An axial-flow fan operates in seal-level air at 1350 rpm and...

An axial-flow fan operates in seal-level air at 1350 rpm and has a blade tip diameter of 3 ft and a root diameter of $2.5 \mathrm{ft} .$ The inlet angles are $\alpha_{1}=55^{\circ}, \beta_{1}=30^{\circ},$ and at the exit $\beta_{2}=60^{\circ} .$ Estimate the flow volumetric flow rate, horsepower, and the outlet angle, $\alpha_{2}.$

Key Concepts

Fan Blade Geometry and Effective Flow Area

The geometry of fan blades, such as the tip and root diameters, determines the effective area through which the fluid flows. This effective flow area is crucial in calculating the volumetric flow rate and assessing the overall performance of the fan. Understanding the blade geometry is key to establishing the relationship between the physical dimensions and the dynamical quantities like velocity or flow rate.

Flow Angle Transformation

Flow angle transformation involves the conversion between the absolute and relative flow angles due to the rotation of the fan blades. The inlet and outlet angles provided in the problem allow for the determination of different velocity components via geometry and trigonometry. Accurately calculating these angles is essential for predicting performance metrics such as the exit flow direction, which directly affects the fan's aerodynamic efficiency and power requirements.

Conservation of Mass (Continuity Equation)

The conservation of mass in fluid dynamics, often expressed as the continuity equation, dictates that the mass flow rate through a control volume remains constant for steady flows. In axial fans, this means that the product of the cross-sectional area and the axial velocity of the air must remain constant, allowing one to estimate the volumetric flow rate based on the geometric parameters of the fan.

Euler’s Turbomachinery Equation

Euler’s turbomachinery equation provides a relationship between the change in angular momentum of the fluid and the work done by the machine. In the context of axial-flow fans, it relates the tangential (swirl) components of the velocity at the inlet and outlet to the energy transfer, which can then be used for estimating the power (or horsepower) requirement of the fan.

Velocity Triangles

Velocity triangles are a graphical representation used in turbomachinery to resolve the absolute, relative, and blade (rotational) velocity components of the flow at various points in the fan or compressor. They allow determination of flow angles and speeds by relating the inlet and outlet flow angles with the rotational velocity of the blades, making them fundamental in analyzing and designing axial?flow machines.

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