GATE 2023 AE – Question 64
A centrifugal compressor has inlet-root diameter 0.25 m, impeller outlet diameter 0.6 m and pressure ratio 5. Air enters at 1 atm and 25°C. Polytropic efficiency is 0.8, slip factor 0.92, $c_p=1.004$ kJ/(kg K) and γ=1.4. Impeller speed (nearest integer RPM) is _____.
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Correct answer: 16002
Explanation
**Energy added by the impeller.** For a centrifugal compressor with a slip factor $\sigma$ and no inlet swirl, the work input per unit mass is
$$\Delta h_0=\sigma\,U_2^2 .$$
**The same work from the pressure ratio** with a polytropic efficiency $\eta_p$ (inlet temperature $T_1=298.15$ K):
$$\Delta h_0=c_pT_1\left[\left(\frac{p_2}{p_1}\right)^{\frac{\gamma-1}{\gamma\eta_p}}-1\right].$$
**Numbers.** The exponent is $\dfrac{0.4}{1.4\times0.8}=0.3571$:
$$5^{0.3571}=1.7768,\qquad\Delta h_0=1004\times298.15\times0.7768=2.325\times10^5\ \text{J/kg}.$$
**Tip speed.**
$$U_2=\sqrt{\frac{\Delta h_0}{\sigma}}=\sqrt{\frac{2.325\times10^5}{0.92}}=502.7\ \text{m/s}.$$
**Rotational speed** with an impeller outlet diameter of 0.6 m ($U_2=\omega\,D_2/2$):
$$N=\frac{60\,U_2}{\pi D_2}=\frac{60\times502.7}{\pi\times0.6}=16\,002\ \text{rpm}.$$