GATE 2023 AE – Question 55
A pump follows $C_P=a\Phi^2$, with $C_P=P/(\rho\omega^3D^5)$ and dimensionless volume-flow coefficient Φ. At constant flow, changing rotation rate to 1.25ω changes power to αP. The value of α is _____.
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Correct answer: 1.24 to 1.26
Explanation
**Express the power in terms of the flow rate.** The flow coefficient is $\Phi=\dfrac{Q}{\omega D^3}$ and the power coefficient is $C_P=\dfrac{P}{\rho\omega^3D^5}=a\Phi^2$.
$$P=a\,\rho\,\omega^3D^5\Phi^2=a\,\rho\,\omega^3D^5\frac{Q^2}{\omega^2D^6}=\frac{a\,\rho\,\omega\,Q^2}{D}.$$
**At constant flow rate $Q$** (and the same pump and fluid), $P\propto\omega$.
When the speed changes to $1.25\,\omega$, the power becomes $1.25\,P$, so
$$\alpha=\mathbf{1.25}.$$