Consider two long, slender rods of the same diameter but...

Consider two long, slender rods of the same diameter but different materials. One end of each rod is attached to a base surface maintained at $100^{\circ} \mathrm{C}$, while the surfaces of the rods are exposed to ambient air at $20^{\circ} \mathrm{C}$. By traversing the length of each rod with a thermocouple, it was observed that the temperatures of the rods were equal at the positions $x_{\mathrm{A}}=0.15 \mathrm{~m}$ and $x_{\mathrm{B}}=0.075 \mathrm{~m}$, where $x$ is measured from the base surface. If the thermal conductivity of rod $\mathrm{A}$ is known to be $k_{\mathrm{A}}=70 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, determine the value of $k_{\mathrm{B}}$ for rod B.

Key Concepts

Thermal Conductivity

Thermal conductivity is a material property that measures a material’s ability to conduct heat. In the context of heat transfer problems involving solids such as rods, a higher thermal conductivity means that heat can be conducted more efficiently along the rod, while a lower conductivity means that the material resists the flow of heat. This concept is key when comparing different materials subjected to the same thermal conditions.

Steady-State Heat Conduction

Steady-state heat conduction refers to a condition in which the temperature distribution within a material does not change with time. In such problems, the heat input, conduction along the material, and any energy losses (such as through convection) are balanced. Understanding this concept is critical for analyzing temperature profiles measured along the length of a heated rod that is also losing heat to the environment.

Convection from Surfaces

Convection is a mode of heat transfer where heat is carried away from a surface by moving fluid (like air). In problems involving heated rods exposed to ambient air, convective heat loss is important because it affects the temperature distribution along the rod. The rate of convective heat transfer is generally characterized by a heat transfer coefficient that depends on the geometry of the object and the nature of the fluid flow around it.

Temperature Distribution in Extended Surfaces (Fin Analysis)

Extended surfaces, commonly known as fins, are designed to enhance heat transfer by increasing the surface area. The temperature distribution along a fin typically follows a hyperbolic (or exponential) profile due to the balance between conduction along the fin and convection from its surface. Analyzing the positions along the rod where the temperature is a specific value involves applying fin theory, which relates the geometry, thermal conductivity, and convective losses.

Please give Ace some feedback