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.

Let us consider the following terms as mentioned here.

XX is the datum line
PA= Pressure at point A
PB = 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. PA – PB.

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
ρ1 = Density of liquid which is contained in left pipe
ρ2 = Density of liquid which is contained in right pipe
ρg = 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. PA – PB.

Pressure in the left limb above the datum line XX = PA + ρ1g (x + h)
Pressure in the right column above the datum line XX = PB + ρ2g y + ρg g h
As pressure is same for the horizontal surface and therefore we will have following equation as mentioned here.
PA + ρ1g (x + h) = PB + ρ2g y + ρg g 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. PA – PB.

Pressure in the left limb above the datum line XX = PA + ρg (x + h)
Pressure in the right column above the datum line XX = PB + ρg x + ρg g h

As pressure is same for the horizontal surface and therefore we will have following equation as mentioned here.

PA + ρg (x + h) = PB + ρg x + ρg g h
PA - PB = ρg x + ρg g h - ρg (x + h)
PA - PB = ρg g h - ρg h

#### PA - PB = g h (ρg - ρ)

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

### Reference:

Fluid mechanics, By R. K. Bansal