A plane wall surface at 200^∘ C is to be cooled with aluminum pin fins of parabolic profile with

step 1 For this one you need to figure out the heat transfer rate from the single aluminum pin fin of p

A plane wall surface at 200^∘ C is to be cooled with...

A plane wall surface at $200^{\circ} \mathrm{C}$ is to be cooled with aluminum pin fins of parabolic profile with blunt tips. Each fin has a length of $25 \mathrm{~mm}$ and a base diameter of $4 \mathrm{~mm}$. The fins are exposed to ambient air at $25^{\circ} \mathrm{C}$, and the heat transfer coefficient is $45 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the thermal conductivity of the fins is 230 $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$, determine the heat transfer rate from a single fin and the increase in the rate of heat transfer per $\mathrm{m}^{2}$ surface area as a result of attaching fins. Assume there are 100 fins per $\mathrm{m}^{2}$ surface area.

Key Concepts

Conduction in Fins

Conduction in fins refers to the transport of thermal energy along the length of the fin due to a temperature gradient. Analyzing conduction within fins involves solving differential equations based on energy conservation, often resulting in a temperature distribution that affects the overall heat transfer performance. This internal conduction is impacted by the fin’s material properties, such as thermal conductivity.

Extended Surfaces (Fins)

Fins are used to increase the effective surface area available for heat transfer between a solid surface and a fluid, thereby enhancing convective cooling. They are critical in applications where the natural surface area of the base is not sufficient to meet thermal performance targets. Analysis of fins involves accounting for both conduction within the fin material and convection from the fin surface to the ambient fluid.

Convection Heat Transfer

Convection is the mode of heat transfer between a surface and a fluid in motion. The rate of convective heat transfer depends on the temperature difference between the surface and the fluid and is characterized by the heat transfer coefficient, which encapsulates the effects of fluid properties and flow conditions. This concept is essential for understanding how the fins dissipate heat to the surrounding air.

Variable Cross-Sectional Geometry

Fins with variable cross-sectional areas, such as those with a parabolic profile, require special treatment because their changing geometry influences the rate of conductive heat transfer along the fin. This concept is crucial for deriving accurate models of fin performance, as the spatial variation in the cross-sectional area affects the temperature distribution and, ultimately, the fin efficiency.

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