A cylindrical furnace has height H and diameter D both 1 m . It is maintained at temperature 360 K . The air gets heated inside the furnace at constant pressure P a and its temperature becomes T = 360

Question

A cylindrical furnace has height H and diameter D both 1 m . It is maintained at temperature 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 enters the furnace. Assume air as an ideal gas, neglect 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 air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ gm s − 1 .

TestHub · Accepted Answer

Applying Bernoulli’s theorem between top and bottom points of furnace :
$P_{a} + \frac{1}{2} ρ_{a} V^{2} = P_{a} + ρ g H + \frac{1}{2} ρ V^{2} . . . (1)$
Also, since $P M = ρ R T$
$\Rightarrow ρ_{a} \times 300 = ρ \times 360$
$\Rightarrow ρ = 1 kg m^{- 3} . . . (2)$
From equation(1), we can write
$\Rightarrow \frac{V^{2}}{2} (0.2) = 1 \times 10 \times 1$
$\Rightarrow V = 10 m s^{- 1}$
Required mass flow rate $= V \times \frac{π D^{2}}{4} \times ρ$
$= 10 \times \frac{3.14}{4} \times 1^{2} \times 1 kg s^{- 1}$
$= 7.85 kg s^{- 1}$
$= 7850 gm s^{- 1}$

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