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 𝑃𝑎 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 𝜌𝑎 = 1.2 kg m−3, pressure 𝑃𝑎 and temperature 𝑇𝑎 = 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

Applying Bernoulli's theorem,
$P_a+\frac{1}{2}\rho_aV^2=P_a+\rho gH+\frac{1}{2}\rho V^2\,\,\,\,...........(1)$
Also, since PM = $\rho$RT
$\rho_{a}\times300=\rho\times360$
$\Rightarrow \rho=1\,kg/m^3\,\,\,..........(2)$
$\frac{v^2}{2}(0.2)=1\times10\times1$
$\Rightarrow v=10\,\,m/s$
Mass flow rate = $V\times \frac{\pi D^2}{4}\times \rho=10\times\frac{3.14}{4}\times\,\,1 kg/s=7.85\,\,kg/s=7850\,gm/s$