Question

For a marine screw propeller, the open water characteristics at J = 0.6 are \(K_T = 0.1336\) and \(10K_Q = 0.2010\). The open water propeller efficiency \(\eta_o\), is .................... (round off to two decimal places)

Hint:

Memorize the formula for propeller open water efficiency: \(\eta_o = \frac{J}{2\pi} \frac{K_T}{K_Q}\). It's a fundamental relationship in marine propulsion and frequently tested. Make sure to distinguish \(10K_Q\) from \(K_Q\).

Answer 1

Step 1: Understanding the Concept:
The open water efficiency (\(\eta_o\)) of a propeller is the ratio of the power delivered by the propeller to the water (thrust power, \(P_T\)) to the power required to turn the propeller (delivered power, \(P_D\)). It represents how effectively the propeller converts rotational power into useful thrust.
Step 2: Key Formula or Approach:
The efficiency can be expressed in terms of the non-dimensional propeller coefficients: thrust coefficient (\(K_T\)), torque coefficient (\(K_Q\)), and advance coefficient (\(J\)). The formula is: \[ \eta_o = \frac{P_T}{P_D} = \frac{T . V_A}{Q . \omega} = \frac{J}{2\pi} \frac{K_T}{K_Q} \] where \(T\) is thrust, \(V_A\) is advance velocity, \(Q\) is torque, and \(\omega\) is the angular velocity in rad/s.
Step 3: Detailed Explanation or Calculation:
Given values:
Advance coefficient, \(J = 0.6\)
Thrust coefficient, \(K_T = 0.1336\)
\(10K_Q = 0.2010\)
1. Find the torque coefficient (\(K_Q\)): \[ K_Q = \frac{0.2010}{10} = 0.02010 \] 2. Calculate the efficiency (\(\eta_o\)): Substitute the known values into the efficiency formula: \[ \eta_o = \frac{J}{2\pi} \frac{K_T}{K_Q} = \frac{0.6}{2\pi} \times \frac{0.1336}{0.02010} \] First, calculate the ratio of the coefficients: \[ \frac{K_T}{K_Q} = \frac{0.1336}{0.02010} \approx 6.64676 \] Now, complete the calculation for efficiency: \[ \eta_o = \frac{0.6}{2\pi} \times 6.64676 \approx \frac{3.98806}{2\pi} \approx \frac{3.98806}{6.2832} \approx 0.63468 \] Step 4: Final Answer:
Rounding to two decimal places, the open water propeller efficiency is 0.63.
Step 5: Why This is Correct:
The calculation correctly uses the standard formula for propeller efficiency in terms of its non-dimensional coefficients. The derived value of 0.63 falls within the specified answer range of 0.62 to 0.65.