/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 58 An ocean-going vessel is to be p... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 58

An ocean-going vessel is to be powered by a rotating circular cylinder. Model tests are planned to estimate the power required to rotate the prototype cylinder. A dimensional analysis is needed to scale the power requirements from model test results to the prototype. List the parameters that should be included in the dimensional analysis. Perform a dimensional analysis to identify the important dimensionless groups.

Short Answer

Expert verified
The parameters required for dimensional analysis are Power, Density, Viscosity, Velocity, Diameter, and Rotational speed. The dimensionless groups derived from these parameters using Buckingham Pi theorem are \( \frac{P} {蟻 V鈦� D鈦祡 \), \( \frac{渭} {蟻 V D} \), and \( \frac{蠅 D} {V} \).

Step by step solution

01

Identifying Parameters

First and foremost, we need to list out which parameters we should include in our analysis. Since we're dealing with a rotating cylinder, the parameters would be: Power (P), Density of fluid (\(蟻\)), Viscosity (\(渭\)), Velocity (\(V\)), Diameter of the cylinder (\(D\)), and Rotational speed (\(蠅\)).
02

Assigning Dimensions to Parameters

Before performing the dimensional analysis, assign dimensions to each parameter. Here are their respective base quantities: Power [M L虏 T鈦宦砞, Density [M L鈦宦砞, Viscosity [M L鈦宦� T鈦宦筣, Velocity [L T鈦宦筣, Diameter [L], Rotational speed [T鈦宦筣.
03

Buckingham Pi Theorem

Use the Buckingham Pi theorem to obtain the dimensionless groups. Our parameters count is 6 (n=6), and the number of base quantities (m) are 3 (M, L, T). According to the theorem, the number of dimensionless groups should be n - m = 3. Let's form them based on this.
04

Dimensionless Groups formation

We can identify three groups as follows: Pi1 = \( \frac{P} {蟻 V鈦� D鈦祡 \), Pi2 = \( \frac{渭} {蟻 V D} \), Pi3 = \( \frac{蠅 D} {V} \). Each of these groups is dimensionless and contains a particular aspect of the problem. Pi1 reflects the effects of power, Pi2 is a reflection of viscous effects, and Pi3 consists of the speed of rotation.

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Most popular questions from this chapter

A washing machine agitator is to be designed. The power, \(\mathscr{P},\) required for the agitator is to be correlated with the amount of water used (indicated by the depth, \(H,\) of the water). It also depends on the agitator diameter, \(D\), height, \(h\) maximum angular velocity, \(\omega_{\max },\) and frequency of oscillations, \(f,\) and water density, \(\rho,\) and viscosity, \(\mu .\) Determine the dimensionless parameters that characterize this problem.

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When a valve is closed suddenly in a pipe with flowing water, a water hammer pressure wave is set up. The very high pressures generated by such waves can damage the pipe. The maximum pressure, \(p_{\max },\) generated by water hammer is a function of liquid density, \(\rho,\) initial flow speed, \(U_{0},\) and liquid bulk modulus, \(E_{v}\). How many dimensionless groups are needed to characterize water hammer? Determine the functional relationship among the variables in terms of the necessary II groups.
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