Pressure Ratio Across Expansion Fan Calculator

Pressure Ratio Across Expansion Fan Formula:

\[ P_{e,r} = \left( \frac{1 + 0.5(\gamma_e - 1)M_{e1}^2}{1 + 0.5(\gamma_e - 1)M_{e2}^2} \right)^{\frac{\gamma_e}{\gamma_e - 1}} \]

Specific Heat Ratio Expansion Wave (γe):

Mach Number Ahead of Expansion Fan (Me1):

Mach Number Behind Expansion Fan (Me2):

Pressure Ratio Across Expansion Fan (Pe,r):

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1. What is Pressure Ratio Across Expansion Fan?

The Pressure Ratio Across Expansion Fan is the ratio of downstream and upstream pressure across an expansion fan in supersonic flow. It quantifies the pressure change that occurs when a supersonic flow expands around a corner or through a diverging section.

2. How Does the Calculator Work?

The calculator uses the expansion fan pressure ratio formula:

\[ P_{e,r} = \left( \frac{1 + 0.5(\gamma_e - 1)M_{e1}^2}{1 + 0.5(\gamma_e - 1)M_{e2}^2} \right)^{\frac{\gamma_e}{\gamma_e - 1}} \]

Where:

\( P_{e,r} \) — Pressure ratio across expansion fan
\( \gamma_e \) — Specific heat ratio expansion wave
\( M_{e1} \) — Mach number ahead of expansion fan
\( M_{e2} \) — Mach number behind expansion fan

Explanation: The equation relates the pressure change across an expansion fan to the specific heat ratio and Mach numbers before and after the expansion.

3. Importance of Pressure Ratio Calculation

Details: Calculating pressure ratio across expansion fans is crucial for analyzing supersonic flow behavior, designing high-speed aerodynamic systems, and understanding shock-expansion theory in compressible flow dynamics.

4. Using the Calculator

Tips: Enter specific heat ratio (typically 1.4 for air), Mach number ahead of expansion fan, and Mach number behind expansion fan. All values must be valid (γe > 1, Me1 > 0, Me2 > 0).

5. Frequently Asked Questions (FAQ)

Q1: What is an expansion fan in supersonic flow?
A: An expansion fan is a region of continuous expansion that occurs when a supersonic flow turns away from itself, typically around a convex corner.

Q2: What are typical values for specific heat ratio?
A: For air at standard conditions, γ = 1.4. For other gases: monatomic gases ≈ 1.67, diatomic gases ≈ 1.4, polyatomic gases ≈ 1.1-1.3.

Q3: How does Mach number affect pressure ratio?
A: Higher Mach numbers generally result in lower pressure ratios across expansion fans, indicating greater pressure drop during expansion.

Q4: What are the limitations of this formula?
A: This formula assumes isentropic flow, perfect gas behavior, and two-dimensional flow. It may not be accurate for very high Mach numbers or complex flow geometries.

Q5: How is this related to Prandtl-Meyer expansion?
A: This pressure ratio formula is derived from Prandtl-Meyer expansion theory, which describes the turning of supersonic flow through an expansion fan.