JEE Advanced 2022 · Paper 1 · Physics Q11

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PhysicsMultiple correct+4 / -2Question 29 of 54

An ideal gas of density $\rho = 0.2\ \mathrm{kg\,m^{-3}}$ enters a chimney of height $h$ at the rate of $\alpha = 0.8\ \mathrm{kg\,s^{-1}}$ from its lower end, and escapes through the upper end as shown in the figure. The cross-sectional area of the lower end is $A_1 = 0.1\ \mathrm m^2$ and the upper end is $A_2 = 0.4\ \mathrm m^2$ . 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 $g = 10\ \mathrm{m\,s^{-2}}$ and the ratio of specific heats of the gas $\gamma = 2$ . Ignore atmospheric pressure.

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

A vertical chimney of height h: a narrow tube at the bottom with cross-section A1 widens through a conical section into a wide cylinder with top cross-section A2. Arrows show gas entering upward at the bottom and flowing up through the top.

The pressure of the gas at the upper end of the chimney is 300 Pa.

The velocity of the gas at the lower end of the chimney is

40\ \mathrm{m\,s^{-1}}

and at the upper end is

20\ \mathrm{m\,s^{-1}}

The height of the chimney is 590 m.

The density of the gas at the upper end is

0.05\ \mathrm{kg\,m^{-3}}

Answer and worked solution

Official answer: B

Adiabatic changes. With $\gamma=2$ , $T\propto\rho^{\gamma-1}=\rho$ .

The temperature halves, so $\rho_2=0.1$ kg m⁻³. (D) is false.
$P\propto\rho T$ , so $P_2=600\times\frac12\times\frac12=150$ Pa. (A) is false.

(B) True. From the mass flow rate:
$v_1=\frac{\alpha}{\rho_1A_1}=\frac{0.8}{0.02}=40\ \text{m s}^{-1},\qquad v_2=\frac{0.8}{0.1\times0.4}=20\ \text{m s}^{-1}.$

(C) False. For adiabatic flow, Bernoulli's equation takes the form $\frac{\gamma}{\gamma-1}\frac P\rho+\frac{v^2}2+gh=$ const:
$2(3000)+800=2(1500)+200+10h\implies h=360\ \text{m},$
not 590 m.

Topics:Bernoulli's principle Continuity equation Thermodynamic processes

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JEE Advanced 2022 · Paper 1, Physics question 11