# Application of Bernoulli’s Equation | Venturimeter | Orifice Meter | Nozzle Meter | Pitot Tube | Prandtl Tube5 min read

In this, We Discuss Venturimeter, Orifice meter, Nozzle meter, Pitot tube, Prandtl tube

## Venturimeter

Venturimeter is a device used for measuring the rate of flow of fluid through pipe Principle: By reducing the cross-sectional area of the flow passage, a pressure difference enables the determination of discharge through the pipe. The geometry of venturimeter is as shown in Figure.

Let at the inlet and throat a, and a be the cross-sectional areas. p, and be the pressures, V, and V are velocities respectively. Then by applying Bernoulli’s equation between sections (1) and (2) we get, be the

$\huge \frac{p_1}{\gamma}-\frac{p_2}{\gamma}=\ \frac{V_2\ ^2}{2g}-\frac{V_1\ ^2}{2g}$

$\huge h = (\ \frac{V_2\ ^2}{2g}-\frac{V_1\ ^2}{2g})$

Then, Qth = Theoretical discharge through pipe, then by continuity equation,

$\huge Q_{th}=\frac{a_1a_2\sqrt{2gh}}{\sqrt{a_1^2-a_2^2}}$

If, Cd is the coefficient of discharge, then, the actual discharge is given by

$\huge Q_{act}=C_d\frac{a_1a_2\sqrt{2gh}}{\sqrt{a_1^2-a_2^2}}$

In general, for fluids of low viscosity, a value of 0.98 is adopted for Cd.

In the venturi meter, as we have a gradual contraction and gradual expansion, there is no flow separation and hence a can be taken as a complete area of the throat.

in the convergent cone, flow is accelerating which may be allowed to take place rapidly in a smaller length. While in divergent cone retardation of flow occurs. If retardation of flow takes place in a small length it may lead to flow separation and energy loss.

Since the separation of flow may occur in the divergent cone of venturimeter, this portion is not used for discharge measurement.

Since, the cross-sectional area of the throat is smaller than that of an inlet section, hence the velocity of flow at the throat is greater, thus, the pressure is reduced at the throat.

If the cross-sectional area of the throat is so much reduced that pressure drops below vapour pressure of flowing fluid, cavitation may occur. Hence, to avoid the phenomenon of cavitation the diameter of the throat can be reduced only up to a certain limited value.

NOTE: Venturimeter can also be used for measuring the discharge through a pipe which is laid either in an
inclined or in a vertical position.

### Venturimeter with Differential Manometer

#### Case-1: Horizontal Venturimeter

Let Sm and S(Sm>S) are the specific gravities of the manometric liquid and the liquid flowing in the venturi meter, respectively

For, Sm>S

$\huge Q=\frac{C_da_1a_2}{\sqrt{a_1^2-a_2^2}}\sqrt{2gx\left(\frac{S_m}{S}-1\right)}$

For, S>Sm

$\huge Q=\frac{C_da_1a_2}{\sqrt{a_1^2-a_2^2}}\sqrt{2gx\left(1-\frac{S_m}{S}\right)}$

#### Case-2: Inclined Venturimeter

For, Sm>S

$\huge Q_(actual)=\frac{C_da_1a_2}{\sqrt{a_1^2-a_2^2}}\sqrt{2gx\left(\frac{S_m}{S}-1\right)}$

For, S>Sm

$\huge Q_(actual)=\frac{C_da_1a_2}{\sqrt{a_1^2-a_2^2}}\sqrt{2gx\left(\frac{S_m}{S}-1\right)}$

## Orifice Meter

This device is used for measuring discharge through pipes, which works on the same principle as a venturimeter.

However, an orifice meter is a cheaper arrangement.

As such, where the space is limited, the orifice meter may be used for the measurement of discharge through pipes.

An orifice meter consists of a flat circular plate with a circular hole called an orifice, which is concentric with the pipe axis.

The diameter of the orifice may vary from 0.2 to 0.85 times the pipe diameter but is generally kept at 0.5 times the pipe diameter.

Coefficient of discharge of orifice meter may be given as:

$\huge C=C_d\frac{\sqrt{1-\frac{a_2^2}{a_1^2}}}{\sqrt{1-C_d^2\frac{a_2^2}{a_1^2}}}$

## Nozzle Meter

This is similar to the venturi meter except in the nozzle meter, the diverging cone is not provided.

In this, a contraction with a well-rounded entrance is placed in the pipeline. This flow nozzle is simpler than the venturi meter and can be installed between the flanges of a pipeline.

The discharge equation for the flow nozzle is the same as for the venturi meter, however, in this case, the coefficient of contraction Cc=1.

$\huge Q =C_d\frac{a_1a_2\sqrt{2gh}}{\sqrt{a_1^2-a_2^2}}$

NOTE: The coefficient for discharge of an orifice meter (C) is smaller than that of venturimeter because in the case of an orifice meter there is no gradual converging and flow passages as that of venturimeter.

## Comparison of Orifice, Flow Nozzle, And Venturi Meter

 Flow Meter Types Head Loss Initial Cost Orifice Meter High Low Flow Nozzle Intermediate Intermediate Venturimeter Low High

## Pitot Tube

the pitot tube is a device for measuring the velocity of flow.

Basic Principle: The velocity of flow at a particular point is reduced to zero, which is known as the stagnation point.

the pressure there is increased due to the conversion of kinetic energy into pressure energy, and by measuring the increase in the pressure energy at this point, the velocity of flow may be determined.

Applying Bernoulli’s equation between points 1 and 2. velocity at point 1,

V1 = √2gh

or

V_actual = Cv √2gh

where Cv= coefficient of velocity= 0.98 for pitot tube.

## Prandtl Tube or Pitot Static Tube

This is also used to measure velocity at a point in fluid flow.

In the case of the pitot tube, the piezometric head at point ‘2’ was measured and it was taken equal to the piezometric head at ‘1’ because streamlines were straight lines and hence the piezometric head at ‘1’ is equal to piezometric head at ‘2’.

However, if the streamlines are curved, the piezometric head at 1 and 2 will not be the same. In that case, it is necessary to find out the piezometric head at the same point at which velocity is to be measured such that the difference of total head and piezometric head at 1 gives the velocity head.

Hence, we use a Prandtl tube or pitot-static tube which satisfies the above purpose.

Pitot static tube or Prandtl tube can also be used if the boundary surface at 2′ is rough such that pressure measurement at ‘2’ can not be correctly done.

The front portion of the tube is rounded to avoid separation of flow and on its shaft, holes are provided at a certain distance where streamlines become parallel.

In the Prandtl pitot tube, the tubes recording static Stagnation pressure and stagnation pressure are combined into one. The static tube (tube-A) surrounds the total head tube or stagnation pressure tube (tube-B).

V1 = √2gh

To account for losses, we use the coefficient of velocity. Hence actual velocity at ‘1’ is

V1 = Cv √2gh : Where, Cv= 0.99