Example Problem with Complete Solution

1E-3 : Pressure in a Tank Using a Complex Manometer     5 pts
The gauge pressure of the air in the tank shown in the figure is measured to be 65 kPa.  Determine the differential height, h, of the mercury column.  
Figure.
                   
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
65
kPa
hoil
0.75
m SGoil
0.72
hwater
0.3
m SGm
13.6
Solution : Begin by writing the manometer equation for each interval between points a and f on the diagram.
Equation. Eqn 1 Equation. Eqn 3
Equation. Eqn 2 Equation. Eqn 4
Equation. Eqn 5
If we add all 5 of these equations together we obtain :
Equation. Eqn 6
The only unknown in this equation is h.  So, the next step is to solve the equation for h.
Equation.
Eqn 7
We know that : g
9.8066
m/s2
gc
1
kg-m/N-s2
ρwater
1000
kg/m3
Also, because Pf = Patm and the definition of gauge pressure : Equation. Eqn 8
Pa-Pf =
65000
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 : Equation. Eqn 9

This helps us simplify Eqn 7 to :
Equation. Eqn 10
     
Equation. Eqn 11
From Eqn 9 : Equation. Eqn 12
 
ρm
13600
kg/m3
Answers : Plugging values into Eqn 11 yields : h
0.487
m
 
48.7
cm