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Question #42099

Difficulty: 8/10 | 🟢 Free

The Question

A saturated liquid of density $\rho_0$ and specific heat capacity $c$ enters a horizontal, thermally insulated nozzle at pressure $P_0$ and boiling temperature $T_0$ with negligible velocity. As the fluid accelerates, its pressure drops to $P$ , causing a small mass fraction to vaporize. Assume the flow is steady and isentropic, and the vapor density $\rho_v$ is constant. Which of the following expressions gives the velocity $v$ of the mixture at pressure $P$ ? (Assume the specific latent heat of vaporization $L$ is constant, and the mass fraction of vapor remains small).

Options

A.
$\sqrt{ \frac{2(P_0 - P)}{\rho_0} + \frac{c T_0}{L^2 \rho_v^2} (P_0 - P)^2 }$
B.
$\sqrt{ \frac{2(P_0 - P)}{\rho_0} }$
C.
$\sqrt{ \frac{2(P_0 - P)}{\rho_0} + \frac{2 c T_0}{L^2 \rho_v^2} (P_0 - P)^2 }$
D.
$\sqrt{ \frac{2(P_0 - P)}{\rho_0} + \frac{c T_0}{L \rho_v} (P_0 - P)^2 }$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Bernoulli’s Energy Balance|Equation of State for Mixtures|Approximations

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the Bernoulli equation by introducing a compressible, two-phase flow scenario.

Students commonly fall into the trap of assuming the fluid remains incompressible (density $\rho_0$ ), leading them to blindly apply the standard algebraic Bernoulli equation and select the incorrect option B.

To solve this, students must recognize that the steady flow energy equation for an isentropic process reduces to the differential form $\int v_{sp} dP + \frac{1}{2}v^2 = 0$ .

They must then couple this mechanical energy balance with thermodynamic principles—specifically the Clausius-Clapeyron relation and isentropic expansion—to derive the pressure-dependent specific volume $v_{sp}(P)$ before integrating.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Bernoulli’s Energy Balance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free