Pressure measurement

The oldest method of measuring low pressures is simple U tube method. If a U shaped small diameter glass tube is half filled with a liquid, e.g. mercury, and a pressure is applied to one end of the tube, the other being open to atmosphere then the liquid will move to balance the pressure.

The weight of liquid so displaced will be proportional to the pressure applied. As the difference in height of the two columns of liquid and the density are known, the pressure can be calculated.

In the above figure a tube with fluid A is attached with the U-tube. Pressure of fluid A is to be measured. Fluid A will exert pressure on U-tube liquid (Mercury) and height of the liquid in Utube

will change until pressure balance is obtained. In the figure point B and C have same pressure as both are on the same plane.

                           ∴PB = PC    -------------------------(1)

Now pressure at B is -

PB = Pressure at A + Pressure due to height h1 of fluid A

                           ∴PB = PA + h1gρ1 --------------------(2)

Where ρ1 is the density of fluid A and g is the acceleration due to gravity.

Now Pressure at C is –

PC = Pressure due to height h2 of mercury + Atmospheric Pressure

                        ∴Pc = h2ρmg + Patm    ---------------(3)

Where ρm is the density of mercury.

Now from equation 1, 2 and 3 we get –

                         PA + h1gρ1 = h2ρmg + Patm

                         PA  = h2ρmg - h1gρ1 + Patm

As we usually consider Gauge pressure instead of Absolute pressure,

                        PA  = h2ρmg - h1gρ1 

By using the above equation we can measure the unknown pressure of a fluid from the liquid displacement or height difference of the liquid in the U-tube manometer.