Question

For which kinds of bodies made of the same material is the lumped system analysis more likely to be applicable: slender ones or well-rounded ones of the same volume? Why?

Short answer

Answer: Lumped system analysis is more likely to be applicable for well-rounded bodies made of the same material. This is because well-rounded bodies have shorter characteristic lengths and higher surface area to volume ratios, leading to lower Biot numbers, which makes the assumption of a uniform temperature distribution within the body valid.

Step-by-step solution

Understand Lumped System Analysis

Lumped system analysis is a method used to simplify the analysis of heat transfer in solid bodies by assuming that the temperature within the solid body is uniform at all times. This means that the heat transfer within the solid body is much faster than the heat transfer across the boundary into or out of the body. It is suitable for objects with low thermal resistance within the object compared to the thermal resistance at the boundary.#tag_end#

Understand the Biot number

The Biot number (Bi) is a dimensionless parameter that helps to determine whether lumped system analysis is applicable to a given situation. It is defined as the ratio of internal thermal resistance to surface thermal resistance. The formula for Biot number is: Bi = (h * Lc) / k where h is the convective heat transfer coefficient at the surface of the body, Lc is the characteristic length of the body, and k is the thermal conductivity of the material. If the Biot number is much less than 1 (typically Bi < 0.1), lumped system analysis is considered to be applicable. This means that the temperature distribution inside the solid body is relatively uniform, and the heat transfer is mainly determined by the surface heat transfer instead of internal heat conduction.#tag_end#

Slender vs. Well-rounded bodies

A slender body is an object that has a large aspect ratio (length / width >>1), meaning it is long and thin. A well-rounded body, on the other hand, has a more compact shape with relatively equal dimensions in all directions (spherical or near-spherical bodies, for example). The heat transfer characteristics of these two types of bodies are different due to their varying surface area to volume ratios and characteristic lengths. Slender bodies typically have a smaller surface area to volume ratio and longer characteristic lengths, while well-rounded bodies have a larger surface area to volume ratio and shorter characteristic lengths. As a result, the surface thermal resistance is higher in slender bodies and lower in well-rounded ones, whereas the internal thermal resistance is higher in well-rounded bodies and lower in slender ones.#tag_end#

Determine the applicability of Lumped System Analysis

Comparing the heat transfer characteristics of slender and well-rounded bodies, we can conclude that lumped system analysis is more likely to be applicable for well-rounded bodies. This is because well-rounded bodies have shorter characteristic lengths and higher surface area to volume ratios, leading to lower Biot numbers. When Bi < 0.1, the lumped system analysis assumption of a uniform temperature distribution within the body is valid, and the heat transfer is mainly determined by the surface heat transfer instead of internal heat conduction. Hence, well-rounded bodies made of the same material are more suitable for lumped system analysis.#tag_end#