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Home » GATE Study Material » Civil Engineering » The Application of Momentum Equation

The Application of Momentum Equation

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The Application of Momentum Equation

Application of the Momentum Equation

In this section we will consider the following examples:

  1. Force due to the flow of fluid round a pipe bend.
  2. Force on a nozzle at the outlet of a pipe.
  3. Impact of a jet on a plane surface.

1. The force due the flow around a pipe bend

Consider a pipe bend with a constant cross section lying in the horizontal plane and turning through an angle of .

 

 

Flow round a pipe bend of constant cross-section

Why do we want to know the forces here? Because the fluid changes direction, a force (very large in the case of water supply pipes,) will act in the bend. If the bend is not fixed it will move and eventually break at the joints. We need to know how much force a support (thrust block) must withstand.
 

Step in Analysis:

  1. Draw a control volume
  2. Decide on co-ordinate axis system
  3. Calculate the total force
  4. Calculate the pressure force
  5. Calculate the body force
  6. Calculate the resultant force


1 Control Volume

 

The control volume is draw in the above figure, with faces at the inlet and outlet of the bend and encompassing the pipe walls.
 

2 Co-ordinate axis system

It is convenient to choose the co-ordinate axis so that one is pointing in the direction of the inlet velocity. In the above figure the x-axis points in the direction of the inlet velocity.
 

3 Calculate the total force

In the x-direction:

 

In the y-direction:

 


 

4 Calculate the pressure force

 



 

5 Calculate the body force

There are no body forces in the x or y directions. The only body force is that exerted by gravity (which acts into the paper in this example - a direction we do not need to consider).
 

6 Calculate the resultant force

 

 


 

And the resultant force on the fluid is given by

 

 

And the direction of application is

 

the force on the bend is the same magnitude but in the opposite direction

 

2. Force on a pipe nozzle

Force on the nozzle at the outlet of a pipe. Because the fluid is contracted at the nozzle forces are induced in the nozzle. Anything holding the nozzle (e.g. a fireman) must be strong enough to withstand these forces.

The analysis takes the same procedure as above:

  1. Draw a control volume
  2. Decide on co-ordinate axis system
  3. Calculate the total force
  4. Calculate the pressure force
  5. Calculate the body force
  6. Calculate the resultant force

1 & 2 Control volume and Co-ordinate axis are shown in the figure below.

 

Notice how this is a one dimensional system which greatly simplifies matters.
 

3 Calculate the total force

 

By continuity, , so

 


 

4 Calculate the pressure force

 

We use the Bernoulli equation to calculate the pressure

 

Is friction losses are neglected,

the nozzle is horizontal,

and the pressure outside is atmospheric,,

and with continuity gives

 



 

5 Calculate the body force

The only body force is the weight due to gravity in the y-direction - but we need not consider this as the only forces we are considering are in the x-direction.
 

6 Calculate the resultant force

 

 

So the fireman must be able to resist the force of

 

3. Impact of a Jet on a Plane

We will first consider a jet hitting a flat plate (a plane) at an angle of 90, as shown in the figure below.

We want to find the reaction force of the plate i.e. the force the plate will have to apply to stay in the same position.

 

 

A perpendicular jet hitting a plane.

The analysis take the same procedure as above:

  1. Draw a control volume
  2. Decide on co-ordinate axis system
  3. Calculate the total force
  4. Calculate the pressure force
  5. Calculate the body force
  6. Calculate the resultant force

1 & 2 Control volume and Co-ordinate axis are shown in the figure below.

 

3 Calculate the total force

 

As the system is symmetrical the forces in the y-direction cancel i.e.

 


 

4 Calculate the pressure force.

The pressure force is zero as the pressure at both the inlet and the outlets to the control volume are atmospheric.
 

5 Calculate the body force

As the control volume is small we can ignore the body force due to the weight of gravity.
 

6 Calculate the resultant force

 

Exerted on the fluid.

The force on the plane is the same magnitude but in the opposite direction

 



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