A cylindrical furnace has height (H) and diameter (D) both 1m . It...

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1m$ . It is maintained at temperature $360K$ . The air gets heated inside the furnace at constant pressure ${P}_{a}$ and its temperature becomes $T = 360K$ . The hot air with density $ρ$ rises up a vertical chimney of diameter $d = 0 . 1m$ and height $h = 9m$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density ${ρ}_{a} = 1 . 2kg{m}^{- 3}$ , pressure ${P}_{a}$ and temperature ${T}_{a} = 300K$ 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 $g = 10m{s}^{- 2} and \pi = 3 . 14$ ]

A cylindrical furnace has height (H) and diameter (D) both 1m . It is maintained at temperature 360K . The air gets

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____

gm{s}^{- 1}

Correct answer

Step-by-step solution

Applying Bernoulli’s theorem between top and bottom points of furnace :
${P}_{a} + \dfrac{1}{2}{ρ}_{a}{V}^{2} = {P}_{a} + ρgH + \dfrac{1}{2}ρ{V}^{2} . . . \left(1\right)$
Also, since $PM = ρRT$
$⇒ {ρ}_{a} \times 300 = ρ \times 360$
$⇒ ρ = 1kg{m}^{- 3} . . . \left(2\right)$
From equation(1), we can write
$⇒ \dfrac{{V}^{2}}{2}\left(0 . 2\right) = 1 \times 10 \times 1$
$⇒ V = 10m{s}^{- 1}$
Required mass flow rate $= V \times \dfrac{\pi{D}^{2}}{4} \times ρ$
$= 10 \times \dfrac{3 . 14}{4} \times {1}^{2} \times 1kg{s}^{- 1}$
$= 7 . 85kg{s}^{- 1}$
$= 7850gm{s}^{- 1}$

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