A 0.4 m diameter fan running at 970 rev min^-1 is tested when the air temperature is 10^∘ C and

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A 0.4 m diameter fan running at 970 rev min^-1 is tested...

A $0.4 \mathrm{~m}$ diameter fan running at $970 \mathrm{rev} \mathrm{min}^{-1}$ is tested when the air temperature is $10^{\circ} \mathrm{C}$ and the barometric pressure is $772 \mathrm{~mm} \mathrm{Hg}$ and the following data are observed: $Q=0.7 \mathrm{~m}^{3} \mathrm{~s}^{-1}$; fan total pressure $=25 \mathrm{~mm}$ of water; shaft power $=250 \mathrm{~W}$ Find the corresponding volume flow, fan total pressure and shaft power of a geometrically similar fan of $1 \mathrm{~m}$ diameter, running at 500 rev $\min ^{-1}$ when the air temperature is $16^{\circ} \mathrm{C}$ and the barometric pressure is $760 \mathrm{~mm} \mathrm{Hg}$. Assume that the fan efficiency is unchanged. $\left[5.64 \mathrm{~m}^{3} \mathrm{~s}^{-1}, 40 \mathrm{~mm}\right.$ of water, $\left.3.22 \mathrm{~kW}\right]$

A $0.4 \mathrm{~m}$ diameter fan running at $970 \mathrm{rev} \mathrm{min}^{-1}$ is tested when the air temperature is $10^{\circ} \mathrm{C}$ and the barometric pressure is $772 \mathrm{~mm} \mathrm{Hg}$ and the following data are observed: $Q=0.7 \mathrm{~m}^{3} \mathrm{~s}^{-1}$; fan total pressure $=25 \mathrm{~mm}$ of water; shaft power $=250 \mathrm{~W}$

Find the corresponding volume flow, fan total pressure and shaft power of a geometrically similar fan of $1 \mathrm{~m}$ diameter, running at 500 rev $\min ^{-1}$ when the air temperature is $16^{\circ} \mathrm{C}$ and the barometric pressure is $760 \mathrm{~mm} \mathrm{Hg}$. Assume that the fan efficiency is unchanged.
$\left[5.64 \mathrm{~m}^{3} \mathrm{~s}^{-1}, 40 \mathrm{~mm}\right.$ of water, $\left.3.22 \mathrm{~kW}\right]$

Key Concepts

Environmental Effects on Performance

Environmental conditions such as temperature and barometric pressure affect air density, which in turn influences the fan's performance. Adjustments based on these ambient conditions are necessary to accurately predict performance parameters under different test scenarios.

Similarity Scaling Laws

Similarity scaling laws, often referred to as the affinity laws in turbomachinery, provide relationships that dictate how different performance parameters change with fan size and speed. For example, flow rate scales with the cube of the diameter and linearly with rotational speed, pressure scales with the square of both diameter and rotational speed, and power scales with the fifth power of the diameter and the cube of the rotational speed.

Fan Efficiency

Fan efficiency, defined as the ratio of useful output power to the input power, is a measure of how well a fan converts energy into airflow. When scaling for different sizes of fans, if efficiency is presumed constant, it simplifies the analysis by directly relating the input shaft power to the aerodynamic work being done.

Dimensional Analysis

Dimensional analysis involves ensuring that equations governing physical systems are consistent in terms of their units. This is critical in determining how performance parameters such as flow, pressure, and power scale with changes in geometric size or operating conditions.

Geometric Similarity

Geometric similarity is the principle that two objects have the same shape but differ in size by a constant factor. In fan performance studies, if fans are geometrically similar, their performance can be scaled up or down using established relationships, enabling the prediction of full?scale behavior from a model.

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