An exhaust fan in a building should be able to move 3 kg / s atmospheric pressure air at 25^∘ C

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An exhaust fan in a building should be able to move 3 kg /...

An exhaust fan in a building should be able to move $3 \mathrm{~kg} / \mathrm{s}$ atmospheric pressure air at $25^{\circ} \mathrm{C}$ through a 0.5 -m-diameter vent hole. How high a velocity must it generate, and how much power is required to do that?

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Key Concepts

Kinetic Energy and Power Calculation

Calculating the kinetic energy per unit mass, which is proportional to the square of the fluid velocity, allows one to determine the energy required to set the fluid in motion. Multiplying this energy by the mass flow rate yields the power necessary to sustain the flow, representing the work done per unit time.

Fluid Density and Environmental Conditions

Fluid density, which varies with atmospheric conditions such as pressure and temperature, directly influences the mass flow rate and the calculated velocity. Accurate determination of density is critical for applying the continuity equation and energy calculations reliably in fluid dynamics.

Mass Flow Rate

Mass flow rate is the measure of the mass of a fluid passing through a cross-section per unit time. It is fundamental in determining the flow characteristics and is used in conjunction with other properties such as density and cross-sectional area to establish relationships in fluid dynamics.

Continuity Equation

The continuity equation asserts that, for an incompressible fluid, the mass flow rate remains constant throughout the flow. This principle connects the fluid’s density, velocity, and cross-sectional area, and is key in calculating required quantities when one of these variables is specified.

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