A fan operates at Q=6.3 m^3 / s, H=0.15 m, and N=1440 rpm. A smaller, geometrically similar fan

step 1 First I am calculating the scaling ratio D2 by D1 and the formula for D2 by D1 is given by N1 ro

A fan operates at Q=6.3 m^3 / s, H=0.15 m, and N=1440 rpm. A...

A fan operates at $Q=6.3 \mathrm{m}^{3} / \mathrm{s}, H=0.15 \mathrm{m},$ and $N=1440$ rpm. A smaller, geometrically similar fan is planned in a facility that will deliver the same head at the same efficiency as the larger fan, but at a speed of 1800 rpm. Determine the volumetric flow rate of the smaller fan.

A fan operates at $Q=6.3 \mathrm{m}^{3} / \mathrm{s}, H=0.15 \mathrm{m},$ and
$N=1440$ rpm. A smaller, geometrically similar fan is planned in a facility that will deliver the same head at the same efficiency as the larger fan, but at a speed of 1800 rpm. Determine the volumetric flow rate of the smaller fan.

Key Concepts

Affinity Laws

The affinity laws describe the relationships between the performance parameters of fans and pumps—such as flow rate, head, and power—when there are changes in operating speed or geometry. These laws state that, for geometrically similar machines, the flow rate is directly proportional to the speed while the head is proportional to the square of the speed. This fundamental concept enables engineers to predict how changes in speed or size affect overall performance.

Geometric Similarity

Geometric similarity means that two devices have the same shape with all corresponding dimensions in proportion. This similarity is essential in scaling studies because it ensures that the performance data from one machine can be reliably used to predict the behavior of another, permitting the use of affinity laws without introducing errors due to differing geometries.

Scaling Relationships

Scaling relationships utilize the affinity laws and the concept of geometric similarity to calculate the expected performance of scaled machines. In particular, when only the speed changes and geometric similarity is maintained, the volumetric flow rate can be scaled directly in proportion to the change in speed. This allows for the calculation of new performance metrics while delivering similar operating conditions, as seen in the determination of the new fan’s flow rate.

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