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
               ρ₁ = Density of liquid at A
               ρ₂ = Density of liquid at B
               ρ₃ = 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

ρ₁ g (h+ x) + P_A  = ρ₁ x g x h + ρ₂ g y + P_B

P_A - P_B  = ρ₃ x g x h + ρ₂ g y - ρ₁ g x

                = h x g ( ρ_{3 -} ρ_{1) +} ρ₂ g y - ρ₁ g x

                = h x g ( ρ₃ - ρ₁) + ρ₂ g y - ρ₁ 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 ρ₁. Then
Pressure above X-X in right limb = Pressure above X-X in right limb

ρ_m x g x  h + ρ₁ g x + P_B  =  ρ₁ x g x (h + x) + P_A

P_A - P_B   =  ρ_m x g x h + ρ₁ g x - ρ₁ g (h + x) = g x h ( ρ_m - ρ₁₎

2.    Inverted U-tube differential manometer:

•    It is used for measuring difference of low fluid pressures.
•    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                 h₁ = Height of liquid in left limb below the datum line X-X
                      h₂ = Height of liquid in right limb
                      h = Difference of light limb
                     ρ₁ = Density of liquid at A
                     ρ₂ = Density of liquid at a B
                     ρ₃  = 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 - ρ₁ x g x h1  =  P_B -ρ₂ x g x h₂ - ρ_s x g x h

P_A - P_B  =  ρ₁ x g x h₁ - ρ₂ x g x h₂ - ρ_s x g x h

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