Tuesday, 19 June 2018

MOMENT OF MOMENTUM EQUATION IN FLUID MECHANICS

We were discussing the various basic concepts such as Euler’s Equation of motionBernoulli’s equation from Euler’s equationderivation of discharge through venturimeter, derivation of discharge through Orifice meter and Pitot tube as well as resultant force exerted by flowing fluid on a pipe bend, in the subject of fluid mechanics, in our recent posts. 

Today we will see here the moment of momentum equation, in the subject of fluid mechanics, with the help of this post.

Moment of momentum equation

According to the concept of moment of momentum equation, Resulting torque acting on a rotating fluid will be equal to the rate of change of moment of momentum.

Let us consider that we have a rotating fluid and let us consider two sections in rotating fluid i.e. section 1 and section 2.

V1 = Velocity of fluid at section 1
r1 = Radius of curvature at section 1
V2 = Velocity of fluid at section 2
r2 = Radius of curvature at section 2
Q = Rate of fluid flow
ρ = Density of fluid

Let us find the momentum of fluid at section 1

Momentum of fluid per second at section 1 = Mass x velocity
Momentum of fluid per second at section 1 = ρ Q x V1
Moment of momentum per second at section 1 = ρ Q x V1 x r1

Similarly, Moment of momentum per second at section 2 = ρ Q x V2 x r2
Rate of change of moment of momentum = ρ Q x V2 x r2 - ρ Q x V1 x r1
Rate of change of moment of momentum = ρ Q [V2 x r2 - V1 x r1]

According to the concept of moment of momentum equation, Resulting torque acting on a rotating fluid will be equal to the rate of change of moment of momentum.

Resultant Torque = Rate of change of momentum    

T = ρ Q [V2 x r2 - V1 x r1]

Above equation will be termed as moment of momentum equation.

Applications of moment of momentum equation

Moment of momentum equation will be applied in following cases.

1. To determine the torque exerted by water on sprinkler
2. For analysis of flow problems in centrifugal pump and turbine

We will now find out the "Momentum equation",in the subject of fluid mechanics, in our next post.

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

Reference:

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

Also read


Continue Reading

Thursday, 14 June 2018

FORCE EXERTED BY A FLOWING FLUID ON A PIPE BEND

We were discussing the various basic concepts such as Euler’s Equation of motionBernoulli’s equation from Euler’s equationderivation of discharge through venturimeter, derivation of discharge through Orifice meter and Pitot tube with the expression of velocity of flow at any point in the pipe or channel, in the subject of fluid mechanics, in our recent posts. 

Today we will see here the resultant force exerted by flowing fluid on a pipe bend.

Force exerted by a flowing fluid on a pipe - bend

 
In order to secure the expression of resultant force exerted by a flowing fluid on a pipe bend, we will use the basic concept of impulse momentum equation. Before going ahead, it is very important to find out and read the concept of the momentum equation.

Let us consider that fluid is flowing through a pipe which is bent as displayed here in following figure. We have considered here two sections i.e. section 1-1 and section 2-2.


V1 = Velocity of fluid flowing at section 1
P1 = Pressure of fluid flowing at section 1
A1 = Area of section 1                                                     
V2 = Velocity of fluid flowing at section 2
P2 = Pressure of fluid flowing at section 2
A2 = Area of section   2
FX = Force exerted by the flowing fluid on the pipe bend in X-direction.
FY = Force exerted by the flowing fluid on the pipe bend in Y-direction. 

As we have considered above that Fx and FY are the forces, exerted by the flowing fluid on the pipe bend in X and Y direction respectively. 

Considering the Newton’s third law of motion, forces exerted by the pipe bend on the flowing fluid will be - FX and - FY in X and Y direction respectively. 

There will be some other forces also acting on the flowing fluid. P1A1 and P2A2 are the pressure forces acting on the flowing fluid at section 1 and section 2 respectively. 

 
Now we will recall the momentum equation and we will have following equation for X direction. 

Net force acting on fluid in X- direction = Rate of change of momentum in X- direction

P1A1 – P2A2 Cos θ – FX = Mass per unit time x change of velocity
P1A1 – P2A2 Cos θ – FX = ρ Q (Final velocity in X-direction – Initial velocity in X-direction)
P1A1 – P2A2 Cos θ – FX = ρ Q (V2 Cos θ – V1)
FX = ρ Q (V1 – V2 Cos θ) + P1A1 – P2A2 Cos θ 

Similarly, we will recall the momentum equation and we will have following equation for Y direction.

