Question

For which solid is the lumped system analysis more likely to be applicable: an actual apple or a golden apple of the same size? Why?

Short answer

Answer: An actual apple is more suitable for a lumped system analysis than a golden apple of the same size because it has a lower thermal conductivity, resulting in a more uniform temperature distribution.

Step-by-step solution

Identify the thermal conductivity values

First, we must identify the thermal conductivities of both an actual apple and a golden apple (gold). The thermal conductivity of an actual apple is approximately k_apple = 0.45 W/m·K, while the thermal conductivity of gold is k_gold = 315 W/m·K.

Compare the thermal conductivities

Now that we have identified the thermal conductivities of both an actual apple and a golden apple, we must compare these values to determine which is more suitable for a lumped system analysis. Since a lower thermal conductivity value results in a more uniform temperature distribution, we must compare the values to see which one is lower: k_apple = 0.45 W/m·K or k_gold = 315 W/m·K. It is clear that the thermal conductivity of an actual apple (k_apple = 0.45 W/m·K) is much lower than that of a golden apple (k_gold = 315 W/m·K).

Determine which solid is more suitable for lumped system analysis

Based on the comparison in Step 2, we can see that an actual apple has a much lower thermal conductivity value than a golden apple. Since lower thermal conductivity values result in more uniform temperature distribution (which is a key assumption for the lumped system analysis), we can conclude that the lumped system analysis is more likely to be applicable for an actual apple than for a golden apple of the same size.

Key concepts

Thermal Conductivity

Thermal conductivity is a measure of a material's ability to conduct heat. It represents how quickly heat can pass through a substance when a temperature difference exists. In the context of our exercise, comparing an apple and a golden apple, the concept of thermal conductivity is crucial for determining the applicability of lumped system analysis.
Thermal conductivity is quantitatively represented by the symbol \( k \) and is expressed in units of watts per meter per Kelvin (W/m\( \cdot \)K). A lower \( k \) value means that the material is a poorer conductor of heat and will not transfer heat as quickly as a material with a higher \( k \) value. Therefore, substances with low thermal conductivity are likely to maintain a more uniform temperature distribution because heat diffuses through them more slowly.
Understanding thermal conductivity helps in predicting the suitability for lumped system analysis. With regard to the exercise, the actual apple (\( k_{apple} = 0.45 \) W/m\( \cdot \)K) is more likely to have a uniform temperature distribution than the golden apple (\( k_{gold} = 315 \) W/m\( \cdot \)K), due to its significantly lower thermal conductivity.

Uniform Temperature Distribution

Uniform temperature distribution within a solid is one of the critical assumptions of lumped system analysis. This principle implies that at any given moment, the temperature variation throughout the object is negligible, hence it can be approximated as being uniform. When heat transfer occurs, if the material has a low thermal conductivity and is sufficiently small, it will quickly reach a state where its entire volume settles at one temperature.
For an object to maintain a uniform temperature, several conditions should be met, including low thermal conductivity, small size, and good internal mixing or low Biot number (the ratio of external to internal resistance to heat flow). The property of a material to resist change in temperature when exposed to a uniform heat source is linked to its thermal mass, which is a function of both its specific heat capacity and density.
In our exercise, the actual apple, with its lower thermal conductivity, tends to exhibit a more uniform temperature distribution compared to the golden apple. This characteristic makes the actual apple better suited for lumped system analysis, which simplifies the complex heat transfer problem by considering the temperature of the object as spatially uniform.

Heat Transfer in Solids

Heat transfer in solids occurs mainly through conduction, which involves the transfer of energy from more energetic particles to less energetic ones due to temperature difference, without any actual movement of the particles. This contrasts with heat transfer in fluids, where convection (the movement of fluid itself) can also play a significant role.
In the context of the exercise, understanding heat transfer in solids like an apple or a golden apple helps us apply the lumped system analysis correctly. Solids with lower thermal conductivity, like the actual apple, transfer heat more slowly, which facilitates the approximation of uniform temperature throughout the object. Conversely, heat rapidly disperses through solids with high thermal conductivity, like gold, often leading to significant temperature gradients within the object.
A lumped system analysis is typically used when the rate of heat transfer to and from an object is slow enough that the object's temperature can be assumed fairly uniform. This is more easily justified in solids like the actual apple, which has a lower thermal conductivity and consequently a slower heat transfer rate, than in high conductivity materials like gold.