# Differential Manometer and its types

The differential manometer is used to measure the difference in pressure between two points in a pipe or in two pipes.

The pressure at these two points is measured using two piezometers.

This differential manometer is also used to measure the pressure difference between two containers.

The differential manometer can be used to measure the flow dynamics of the gas by comparing the pressure at different points in the pipe.

A simple differential manometer is U-shaped tubes with both ends nearly at the same height.

The fluid used on the differential manometer is called the manometric fluid.

### Types Of Differential Manometer

**There are mainly three types of differential manometer:**

i) U-tube upright differential manometer.

ii) U-tube inverted differential manometer.

iii) Micro Manometer of differential type.

**i) U-tube upright differential manometer:**

U-tube upright differential manometer is used to measure pressure difference at two points in a pipe or between two pipes at different levels.

To measure the pressure difference between two pints, the U-tube differential manometer is placed between the points at which the pressure has to be measured.

U tube differential manometer is used when the pressure difference between two points is considerably high or moderate.

It cannot be used to measure the low-pressure differences accurately.

As we can see in the diagram, the lower level of the two limbs is taken as reference axis XX.

We know that below the reference axis XX, both side have the same liquid. So, the pressure on both side below the reference axis XX will be same.

Also the pressure at both the limbs above the reference axis will also be equal.

Let the mass density of the manometric fluid be **ρ _{m}**

and the mass density of the liquid at left limb be

**ρ**

_{a}and the mass density of the liquid at left limb be

**ρ**

_{b}Here

**ρ**>>

_{m}**ρ**and

_{a}**ρ**>>

_{m}**ρ**

_{b}**Now for reference axis (XX):-**

Pressure at left limb above the reference plane = Press at right limb above the reference plane

P

_{A}+ ρ

_{A}x g x h1 = P

_{B}+ ρ

_{B}x g x (h3 – h2) + ρ

_{M}x g x h2

**=> P**

_{A }– P_{B}= + ρ_{B}x g x (h3 – h2) + ρ_{M}x g x h2 – ρ_{A}x g x h1**P**is the pressure difference between two points at A and B.

_{A }– P_{B}**ii) U-tube Inverted differential manometer:**

As the U-tube upright differential manometer cannot measure the difference between two points when the difference is low.

So another kind of differential manometer is needed to measure low pressure difference and this is done by** U-tube Inverted differential manometer**.

The **U**-tube Inverted differential manometer can be used to measure a very small pressure difference between two points.

This **U**-tube Inverted differential manometer is just the inverted version of the U-tube upright differential manometer.

If **P _{A} – P_{B}**

_{ }is very small, then

**U-tube Inverted differential manometer**is used to measure the pressure difference.

In this inverted manometer, the manometric fluid used has very little mass density due to which a large column height is obtained even for a small pressure difference. Hence, it will be convenient to read the column height for small pressure differences.

If the pressure difference is very low then gas can also be used as the manometric fluid.

In U-tube inverted differential manometer, higher level of manometric fluid is taken as the reference axis.

Let the mass density of the manometric fluid be **ρ _{m}**.

Let the mass density of the fluid in the left and right limb be

**ρ**and

_{a}**ρ**respectively.

_{b}In a U-tube Inverted differential manometer,

**ρ**

_{m}<ρ_{a }and ρ_{m }<ρ_{b}**Now,**

**applying Pascal Law in plane XX, we have**Pressure at the left limb below the reference axis = pressure at right limb below the reference axis

P_{A} + ρ_{A} x g x h1 = P_{B} + ρ_{B} x g x (h3 – h2) + ρ_{M} x g x h2** => P _{A }– P_{B} = + ρ_{B} x g x (h3 – h2) + ρ_{M} x g x h2 – ρ_{A} x g x h1**

**P _{A }– P_{B}** is the pressure difference between two points at A and B.

**Micro Manoneter of differential type:**

A micro manometer of differential type is a micrometer that measures the pressure difference between two points.

It has two limbs like the normal U-tube manometer. But there is an extra construction in the micro manometer which is known as **well**.

The cross-section area of the well is ‘**A**‘ and the cross-section area of the tube is ‘**a**‘.

The diameter of the tube in any type of manometer is kept more than 10 mm because if we use a diameter less than 10 mm then a capillary effect will be seen due to which the accuracy of the reading will be affected.

The micro manometer is used to measure very small pressure differences.

There are two types of liquid used in the micromanometer apart from the two liquids at the points whose pressure difference is to be measured and that are:**i) Manometric Liquidii) Gauge Liquid**

The

**gauge liquid**is very beneficial in the micro manomter as we get a large manometric column even for a very small pressure difference.

In the diagram of the micro manometer, we have reference axis XX which is the lower level of manometric fluid.

Let both the limbs, left limb and right limb be filled with water and water density be ρw.

Let the density of the gauge liquid be ρg and the density of the manometric fluid be ρm.

Applying Pascal Law in plane** XX, we have **

Pressure above XX in left limb = Pressure above XX in right limb

P_{1 }+ ρ_{w} x g x (h + Δz) + ρ_{g }x g x (z – Δz + y/2 ) = P_{2} + ρ_{w }x g x ( h – Δz ) + ρ_{g }x g x ( z + Δz –y/2 ) + ρ_{m }x g x y**=>** P_{1 }+ ρ_{w} x g x h + ρ_{w} x g x Δz + ρ_{g }x g x z – ρ_{g }x g x Δz + ρ_{g }x g x y/2 = P_{2} + ρ_{w }x g x h – ρ_{w }x g x Δz + ρ_{g }x g x z + ρ_{g }x g x Δz – ρ_{g }x g x y/2 + ρ_{m }x g x y**=> **P_{1} – P_{2} = ρ_{m }x g x y + 2 x ρ_{g }x g x Δz – 2 x ρ_{g }x g x y/2 -2 ρ_{w }x g x Δz …………………. (1)

Using continuity equation in gauge liquid,

Volume of gauge liquid displaced from the well = Volume of gauge liquid move in the tube

A x Δz = a x y/2

** If a << A** , then

Δz will be negligible, So eq (1) becomes,

P_{1} – P_{2} = ρ_{m }x g x y – ρ_{g }x g x y