Differential Manometer , at same level , at different level , and Inverted type

Differential Manometer :

- it measured the difference of pressure between any two points in  a fluid contained in a pipe or container or between two different pipes .

- it consist of a U-tube containing a heavy liquid , whose two ends are connected to the points , whose difference of pressure is to be measured  . 

U-tube differential manometer at the same level :

Pressure in the left limb above the datum line .

XX =  Pa + (rho)g( x + h )

Pressure in the right column above the datum line :

XX = Pb + (rho)gx + (rhog)gh

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

Pa + (rho)g( x + h ) = Pb + (rho)gx + (rhog)gh

Pa - Pb  = (rho)g + (rhog)gh - (rho)g(x+h)

Pa - Pb = (rhog)gh - (rho)gh 

U-tube differential manometer at the different level :

Pressure in the left limb above the datum line 

XX =  Pa + (rho1)g(x+h)

Pressure in the right column above the datum line 

XX = Pb + (rho2)gy + (rhog)gh

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

Pa + (rho1)g( x + h ) = Pb + (rho2)gy + (rhog)gh

Pa -  Pb = (rho2)gy + (rhog)gh - (rho1)g(x+h)

Inverted U-tube differential manometer :

-  The inverted U-tube differential manometer is reciprocal of U-tube differential manometer at the different level .

- It Consist of an Inverted U-tube containing a light liquid .

-  the two ends of the tube are connected to the point whose difference of pressure is to be measured .

Pa - Pb  =  (rho1).gh1 - (rho2)gh2 - (rhos)gh

Pa =  Pressure at a  .

Pb =  Pressure at b .

h1 =  height of liquid in the left limb below the datum XX .

h2 = height of liquid in right limb .

h =  difference in height of liquid in right and left limb .

(rho1) =  density of liquid a .

(rho2) =  density of liquid b .

(rhos) = density of light liquid .