Choked-Flow Threshold – Critical Pressure Ratio for Ideal Gases (Gamma ≈ 1.4) For an ideal gas with specific heat ratio around gamma = 1.4 (e.g., air), the critical pressure ratio (downstream-to-upstream static pressure at sonic throat) is approximately which of the following values?

Introduction / Context:
In nozzles and compressible-duct flows, choked flow occurs when the Mach number at the minimum area (throat) reaches 1. The pressure ratio at which this happens is critical for sizing nozzles, valves, and gas-discharging systems. Engineers often memorize an approximate value for air (gamma ≈ 1.4).

Given Data / Assumptions:

• Ideal-gas behavior with constant specific heats.
• Isentropic, one-dimensional flow through a smoothly contoured nozzle.
• Upstream reservoir conditions well defined; throat is the minimum area.

Concept / Approach:

The isentropic relation linking static pressure and Mach number yields the critical pressure ratio at M = 1 as p*/p0 = (2/(gamma + 1))^(gamma/(gamma − 1)). For gamma = 1.4, this evaluates numerically to about 0.528. If the downstream static pressure falls below p* = 0.528 * p0, the mass flow rate remains fixed (choked) and further reductions in back pressure do not increase mass flow.

Step-by-Step Solution:

Verification / Alternative check:

Design charts and gas-dynamics tables list 0.528 for air. CFD and experiments corroborate that choking begins near this ratio for gamma ≈ 1.4 in smooth nozzles without significant losses.

Why Other Options Are Wrong:

Values 0.546, 0.577, and 0.582 are off the theoretical value for gamma = 1.4. The slight differences might correspond to different gamma values or non-isentropic effects but are not the standard air value.

Common Pitfalls:

Confusing p*/p0 with p0/p* (the inverse); forgetting that gamma changes with temperature, slightly shifting the ratio. Also, mixing total (stagnation) vs. static pressures.

Final Answer:

0.528