Example Problem with Complete Solution

1E-3 : Pressure in a Tank Using a Complex Manometer 5 pts
The pressure gauge on the air in the tank shown below reads 87 kPa.  Determine the manometer reading, h2, in cm. 
h1 = 25 cm and h3 = 65 cm
SGmercury = 13.6, SGoil = 0.75, ρwater = 1000 kg/m3
Read: The density of the air is so much lover than the density of the liquids in this problem that the weight of the air can be considered negligible in the force balances we will write in this problem.
Given: Pgauge 87 kPa
hoil 0.65 m SGoil 0.7
hwater 0.25 m SGm 13.6
Find: h2 ??? cm
Assumptions: 1- The fluids in the system are completely static.
2- The densities of the liquids are uniform and constant.
3- The reference density of water used to determine specific gravity is: rwater 1000 kg/m3
4- The acceleration of gravity is: g 9.8066 m/s2
gc 1 kg-m/N-s2
Equations / Data / Solve:
Begin by writing the Manometer Equation for each interval between points A and F on the diagram.
Eqn 1
Eqn 3
Eqn 2
Eqn 4
Eqn 5
If we add all 5 of these equations together we obtain :
Eqn 6
The only unknown in this equation is h.  So, the next step is to solve the equation for h.
Eqn 7
Also, because Pf = Patm and the definition of gauge pressure we can use:
Eqn 8
Pa-Pf = 87000 N/m2
All we need to do is convert specific gravity into density and we are ready to plug values into Eqn 7.
The definition of specific gravity is :
Eqn 9
This helps us simplify Eqn 7 to :
Eqn 10
Eqn 11
From Eqn 9 :
Eqn 12
rm 13600 kg/m3
Plugging values into Eqn 11 yields : h 0.652 m
65.2 cm
Answers: h 65.2 cm