An ideal gas with specific heat ratio $\gamma = 1.4$ flows through a frictionless constant-area duct with heat addition (Rayleigh flow). The Mach number at which the static temperature $T$ reaches its maximum possible value along the Rayleigh line is:

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### 1. Strategy & Theoretical Principles In 1D frictionless flow with heat addition in a constant-area duct (Rayleigh flow), the static temperature ratio relative to sonic state ($M^* = 1$) is given by: $$\frac{T}{T^*} = \left[\frac{(1 + \gamma) M}{1 + \gamma M^2}\right]^2$$ ### 2. Analytical Formulation To find the Mach number where static temperature $T$ reaches its maximum, take the derivative with respect to $M$ and set it to zero: $$\frac{d}{dM}\left(\frac{T}{T^*}\right) = 0$$ $$\frac{d}{dM} \left[\frac{M}{1 + \gamma M^2}\right] = \frac{(1 + \gamma M^2)(1) - M(2\gamma M)}{(1 + \gamma M^2)^2} = \frac{1 - \gamma M^2}{(1 + \gamma M^2)^2} = 0$$ ### 3. Step-by-Step Calculation Setting numerator to zero: $$1 - \gamma M^2 = 0 \implies M^2 = \frac{1}{\gamma} \implies M = \frac{1}{\sqrt{\gamma}}$$ For air with $\gamma = 1.4$: $$M = \frac{1}{\sqrt{1.4}} = \frac{1}{1.183216} \approx 0.84515 \approx 0.845$$ Note that stagnation temperature $T_0$ reaches its maximum at $M = 1.0$, whereas static temperature $T$ peaks in the subsonic regime at $M = 1/\sqrt{\gamma} \approx 0.845$. ### 4. Distractor Forensics - Candidates frequently confuse static temperature maximum ($M = 1/\sqrt{\gamma} = 0.845$) with stagnation temperature maximum ($M = 1.0$). ### 5. Exam Trap & Standard Reference - **Trap Alert:** Distinguish static temperature peak ($M = 1/\sqrt{\gamma}$) from sonic point choking condition ($M = 1$). - **Standard Reference:** Anderson, J.D., *Modern Compressible Flow*, 4th Edition, McGraw-Hill, Chapter 3 (Rayleigh Flow).

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