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Chapter 8: Problem 65

A large centrifugal fan generates a flow of \(7 \mathrm{~m}^{3} / \mathrm{s}\) with a motor speed of \(1140 \mathrm{rpm}\). A smaller geometrically similar fan has a motor speed of \(1725 \mathrm{rpm}\), operates at the same efficiency as the larger fan, and generates the same pressure increase. What flow rate is generated by the smaller fan?

Short Answer

Expert verified
The smaller fan generates approximately 10.592 m鲁/s of flow.

Step by step solution

01

Understanding Fan Laws

The problem refers to geometrically similar fans, which means we can apply the affinity laws for fans. The affinity laws relate the performance of geometric similar fans' speed, flow, and diameter. The flow rate (\( Q \)), pressure, and power of a fan are related to its speed (\( N \)) by:\[ Q_1/Q_2 = N_1/N_2 \]This formula will allow us to calculate the new flow rate for the smaller fan given that both fans operate at the same efficiency.
02

Extract Given Values

From the problem, we have:- The flow rate of the large fan, \( Q_1 = 7 \, \mathrm{m^3/s} \)- The motor speed of the large fan, \( N_1 = 1140 \, \mathrm{rpm} \)- The motor speed of the smaller fan, \( N_2 = 1725 \, \mathrm{rpm} \)Our goal is to find the flow rate of the smaller fan, \( Q_2 \).
03

Apply the Affinity Law for Flow Rate

Using the affinity law for flow rate:\[ \frac{Q_1}{Q_2} = \frac{N_1}{N_2} \]Substitute the known values into the equation:\[ \frac{7}{Q_2} = \frac{1140}{1725} \]
04

Solve for the New Flow Rate

To find \( Q_2 \), rearrange the equation:\[ Q_2 = Q_1 \times \frac{N_2}{N_1} \]Substitute the values:\[ Q_2 = 7 \times \frac{1725}{1140} \]Calculate \( Q_2 \):\[ Q_2 \approx 7 \times 1.5132 \approx 10.592 \mathrm{\ m^3/s} \]Therefore, the flow rate generated by the smaller fan is approximately \( 10.592 \mathrm{\ m^3/s} \).

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Key Concepts

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

Flow Rate Calculation
Flow rate is a crucial parameter in the evaluation of fan performance. For fans that are geometrically similar, we can utilize the Affinity Laws to determine changes in flow rate as motor speeds vary. These laws are particularly useful when comparing different fan sizes that maintain proportional dimensions.

The Affinity Laws offer a simple relationship for calculating the flow rate. The formula:
  • Flow rate (\( Q \)): directly proportional to the rotational speed (\( N \)).
  • Given by: \(\frac{Q_1}{Q_2} = \frac{N_1}{N_2}\)
  • This allows us to compute the new flow rate for a fan by knowing the initial flow rate and speed ratios.
In our example, the large fan moves air at\( 7 \ m^3/s\) with a motor speed of\( 1140 \ rpm\). The smaller fan, rotating faster at\( 1725 \ rpm\) while maintaining geometric similarity, has a modified flow rate due to this increase in speed.

Applying the above formula and substituting the known values, we calculate the smaller fan's flow rate:\[ Q_2 = Q_1 \times \frac{N_2}{N_1} \]Substitute the numbers to find:\[ Q_2 = 7 \times \frac{1725}{1140} \approx 10.592 \ m^3/s \]This calculation shows the increased flow rate attributed to the higher speed of the smaller fan.
Geometrically Similar Fans
Geometrically similar fans are those which share the same ratio among their dimensions, despite their overall size differences. When fans are geometrically similar, changes in their operating conditions can be predicted accurately using the Affinity Laws.
Some features of geometrically similar fans include:
  • All linear dimensions (diameters, widths, etc.) are proportionally scaled.
  • A characteristic that makes performance prediction consistent across different sizes.
In practical settings, engineers leverage the geometrically similar nature of fans when scaling designs up or down to suit specific requirements, such as adapting to space constraints without redesigning from scratch.
In our exercise, both the large and smaller fans are constructed with these similar geometries. This attribute simplifies the application of mathematical formulas to predict performance metrics like flow rate, as they maintain these consistent dimensional ratios.
Centrifugal Fan Performance
Centrifugal fans are widely used for ventilation, cooling, and various industrial purposes due to their reliable airflow characteristics and performance robustness. Their efficiency and performance can often be predicted based on the fundamental principles of fluid dynamics and affinity laws.
Key points on centrifugal fan performance include:
  • Flow Rate: Directly influenced by fan speed and impeller design.
  • Pressure Increase: Generally constant for similar fans provided the speed and airflow are adjusted correctly.
  • Efficiency: Closely relates to how effectively the fan uses power to produce airflow.
By understanding changes in rotational speed and the use of dimensional similarity, as referenced in our exercise, engineers can calculate fan performance outcomes under different conditions. The ability to maintain consistent pressure increases across different fan sizes, while adjusting for speed, is highlighted as these fans are geometrically similar and operate under comparable efficiencies.
In conclusion, awareness of these performance fundamentals allows for predictable modifications to fan systems, ensuring optimal operation and energy use across varied scales.