Net force acting on fluid in Y- direction = Rate of change of momentum in Y direction
– P2A2 Sin θ – FY = Mass per unit time x change of velocity
– P2A2 Sin θ – FY = ρ Q (Final velocity in Y-direction – Initial velocity in Y-direction)
– P2A2 Sin θ – FY = ρ Q (V2 Sin θ – 0)
– P2A2 Sin θ – FY = ρ Q (V2 Sin θ – 0)
FY = ρ Q (-V2 Sin θ) – P2A2 Sin θ 

Let us determine the resultant force (FR) acting on pipe bend and angle bend by the resultant force (FR) with horizontal direction.
                                
We will now find out the "Moment of momentum equation", in the subject of fluid mechanics, in our next post.

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

 

Reference:

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

Also read

Closed cycle gas turbine engine 
Continue Reading

Monday, 11 June 2018

MOMENTUM EQUATION FLUID MECHANICS

We were discussing the various basic concepts such as Euler’s Equation of motionBernoulli’s equation from Euler’s equation, derivation of discharge through venturimeter, derivation of discharge through Orifice meter and Pitot tube with the expression of velocity of flow at any point in the pipe or channel in the subject of fluid mechanics, in our recent posts. 

Today we will see here the concept of the momentum equation, in the subject of fluid mechanics, with the help of this post.

Momentum equation

Momentum equation is based on the law of conservation of momentum or on the momentum principle.

 
According to the law of conservation of momentum, net force acting on a fluid mass will be equivalent to the change in momentum of flow per unit time in that direction.

Force acting on a fluid mass (m) will be given by Newton’s second law of motion and we will have following equation as mentioned here.
F = m x a
Where, a is the acceleration of the fluid flow acting in the same direction as force F.

As we know that acceleration could be defined as the rate of change of velocity or we can write as mentioned here.

Above equation is termed as the law of conservation of momentum or on the momentum principle.
We can also write the law of conservation of momentum or on the momentum principle as mentioned here.
F. dt = d (mv)
 
Above equation will be termed as the impulse-momentum equation.

After considering above equation we can say that impulse of a force F acting on a fluid of mass m in a short duration of time dt will be equal to the change of momentum in the direction of force.

We will now find out the force exerted by a flowing fluid on a pipe bend, in the subject of fluid mechanics, in our next post.

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

 

Reference:

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

Also read

Continue Reading

PITOT TUBE WORKING AND PRINCIPLE

We were discussing the various basic concepts such as Euler’s Equation of motionBernoulli’s equation from Euler’s equation, derivation of discharge through venturimeter and derivation of discharge through Orifice meter, in the subject of fluid mechanics, in our recent posts. 

We had already seen the application of Bernoulli’s equation in the working principle of Venturimeter and orifice meter. Now we will go ahead to find out the other practical applications of Bernoulli’s equation, in the subject of fluid mechanics, with the help of this post.

Today we will see here the basic concept of Pitot tube and also we will secure here the expression of velocity of flow at any point in the pipe or channel with the help of this post.

Pitot Tube 

 
Pitot tube is basically defined as a device which is used for measuring the velocity of flow at any point in the pipe or a channel.

Working Principle of Pitot tube

Pitot tube works on the principle of Bernoulli’s equation. If the velocity of flow at a point decreases, pressure will be increased at that point due to the conversion of kinetic energy in to pressure energy.
Pitot tube will be made of a glass tube bent at right angle as displayed here in following figure. Lower end of Pitot tube will be bent at right angle and will be directed in upstream direction as displayed here.

Due to conversion of kinetic energy in to pressure energy, liquid will rise up in the glass rube. Rise of liquid level will provide the velocity of flow at any point in the pipe or a channel.

Derivation of veloctiy of flow through pitot tube

Let us consider one pitot tube as displayed here in following figure. Let us say that water is flowing through the horizontal pipe.
P1 = Pressure at section 1 (Inlet section)
v1 = Velocity of fluid at section 1 (Inlet section)
A1 = Area of pipe at section 1 (Inlet section)

P2 = Pressure at section 2
v2 = Velocity of fluid at section 2
A2 = Area at section 2
H = Depth of tube in the liquid
h = Rise of kiquid in the tube above the free surface.

