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Chapter 7: Problem 22

The input power, \(\dot{W}\), to a large industrial fan depends on the fan impeller diameter \(D\), fluid viscosity \(\mu\), fluid density \(\rho\), volumetric flow \(Q,\) and blade rotational speed \(\omega .\) What are the appropriate dimensionless parameters?

Short Answer

Expert verified
The appropriate dimensionless parameters are: \(\pi_{1}=\frac{\dot{W}}{\rho D^5 \omega^3}\), \(\pi_{2}=\frac{Q}{D^3 \omega}\), \(\pi_{3}=\frac{\mu}{\rho D \omega}\)

Step by step solution

01

Assign Dimensions to given Parameters

Dimensions of various quantities are assigned as follows: Power \(\dot{W}\): \([M L^2 T^{-3}]\), Diameter \(D\): \([L]\), Viscosity \(\mu\): \([M L^{-1}T^{-1}]\), Density \(\rho\): \([M L^{-3}]\), Volumetric flow \(Q\): \([L^3 T^{-1}]\), Rotational speed \(\omega\): \([T^{-1}]\).
02

Identify the repeating variables

In this case, let's use Diameter \(D\), Density \(\rho\) and the rotational speed \(\omega\) as our repeating variables.
03

Form the Dimensionless Parameters

According to the Buckingham's \(\pi\) theorem, we can form the dimensionless groups as follows: \(\pi_{1}=\frac{\dot{W}}{\rho D^5 \omega^3}\), \(\pi_{2}=\frac{Q}{D^3 \omega}\), \(\pi_{3}=\frac{\mu}{\rho D \omega}\).
04

Check the dimensions of formed parameters to be sure

Each of the groups formed should be dimensionless. If you substitute the dimensions of each variable into the groups, you will find that all dimensions cancel out, verifying that these are indeed dimensionless groups.

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Most popular questions from this chapter

The dimensional parameters used to describe the operation of a ship or airplane propeller (sometimes called a screw propeller) are rotational speed, \(\omega,\) diameter, \(D,\) fluid density, \(\rho\) speed of the propeller relative to the fluid, \(V\), and thrust developed, \(T .\) The common dimensionless groups are called the thrust coefficient and the advance ratio. Propose appropriate definitions for these groups.

A coach has been trying to evaluate the accuracy of a baseball pitcher. After two years of studying, he proposes a function that can be presented as the accuracy of any pitcher: \\[ \mathrm{Acc}=f(V, a, m, \rho, p, z) \\] where Acc is the dimensionless accuracy, \(V\) is the velocity of the ball, \(a\) is the age of the pitcher, \(m\) is the mass of the pitcher, \(\rho\) is the density of air where the game is played, \(p\) is the pressure where the game is played (which varies with elevation), and \(z\) is the elevation above sea level. Find the dimensionless groups for this function.

The pressure drop, \(\Delta p,\) along a straight pipe of diameter \(D\) has been experimentally studied, and it is observed that for laminar flow of a given fluid and pipe, the pressure drop varies directly with the distance, \(\ell\), between pressure taps. Assume that \(\Delta p\) is a function of \(D\) and \(\ell\), the velocity, \(V\), and the fluid viscosity, \(\mu\). Use dimensional analysis to deduce how the pressure drop varies with pipe diameter.

The fluid dynamic characteristics of an airplane flying \(240 \mathrm{mph}\) at \(10,000 \mathrm{ft}\) are to be investigated with the aid of a 1: 20 scale model. If the model tests are to be performed in a wind tunnel using standard air, what is the required air velocity in the wind tunnel? Is this a realistic velocity?

Assume that the flowrate, \(Q\), of a gas from a smokestack is a function of the density of the ambicnt air, \(\rho_{\alpha}\), the density of the gas, \(\rho_{x},\) within the stack, the acceleration of gravity, \(g,\) and the height and diameter of the stack, \(h\) and \(d\), respectively. Use \(\rho_{c}, d,\) and \(g\) as repeating variables to develop a set of pi terms that could be used to describe this problem.
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