1E1 :  Pressure Measurement Using a MultiFluid Manometer  6 pts 

A pressurized vessel contains water with some air above it, as shown below. A multifluid manometer system is used to determine the pressure at the airwater interface, point F. Determine the gage pressure at point F in kPa gage.  


Data: h_{1} = 0.24 m, h_{2} = 0.35 m and h_{3} = 0.52 m Assume the fluid densities are water: 1000 kg/m^{3}, oil: 790 kg/m^{3} and mercury(Hg): 13,600 kg/m^{3}. 

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:  h_{1}  0.24  m  r_{w}  1000  kg/m^{3}  
h_{2}  0.35  m  r_{oil}  790  kg/m^{3}  
h_{3}  0.52  m  r_{Hg}  13600  kg/m^{3}  
P_{2}  101.325  kPa  
Find:  P_{1,gauge}  ???  kPa gauge  
Assumptions:  1 The fluids in the system are completely static.  
2 The densities of the liquids are uniform and constant.  
3 The acceleration of gravity is:  g  9.8066  m/s^{2}  
g_{C}  1  kgm/Ns^{2}  
Equations / Data / Solve:  
Gage pressure is defined by : 

Eqn 1  
If we assume that P_{2} is atmospheric pressure, then Eqn 1 becomes : 

Eqn 2  
The key equation is the Barometer Equation : 

Eqn 3  
Now, apply Eqn 1 repeatedly to work our way from point 1 to point 2. 

Eqn 4  
Some key observations are: 

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.  
P_{A} > P_{2}, therefore : 

Eqn 6  
P_{E} > P_{1}, therefore : 

Eqn 7  
P_{B} > P_{C}, therefore : 

Eqn 8  
Combine Eqns 2, 5 & 6 to get : 

Eqn 9  
Use Eqns 3 & 5 to eliminate P_{C} from Eqn 7 :  

Eqn 10  
Now, solve for P_{1}  P_{2} : 

Eqn 11  
Combining Eqns 10 & 2 yields : 

Eqn 12  
Plugging values into Eqn 12 yields :  P_{1,gage}  64287  Pa gage  
P_{1,gage}  64.29  kPa gage  
Answers:  P_{1,gage}  64.3  kPa gage  
If you are curious :  P_{1}  165.61  kPa  P_{A} = P_{B}  170.68  kPa  
P_{2}  101.325  kPa  P_{C} = P_{D} = P_{E}  167.97  kPa  