The ducted fan unit of mass m is supported in the vertical...

The ducted fan unit of mass $m$ is supported in the vertical position on its flange at $A$. The unit draws in air with a density $\rho$ and a velocity $u$ through section $A$ and discharges it through section $B$ with a velocity $v .$ Both inlet and outlet pressures are atmospheric. Write an expression for the force $R$ applied to the flange of the fan unit by the supporting slab.

The ducted fan unit of mass $m$ is supported in the vertical position on its flange at $A$. The unit draws in air with a density $\rho$ and a velocity $u$ through section $A$ and discharges it through section $B$ with a velocity
$v .$ Both inlet and outlet pressures are atmospheric. Write an expression for the force $R$ applied to the flange of the fan unit by the supporting slab.

Key Concepts

Control Volume Analysis

This concept involves selecting a fixed region in space over which the fluid flow is analyzed. By applying conservation laws to the control volume, one can account for the fluid’s momentum and mass as it enters and exits through various sections, leading to insights into the forces acting on the system.

Conservation of Mass

In fluid dynamics, the conservation of mass, often applied in the form of the continuity equation, ensures that the mass flow rate entering a system equals the mass flow rate leaving it under steady conditions. This principle is critical in relating the velocities and cross-sectional areas at different points in a ducted flow.

Conservation of Momentum

This principle involves applying Newton's second law to a fluid system by equating the net momentum flux (the difference between the momentum carried by the inflow and outflow) to the resultant force acting on the control volume. In engineering applications, this analysis enables the determination of reactive forces due to fluid acceleration or deceleration.

Reaction Forces and Newton's Third Law

Newton's third law states that every action has an equal and opposite reaction. When the fluid is accelerated by the fan, it exerts a force on the structure that supports the fan. This reaction force is crucial for maintaining equilibrium and is counteracted by the supporting slab or flange.

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