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Most popular questions from this chapter

A pump is to be used to withdraw water from a reservoir at a rate of \(1500 \mathrm{~L} / \mathrm{s}\). When operating at this flow rate, the head loss in the suction pipe is estimated to be \(2.3 \mathrm{~m}\), and the pump specifications give the required net positive suction head as \(2.9 \mathrm{~m}\). Standard sea-level atmospheric conditions are expected at the site, and under worstcase conditions, the temperature of the water in the reservoir is \(25^{\circ} \mathrm{C}\). What is the maximum allowable elevation of the suction side of the pump above the reservoir water surface?

A proposed hydropower plant is to be located at a site with an available head of \(9 \mathrm{~m}\), and it is desired to obtain a (shaft) power of \(35 \mathrm{MW}\) from this site. The axial-flow turbine units under consideration operate at an angular speed of \(150 \mathrm{rpm}\), have a specific speed of \(5,\) and have an estimated maximum efficiency of \(80 \%\). (a) How may of these units are required? (b) What total flow rate must be available to generate the desired power? Assume water at \(20^{\circ} \mathrm{C}\).

A Francis turbine is located at an elevation of \(2000 \mathrm{~m}\) above sea level. The discharge from the runner is \(5 \mathrm{~m}\) above the tailwater pool, the water flowing through the runner is at a temperature of \(15^{\circ} \mathrm{C}\), and the critical cavitation parameter of the runner is given by the manufacturer as 0.25 . The head loss in the draft tube is negligible. Estimate the maximum available head that can be accommodated without the occurrence of cavitation.

Water is pumped from a lower reservoir to a higher reservoir. The water surface in the higher reservoir is \(10 \mathrm{~m}\) above the water surface in the lower reservoir. The piping system consists of 200 -mm-diameter ductile iron pipe with a length of \(2 \mathrm{~km}\) and minor losses equal to 6.2 times the velocity head. The pump characteristics are shown in Table 8.3. (a) Determine the expected flow rate through the system, assuming fully turbulent flow. (b) What is the power of the motor required to drive the pump? $$ \begin{array}{l|c|c|c|c|c|c} \text { Discharge (L/s) } & 0 & 10 & 20 & 30 & 40 & 50 \\ \hline \text { Total head (m) } & 25 & 23.2 & 20.8 & 17.0 & 12.4 & 7.3 \\ \hline \text { Efficiency (\%) } & -1 & 45 & 65 & 71 & 65 & 45 \end{array} $$

Water at \(20^{\circ} \mathrm{C}\) is to be pumped out of a reservoir at a rate of \(20 \mathrm{~L} / \mathrm{s}\) through a vertical 150 -mm-diameter ductile iron pipeline with an estimated roughness height of \(0.3 \mathrm{~mm}\). Appurtenances installed in the intake pipe are expected to contribute to a total local head loss coefficient of 12 . The pump being considered for installation has a required net positive suction head of \(5.5 \mathrm{~m}\). (a) What is the maximum elevation of the pump relative to the water surface in the reservoir? (b) If the total head loss coefficient can be reduced to 1.2 , how much higher can the pump be placed? Assume standard atmospheric conditions.
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