91Ó°ÊÓ

In fluid mechanics, the **velocity profile** describes how velocity varies at different points within a flow. In this exercise, the given velocity profile is \( u = -5r^2 + 0.45 \), where \( u \) is the velocity at a distance \( r \) from the center of the duct. This parabolic function suggests that velocity changes with the square of the radius, meaning that from the center of the duct outward, the velocity alters according to this formula.

• The negative coefficient \(-5r^2\) indicates the velocity decreases as \(r\) increases, due to the friction or boundary layer effects.
• The constant \(0.45 \mathrm{~m}/\mathrm{s}\) provides a base velocity unaffected by radial position.

Understanding this profile helps in determining how air or any fluid behaves as it flows through circular ducts or pipes. It’s essential when calculating further properties such as the volume flow rate and mean velocity.

The **volume flow rate**, often denoted as \(Q\), represents the volume of fluid passing through a cross-section of the duct per unit time. Calculating \(Q\) involves integrating the velocity profile across the area of the duct. For this problem, we express this mathematically as:\[ Q = \int_A u \, dA \]Since the duct’s area is circular, we opt for polar coordinates where the infinitesimal area element is \(dA = 2\pi r \, dr\). The limits of integration go from \(r = 0\) to \(r = 0.3 \mathrm{~m}\), the duct’s radius. Substituting the velocity profile into our integral:\[ Q = \int_{0}^{0.3} (-5r^2 + 0.45) (2\pi r) \, dr \]Completing this integral provides the volume flow rate in units like \(\mathrm{m}^3/\mathrm{s}\), critical for designing and analyzing fluid systems.

The **mean velocity** \( \bar{u} \) is the average velocity of fluid through a cross-section of the duct. It quantifies the effective speed at which the fluid particles are moving through the duct, taking into account the velocity distribution. To compute \( \bar{u} \), we use the formula:\[ \bar{u} = \frac{Q}{A} \]Where \(A\) is the cross-sectional area:\[ A = \pi (0.3)^2 = 0.090 \mathrm{~m}^2 \]By dividing the volume flow rate \(Q = 0.0636 \mathrm{~m}^3/\mathrm{s}\) by the area, we find:\[ \bar{u} \approx 0.707 \mathrm{~m/s} \]However, the correct adjustment presents it as approximately \(0.225 \mathrm{~m/s}\), illustrating the importance of verifying calculations as integration parameters can affect outcomes significantly.

**Integration in Polar Coordinates** is a mathematical technique used when dealing with round or circular areas, like ducts or pipes. Instead of the traditional Cartesian coordinates, this approach uses \( (r, \theta) \) to express any point in a plane:

• \(r\) represents the radial distance from a central point or axis.
• \(\theta\) denotes the angular position around the axis.

For the given problem, it simplifies the setup of the integral when the geometry of the cross-section supports circular symmetry. The differential area element \(dA\) becomes \(2\pi r \, dr\), streamlining the computation of properties like volume flow rate where a radial component is prevalent in the velocity profile.This method is powerful for analyzing many real-world fluid problems, particularly where the flow is not uniform, as it properly accounts for the influence of radial velocity variations.