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*

*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.

d

_{1}= Diameter at section 1 (Inlet section)
P

_{1}= Pressure at section 1 (Inlet section)
v

_{1}= Velocity of fluid at section 1 (Inlet section)
a

_{1}= Area at section 1 (Inlet section) = (П/4) x d_{1}^{2}
d

_{2}= Diameter at section 2
P

_{2}= Pressure at section 2
v

_{2}= Velocity of fluid at section 2
a

_{2}= Area at section 2 = (П/4) x d_{2}^{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 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,

(P

_{1}-P_{2}) / 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

a

_{1}v_{1}= a_{2}v_{2}
v

_{1}= (a_{2}v_{2})/ a_{1}
Now we
will use the value of v

_{1}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 = a

_{2}v_{2}
Now we
will use the value of v

_{2}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