Recent Updates

Friday, 1 February 2019

FORCE EXERTED BY A JET ON STATIONARY CURVED PLATE

We have already seen the derivation of continuity equationBernoulli’s equationmomentum equationvelocity of sound in an isothermal process, velocity of sound in an adiabatic process, fundamentals of stagnation properties i.e. stagnation pressure, stagnation temperature and stagnation density for compressible fluid flow in our previous posts. 

We have also discussed the fundamentals of impact of jets, force exerted by a jet on vertical flat plate and force exerted by a jet on stationary inclined flat plate in our recent post. 

Now we will see here the derivation of expression of force exerted by a jet on stationary curved plate with the help of this post. Let us first brief here the basic concept of impact of jets and after that we will derive the expression of force exerted by a jet on stationary curved plate. 

Impact of jets 

Let us consider that we have one pipe through which liquid is flowing under pressure. Let us assume that a nozzle is fitted at outlet of pipe. Liquid which will come through the outlet of nozzle will be in the form of jet. 

If a plate, which may be moving or fixed, is placed in the path of jet, there will be one force which will be exerted by the jet over the surface of plate. The force which will be exerted by the jet over the surface of plate, which might be moving or fixed, will be termed as impact of jet. 

In order to determine the expression of force exerted by the jet over the surface of plate i.e. impact of jet, we will use the basic concept of Newton’s second law of motion and impulse-momentum equation. 

Force exerted by jet on stationary curved plate

We will see here three conditions as mentioned here.
  1. Jet strikes the curved plate at the center
  2. Jet strikes the curved plate at one end tangentially when the plate is symmetrical
  3. Jet strikes the curved plate at one end tangentially when the plate is symmetrical 


First we will see the case of Jet striking the curved plate at the center and we will secure the expression for force exerted by jet. 

Jet strikes the curved plate at the center 

Let us consider that a jet of water strikes a fixed stationary curved plate at its center as displayed here in following figure. 

Let us assume the following data from above figure. 

V = Velocity of the jet
d = Diameter of the jet
a = Area of cross-section of the jet = (π/4) x d2
θ = Angle made by jet with x- axis 

Let us assume that the plate is smooth and there is no loss of energy due to impact of water jet. Water jet, after striking the stationary curved plate, will come with similar velocity in a direction tangentially to the curved plate. 

We will resolve the velocity at the outlet of curved plate in its two components i.e. in the direction of jet and in a direction perpendicular to the jet. 

Component of velocity of water jet in the direction of jet = - V Cos θ 

We have taken negative sign because velocity at the outlet is in the opposite direction of the water jet coming out from nozzle. 

Component of velocity of water jet perpendicular to the jet = V Sin θ 

Force exerted by the water jet in the direction of jet 

FX = Mass per second x [V1x –V2x

Where,
V1x = Initial velocity in the direction of jet = V
V2x = Final velocity in the direction of jet = - V Cos θ 

FX = ρ a V x [V + V Cos θ]
FX = ρ a V2 x [1+ Cos θ] 

Force exerted by the water jet in the direction perpendicular to the jet 

FY = Mass per second x [V1y –V2y

Where,
V1y = Initial velocity in Y direction = 0  
V2y = Final velocity in Y direction = V Sin θ 

FY = ρ a V x [0 - V Sin θ] 
FY = - ρ a V2 Sin θ 

Jet strikes the curved plate at one end tangentially when the plate is symmetrical 

Let us consider that a jet of water strikes a fixed stationary curved plate at its one end tangentially, when the plate is symmetrical, as displayed here in following figure. 

Let us assume the following data from above figure. 
V = Velocity of jet
θ = Angle made by jet with x - axis at inlet tip of the curved plate 

Let us assume that the plate is smooth and there is no loss of energy due to impact of water jet. Water jet, after striking the stationary curved plate, will come at the outlet tip of the curved plate with similar velocity i.e. V in a direction tangentially to the curved plate. 

Force exerted by the water jet in the x - direction
FX = Mass per second x [V1x –V2x

Where,
V1x = Initial velocity in the x direction = V Cos θ
V2x = Final velocity in the x direction = - V Cos θ 

FX = ρ a V x [V Cos θ + V Cos θ] 
FX = 2 ρ a V2 Cos θ 

Force exerted by the water jet in the Y- direction 
FY = Mass per second x [V1y –V2y

Where,
V1y = Initial velocity in Y direction = V Sin θ
V2y = Final velocity in Y direction = V Sin θ 

FY = ρ a V x [V Sin θ - V Sin θ]
FY = 0 

Jet strikes the curved plate at one end tangentially when the plate is unsymmetrical 

Let us consider that a jet of water strikes a fixed stationary curved plate at its one end tangentially, when the plate is unsymmetrical, as displayed here in following figure. 

V = Velocity of jet
θ = Angle made by tangent with x - axis at inlet tip of the curved plate
ϕ = Angle made by tangent with x - axis at outlet tip of the curved plate 

Let us assume that the plate is smooth and there is no loss of energy due to impact of water jet. Water jet, after striking the stationary curved plate, will come at the outlet tip of the curved plate with similar velocity i.e. V in a direction tangentially to the curved plate. 

Force exerted by the water jet in the x - direction
FX = Mass per second x [V1x –V2x

Where,
V1x = Initial velocity in the x direction = V Cos θ
V2x = Final velocity in the x direction = - V Cos ϕ  

FX = ρ a V x [V Cos θ + V Cos ϕ]
FX = ρ a V2 [Cos θ + Cos ϕ] 

Force exerted by the water jet in the Y- direction
FY = Mass per second x [V1y –V2y

Where,
V1y = Initial velocity in Y direction = V Sin θ
V2y = Final velocity in Y direction = V Sin ϕ   

FY = ρ a V x [V Sin θ - V Sin ϕ]
FY = ρ a V2 x [Sin θ - Sin ϕ] 

Above equation, derived here, provides the components of force exerted by the liquid jet on the stationary curved plate for above three cases. 

Do you have any suggestions? Please write in comment box. 

Reference: 

Fluid mechanics, By R. K. Bansal 
Image courtesy: Google  

Also read  

No comments:

Post a Comment