Derive the conduction of heat through hollow sphere of material steel with 2 outer layers (Asbestos and rubber).
To derive the conduction of heat through a hollow sphere with multiple layers, we need to apply the principles of heat transfer and conduction.
1. First, we'll assume steady-state heat conduction, meaning that the temperature within the sphere remains constant over time.
2. Next, we'll consider that the heat is transferred through each layer independently. This means that the temperature gradient within the sphere is negligible.
3. To calculate the overall heat conduction, we'll use the thermal conductivity (k) of each material. The thermal conductivity is a measure of a material's ability to conduct heat. Higher thermal conductivity represents better heat conduction.
Let's denote the thermal conductivity for each material as follows:
- Steel: k_s
- Asbestos: k_a
- Rubber: k_r
To derive the conduction of heat through the hollow sphere, we'll use the following steps:
Step 1: Calculate the thermal resistance of each layer
- The thermal resistance (R) of a material is given by the formula: R = (thickness of layer) / (thermal conductivity of layer)
- For the hollow sphere, we need to find the thermal resistance of steel (R_s), the asbestos layer (R_a), and the rubber layer (R_r).
Step 2: Calculate the overall thermal resistance
- The overall thermal resistance (R_total) is the sum of the thermal resistances of each layer. Therefore, R_total = R_s + R_a + R_r.
Step 3: Calculate the overall thermal conductivity
- The overall thermal conductivity (k_total) is the reciprocal of the overall thermal resistance. Therefore, k_total = 1 / R_total.
The overall thermal conductivity (k_total) gives us an indication of how well heat is conducted through the hollow sphere with multiple layers.
Note: The above steps assume that the layers are in contact and don't have any air gaps between them. If there are air gaps, additional calculations might be required to account for any thermal resistance introduced by the gaps.
To obtain more accurate values for thermal conductivities, you can look up the specific thermal conductivity values for steel, asbestos, and rubber from reliable sources such as engineering handbooks or material property databases.