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Question #42247

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The Question

An ideal gas of molar mass $M$ and adiabatic exponent $\gamma$ is kept in a large insulated stagnation tank at a constant pressure $P_0$ and temperature $T_0$ . The gas flows out steadily through a small horizontal converging nozzle of exit area $A$ into the atmosphere at pressure $P$ (assume $P$ is high enough such that the flow is not choked). Which of the following expressions correctly represents the mass flow rate $\dot{m}$ of the gas?

Options

A.
$A P_0 \sqrt{\frac{2 \gamma M}{(\gamma - 1) R T_0} \left[ \left(\frac{P}{P_0}\right)^{\frac{2}{\gamma}} - \left(\frac{P}{P_0}\right)^{\frac{\gamma+1}{\gamma}} \right]}$
B.
$A P_0 \sqrt{\frac{2 M}{R T_0} \left( 1 - \frac{P}{P_0} \right)}$
C.
$A P_0 \sqrt{\frac{2 \gamma M}{(\gamma - 1) R T_0} \left[ 1 - \left(\frac{P}{P_0}\right)^{\frac{\gamma-1}{\gamma}} \right]}$
D.
$A P \sqrt{\frac{2 \gamma M}{(\gamma - 1) R T_0} \left[ 1 - \left(\frac{P}{P_0}\right)^{\frac{\gamma-1}{\gamma}} \right]}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Equation of Continuity|Ideal Gas Assumptions|Temperature and Kinetic Energy

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between incompressible and compressible flow.

Students often incorrectly apply the standard Bernoulli's equation ( $P + \frac{1}{2}\rho v^2 = \text{const}$ ) and the incompressible continuity equation ( $A_1 v_1 = A_2 v_2$ ) to gases, leading to Option B.

For a gas undergoing significant pressure changes, density is not constant.

The correct approach requires using the steady flow energy equation (conservation of enthalpy) to find velocity, and the ideal gas law with adiabatic relations to find the local exit density, synthesizing thermodynamics and fluid dynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Equation of Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free