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Chapter 7: Problem 91

Consider again Problem \(7.51 .\) Experience shows that for ship-size propellers, viscous effects on scaling are small. Also, when cavitation is not present, the nondimensional parameter containing pressure can be ignored. Assume that torque, \(T,\) and power, \(\mathscr{P},\) depend on the same parameters as thrust. For conditions under which effects of \(\mu\) and \(p\) can be neglected, derive scaling "laws" for propellers, similar to the pump "laws" of Section \(7.6,\) that relate thrust, torque, and power to the angular speed and diameter of the propeller.

Short Answer

Expert verified
The scaling laws for propeller are: Thrust \( F \sim D^2 \omega^2 \), Torque \( T \sim D^5 \omega^2 \), and Power \( \mathscr{P} \sim D^5 \omega^3 \).

Step by step solution

01

Identify the Parameters

The variables under consideration here are: Thrust (F), Torque (T), Power (\(\mathscr{P}\)), Angular speed (\(\omega\)), and Diameter of the propeller (D). Since the exercise tells us to ignore the effects of \(\mu\) and \(p\), our variables are F, T, \(\mathscr{P}\), \(\omega\), and D.
02

Formulate the Scaling Laws

Assuming similarity, the propeller parameters can be grouped in the following way: \( F = k_1 D^a \omega^b \), \( T = k_2 D^c \omega^d \), \( \mathscr{P} = k_3 D^e \omega^f \), In the above equations, a-f are scaling exponents to be determined by a dimensional analysis and k1-k3 are dimensionless constants.
03

Dimensional Analysis

First, we associate the dimensions: [F] = [M L T^(-2)], [T] = [M L^2 T^(-2)], [\(\mathscr{P}\)] = [M L^2 T^(-3)], [\(\omega\)] = [T^(-1)], and [D] = [L]. We then replace the dimensions into the scaling laws and solve for the exponents a-f. Doing this gives us the equations: \( F = k_1 D^2 \omega^2 \), \( T = k_2 D^5 \omega^2 \), and \( \mathscr{P} = k_3 D^5 \omega^3 \).
04

Finalization of the Scaling Laws

We can now summarize the scaling laws finally as follows: Thrust: \( F \sim D^2 \omega^2 \) Torque: \( T \sim D^5 \omega^2 \) Power: \( \mathscr{P} \sim D^5 \omega^3 \) These are the required scaling laws for propellers, which show how thrust, torque, and power scale with the diameter and angular speed of the propeller.

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Most popular questions from this chapter

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