Temperature Ratio Across Expansion Fan Calculator

Temperature Ratio Across Expansion Fan Formula:

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

Specific Heat Ratio Expansion Wave (γe):

Mach Number Ahead of Expansion Fan (Me1):

Mach Number Behind Expansion Fan (Me2):

Temperature Ratio Across Expansion Fan (Te,r):

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1. What is Temperature Ratio Across Expansion Fan?

The Temperature Ratio Across Expansion Fan is the ratio of downstream temperature to upstream temperature across the expansion fan in compressible flow. It quantifies the temperature change that occurs when a supersonic flow expands through an expansion wave.

2. How Does the Calculator Work?

The calculator uses the expansion fan temperature ratio formula:

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

Where:

\( T_{e,r} \) — Temperature 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 formula calculates the temperature change across an expansion fan based on the specific heat ratio and Mach numbers before and after the expansion.

3. Importance of Temperature Ratio Calculation

Details: Calculating temperature ratio across expansion fans is crucial for analyzing compressible flow behavior, designing supersonic nozzles, and understanding aerodynamic heating effects in high-speed flows.

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 fluid dynamics?
A: An expansion fan is a series of Mach waves that form when a supersonic flow expands around a convex corner, causing the flow to accelerate and temperatures to drop.

Q2: Why does temperature decrease across an expansion fan?
A: Temperature decreases because the expansion process converts thermal energy into kinetic energy as the flow accelerates.

Q3: 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, triatomic gases ≈ 1.3.

Q4: How does Mach number affect temperature ratio?
A: Higher Mach numbers result in more significant temperature changes across the expansion fan due to greater flow acceleration.

Q5: Can this formula be used for compression waves?
A: No, this specific formula applies only to expansion fans. Compression waves (shock waves) have different governing equations.