Air flows through a wide duct at 50^∘ C temperature. Determine the required dimension a of the d

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Air flows through a wide duct at 50^∘ C temperature....

Air flows through a wide duct at $50^{\circ} \mathrm{C}$ temperature. Determine the required dimension $a$ of the duct at $x=5 \mathrm{~m}$ so that the central 250 -mm core flow velocity maintains the constant free-stream velocity of $0.4 \mathrm{~m} / \mathrm{s}$.

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Key Concepts

Core Flow and Free-Stream Velocity

In many duct flows, the central or core region is less affected by wall friction and viscous effects that dominate in the boundary layers along the walls. The free-stream velocity is defined as the velocity of the fluid in this undisturbed core region. Maintaining this velocity, even when changes occur in the duct geometry, is crucial in applications where a consistent velocity profile is required over a specific section of the flow.

Duct Geometry and Flow Control

Duct geometry plays a key role in controlling and shaping the flow field. Altering the dimensions of a duct, such as its width or height, directly impacts the local flow velocity due to mass conservation. In engineering applications, adjusting these dimensions allows the designer to target specific velocity profiles, ensuring that regions like the core flow maintain a constant free-stream velocity.

Continuity Equation

The continuity equation, a fundamental principle in fluid mechanics, states that for an incompressible and steady flow, the mass flow rate through different cross-sections of a duct or channel remains constant. This means that any change in the cross-sectional area must be compensated by a corresponding change in the fluid velocity, which is critical when designing ducts to maintain desired flow speeds.

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