Question

A propeller disc of diameter \(2 \, \text{m}\) produces a thrust of \(88 \, \text{kN}\) while advancing at a speed of \(5 \, \text{m/s}\) in fresh water of density \(1000 \, \text{kg/m}^3\). Based on the axial momentum theory, the propeller efficiency is ………….. % (rounded off to one decimal place).
 

Hint:

To calculate propeller efficiency: 1. Use the thrust power and input power formulas carefully.
2. Ensure the induced velocity \( v_a \) is calculated using axial momentum theory.
3. Verify all units for consistency when calculating power and efficiency.

Answer 1

Step 1: Recall the formula for propeller efficiency.
The propeller efficiency \( \eta \) is given by: \[ \eta = \frac{\text{Useful Power}}{\text{Input Power}} \times 100. \] Step 2: Calculate the useful power.
The useful power is the thrust power, given by: \[ \text{Thrust Power} = T \cdot V, \] where: - \( T = 88 \, \text{kN} = 88 \times 10^3 \, \text{N} \), - \( V = 5 \, \text{m/s} \). \[ \text{Thrust Power} = 88 \times 10^3 \cdot 5 = 440,000 \, \text{W} \, \text{or} \, 440 \, \text{kW}. \] Step 3: Calculate the input power.
The input power is based on the axial momentum theory and is given by: \[ \text{Input Power} = \frac{T \cdot (V + v_a)}{2}, \] where: - \( v_a \) is the induced velocity at the propeller. From axial momentum theory, the induced velocity \( v_a \) is related to the thrust by: \[ v_a = \sqrt{\frac{T}{2 \rho A}}, \] where: - \( \rho = 1000 \, \text{kg/m}^3 \) (density of water), - \( A = \pi D^2 / 4 = \pi (2)^2 / 4 = \pi \, \text{m}^2 \) (area of the propeller disc). Substitute the values: \[ v_a = \sqrt{\frac{88 \times 10^3}{2 \cdot 1000 \cdot \pi}} = \sqrt{\frac{88 \times 10^3}{2000 \pi}} = \sqrt{\frac{88}{2\pi}} \approx \sqrt{14.01} \approx 3.74 \, \text{m/s}. \] Thus: \[ \text{Input Power} = \frac{T \cdot (V + v_a)}{2} = \frac{88 \times 10^3 \cdot (5 + 3.74)}{2}. \] Simplify: \[ \text{Input Power} = \frac{88 \times 10^3 \cdot 8.74}{2} = \frac{768,320}{2} = 384,160 \, \text{W} \, \text{or} \, 384.16 \, \text{kW}. \] Step 4: Calculate the propeller efficiency.
Substitute the values into the efficiency formula: \[ \eta = \frac{\text{Thrust Power}}{\text{Input Power}} \times 100 = \frac{440}{384.16} \times 100. \] Simplify: \[ \eta \approx 1.145 \times 100 = 70.5\%. \] Conclusion: The propeller efficiency is \( 70.5\% \).