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. |
|
|
|
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. |
|
|
|
|
|
|
|
|
|
|
|
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 : |
|
|
|
|
Eqn 1 |
|
|
|
|
|
|
|
|
|
|
|
If we assume that P2 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. |
|
|
|
|
|
|
|
|
|
|
|
PA > P2, therefore : |
|
|
|
|
Eqn 6 |
|
|
|
|
|
|
|
|
|
|
|
PE > P1, therefore : |
|
|
|
|
Eqn 7 |
|
|
|
|
|
|
|
|
|
|
|
PB > PC, therefore : |
|
|
|
|
Eqn 8 |
|
|
|
|
|
|
|
|
|
|
|
Combine Eqns 2, 5 & 6 to get : |
|
|
|
|
|
Eqn 9 |
|
|
|
|
|
|
|
|
|
|
|
Use Eqns 3 & 5 to eliminate PC from Eqn 7 : |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Eqn 10 |
|
|
|
|
|
|
|
|
|
|
|
Now, solve for P1 - P2 : |
|
|
Eqn 11 |
|
|
|
|
|
|
|
|
|
|
|
Combining Eqns 10 & 2 yields : |
|
|
|
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 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|