Let us recall the Bernoulli’s equation and applying at section 1 and section 2.

According to Bernoulli’s theorem.....

In an incompressible, ideal fluid when the flow is steady and continuous, the sum of pressure energy, kinetic energy and potential energy will be constant along a stream line.

Assumptions

 
Assumptions made for deriving the expression for velocity of flow at any point in the pipe or channel is as mentioned here.
1. Fluid is ideal, i.e. inviscid and incompressible.
2. Fluid flow is steady and continuous
3. Fluid flow is irrotational
4. Frictionless inner surface

We will have following equation after applying Bernoulli’s equation at section 1 and section 2.
We will now find out the momentum equation, in the subject of fluid mechanics, in our next post.

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

 

Reference:

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

Also read

Continue Reading

Sunday, 10 June 2018

DISCHARGE THROUGH ORIFICE METER

We were discussing the various basic concepts such as Euler’s Equation of motion, Bernoulli’s equation from Euler’s equation and derivation of discharge through venturimeter, in the subject of fluid mechanics, in our recent posts. 

We had already seen the application of Bernoulli’s equation in working principle of Venturimeter. Now we will go ahead to find out the other practical applications of Bernoulli’s equation, in the subject of fluid mechanics, with the help of this post.
 
Today we will see here the basic concept of Orifice meter and also the derivation of discharge through orifice meter with the help of this post.

 

Orifice meter or Orifice plate

Orifice meter is basically defined as a device which is used for measuring the rate of flow of fluid flowing through a pipe. Orifice meter is also known as Orifice plate.

Orifice meter works on the principle of Bernoulli’s equation and continuity equation.

Orifice meter is less costly as compared to the venturimter. Venturimeter is also very reliable flow measuring device. There are some pressure losses in venturimeter and it is usually used in larger volume liquid and gas flows. Venturimeter is expensive due to complexity of its design. Therefore, in order to determine the rate of flow of fluid through small pipe lines, orifice meter is better to use as compared to venturimeter.

Orifice meter consists of one flat circular plate and this circular plate will have one circular sharp edge hole bored in it. The circular sharp edge hole is termed as orifice.

Diameter of orifice will be 0.5 times of diameter of pipe through which fluid is flowing, though it may vary from 0.4 to 0.8 times of diameter of pipe.

Orifice plate is installed in pipe between two flanges of pipe. Orifice will restrict the flow of fluid and will reduce the cross sectional area of flow passage. A differential pressure will be developed across the orifice plate. Due to creation of pressure difference, we will be able to determine the rate of fluid flow through the pipe.

Types of Orifice meter

 
Orifices are usually of concentric types i.e. orifice will be concentric with the pipe line. But, there are few more designs available as mentioned here
1. Eccentric orifice plate
2. Sharp edge orifice plate
3. Segmental orifice plate
4. Conical orifice plate

Derivation of rate of flow through Orifice meter

Let us consider one orifice meter fitted in a horizontal pipe as displayed here in following figure. Let us say that water is flowing through the horizontal pipe.

Let us consider two sections i.e. section 1 and section 2 as displayed here in following figure. A differential manometer will be connected as displayed in figure at section 1, which will be approximate 1.5 to 2 times the diameter of pipe upstream from the orifice, and at section 2, which will be approximate 0.5 times the diameter of the orifice on the downstream side from the orifice plate.
d1 = Diameter at section 1 (Inlet section)
P1 = Pressure at section 1 (Inlet section)
v1 = Velocity of fluid at section 1 (Inlet section)
A1 = Area of pipe at section 1 (Inlet section)
d2 = Diameter at section 2
P2 = Pressure at section 2
v2 = Velocity of fluid at section 2
A2 = Area at section 2

Let us recall the Bernoulli’s equation and applying at section 1 and section 2.

According to Bernoulli’s theorem.....

In an incompressible, ideal fluid when the flow is steady and continuous, the sum of pressure energy, kinetic energy and potential energy will be constant along a stream line.

