Q47 / 54 · JEE Advanced 2022 (Paper 1) · Physics

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 A1=0.1 m2 and the upper end is A2=0.4 m2. 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 m s-2 and the ratio of specific heats of the gas γ=2. Ignore atmospheric pressure.

Figure

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.
  • Correct answer
    B
    The velocity of the gas at the lower end of the chimney is 40 m s-1 and at the upper end is 20 m s-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

Here the gas undergoes adiabatic expansion, 

P1−γTγ=constant

⇒P2P1=T1T2γ1−γ

⇒P2=30015021−2×600

⇒P2=6004=150 Pa

Thus, pressure of the gas at the upper end of the chimney is P2=150 Pa

From ideal gas equation, ρ=PMRT⇒ρ∝PT

Then, ρ1ρ2=P1P2T1T2=150600300150=12

⇒ρ2=0.212=0.1 kg m-3

Rate of mass flow α=0.8 kg s-1

⇒ Volume flow rate at top =0.80.1=8 m3 s-1

Velocity of gas at lower end v1=V1A1=0.80.2×0.1=40 m s-1

Velocity of gas at upper end v2=0.8×20.2×0.4=20 m s-1

Work done  Won gas=∆K+∆Ug+internal energy

P1A1∆x1-P2A2∆x2=12∆mv12-12∆mv22+mg∆h+f2P2∆V2-P1∆V1

Solving we get h=360 m

Hence, the correct answer is option: 'B'.

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