Example Problem with Complete Solution

1E-1 : Pressure Measurement Using a Multi-Fluid Manometer 6 pts
The water in a tank is pressurized by air, and the pressure is measured by a multi-fluid manometer, as shown in the figure. Determine the gage pressure of air in the tank if h1 = 0.2 m, h2 = 0.3 m and h3 = 0.46 m.
  Problem 1E-1: Manometer System Diagram  
   
   
   
   
   
   
   
   
   
   
   
   
               
Take the densities of water, oil and mercury to be 1000 kg/m3, 850 kg/m3 and 13,600 kg/m3, respectively.
               
Read : Use the barometer equation to work your way through the different fluids from point 1 to point 2.
Remember that gage pressure is the difference between the absolute pressure and atmospheric pressure.
Given : h1 0.20 m ρw 1000 kg/m3
h2 0.30 m ρoil 850 kg/m3
h3 0.46 m ρHg 13600 kg/m3
P2 101.325 kPa
Find : P1,gage ???
Solution : Gage pressure is defined by : eqn-1 Eqn 1
If we assume that P2 is atmospheric pressure, then Eqn 1 becomes :
eqn-2 Eqn 2
The key equation is the Barometer Equation :
eqn-3
Eqn 3
Now, apply Eqn 1 repeatedly to work our way from point 1 to point 2.
eqn-4
Some key observations are:  Eqn 4
eqn-5 Eqn 5
These are true because the points are connected by open tubing, the fluid is not flowing in this system and no change in the composition of the fluid occurs between A & B or C & D or D & E.
eqn-6
PA > P2, therefore : Eqn 6
eqn-7
PE > P1, therefore : Eqn 7
eqn-8
PB > PC, therefore : Eqn 8
Combine Eqns 2, 5 & 6 to get :
eqn-9
Eqn 9
Use Eqns 3 & 5 to eliminate PC from Eqn 7 :
eqn-10
Eqn 10
Now, solve for P1 - P2 :
eqn-11
Eqn 11
Combining Eqns 10 & 2 yields :
eqn-12
Eqn 12
Plugging values into Eqn 11 yields :
g 9.8066 m/s2 P1,gage 56888 Pa gage
gC 1 kg-m/N-s2 P1,gage 56.89 kPa gage
Answers :     P1,gage 56.9 kPa gage
If you are curious : P1 158.21 kPa
  P2 101.325 kPa
  PA = PB 162.68 kPa
  PC = PD = PE 160.17 kPa