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Chapter 6: Problem 67

A manufacturer makes several fans in different sizes, but all have the same shape (i.e., they are geometrically similar). Such a series of models is called a homologous series. Two fans in the series are to be operated under dynamically similar conditions. The first fan has a diameter of \(260 \mathrm{~mm}\), has a rotational speed of \(2500 \mathrm{rpm},\) and produces an airflow rate of \(0.9 \mathrm{~m}^{3} / \mathrm{s}\). The second fan is to have a rotational speed of \(1500 \mathrm{rpm}\) and produce a flow rate of \(3 \mathrm{~m}^{3} / \mathrm{s}\). What diameter should be selected for the second fan?

Short Answer

Expert verified
The second fan should have a diameter of approximately 459 mm.

Step by step solution

01

Understanding the Fan Affinity Laws

Fan affinity laws are used to relate the performance of geometrically similar fans operating under different conditions. For dynamically similar fans, the following equation applies to the flow rate: \( \frac{Q_1}{Q_2} = \left( \frac{D_1}{D_2} \right)^3 \left( \frac{N_1}{N_2} \right) \), where \(Q\) is the flow rate, \(D\) is the diameter, and \(N\) is the rotational speed.
02

Substituting the Known Values

Here, \(Q_1 = 0.9\, \text{m}^3/\text{s}\), \(N_1 = 2500\, \text{rpm}\), \(D_1 = 260\, \text{mm} = 0.26\, \text{m}\); \(Q_2 = 3\, \text{m}^3/\text{s}\), and \(N_2 = 1500\, \text{rpm}\). Substitute these into the affinity law equation: \[\frac{0.9}{3} = \left( \frac{0.26}{D_2} \right)^3 \left( \frac{2500}{1500} \right)\]
03

Solving for the Diameter of the Second Fan

Rearrange the equation from Step 2 to solve for \(D_2\):\[\frac{0.9}{3} \times \frac{1500}{2500} = \left( \frac{0.26}{D_2} \right)^3\]Simplify and solve for \(\left( \frac{0.26}{D_2} \right)^3\):\[\frac{0.9 \times 1500}{3 \times 2500} = \left( \frac{0.26}{D_2} \right)^3\]\[0.18 = \left( \frac{0.26}{D_2} \right)^3\]Take the cube root on both sides:\[\left( \frac{0.26}{D_2} \right) = 0.18^{1/3}\]Calculate \(0.18^{1/3} \approx 0.5657\):\[\frac{0.26}{D_2} = 0.5657\]
04

Calculating the Final Diameter

Rearrange the equation from Step 3 to solve for \(D_2\):\[D_2 = \frac{0.26}{0.5657}\]Calculate \(D_2 \approx 0.459\, \text{m}\).Convert to mm: \(D_2 \approx 459\, \text{mm}\).

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Key Concepts

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

Geometrical Similarity
Geometrical similarity is an important concept when studying homologous series of models, such as fans. It means that all fans in the series have the same shape but may differ in size. The principle of geometrical similarity allows engineers to predict the performance of one fan based on the known performance of another. For fans, this involves consistency in proportions and angles, which ensures that the aerodynamic properties are retained as size changes.
Geometrically similar fans allow for the use of affinity laws, as these laws are derived based on the assumption that the geometric proportions are held constant. This concept helps in designing fans that operate efficiently across various sizes while maintaining similar functionality.
Dynamically Similar Conditions
Fans operating under dynamically similar conditions refer to when they have scaled versions of the same operational parameters, such as velocity, pressure, and flow patterns. In dynamic similarity, not only are the fans geometrically alike but the forces acting upon them scale uniformly with changes in size.
This ensures consistency in performance characteristics across different scales. In practical terms, if two fans are dynamically similar, their flow, speed, and pressure relationships can be correlated using equations like the fan affinity laws. Such similarity guarantees that predictions and calculations made for one fan can be applied to another, simplifying the analysis and design process significantly.
Flow Rate Calculation
Flow rate is a crucial factor in fan operation, representing the volume of air moved by the fan per unit time. It is directly influenced by the fan's size (diameter) and speed.
To find the unknown flow rate or diameter, we employ the fan affinity laws, which relate the flow rate of two geometrically similar fans operating under dynamically similar conditions. According to the law:
  • \[ \frac{Q_1}{Q_2} = \left( \frac{D_1}{D_2} \right)^3 \left( \frac{N_1}{N_2} \right) \]

In this equation, \(Q\) is the flow rate, \(D\) the diameter, and \(N\) the rotational speed. By rearranging this formula and inserting known values, you can solve for unknown parameters such as the diameter in this case study.
Rotational Speed
Rotational speed, typically measured in revolutions per minute (rpm), is a fundamental characteristic of fans that influences their performance. The speed at which a fan rotates determines how much air it can move and affects the pressure it can deliver.
In the context of dynamically similar fans, rotational speed is not only a measure of performance but also a factor in the affinity laws, which allow us to predict how changes in speed will affect other parameters like flow rate and diameter. By maintaining a balance between flow rate and speed, fans are optimized for efficiency. When calculating or predicting fan performance based on affinity laws, maintaining proportional changes in speed across geometrically similar fans ensures that predictions remain valid and representative.

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Most popular questions from this chapter

The energy per unit mass (of fluid), \(e\), added by a pump of a given shape depends on the pump size, \(D,\) volume flow rate, \(Q,\) speed of the rotor, \(\omega,\) density of the fluid, \(\rho\), and dynamic viscosity of the fluid, \(\mu\). This functional relation can be stated as $$e=f(D, Q, \omega, \rho, \mu)$$ Express this as a relationship between dimensionless groups. What is gained by expressing the pump performance as an empirical relationship between dimensionless groups versus expressing the pump performance as a relationship between the given dimensional variables?

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