We
were discussing the basic definition and significance of Pascal’s
Law along with its derivation , Vapour
pressure and cavitation, Absolute
pressure, Gauge pressure, Atmospheric pressure and Vacuum pressure, pressure
measurement, Piezometer,
Single column manometer and also the basic concept of U-tube
manometer in our previous posts.

Today
we will understand here the basic concept of differential manometer to measure
the difference of pressure between two points in the fluid, in the subject of
fluid mechanics, with the help of this post.

###
**Differential manometer**

Differential
manometer is also one type of pressure measuring device which is basically used
to measure the difference of pressure between two points in a pipe or between
two different pipes.

Let us consider that we have two pipes as displayed here in following figure. A is
one point in first pipe and B is other point in second pipe. There will be one
U-tube contained with heavy liquid and two ends of this U-tube manometer will
be connected with these two points i.e. point A and point B.

Pressure
difference between point A and point B will be measured with the help of
differential manometer.

###
**Types of differential
manometers**

U-tube
differential manometer

Inverted
U-tube differential manometer

Let
us first understand here the basic concept of U-tube differential manometer and
further we will understand the basic concept of inverted U-tube differential
manometer.

###
**U-tube differential
manometer**

Let
us consider that U tube differential manometer connected with two points in
two pipes as displayed here in following figure. These two pipes are filled
with different specific gravity liquid.

Differential manometers are basically used to measure the pressure above atmospheric pressure.

As displayed here in figure, point A and point B are at different level.

Differential manometers are basically used to measure the pressure above atmospheric pressure.

As displayed here in figure, point A and point B are at different level.

Let
us consider the following terms as mentioned here.

XX
is the datum line

P

_{A}= Pressure at point A
P

_{B }= Pressure at point B
We need to measure the pressure difference between point A and point B or we need
to find out the expression for difference of pressure i.e. P

_{A}– P_{B}.
h

_{ }= Difference of mercury level in the U-Tube
x
= Distance between point A and mercury level in right limb of manometer

y
= Distance between point B and mercury level in right limb of manometer

ρ

_{1}= Density of liquid which is contained in left pipe
ρ

_{2}= Density of liquid which is contained in right pipe
ρ

_{g}= Density of heavy liquid i.e. density of mercury
We
will now determine the pressure above datum line in left limb of manometer and
also we will secure here the value of pressure above the datum line in right
limb of manometer and we will equate these two pressures to find out the
pressure difference i.e. P

_{A}– P_{B}.
Pressure
in the left limb above the datum line XX = P

_{A}+ ρ_{1}g (x + h)
Pressure
in the right column above the datum line XX = P

_{B}+ ρ_{2}g y + ρ_{g }g h
As
pressure is same for the horizontal surface and therefore we will have
following equation as mentioned here.

P

_{A}+ ρ_{1}g (x + h) = P_{B}+ ρ_{2}g y + ρ_{g }g h####
**P**_{A}
-** ****P**_{B}**
= ****ρ**_{2}g y + ρ_{g }g
h - **ρ**_{1}g
(x + h)

_{A}-

_{B}

_{2}g y + ρ

_{g }g h -

_{1}g (x + h)

### Case : 2

Let us consider the second
case where point A and point B are at same level as displayed in above figure
part B.

Let us consider that both pipes
are contained with the same liquid of density ρ and x is the distance between point
A and mercury level in right limb of manometer. h is the difference of mercury
level in U-tube.

We will now determine the
pressure above datum line in left limb of manometer and also we will secure
here the value of pressure above the datum line in right limb of manometer and we
will equate these two pressures to find out the pressure difference i.e. P

_{A}– P_{B}.
Pressure in the left limb
above the datum line XX = P

_{A}+ ρg (x + h)
Pressure in the right
column above the datum line XX = P

_{B}+ ρg x + ρ_{g }g h
As pressure is same for the
horizontal surface and therefore we will have following equation as mentioned
here.

P

_{A}+ ρg (x + h) = P_{B}+ ρg x + ρ_{g }g h
P

_{A}- P_{B}= ρg x + ρ_{g }g h - ρg (x + h)
P

_{A}- P_{B}= ρ_{g }g h - ρg h####
**P**_{A} - P_{B}
= g
h (ρ_{g} - **ρ)**

_{A}- P

_{B}= g h (ρ

_{g}-

Do
you have any suggestions? Please write in comment box.

###
**Reference:**

Fluid mechanics, By R. K. Bansal

Image
Courtesy: Google

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