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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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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Buckingham's Pi Theorem
The Buckingham's Pi Theorem is a powerful mathematical tool used in the field of dimensional analysis. It helps us break down physical equations into dimensionless forms, making it easier to understand and compare different systems. The theorem states that if a problem involves \( n \) variables and \( m \) fundamental dimensions (such as mass, length, and time), you can restate the problem with \( n - m \) dimensionless parameters, known as Pi terms. This reduction simplifies the complexity of physical problems.
  • Identify the relevant variables and their dimensions.
  • Choose a set of repeating variables which cover all the fundamental dimensions present.
  • Create dimensionless groups from the remaining variables by combining them with the repeating variables.
Buckingham's Pi Theorem is particularly useful in engineering and physics. It provides a structured way to transform complex dimensional equations into simpler forms, which can then be experimentally or analytically analyzed.
Dimensionless Parameters
Dimensionless parameters are key to comparing different systems or scenarios without the need for specific units. These parameters arise from the dimensional analysis process, ensuring that each one is independent of the scale of measurement. In fluid mechanics, for example, dimensionless parameters allow engineers to extrapolate results from a model (such as a wind tunnel) to a full-scale application, like an actual aircraft wing. By maintaining similarity using dimensionless parameters, predictions become more reliable.
  • Examples include the Reynolds number, which compares inertial forces to viscous forces, and the Froude number, which compares inertial forces to gravitational forces.
  • They help in turning dimensional equations into forms where the governing equations remain consistent across varying conditions.
Dimensionless parameters play a critical role in experiments and simulations, where they offer insights into the importance and influence of different forces and effects, independent of physical dimensions.
Fluid Mechanics
Fluid mechanics is the study of fluids (liquids and gases) and the forces acting upon them. It encompasses the behavior of fluids in motion (fluid dynamics) and at rest (fluid statics). Fluid mechanics is crucial for understanding natural phenomena and designing machines that involve fluid flow, such as pumps, turbines, and aircraft.In the context of the exercise, the interaction of the fan impeller's movement with fluid dynamics is of interest. Parameters such as viscosity and density play significant roles in determining the system's behavior.
  • Density \( \rho \) affects buoyancy and the mass flow rate.
  • Viscosity \( \mu \) provides a measure of a fluid's resistance to gradual deformation.
  • Volumetric flow \( Q \) and rotational speed \( \omega \) describe how the fluid moves through and around objects.
Fluid mechanics principles, like continuity equations and Bernoulli's principle, help in solving engineering problems involving fluid flow. By understanding these concepts, engineers can design efficient systems that maximize performance and safety.

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Most popular questions from this chapter

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?

The pressure rise, \(\Delta p=p_{2}-p_{1},\) across the abrupt expansion of Fig. \(\mathrm{P} 7.38\) through which a liquid is flowing can be expressed as $$\Delta p=f\left(A_{1}, A_{2}, \rho, V_{1}\right)$$ where \(A_{1}\) and \(A_{2}\) are the upstream and downstream cross-sectional areas, respectively, \(\rho\) is the fluid density, and \(V_{1}\) is the upstream velocity. Some experimental data obtained with \(A_{2}=1.25 \mathrm{ft}^{2}\) \(V_{1}=5.00 \mathrm{ft} / \mathrm{s},\) and using water with \(\rho=1.94\) slugs/ft \(^{3}\) are given in the following table: $$\begin{array}{l|l|l|l|l|r} A_{1}\left(\mathrm{ft}^{2}\right) & 0.10 & 0.25 & 0.37 & 0.52 & 0.61 \\ \hline \Delta p\left(\mathrm{lb} / \mathrm{ft}^{2}\right) & 3.25 & 7.85 & 10.3 & 11.6 & 12.3 \end{array}$$ Plot the results of these tests using suitable dimensionless parameters. With the aid of a standard curve fitting program determine a general equation for \(\Delta p\) and use this equation to predict \(\Delta p\) for water flowing through an abrupt expansion with an area ratio \(A_{1} / A_{2}=0.35\) at a velocity \(V_{1}=3.75 \mathrm{ft} / \mathrm{s}\).

A mixing basin in a sewage filtration plant is stirred by a mechanical agitator with a power input \(\dot{W} \doteq F \cdot L / T\). Other parameters describing the performance of the mixing process are the fluid absolute viscosity \(\mu \doteq F \cdot T / L^{2},\) the basin volume \(V \doteq L^{3}\) and the velocity gradient \(G \doteq 1 / T\). Determine the form of the dimensionless relationship.

A screw propeller has the following relevant dimensional parameters: axial thrust, \(F\), propeller diameter, \(D\), fluid kinematic viscosity, \(v,\) fluid density, \(\rho,\) gravitational acceleration, \(g,\) advance velocity, \(V,\) and rotational speed, \(N .\) Find appropriate dimensionless parameters to present the test data.

The pressure rise, \(\Delta p,\) across a pump can be expressed as \\[ \Delta p=f(D, \rho, \omega, Q) \\] where \(D\) is the impeller diameter, \(\rho\) the fluid density, \(\omega\) the rotational speed, and \(Q\) the flowrate. Determine a suitable set of dimensionless parameters.
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