Question

The operating characteristics of a pump are measured as $C_p = a \Phi^2$, where \[ C_p = \frac{P}{\rho \omega^3 D^5}, \Phi = \text{flow coefficient}, a=\text{constant}. \] If $\omega$ increases by 25% (i.e. $\omega \to 1.25\omega$) and the flow coefficient $\Phi$ is constant, determine $\alpha$ such that $P$ becomes $\alpha P$. (round off to two decimal places)

Hint:

In turbomachinery similarity laws, if flow coefficient remains constant, power scales as cube of speed: $P \propto \omega^3 D^5$.

Answer 1

Step 1: Expression for power.
\[ P = C_p \rho \omega^3 D^5 \] But $C_p = a\Phi^2$ (constant since $\Phi$ same). So, \[ P \propto \omega^3 \]

Step 2: Ratio of powers.
If $\omega \to 1.25 \omega$: \[ \frac{P_{new}}{P_{old}} = \left(\frac{1.25\omega}{\omega}\right)^3 = (1.25)^3 \] \[ = 1.953 \approx 1.95 \] Wait — but note: $C_p = a\Phi^2$, but $\Phi$ scales with $\dfrac{Q}{\omega D^3}$. If flow coefficient remains same, $Q \propto \omega$. Then $P \propto \omega^5$.

Step 3: Correct scaling.
Since $\Phi$ constant: \[ C_p \text{ constant} \Rightarrow P \propto \omega^3 \] But if $Q \propto \omega$, $\Phi$ constant indeed. Correction: $P \propto \omega^3$, confirmed. So, \[ \alpha = (1.25)^3 = 1.953 \approx 1.95 \] Final: \[ \boxed{1.95} \]