## Differential Manometer

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# Differential Manometer

• Differential manometers are the devices used for measuring the difference of pressure between two points in the same pipe or in two different pipes.

• A differential manometer of a U-tube are connected to the points, whose difference of pressure is to be measured.

• Most commonly types of differential manometers are:

1. U-tube differential manometer 2. Inverted U-tube differential manometer

• A differential manometer of a U-tube are connected to the points, whose difference of pressure is to be measured.

• Most commonly types of differential manometers are:

1. U-tube differential manometer 2. Inverted U-tube differential manometer

## 1. U-tube differential manometer:

**(a) Two pipes at different levels (b) A and B are at the same level**

Case a : Two pipes at different levels

Let the two points A and B be at different level and also contains liquids of different specific gravity. These point are connected to the U-tube differential manometer. Let the pressure at A and B be PA and PB

Let h = Difference f mercury level in the U-tube

y = Distance of the center of B, from the mercury level in the right limb

x = Distance of the center of A, from the mercury level in the right limb

ρ

_{1}= Density of liquid at A

ρ

_{2}= Density of liquid at B

ρ

_{3}= Density of heavy liquid or mercury

Taking datum line at X-X

Pressure above X-X in the left limb = Pressure above X-X in the right limb

ρ

_{1 }g (h+ x) + P

_{A}= ρ

_{1}x g x h + ρ

_{2 }g y + P

_{B}

P

_{A}- P

_{B}= ρ

_{3 }x g x h + ρ

_{2 }g y - ρ

_{1 }g x

= h x g ( ρ

_{3 - }ρ

_{1) + }ρ

_{2 }g y - ρ

_{1 }g x

= h x g ( ρ

_{3 }- ρ

_{1}) + ρ

_{2}g y - ρ

_{1 }g x

Difference of pressure at A and B

## Case b: Two pipes are at the same level

In A and B are the same level and contains the same liquid of density ρ_{1}. Then

Pressure above X-X in right limb = Pressure above X-X in right limb

ρ

_{m}x g x h + ρ

_{1 }g x + P

_{B }= ρ

_{1 }x g x (h + x) + P

_{A}

P

_{A}- P

_{B}= ρ

_{m}x g x h + ρ

_{1 }g x - ρ

_{1 }g (h + x) = g x h ( ρ

_{m}- ρ

_{1)}

## 2. Inverted U-tube differential manometer:

• It consists of an inverted U-tube, containing a light liquid.

• The two ends of the tube are connected to the points whose pressure different is to measured.

an inverted U-tube differential manometer connected to the two points A and B. Let the pressure at A is more than the pressure at B

Let h

_{1}= Height of liquid in left limb below the datum line X-X

h

_{2}= Height of liquid in right limb

h = Difference of light limb

ρ

_{1}= Density of liquid at A

ρ

_{2}= Density of liquid at a B

ρ

_{3}= Density of light liquid

P

_{A}= Pressure at A

P

_{B}= Pressure at B

Taking datum line at X-X, by using static equilibrium

Pressure above X-X in the lift limb = Pressure above X-X in the right limb

P

_{A}- ρ

_{1 }x g x h1 = P

_{B}-ρ

_{2 }x g x h

_{2}- ρ

_{s }x g x h

P

_{A}- P

_{B}= ρ

_{1}x g x h

_{1}- ρ

_{2 }x g x h

_{2}- ρ

_{s }x g x h

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