JEE Advanced 2022 — Paper 1 — PHY-II solution

JEE Advanced2022Paper 1PHY-II

An ideal gas of density $ρ = 0.2$ kg m $^{- 3}$ enters a chimney of height $h$ at the rate of $α = 0.8$ kg s $^{- 1}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_{1} = 0.1$ m $^{2}$ and the upper end is $A_{2} = 0.4$ m $^{2}$ . The pressure and the temperature of the gas at the lower end are 600 Pa and 300 K, respectively, while its temperature at the upper end is 150 K. The chimney is heat insulated so that the gas undergoes adiabatic expansion. Take $g = 10 ms^{- 2}$ and the ratio of specific heats of the gas $γ = 2$ . Ignore atmospheric pressure.

Which of the following statement(s) is(are) correct?

A
The pressure of the gas at the upper end of the chimney is 300 Pa.
B
The velocity of the gas at the lower end of the chimney is 40 ms $^{- 1}$ and at the upper end is 20 ms $^{- 1}$ .
C
The height of the chimney is 590 m.
D
The density of the gas at the upper end is 0.05 kg m $^{- 3}$ .

Solution

Density at top (adiabatic): $\frac{ρ _{1}}{ρ _{2}} = (\frac{T _{1}}{T _{2}})^{1/ (γ - 1)}$ . With $γ = 2$ , $\frac{ρ _{1}}{ρ _{2}} = \frac{T _{1}}{T _{2}} = 2$ . So $ρ_{2} = 0.1$ kg/m $^{3}$ , ruling out (D).

Pressure at top: $\frac{P _{2}}{P _{1}} = (\frac{ρ _{2}}{ρ _{1}})^{γ} = (1/2)^{2} = 1/4$ . So $P_{2} = 150$ Pa, ruling out (A).

Velocities from mass flow rate $\overset{m}{˙} = ρ v A$ :

$v_{1} = \frac{0.8}{0.2 \times 0.1} = 40$ m/s. $v_{2} = \frac{0.8}{0.1 \times 0.4} = 20$ m/s. (B) is correct.

Bernoulli (energy conservation per unit volume):

$\frac{1}{2} ρ_{1} v_{1}^{2} + P_{1} = \frac{1}{2} ρ_{2} v_{2}^{2} + P_{2} + ρ_{2} g h$ .

$\frac{1}{2} (0.2) (1600) + 600 = \frac{1}{2} (0.1) (400) + 150 + (0.1) (10) h$ .

$160 + 600 = 20 + 150 + h \Rightarrow h = 590$ m. (C) is correct.

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