Air at standard conditions flows through a sudden expansion...

Air at standard conditions flows through a sudden expansion in a circular duct. The upstream and downstream duct diameters are $75 \mathrm{mm}$ and $225 \mathrm{mm}$, respectively. The pressure downstream is $5 \mathrm{mm}$ of water higher than that upstream. Determine the average speed of the air approaching the expansion and the volume flow rate.

Key Concepts

Volumetric Flow Rate Calculation

Volumetric flow rate is defined as the volume of fluid passing through a given cross-sectional area per unit time. It is calculated by multiplying the average flow velocity by the cross-sectional area, linking the dynamic aspects of fluid motion to the physical geometry of the duct.

Relationship between Pressure, Velocity, and Energy

This concept involves applying principles such as Bernoulli’s equation, where the pressure energy is converted into kinetic energy, allowing one to relate a measured pressure difference to the fluid’s velocity. It is crucial when determining how pressure recoveries or drops translate into changes in flow speed.

Continuity Equation

The continuity equation is a fundamental principle stating that for a steady, incompressible flow, the mass flow rate must remain constant along a streamline. In duct flows, this relates the fluid velocity to the cross-sectional area, ensuring that a change in duct diameter results in corresponding changes in velocity to conserve mass.

Pressure Difference and Unit Conversion

This concept involves understanding that pressure differences drive fluid motion and that they must be consistently converted to standard units (such as converting mm of water to Pascals) before applying fluid flow equations. It ensures that the analysis accurately accounts for the energy available to accelerate the fluid.

Flow Through a Sudden Expansion

Sudden expansions occur when a fluid flows from a smaller to a larger cross-sectional area abruptly, often leading to flow separation, turbulence, and pressure recovery issues. Understanding this helps in estimating how the expanding geometry affects the velocity field and energy losses across the expansion.

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