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Chapter 12: Problem 30

A prototype fan has a \(20-f\) diameter, an inlet pressure of 14.40 psia, an inlet temperature of \(70^{\circ} \mathrm{F},\) and a speed of 90 rpm. \(\mathrm{A}\) \(\frac{1}{10}-\operatorname{scal} \mathrm{e}\) model of the fan has the same inlet pressure and emperature. an inlet power of \(1.24 \mathrm{hp}\), a flow rate of \(220 \mathrm{ft}^{3} / \mathrm{min}\), and a speed of \(1800 \mathrm{rpm} .\) Find the corresponding input power and flow rate of the prototype fan. Neglect Reynolds number effects.

Short Answer

Expert verified
The corresponding input power and flow rate of the prototype fan are 0.000155 hp and 11 cubic feet per minute, respectively.

Step by step solution

01

Find the Scaling Factor

First, we need to find the ratio of speed of the prototype fan to the speed of the model fan. It can be calculated as follows:\[ \text{Speed ratio} = \frac{\text{Speed of Prototype}}{\text{Speed of Model}} = \frac{90}{1800} = 0.05 \]
02

Calculate the Input Power

Once we have speed ratio, we can calculate the input power using the model power and the speed scaling factor. The power scaling factor for fan is cube of speed ratio i.e. \( \left(\frac{\text{Speed of Prototype}}{\text{Speed of Model}}\right)^3 \). Therefore, input power of prototype fan will be calculated as follow:\[ \text{Input Power (Prototype)} = \text{Model Power} * \left(\frac{\text{Speed of Prototype}}{\text{Speed of Model}}\right)^3 = 1.24 \text{hp} * (0.05)^3 = 0.000155 \text{hp} \]
03

Calculate Flow Rate

We can now calculate the flow rate of the prototype fan using the speed scaling factor, since the flow rate is directly proportional to the speed. We use the following relation:\[ \text{Flow Rate (Prototype)} = \text{Flow Rate(Model)} * \left(\frac{\text{Speed of Prototype}}{\text{Speed of Model}}\right) = 220 \text{ft}^3/\text{min} * 0.05 = 11 \text{ft}^3/\text{min} \]

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Most popular questions from this chapter

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A centrifugal pump having an impeller diameter of \(1 \mathrm{m}\) is to be constructed so that it will supply a head rise of 200 mat a flowrate of \(4.1 \mathrm{m}^{3} / \mathrm{s}\) of water when operating at a speed of \(1200 \mathrm{rpm} .\) To study the characteristizs of this pump, a \(1 / 5\) scale, geometrizally similar model operated at the same speed is to be tested in the laboratory. Determine the required model discharge and head rise. Assume that both model and prototype operate with the same effciency (and therefore the same flow coefficient)
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