GATE 2022 AE – Question 65
A multistage axial compressor receives air at 1 atm,300 K and compresses it to at least 5 atm. Mean blade speed U=245 m/s and stage work coefficient $\Delta h_0/U^2=0.55$. Use $c_p=1005$ J/(kg K),R=287 J/(kg K),γ=1.4. For isentropic compression, required number of stages is (round to the next higher integer).
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Correct answer: 6
Explanation
**Isentropic work per kg for the whole compressor** (pressure ratio 5, inlet at 300 K):
$$\Delta h_{0,total}=c_pT_1\left[\left(\frac{p_2}{p_1}\right)^{\frac{\gamma-1}{\gamma}}-1\right]=1005\times300\times\left[5^{0.2857}-1\right].$$
$$5^{0.2857}=1.5838\;\Rightarrow\;\Delta h_{0,total}=301\,500\times0.5838=176\,000\text{ J/kg}.$$
**Work per stage.** The work coefficient is $\Delta h_0/U^2=0.55$:
$$\Delta h_{0,stage}=0.55\times245^2=0.55\times60\,025=33\,014\text{ J/kg}.$$
**Number of stages:**
$$N=\frac{176\,000}{33\,014}=5.33 .$$
Since a whole number of stages is needed and the pressure ratio must be at least 5, round up: **6 stages**.