Problem Statement
In the figure, gage A reads \( 290 \, \text{kPa abs} \). What is the height of water \( h \)? What does gage B read?
Solution
Given:
- Specific weight of water (\( \gamma \)) = \( 9.81 \, \text{kN/m}^3 \)
- Specific weight of mercury (\( \gamma_{\text{m}} \)) = \( 13.6 \times 9.81 = 133.416 \, \text{kN/m}^3 \)
- Pressure at gage A (\( P_A \)) = \( 290 \, \text{kPa abs} \)
Height of Water (\( h \)):
Using the pressure equation at A:
\( P_A = 175 + \gamma h + \gamma_{\text{m}} h_{\text{m}} \)
Substitute the values:
\( 290 = 175 + 9.81 h + 133.416 \times 0.7 \)
Simplify:
\( 290 = 175 + 9.81 h + 93.3912 \)
\( 290 = 268.3912 + 9.81 h \)
Solve for \( h \):
\( h = \frac{290 – 268.3912}{9.81} \)
\( h = 2.2 \, \text{m} \)
Pressure at Gage B (\( P_B \)):
Using the pressure equation at B:
\( P_B = 175 + \gamma (h + 0.7) \)
Substitute the values:
\( P_B = 175 + 9.81 \times (2.2 + 0.7) \)
Simplify:
\( P_B = 175 + 9.81 \times 2.9 \)
\( P_B = 175 + 28.449 \)
\( P_B = 203.4 \, \text{kN/m}^2 \)
Explanation
This problem involves determining the height of water and the pressure at gage B using hydrostatic principles:
- The pressure at A is given in absolute terms and is used to calculate the height of water in the tank by subtracting other pressure contributions (mercury).
- The pressure at B is calculated by adding the contributions from the water column and the additional height of 0.7 m.
Physical Meaning
- Height of Water (\( h \)): The calculated height represents the level of water in the tank that corresponds to the given pressure at gage A.
- Pressure at Gage B (\( P_B \)): This reflects the total pressure exerted by the water column and the additional contribution from the specific setup of the tank.
