## What is a Differential Manometer?

*A differential manometer consists of a U tube containing a heavy liquid, whose two ends are connected to the two points of same pipe line or different pipe lines, whose difference of pressures is required to be measured.*

### Manometer for pipes at same level

Consider a differential manometer whose two ends are connected to two different points A and B at the same level of same pipe line as shown in figure.

Assume that the *pressure* at point A is more than that at point B.

Then, the greater pressure at point A will force the heavy liquid in the U tube to move downwards. This downward movement of the heavy liquid in the left limb, will cause a corresponding rise of the heavy liquid in the right limb as shown in figures.

Taking the horizontal line ZZ as the datum line.

#### Reading Calculations

Let –

- ( h ) is reading of manometer. It is the difference of the heavy liquid levels in the right and left limb in cm.
- ( s_1 ) is specific gravity of the light liquid in the pipe.
- ( s ) is specific gravity of the heavy manometer liquid in U tube.
- ( h_A ) is pressure head at point A.
- ( h_B ) is pressure head at point B.

Two ends of U tube are at same level.

Therefore, effective pressure in left limb above the datum line is –

( h_A + s_1 h )

And effective pressure in right limb above the datum line is –

( h_B + s h )

For *equilibrium* of liquids, the pressure in the left limb and right limb above the datum line are equal.

Therefore, \quad ( h_A + s_1 h ) = ( h_B + s h )

Or, \quad \left ( h_A - h_B \right ) = h \left ( s - s_1 \right )

Hence, difference of pressures between the two points A and B are –

h \left ( s - s_1 \right ) cm of water column.

- This is also called difference of
*pressure head*in terms of water head.

### Manometer for pipes at different level

Sometimes, a differential manometer is connected between two different pipes or two different points of same pipe line which are at different levels as shown in figure.

In this case, the two ends of manometer are not at the same level and at the same time the liquids flowing in the two pipes may be different.

Consider a differential manometer whose two ends are connected to two different points A and B at different levels and containing different liquids as shown in figure. Let, us assume that, the pressure at the point A is more than that at the point B.

The greater pressure at A will force the heavy liquid to move downwards. This downward movement of the liquid in the left limb will cause a corresponding rise of the heavy liquid in the right limb.

#### Reading Calculations

Let,

- ( h_1 ) is height of liquid in the left limb above the datum line in cm.
- ( h ) is reading of manometer. It is the difference of levels of the heavy liquid in the right and left limb in cm.
- ( h_2 ) is height of the liquid in the right limb above the datum line in cm.
- ( h_A ) is pressure in the pipe A, expressed in terms of head of the liquid in cm.
- ( h_B ) is pressure in the pipe B, expressed in terms of head of the liquid in cm.
- ( s_1 ) is specific gravity of the liquid in left pipe (A).
- ( s ) is specific gravity of the heavy liquid in manometer U tube.
- ( s_2 ) is specific gravity of the liquid in right pipe (B).

We know that the pressures in the left limb and right limb above the datum line are equal.

Pressure in the left limb above the datum line is –

( h_A + s_1 h_1 ) cm of water ……. (1)

And pressure in the right limb above the datum line is –

( s_2 h_2 + s h + h_B ) cm of water ……. (2)

Equating these two pressures we get –

( h_A + s_1 h_1 ) = ( s_2 h_2 + s h + h_B )

From the above equation, the value of ( h_A ), \ ( h_B ) or their difference can be calculated.

### Use of differential manometer

- Differential manometer is used for measuring the difference of pressures between two points in a pipe line.
- It is also used for comparing pressure of fluids flowing through two different pipes either at same level or at different levels.

## Inverted Differential Manometer

*An inverted differential manometer is a particular type of differential manometer, in which an inverted U tube is used.*

It consists of an inverted U tube containing a lighter liquid compared to the liquids flowing in the pipe line. Two ends are connected to the two different points whose difference of pressures has to measure.

### Working of inverted manometer

Consider an inverted differential manometer whose two ends are connected to two different points A and B as shown in figure.

Let us assume that, the pressure at point A is more than that at point B of the manometer.

The greater pressure at A will force the light liquid in the inverted U tube to move it upwards. This upward movement of liquid in the left limb will cause a corresponding fall of the light liquid in the right limb as shown in figure. Take ZZ as the datum line.

#### Reading Calculations

Let,

- ( h_1 ) is height of liquid in the left limb below the datum line in cm.
- ( h ) is manometer reading. It is the difference of levels of the light liquid in the right and left limbs in cm.
- ( h_2 ) is height of manometer liquid in the right limb below the datum line in cm.
- ( h_A ) is pressure in the pipe A expressed in terms of head of the liquid in cm.
- ( h_B ) is pressure in the pipe B expressed in terms of head of the liquid in cm.
- ( s_1 ) is specific gravity of the liquid in the left limb.
- ( s ) is specific gravity of the manometer liquid.
- ( s_2 ) is specific gravity of the liquid in the right limb.

We know that, the pressures in the left limb and right limb below the datum line are equal.

Pressure in the left limb below the datum line is –

( h_A - s_1 h_1 ) …… (1)

Pressure in the right limb below the datum line is –

( h_B - s_2 h_2 - s h ) …… (2)

Equating these two pressures we get –

( h_A - s_1 h_1 ) = ( h_B - s_2 h_2 - s h ) ……. (3)

From the above equation the value of ( h_A ), \ ( h_B ) or their difference may be calculated.

### Use of inverted manometer

- In this type of manometer, a lighter liquid is used in U tube as gauge liquid. Hence, it is suitable for measuring of difference of pressure between two points of pipe line containing heavy liquid.
- An inverted differential manometer is used for measuring difference of low pressures with more accuracy.

**TO BE NOTED –**

(1) In differential manometer, a liquid of higher specific gravity (i.e. mercury) is used in U tube as a gauge liquid. Liquid in the pipe line has low specific gravity. Hence, it is suitable for comparing pressures of lighter liquids.

(2) In inverted differential manometer, a liquid of lower specific gravity is used in U tube as a gauge liquid. Liquid in the pipe line has high specific gravity. Hence, it is suitable for comparing pressures of heavy liquids.