Question

A cylindrical furnace has height (𝐻) and diameter (𝐷) both 1 m. It is maintained at a temperature of 360 K. The air gets heated inside the furnace at constant pressure 𝑃_𝑎 and its temperature becomes 𝑇 = 360 𝐾. The hot air with density 𝜌 rises up a vertical chimney of diameter 𝑑 = 0.1 m and height ℎ = 9 m above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density 𝜌_𝑎 = 1.2 kg m⁻³, pressure 𝑃_𝑎 and temperature 𝑇_𝑎 = 300 K enter the furnace. Assume air as an ideal gas, neglecting the variations in 𝜌 and 𝑇 inside the chimney and the furnace. Also, ignore the viscous effects. [Given: The acceleration due to gravity 𝑔 = 10 m s⁻² and 𝜋 = 3.14] Considering the airflow to be streamlined, the steady mass flow rate of air exiting the chimney is _______ gm s⁻¹.

Answer 1

Correct Answer: 7850

Application of Bernoulli's Theorem and Mass Flow Rate Calculation 

Applying Bernoulli's theorem: \[ P_a + \frac{1}{2} \rho_a V^2 = P_a + \rho g H + \frac{1}{2} \rho V^2 \quad \text{(1)} \]

Also, using the ideal gas law \( PM = \rho RT \): \[ \rho_a \times 300 = \rho \times 360 \] \[ \Rightarrow \rho = 1 \, \text{kg/m}^3 \quad \text{(2)} \]

Solving for \( v \): \[ \frac{v^2}{2} \times (0.2) = 1 \times 10 \times 1 \] \[ \Rightarrow v = 10 \, \text{m/s} \]

Mass flow rate is given by: \[ \text{Mass Flow Rate} = V \times \frac{\pi D^2}{4} \times \rho = 10 \times \frac{3.14}{4} \times 1 \, \text{kg/s} = 7.85 \, \text{kg/s} = 7850 \, \text{gm/s} \]