Question

Under what conditions can a plane wall be treated as a semi-infinite mediaum?

Short answer

Answer: A plane wall can be treated as a semi-infinite medium when the thermal penetration depth (δ) is significantly smaller than the actual thickness (L) of the material. This occurs when the disturbance, such as a temperature change, does not penetrate significantly into the plane wall's interior, allowing heat transfer and thermal behavior near the surface to be almost independent of the interior of the wall.

Step-by-step solution

Understanding the thermal penetration depth

Thermal penetration depth is the distance a temperature change can penetrate into a material during a specific period of time. It is essentially a measure of how far the temperature variation can be felt inside the material from the exposed surface. The thermal penetration depth (δ) can be calculated using the following formula: δ = \sqrt{αt} Where α is the thermal diffusivity of the material and t is the time since the temperature change occurred. Thermal diffusivity is a measure of a material's ability to conduct heat relative to its ability to store heat and is given by the ratio of thermal conductivity (k) to the product of density (ρ) and specific heat (c_p): α = k / (ρc_p)

Determining if the plane wall can be treated as semi-infinite

A plane wall can be considered semi-infinite if the thermal penetration depth is much less than the actual thickness of the wall or material (L). Mathematically, this condition can be written as: δ << L In other words, when the disturbance (e.g., temperature change) does not penetrate significantly into the plane wall's interior, we can treat this wall as a semi-infinite medium. This is because the heat transfer and thermal behavior near the surface will be nearly independent of the interior of the wall.

Evaluating given conditions

To decide if the plane wall can be treated as a semi-infinite medium under specific circumstances, compare the thermal penetration depth (δ) to the actual thickness (L) of the wall. If the wall's thickness is significantly larger than the calculated δ value, it can be considered semi-infinite. In summary, a plane wall can be treated as a semi-infinite medium when the thermal penetration depth is significantly smaller than the actual thickness of the material. This allows us to focus on the heat transfer and thermal behavior at the surface, which is independent of the wall's interior under these conditions.