1E-1 : | Pressure Measurement Using a Multi-Fluid Manometer | 6 pts |
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A pressurized vessel contains water with some air above it, as shown below. A multi-fluid manometer system is used to determine the pressure at the air-water interface, point F. Determine the gage pressure at point F in kPa gage. | ||||||||||||||
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Data: h1 = 0.24 m, h2 = 0.35 m and h3 = 0.52 m Assume the fluid densities are water: 1000 kg/m3, oil: 790 kg/m3 and mercury(Hg): 13,600 kg/m3. |
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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 | 0.24 | m | rw | 1000 | kg/m3 | ||||||||
h2 | 0.35 | m | roil | 790 | kg/m3 | |||||||||
h3 | 0.52 | m | rHg | 13600 | kg/m3 | |||||||||
P2 | 101.325 | kPa | ||||||||||||
Find: | P1,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/s2 | |||||||||||
gC | 1 | kg-m/N-s2 | ||||||||||||
Equations / Data / Solve: | ||||||||||||||
Gage pressure is defined by : |
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Eqn 1 | ||||||||||||
If we assume that P2 is atmospheric pressure, then Eqn 1 becomes : |
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Eqn 2 | ||||||||||||
The key equation is the Barometer Equation : |
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Eqn 3 | ||||||||||||
Now, apply Eqn 1 repeatedly to work our way from point 1 to point 2. |
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Eqn 4 | ||||||||||||
Some key observations are: |
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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. | ||||||||||||||
PA > P2, therefore : |
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Eqn 6 | ||||||||||||
PE > P1, therefore : |
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Eqn 7 | ||||||||||||
PB > PC, therefore : |
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Eqn 8 | ||||||||||||
Combine Eqns 2, 5 & 6 to get : |
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Eqn 9 | ||||||||||||
Use Eqns 3 & 5 to eliminate PC from Eqn 7 : | ||||||||||||||
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Eqn 10 | |||||||||||||
Now, solve for P1 - P2 : |
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Eqn 11 | ||||||||||||
Combining Eqns 10 & 2 yields : |
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Eqn 12 | ||||||||||||
Plugging values into Eqn 12 yields : | P1,gage | 64287 | Pa gage | |||||||||||
P1,gage | 64.29 | kPa gage | ||||||||||||
Answers: | P1,gage | 64.3 | kPa gage | |||||||||||
If you are curious : | P1 | 165.61 | kPa | PA = PB | 170.68 | kPa | ||||||||
P2 | 101.325 | kPa | PC = PD = PE | 167.97 | kPa | |||||||||