INVERTED U TUBE DIFFERENTIAL MANOMETER

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 inverted U-tube differential manometer to measure the difference of low pressures between two points in the fluid, in the subject of fluid mechanics, with the help of this post.

Let us brief first 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. Differential manometers are basically used to measure the pressure above atmospheric pressure.

Types of differential manometers

U-tube differential manometer

Inverted U-tube differential manometer

We have already discussed the basic concept of U-tube differential manometer, now it’s time to go ahead to understand the basic concept of inverted U-tube differential manometer.

Inverted U-tube differential manometer

Inverted U-tube differential manometer will be used for measuring the vacuum pressure. Inverted U-tube differential manometer will have one inverted U-tube contained with light liquid.

Let us consider that inverted U tube differential manometer is connected with two points in two pipes as displayed here in following figure. These two pipes are filled with different specific gravity liquid. 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 light liquid level in the U-Tube

h₁ = Height of liquid level in left limb of manometer below the datum line

h₂ = Height of liquid level in right limb of manometer below the datum line

ρ₁ = Density of liquid which is contained in pipe A

ρ₂ = Density of liquid which is contained in pipe B

ρ_L = Density of light liquid

We will now determine the pressure below datum line in left limb of manometer and also we will secure here the value of pressure below 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 below the datum line XX = P_A - ρ₁g h₁

Pressure in the right limb below the datum line XX = P_B - ρ₂g h₂ - ρ_Lg h

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

P_A - ρ₁g h₁= P_B - ρ₂g h₂ - ρ_Lg h

P_A - P_B = ρ₁g h₁- ρ₂g h₂ - ρ_Lg h

We will find out the basic definition and derivation of total pressure and centre of pressure, in the subject of fluid mechanics, in our next post.

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