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, C_d = 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

Streamline and equipotential line, 

Dimensional homogeneity, 

Buckingham pi theorem, 

Difference between model and prototype, 

Basic principle of similitude i.e. types of similarity, 

Various forces acting on moving fluid 

Model laws or similarity laws