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Question #42550

Difficulty: 8/10 | 🟢 Free

The Question

An ideal gas flows steadily through a horizontal nozzle. At the inlet, the gas pressure is $P_1 = 8.0 \times 10^4 \text{ Pa}$ , its density is $\rho_1 = 6.0 \text{ kg/m}^3$ , and its velocity is negligible ( $v_1 \approx 0$ ). The gas undergoes a polytropic expansion defined by the relation $P \rho^{-n} = \text{constant}$ , with a polytropic index $n = 1.5$ . If the pressure at the exit of the nozzle is $P_2 = 1.0 \times 10^4 \text{ Pa}$ , what is the exit velocity $v_2$ of the gas? (Assume the flow is frictionless and one-dimensional).

Options

A.
$200 \text{ m/s}$
B.
$100\sqrt{\frac{7}{3}} \text{ m/s}$
C.
$100\sqrt{\frac{28}{3}} \text{ m/s}$
D.
$200\sqrt{\frac{2}{3}} \text{ m/s}$

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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

Polytropic Process|Equation of Continuity|Mechanical Properties of Fluids

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the principles of fluid dynamics to compressible flows, specifically distinguishing between variable-density gas dynamics and incompressible liquid flows.

A common trap is reflexively applying Bernoulli's equation ( $P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2$ ), which assumes a constant density $\rho$ .

In a nozzle undergoing a polytropic expansion, the gas density drops significantly due to the pressure gradient.

Students falling for this trap will incorrectly use the inlet density $\rho_1$ to find $v_2 = 100\sqrt{7/3} \text{ m/s}$ , completely ignoring the thermodynamic coupling required to integrate Euler's equation ( $v \, dv = -dP/\rho$ ) with the polytropic state equation $P \rho^{-n} = \text{constant}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Polytropic Process
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free