The axial-flow fan C pumps air through the duct of circular...

The axial-flow fan $C$ pumps air through the duct of circular cross section and exhausts it with a velocity $v$ at $B .$ The air densities at $A$ and $B$ are $\rho_{A}$ and $\rho_{B}$, respectively, and the corresponding pressures are $p_{A}$ and $p_{B}$. The fixed deflecting blades at $D$ restore axial flow to the air after it passes through the propeller blades $C .$ Write an expression for the resultant horizontal force $R$ exerted on the fan unit by the flange and bolts at $A$.

The axial-flow fan $C$ pumps air through the duct of circular cross section and exhausts it with a velocity
$v$ at $B .$ The air densities at $A$ and $B$ are $\rho_{A}$ and $\rho_{B}$, respectively, and the corresponding pressures are $p_{A}$ and $p_{B}$. The fixed deflecting blades at $D$ restore axial flow to the air after it passes through the propeller blades
$C .$ Write an expression for the resultant horizontal force $R$ exerted on the fan unit by the flange and bolts at $A$.

Key Concepts

Pressure Forces

Pressure forces are generated by the fluid exerting pressure on the surfaces of the control volume boundaries. These forces act over the areas at the inlet and outlet, and their differences contribute to the net force experienced by the fan assembly. The pressures at different points, combined with the cross-sectional areas, determine the magnitude and direction of the resultant force.

Control Volume Analysis

Control volume analysis involves selecting a defined region in space through which fluid flows and applying conservation laws, such as conservation of mass and momentum, to that region. This method helps in relating the forces acting on the boundaries (due to pressures and fluid momentum) to the changes in the fluid properties as it passes through the region of interest.

Axial Momentum Balance

The axial momentum balance is the application of Newton’s second law along the axis of the flow. It equates the net force acting on the fluid in the axial direction to the change in momentum flux of the fluid across the control volume. In this context, the balance accounts for pressure forces entering and leaving the control volume and the acceleration of the fluid by the fan.

Momentum Flux

Momentum flux refers to the rate at which momentum is carried by the fluid through a given surface. It is calculated as the product of the mass flow rate and the velocity. Changes in momentum flux between the inlet and outlet sections of the control volume are directly related to the forces produced by modifications in flow velocity as the fluid passes through the fan system.

Newton's Third Law and Reaction Forces

Newton’s third law, stating that every action has an equal and opposite reaction, is essential in determining the reaction forces on the fan unit. The forces that the fluid exerts on the fan blades and duct result in an equal and opposite reaction force on the fan assembly. This reaction force, transmitted through components such as the flange and bolts, must balance the momentum change occurring in the fluid.

Transcript

Please give Ace some feedback