1E1 : 
Pressure Measurement Using a MultiFluid Manometer 
6 pts 








The water in a tank is pressurized by air, and the pressure is measured by a multifluid manometer, as shown in the figure. Determine the gage pressure of air in the tank if h_{1} = 0.2 m, h_{2} = 0.3 m and h_{3} = 0.46 m. 



































Take the densities of water, oil and mercury to be 1000 kg/m^{3}, 850 kg/m^{3} and 13,600 kg/m^{3}, respectively. 
















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} 
0.20 
m 

ρ_{w} 
1000 
kg/m^{3} 

h_{2} 
0.30 
m 

ρ_{oil} 
850 
kg/m^{3} 

h_{3} 
0.46 
m 

ρ_{Hg} 
13600 
kg/m^{3} 









P_{2} 
101.325 
kPa 












Find : 
P_{1,gage} 
??? 













Solution : 
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. 









Some key observations are: 


Eqn 4 













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 11 yields : 












g 
9.8066 
m/s^{2} 

P_{1,gage} 
56888 
Pa gage 

g_{C} 
1 
kgm/Ns^{2} 
P_{1,gage} 
56.89 
kPa gage 








Answers : 


P_{1,gage} 
56.9 
kPa gage 










If you are curious : 
P_{1} 
158.21 
kPa 





P_{2} 
101.325 
kPa 





P_{A} = P_{B} 
162.68 
kPa 





P_{C} = P_{D} = P_{E} 
160.17 
kPa 
