Question

A single stage impulse turbine with a diameter of 1.2 m runs at 3000 rpm. If the blade speed ratio is 0.42, the inlet velocity of steam will be (in $\text{ms}^{-1}$)

• A. \(450 \)
• B. \(900 \)
• C. \(200 \)
• D. \(600 \)

Hint:

To perform the calculation quickly without a calculator, approximate $\pi \approx \frac{22}{7}$ or $3.14$. - $u = 60 \times 3.14 = 188.4 \text{ m/s}$. - Then, $V_1 = \frac{188.4}{0.42} \approx \frac{188.4}{0.4} = 471 \text{ m/s}$. Since $0.42$ is slightly larger than $0.4$, the true value must be slightly less than $471$, pointing directly to $450 \text{ m/s}$ (Option A).

Answer 1

The Correct Option is A

Step 1: Convert the rotor speed into revolutions per second.
The rotor turns at \(N = 3000\) rpm, which in revolutions per second is:\[ n = \frac{3000}{60} = 50 \text{ rev/s} \]

Step 2: Get the blade tip speed from the circumference travelled per second.
Each revolution, a point on the rotor rim of diameter \(D = 1.2\) m travels a distance equal to the circumference \(\pi D\). So the blade linear speed is:\[ u = \pi D \times n = \pi \times 1.2 \times 50 = 60\pi \approx 188.5 \text{ m/s} \]

Step 3: Use the blade speed ratio as a proportion to back out the jet velocity.
The blade speed ratio tells us the blade speed is 0.42 times the steam jet's inlet velocity, so the jet velocity must be the blade speed scaled up by the reciprocal of 0.42:\[ V_1 = \frac{u}{\rho} = \frac{188.5}{0.42} \approx 448.8 \text{ m/s} \]Rounding to the nearest option gives:\[ \boxed{V_1 \approx 450 \text{ m/s}} \]
This matches option (A).