A mercury manometer (ρ = 13,600 kg/m³) is connected to an air duct to measure the pressure inside. The difference in the manometer levels is 30 mm, and the atmospheric pressure is... A mercury manometer (ρ = 13,600 kg/m³) is connected to an air duct to measure the pressure inside. The difference in the manometer levels is 30 mm, and the atmospheric pressure is 100 kPa. (a) Judging from the information, determine if the pressure in the duct is above or below atmospheric pressure. (b) Determine the absolute pressure in the duct.
Understand the Problem
The question involves using a mercury manometer to determine the pressure inside an air duct in relation to atmospheric pressure. It provides data such as the density of mercury, the difference in manometer levels, and atmospheric pressure, asking to confirm if the duct pressure is above or below atmospheric pressure and to calculate the absolute pressure in the duct.
Answer
The duct pressure is above atmospheric, with an absolute pressure of $104.00 \, \text{kPa}$.
Answer for screen readers
(a) The pressure in the duct is above atmospheric pressure.
(b) The absolute pressure in the duct is $104.00 , \text{kPa}$.
Steps to Solve
- Convert height difference to meters
Given the difference in manometer levels is 30 mm, convert this to meters: $$ h = 30 \text{ mm} = 0.030 \text{ m} $$
- Calculate the pressure difference using the hydrostatic pressure formula
Use the formula for hydrostatic pressure to find the pressure difference ($\Delta P$) due to the height of the mercury column: $$ \Delta P = \rho g h $$ where:
- $\rho = 13,600 , \text{kg/m}^3$ (density of mercury)
- $g \approx 9.81 , \text{m/s}^2$ (acceleration due to gravity)
Plugging in the values: $$ \Delta P = 13600 \times 9.81 \times 0.030 $$
- Determine the calculated pressure difference
Calculate $\Delta P$: $$ \Delta P = 13600 \times 9.81 \times 0.030 \approx 4004.88 , \text{Pa} $$
In kilopascals, this is: $$ \Delta P \approx 4.00 , \text{kPa} $$
- Assess the pressure in the duct
Determine if the duct pressure ($P_{\text{duct}}$) is above or below atmospheric pressure. Assuming the level in the manometer connected to the duct is lower, the duct pressure is: $$ P_{\text{duct}} = P_{\text{atm}} + \Delta P $$ where $P_{\text{atm}} = 100 , \text{kPa}$.
Calculate: $$ P_{\text{duct}} = 100 + 4.00 = 104.00 , \text{kPa} $$
- Determine the absolute pressure in the duct
As the calculation indicates the duct pressure is above atmospheric pressure, the absolute pressure in the duct is: $$ P_{\text{absolute}} = P_{\text{duct}} = 104.00 , \text{kPa} $$
(a) The pressure in the duct is above atmospheric pressure.
(b) The absolute pressure in the duct is $104.00 , \text{kPa}$.
More Information
Understanding manometers is essential in fluid mechanics, as they indicate pressure differences. The pressure readings can inform HVAC systems or other engineering applications where air duct pressures are critical.
Tips
- Mistaking the pressure direction: It's crucial to correctly understand whether the measured pressure is above or below atmospheric pressure based on the manometer's levels. Make sure to consider whether the level rises or falls.
- Converting units incorrectly: Ensure that all measurements are consistent, especially when converting mm to meters.
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