Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is ________ gm .??[2023]

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 $π = 3.14$ ]

Q. Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is ________ gm $s^{- 1}$ . [2023]

[Given the acceleration due to gravity $g = 10 {m s}^{- 2}$ and $π = 3.14$ ]

Q. Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is ________ gm $s^{- 1}$ . [2023]

Smart Achievers · Accepted Answer

(47.10)

$Since, pressure P = constant$ $ρ_{a} T_{a} = ρ T$

$\Rightarrow 1.2 \times 300 = ρ (360)$

$∴$ $ρ = 1 {kg/m}^{3}$

$Applying Bernoulli's theorem between upper and bottom point$

$Assuming velocity of hot air inside the furnace ≃ 0$

$P_{a} + 0 + 0 = P_{a} - ρ_{a} g (h) + ρ g (h) + \frac{1}{2} ρ V^{2}$

$∴$ $V = \sqrt{\frac{2 (ρ_{a} - ρ) g \times 9}{ρ}} = \sqrt{2 (0.2) 90} = 6$

$Therefore the steady mass flow rate of air existing the chimney$

$Q = ρ π (\frac{d^{2}}{4}) V$ $= 1 \times 3.14 \times \frac{{(0.1)}^{2}}{4} \times 6$

$= 0.0471 kg/s = 47.10 {gms}^{- 1}$

Solution

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