GATE 2022 AE – Question 53
A converging duct has inlet area 1 m² and speed 15 m/s at 300 K. Air is inviscid and isothermal with R=287 J/(kg K), γ=1.4. Minimum exit area for M≤0.3 (two decimal places m²) is ________.
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Correct answer: 0.13 to 0.15
Explanation
**Maximum speed allowed.** The Mach number must stay at or below 0.3 everywhere. With an isothermal flow at 300 K the speed of sound is
$$a=\sqrt{\gamma RT}=\sqrt{1.4\times287\times300}=347.2\text{ m/s},$$
so the maximum speed is $V_{max}=0.3\times347.2=104.2$ m/s.
**Continuity.** The flow is isothermal and at low Mach number, so the density is taken as constant (a Mach number of at most 0.3 means density changes of under about 5%), and $A_1V_1=A_2V_2$.
**Smallest exit area** (largest exit speed):
$$A_{exit}=\frac{A_1V_1}{V_{max}}=\frac{1\times15}{104.2}=\mathbf{0.14\ m^2}.$$