Assumptions

Assumptions made for deriving the expression of discharge through the orifice meter is as mentioned here.
1. Fluid is ideal, i.e. inviscid and incompressible.
2. Fluid flow is steady and continuous
3. Fluid flow is irrotational
4. Frictionless inner surface

We will have following equation after applying Bernoulli’s equation at section 1 and section 2.
Let A0 is the area of the orifice
Co-efficient of contraction, CC = A2/A0
 
Let us recall the continuity equation and we will have following equation
Thus we will use the value of CC in above equation of discharge Q and we will have following result for rate of flow or discharge through orifice meter.
Co-efficient of discharge of the orifice meter will be quite small as compared to the co-efficient of discharge of the venturimeter.

We will now find out the Basic principle of Pitot-tube, in the subject of fluid mechanics, in our next post.

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

Reference:

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

Also Read

Continue Reading

Saturday, 9 June 2018

DERIVATION OF DISCHARGE THROUGH VENTURIMETER

We were discussing the various basics concepts such as Euler’s Equation of motion and Bernoulli’s equation from Euler’s equation, in the subject of fluid mechanics, in our recent posts.

Now we will go ahead to find out the various practical applications of Bernoulli’s equation, in the subject of fluid mechanics, with the help of this post. We will start here our discussion with the basic concept and working principle of “Venturimeter”.

Venturimeter

 
Venturimeter is basically defined as a device which is used for measuring the rate of flow of fluid flowing through a pipe.

Venturimeter works on the principle of Bernoulli’s equation and continuity equation.

Venturimeter consist of three parts as mentioned here

1. Converging Part
2. Throat
3. Diverging part

Velocity of accelerated fluid flow will be increased with the decrease in cross-sectional area of flow passage. Therefore, pressure will be reduced at this section and pressure difference will be created. Due to creation of pressure difference, we will be able to determine the rate of fluid flow through the pipe.

Types of Venturimeter

There are basically three types of venturimeter as mentioned here
1. Horizontal venturimeter
2. Inclined venturimeter
3. Vertical venturimeter

Derivation of rate of flow through Venturimeter

Let us consider one venturimeter fitted in a horizontal pipe as displayed here in following figure. Let us say that water is flowing through the horizontal pipe.
Let us consider two sections i.e. section 1 and section 2 as displayed here in following figure.
d1 = Diameter at section 1 (Inlet section)
P1 = Pressure at section 1 (Inlet section)
v1 = Velocity of fluid at section 1 (Inlet section)
a1 = Area at section 1 (Inlet section) = (П/4) x d12
d2 = Diameter at section 2
P2 = Pressure at section 2
v2 = Velocity of fluid at section 2
a2 = Area at section 2 = (П/4) x d22
Let us recall the Bernoulli’s equation and applying at section 1 and section 2.

According to Bernoulli’s theorem.....

In an incompressible, ideal fluid when the flow is steady and continuous, the sum of pressure energy, kinetic energy and potential energy will be constant along a stream line.

Assumptions

 
Assumptions made for deriving the Bernoulli’s equation from Euler’s equation of motion is as mentioned here.
1. Fluid is ideal, i.e. inviscid and incompressible.
2. Fluid flow is steady and continuous
3. Fluid flow is irrotational
4. Frictionless inner surface

We will have following equation after applying Bernoulli’s equation at section 1 and section 2.
Where,
(P1-P2) / ρg = h = Difference of pressure head at section 1 and section 2

Let us recall the continuity equation and applying at section 1 and section 2
a1v1 = a2v2
v1 = (a2v2)/ a1

Now we will use the value of v1 in above equation no. 1 and we will have following result as mentioned here

Rate of flow of fluid i.e. discharge

Rate of flow of fluid i.e. discharge will be determined with the help of following equation
Q = a2v2
Now we will use the value of v2 in above equation and we will have the equation for the rate of flow of fluid or discharge
Above equation is termed as equation for theoretical discharge. Actual discharge will be less than the theoretical discharge.
Where, Cd = Co-efficient of venturimeter and its value will be lower than the 1.

 
We will now find out the Basic principle of Orifice Meter, in the subject of fluid mechanics, in our next post.

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

Reference:

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

Also Read

Continue Reading

Archivo del blog

Copyright © HKDIVEDI.COM | Powered by Blogger | Designed by Dapinder