An inclined manometer is used to measure the pressure of gas in a reservoir with mercury as the fluid, which has a density of 845 lb/ft³. The fluid rises 5 inches at an angle of 30... An inclined manometer is used to measure the pressure of gas in a reservoir with mercury as the fluid, which has a density of 845 lb/ft³. The fluid rises 5 inches at an angle of 30° from the horizontal. Calculate the gas pressure.
Understand the Problem
The question is asking for the calculation of gas pressure in a reservoir using an inclined manometer. Key concepts include the density of the manometer fluid (mercury), the distance the fluid rises, the angle of inclination, standard atmospheric pressure, and gravity.
Answer
The total pressure in the reservoir is \( 53.9 \text{ psi} \).
Answer for screen readers
The total pressure in the reservoir is approximately ( 53.9 \text{ psi} ).
Steps to Solve
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Convert Units for Consistency First, convert the distance from inches to feet since the density of the fluid is in lb/ft³.
Given: [ d = 5 \text{ inches} = \frac{5}{12} \text{ feet} \approx 0.41667 \text{ feet} ]
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Calculate the Height Rise in Vertical Direction Calculate the vertical rise of the manometer fluid using the angle of inclination. The vertical height ( h ) can be calculated as: [ h = d \sin(\theta) ] Where ( \theta = 30^\circ ).
[ h = 0.41667 \sin(30^\circ) = 0.41667 \cdot 0.5 \approx 0.20833 \text{ feet} ]
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Calculate the Pressure Difference The pressure difference caused by the manometer fluid can be calculated using the formula: [ \Delta P = \rho g h ] Where ( \rho = 845 \text{ lb/ft}^3 ) (density), ( g = 32.2 \text{ ft/s}^2 ) (acceleration due to gravity), and ( h ) is calculated from the previous step.
[ \Delta P = 845 \cdot 32.2 \cdot 0.20833 \approx 5637.5 \text{ lb/ft}^2 ]
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Convert Pressure Difference to psi Convert the pressure difference from lb/ft² to psi (pound per square inch) using the conversion factor ( 1 \text{ psi} = 144 \text{ lb/ft}^2 ).
[ \Delta P \text{ (in psi)} = \frac{5637.5}{144} \approx 39.2 \text{ psi} ]
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Calculate Total Pressure in Reservoir Finally, to find the absolute pressure in the reservoir, add atmospheric pressure to the pressure difference:
Given atmospheric pressure: [ P_{atm} = 14.7 \text{ psi} ]
[ P_{total} = P_{atm} + \Delta P = 14.7 + 39.2 \approx 53.9 \text{ psi} ]
The total pressure in the reservoir is approximately ( 53.9 \text{ psi} ).
More Information
This calculation provides the pressure in the gas reservoir using an inclined manometer. Understanding the relationship between fluid height and pressure difference is crucial in applications such as HVAC and fluid mechanics.
Tips
- Not converting units properly (e.g., inches to feet).
- Miscalculating the vertical rise using the sine function.
- Forgetting to convert pressure units when needed.
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