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Applications of Bernoulli’s Equation

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Bernoulli’s equation is used to find out the measurement of various fluids’ (liquid and air) flowing rate with the assistance of certain flow rate measuring devices. Because of this aid, application of this equation is utilized in many fields.

3 of the most important measurement devices used for a fluid’s flow rate measurement are described here.

  1. Pitot tube

For the measurement of the flow rate of a flowing fluid moving with a certain velocity, this device (Pitot tube) is used. This device has an ‘L’ shaped structure whose one end is open and is positioned above the fluid in the air. You will find the other side of the pitot tube kept immersed in that fluid. The positioning of the other side of the tube is about middepth point of it. This is done to measure the accurate velocity of that liquid or fluid.

To understand the working of this device in a better way, we will go through these following equations.

For that, we have to assume 2 points inside the flowing liquid before the application of Bernoulli’s equation. As stated, one point will be in the liquid, and the other will be at the mouth of the submerged sectionof the Pitot tube.

From that, we will get the equation,

P1 / p g + 1v2 / 2 g + z1 = p2 /p g + 2v2 / 2 g + z2

Now, at the beginning pointof the mouth of the tube inside the liquid, thekinetic head is converted so that it becomes pressure head.

So after conversion, its equation stands to be,

P1 /p g + 1v2 / 2g = p2 / p g

This equation can also be shown as,

H + v12/ 2 g = H + h

According to this equation, v1 = 2 g h

Here the action of ‘v’ is C v v1= cv 2gh

In this case, ‘cv’ is the coefficient of velocity for Pitot tube

Now, Oact in equational term will be,

Q act = Cv a 2 g h

Where, ‘a’ means cross sectional area of that Pitot tube.

Venturimeter

This device is also utilized for measuring the rate of flow of any fluid that flows through a pipe. Its unit is m3/s. Venturimeter is made up of 3 parts.

  • The initial is a short converging part
  • The second part is called throat, and
  • The third is known as diverging part

With the help of a diagram depicting the Venturimeter, we will apply Bernoulli’s equation to 2 portions of this device. The first is in the inlet section of the pipe, and the second portion is the divergent part. Throat becomes the barrier that divides both the portions.

On application of this Bernoulli’s equation in both sections we will get,

P1 / p g + v1 / 2 g + z1 = p2 / p g + v2 / 2 g + z2

Suppose, this Venturimeter is kept is a horizontal position.

Because of this positioning, z1 = z2

So the equation for this horizontal position will be,

P1 / p g + v1 /2 g = p2 p g + v2 2 g

P1 –p2 / p g = v2 / p g + v2 / 2g

h = v2 – v2 / 2 g

Now from continuity equation,

a1 v1 = a2 v2

v2 = a1 / a2 v1 (equation 5.4)

Taking the value of v2 from the above equation we will get,

= (1a2 – 2a2 / 2a2) 1v2 / 2 g

1V2 = 2a2 / 1a22 a2

Its discharge value is calculated as,

Q th = a1 v1

Q th = a1 v2 / 1a2

= 2a2

= 2 g h

It is through the equation of this expression that you can find out the value of theoretical discharge.

This value is of theoretical discharge because in case of actual discharge there will be some type of losses.

This loss is represented by,

Q act = Cd x Q th

where, Cd is known as the coefficient of discharge for a Venturimeter.In this case, the value of Cd is less than 1.

Now, in case of using a different manometer, expression for this device’s horizontal positioning is,

h = x [S h / So – 1]

In this different manometer, certain values describe the workings and equation of this Venturimeter.

S h = special gravity of heavier liquid

x = difference of heavier liquid inside the u-tube

S0 = special gravity of liquid flowing through pipe of Venturimeter

In this case, S0is greater than S h.

Now, in a certain instance if S h is greater than S0, then expression of ‘h’ will change, and its new expression will be,

h = x [1 – St / So]

Here,

x = difference of lighter liquid inside the u-tube.

S1 = special gravity of the lighter liquid

If this Venturimeter is inclined to a certain position, then,

z1 = z2

h = (p1 / p g + z1) – (p2 / p g + z2)

= x [S h / So – 1]

Orit can also be stated as,

[1 – St / So] x

Orifice Plate or Orifice meter

In case you need to measure the rate of flow of a fluid which flows with restriction or with a reduced pressure, an orifice plate or orifice meter is an apt device. With it, you can determine its mass flow rate or volumetric rate.

For an effective measurement, a circular sharp edge plate is generally utilized in this device.

Before we apply Bernoulli’s equationto this device, we will divide it into 2 sections.

  1. Minimum cross section at vena contracta, which is the half circular lower section of this device
  2. Upper half of this device from where the fluid flows. It is basically that section where the orifice plate is positioned.

After the application of Bernoulli’s equation in both the sections, the equation stands at,

P1 / p g + 1v2 / 2 g + z1 = p2 / p g + 2v2 / 2 g + z2

On simplifying this equation,

(p1 / pg + z1) – (p2 / p g + z2) = 2v2 /2 g – 1v2 / 2g

Now,

h = 2v2 / 2 g – 1v2 / 2 g

a2 v2 = a1 v1

v2 = a1 / Cc a o v1

Here C represents coefficient of contraction

From the above, the new equation will be,

h = (a12/(Cc2 .  a02)-1)  v12/(2 g) = ((a12-Cc2-Cc2a02)/(Cc2 .  a02))  v12/(2 g)

v12 = ((Cc2 .  a02)/(a12- Cc2 .  a02))2 gh

v1 = ((Cc .  a0)/√(a12- Cc2 .  a02)) √(2 g h)

Q = a1v1

= a1 x ((Cc .  a0)/√(a12- Cc2 .  a02)) √(2 g h)

Q = ((a1 .  Cc .  a0)/√(a12- Cc2 .  a02)) √(2 g h)

On simplifying the entire equation with the help of the relation between Cd and C, we will get the following equation.

√(a12 – Cc .  a02) =  Cc/Cd √(a12-a02)

So, Q = (Cd .  a0 .  a1  √(2 g h))/√(a12-a02)

Here, Cd (orifice) is way less than Cd (Venturimeter).

It is represented as Cdorifice<< Cd Venturimeter.

Cd is the coefficient of discharge for this device (orificemeter).

 

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