersing pardeshi

5.Yash thorat

6.Devashri zope