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Chapter 7: Problem 41

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.

Short Answer

Expert verified
There are 5 dimensionless groups that can be derived from the given parameters. The exact form of these groups cannot be determined without specific information about the system at hand, but these groups will represent combinations of the given variables that do not depend on the units of measurement.

Step by step solution

01

Identifying the Variables

The important parameters given are power \(\mathscr{P}\), depth of water \(H\), diameter of the agitator \(D\), height of the agitator \(h\), maximum angular velocity \(\omega_{\max}\), frequency of oscillations \(f\), water density \(\rho\), and water viscosity \(\mu\).
02

Determining the Fundamental Dimensions

These parameters can be expressed in terms of the fundamental dimensions: mass \(M\), length \(L\), and time \(T\). They are \(\mathscr{P} = ML^2T^{-3}\), \(H = L\), \(D = L\), \(h = L\), \(\omega_{\max} = T^{-1}\), \(f = T^{-1}\), \(\rho = ML^{-3}\), and \(\mu = ML^{-1}T^{-1}\).
03

Applying the Buckingham Pi Theorem

The Buckingham Pi theorem states that if there are \(n\) physical variables in a problem and they can be expressed in terms of \(m\) fundamental dimensions, then the equation relating all the variables can be rewritten in terms of \(n-m\) dimensionless parameters. Here we have \(n = 8\) variables and \(m = 3\) fundamental dimensions, so we can form \(n-m = 8 - 3 = 5\) dimensionless groups.
04

Creating the Dimensionless Parameters

As we have 5 dimensionless groups, we need to attempt various combinations to create 5 dimensionless parameters, often taking the form of existing dimensionless numbers like the Reynolds number or the Froude number. The exact form of these will depend on the specific physical characteristics of the system being studied.

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Most popular questions from this chapter

A 1: 50 -scale model of a submarine is to be tested in a towing tank under two conditions: motion at the free surface and motion far below the surface. The tests are performed in freshwater. On the surface, the submarine cruises at 24 knots. At what speed should the model be towed to ensure dynamic similarity? Far below the surface, the sub cruises at 0.35 knot. At what speed should the model be towed to ensure dynamic similarity? What must the drag of the model be multiplied by under each condition to give the drag of the full-scale submarine?

Experiments show that the pressure drop for flow through an orifice plate of diameter \(d\) mounted in a length of pipe of diameter \(D\) may be expressed as \(\Delta p=p_{1}-p_{2}=\) \(f(\rho, \mu, \bar{V}, d, D),\) You are asked to organize some experimental data. Obtain the resulting dimensionless parameters.

Your favorite professor likes mountain climbing, so there is always a possibility that the professor may fall into a crevasse in some glacier. If that happened today, and the professor was trapped in a slowly moving glacier, you are curious to know whether the professor would reappear at the downstream drop-off of the glacier during this academic year. Assuming ice is a Newtonian fluid with the density of glycerine but a million times as viscous, you decide to build a glycerin model and use dimensional analysis and similarity to estimate when the professor would reappear. Assume the real glacier is \(15 \mathrm{m}\) deep and is on a slope that falls \(1.5 \mathrm{m}\) in a horizontal distance of \(1850 \mathrm{m}\). Develop the dimensionless parameters and conditions expected to govern dynamic similarity in this problem. If the model professor reappears in the laboratory after 9.6 hours, when should you return to the end of the real glacier to provide help to your favorite professor?

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.

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.
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