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

Concept: An impulse turbine extracts kinetic energy from a high-velocity fluid jet exiting a nozzle. To evaluate its performance, we use two key velocity parameters:
• Blade Linear Velocity ($u$): The tangential speed of the rotating blades, calculated from the rotor diameter ($D$) and rotational speed ($N$ in rpm) as: \[ u = \frac{\pi \cdot D \cdot N}{60} \]
• Blade Speed Ratio ($\rho$): The ratio of the blade linear velocity to the absolute inlet velocity of the steam jet ($V_1$) exiting the nozzle: \[ \rho = \frac{u}{V_1} \] By determining the blade linear speed from the physical dimensions and rotational rate, we can use the blade speed ratio to find the absolute inlet velocity of the steam.

Step 1: Extract the given parameters and convert them to standard SI units.
The values specified in the problem statement are:
• Mean Diameter of the turbine rotor, \(D = 1.2 \text{ m}\)
• Rotational Speed, \(N = 3000 \text{ rpm}\)
• Blade Speed Ratio, \(\rho = 0.42\)

Step 2: Compute the linear peripheral velocity of the turbine blades ($u$).
Using the standard rotational-to-linear conversion formula: \[ u = \frac{\pi \cdot D \cdot N}{60} \] Substitute the values into the equation: \[ u = \frac{\pi \times 1.2 \times 3000}{60} \] Simplify the fraction by dividing 3000 by 60: \[ \frac{3000}{60} = 50 \] Now substitute this back into the product: \[ u = \pi \times 1.2 \times 50 \] \[ u = 1.2 \times 50 \times \pi = 60\pi \text{ m/s} \] Using the standard approximation for $\pi \approx 3.14159$: \[ u = 60 \times 3.14159 = 188.495 \text{ m/s} \]

Step 3: Calculate the absolute inlet velocity of the steam ($V_1$).
From the definition of the blade speed ratio: \[ \rho = \frac{u}{V_1} \quad \Rightarrow \quad V_1 = \frac{u}{\rho} \] Substitute the calculated blade velocity ($u = 188.495 \text{ m/s}$) and the given speed ratio ($\rho = 0.42$): \[ V_1 = \frac{188.495}{0.42} \] Let's calculate this division step-by-step: \[ V_1 \approx 448.798 \text{ m/s} \] Rounding this value to the nearest integer option gives approximately $450 \text{ m/s}$. This matches Option (A).