Venturimeter Working , Application and its flow discharge formula derivation

Venturi meter

What is Venturimeter ?
A Venturi meter, also known as a Venturi tube, is a flow measurement device used to determine the flow rate of fluid in a pipe. It consists of a tapered tube with a constriction in the middle, creating a region of reduced cross-sectional area.

How Venturimeter Works ?
The basic principle behind the Venturi meter is the Bernoulli's equation, which states that as the fluid's velocity increases, the pressure decreases.

When fluid flows through a Venturi meter, it accelerates as it passes through the constricted section, resulting in a decrease in pressure. By measuring the pressure difference between the narrowest section (throat) and the upstream section, the flow rate can be determined.

Application of Venturimeter 
Venturi meters are commonly used in various applications, including water supply systems, HVAC systems, and industrial processes, to accurately measure the flow rate of liquids or gases. They offer several advantages such as low pressure drop, high accuracy, and minimal maintenance requirements.

Some Inside about Venturimeter Design 

When the length of the exit cone of a Venturi meter is larger than the length of the entrance cone, it is referred to as a divergent or expanding Venturi meter. This design variation aims to minimize the pressure recovery loss that occurs in the standard convergent Venturi meter.

In a divergent Venturi meter, the flow passage gradually expands after the throat section, allowing the fluid to transition smoothly to the original pipe diameter. The purpose of the divergent cone is to recover some of the pressure that was lost in the convergent section, resulting in a higher pressure at the exit compared to the convergent Venturi meter.

The longer length of the exit cone allows for a more gradual pressure recovery process, reducing the intensity of the pressure recovery shock and minimizing disturbances in the flow. This design variation helps to prevent flow separation and turbulence, resulting in improved accuracy and efficiency of the flow measurement.

Divergent Venturi meters are commonly used in applications where pressure recovery is crucial, such as in systems with sensitive downstream equipment or where energy conservation is a priority. By incorporating a longer exit cone, these meters can provide more accurate and reliable flow rate measurements while minimizing energy losses.

It's important to note that the design and dimensions of a Venturi meter, including the length of the entrance and exit cones, are carefully engineered based on the specific application requirements and desired performance characteristics. Different designs may be employed depending on factors such as fluid properties, flow rates, and system conditions.

Venturimeter Formula Derivation

Let `d_{1}` and `d_{2}` be the inside diameter of venturimeter at inlet and outlet respectively.

Let `p_{1}` and `p_{2}` be the pressure at the section 1 and 2 respectively.
Let `v_{1}` and `v_{2}` be the velocity of the fluid at section 1 and 2 respectively.
Let `a` be the area at the section (1) be `\frac{\pi}{4}d_{1}^{2}`

Apply Bernoulli's equation at section (1) and section (2) , we get
`\frac{p_{1}}{\rho g} + \frac{v_{1}^{2}}{2g}+ z_{1} = \frac{p_{2}}{\rho g} + \frac{v_{2}^{2}}{2g}+ z_{2}`

As Pipe is Horizontal , hence `z_{1}` = `z_{2}`
`\frac{p_{1}}{\rho g} + \frac{v_{1}^{2}}{2g} = \frac{p_{2}}{\rho g} + \frac{v_{2}^{2}}{2g}` or `\frac{p_{1}-p_{2}}{\rho g} = \frac{v_{1}^{2}-v_{2}^{2}}{2g}`  ....(1)

But `\frac{p_{1}-p_{2}}{\rho g}` it is the pressure head which can be donate by `h`

The equation 1 can be now written as 
`h = \frac{v_{1}^{2}-v_{2}^{2}}{2g}` ....(2)

According to equation of continuity volume flow rate is constant
 `a_{1}v_{1}=a_{2}v_{2} `  or `v_{1} =\frac{ a_{2}v_{2}}{a_{1}}`

Put this value in equation (2) we get
`h = \frac {v_{2}^{2}}{2g}-\frac{(\frac{a_{2}v_{2}}{a_{1}})^{2}}{2g}`
= `\frac {v_{2}^{2}}{2g} \left[ 1- \frac{a_{2}^{2}}{a_{1}^{2}} \right]= \frac {v_{2}^{2}}{2g} \left[  \frac{a_{1}^{2}-a_{2}^{2}}{a_{1}^{2}}  \right]`

= `v_{2}^{2} = 2gh \frac{a_{1}^{2}}{a_{1}^{2}- a_{2}^{2}}`
`v_{2}  = \sqrt{2gh \frac{a_{1}^{2}}{a_{1}^{2}- a_{2}^{2}}} =  \frac{a_{1}}{\sqrt{a_{1}^{2}- a_{2}^{2}}}\sqrt{2gh}`

`Q = a_{2}v_{2}`
=`a_{2}\frac{a_{1}}{\sqrt{a_{1}^{2}- a_{2}^{2}}}\sqrt{2gh} =\frac{a_{1}a_{2}}{\sqrt{a_{1}^{2}- a_{2}^{2}}}\sqrt{2gh} `

In real life actual discharge rate will be less than theoretical discharge due to known and unkown reasons So 
`Q_{act} = C_{d} \frac{a_{1}a_{2}}{\sqrt{a_{1}^{2}- a_{2}^{2}}}\sqrt{2gh}`

Here `C_{d}` is coefficient of  Venturimeter and its value is less than 1.

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