Air enters a square duct through a 1 -ft opening as is shown in Fig. P 9.20 . Because the bounda

step 1 For this exercise we have 2 feet cubed over s equals 2 feet per second. D minus 2 times 0 .0152

Air enters a square duct through a 1 -ft opening as is shown...

Air enters a square duct through a 1 -ft opening as is shown in Fig. $\mathrm{P} 9.20 .$ Because the boundary layer displacement thickness increases in the direction of flow, it is necessary to increase the cross-sectional size of the duct if a constant $U=2 \mathrm{f}$ Us velocity is to be maintained outside the boundary layer. Plot a graph of the duct size, $d,$ as a function of $x$ for $0 \leq x \leq 10$ ft if $U$ is to remain constant. Assume laminar flow.

Key Concepts

Laminar Flow

Laminar flow is a regime of fluid motion characterized by smooth, orderly layers where viscous forces dominate over inertial effects. In this flow type, fluid particles move in parallel paths without turbulent mixing, which simplifies the analysis of velocity profiles and boundary layer development.

Boundary Layer Theory

Boundary layer theory describes how fluid flows along a surface, developing a region near the wall where viscous forces cause the velocity to change from zero (at the wall) to the free-stream value. This theory is fundamental in predicting how the layer grows and how it affects overall flow characteristics, such as drag and flow area.

Displacement Thickness

Displacement thickness quantifies the effective reduction in the flow cross-sectional area caused by the growth of the boundary layer. It represents the distance by which the outer potential flow is displaced outward because of the momentum deficit in the boundary layer, impacting the design of ducts to maintain a desired velocity profile.

Mass Conservation in Duct Flows

The principle of mass conservation, embodied in the continuity equation, ensures that the mass flow rate remains constant along a duct. When the boundary layer grows, reducing the effective area available for the free stream, the duct must be enlarged accordingly to sustain a constant free-stream velocity, making this concept essential for duct design adjustments.

Transcript

Please give Ace some feedback