Let's break down the problem step by step to understand how to find the effective thermal conductivity of a compound slab made of two different materials. We have two materials with equal lengths and thicknesses, and their thermal conductivities are K and 2K. We want to determine the overall thermal conductivity of the slab.
Understanding Thermal Conductivity
Thermal conductivity (K) is a measure of a material's ability to conduct heat. When two materials are combined, their thermal conductivities affect the overall heat transfer through the compound slab. In this case, we have:
- Material 1: Thermal conductivity = K
- Material 2: Thermal conductivity = 2K
Setting Up the Problem
Since both materials have the same length (L) and thickness (d), we can analyze the heat transfer through each material separately. The heat transfer rate (Q) through a material can be described by Fourier's law:
Q = (K * A * ΔT) / d
Where:
- Q = heat transfer rate
- K = thermal conductivity of the material
- A = cross-sectional area
- ΔT = temperature difference across the material
- d = thickness of the material
Calculating Effective Thermal Conductivity
For two materials in series, the total thermal resistance (R_total) is the sum of the individual resistances:
R_total = R1 + R2
Where:
- R1 = d / (K * A)
- R2 = d / (2K * A)
Now, substituting these into the equation:
R_total = (d / (K * A)) + (d / (2K * A))
To combine these fractions, we need a common denominator:
R_total = (2d / (2K * A)) + (d / (2K * A)) = (3d / (2K * A))
Finding the Effective Conductivity
The effective thermal conductivity (K_eff) can be found from the total resistance:
K_eff = d / (R_total * A)
Substituting R_total into this equation gives:
K_eff = d / ((3d / (2K * A)) * A) = (2K / 3)
Final Result
Thus, the effective thermal conductivity of the compound slab is:
K_eff = 2K / 3
This matches the answer you mentioned, which is option C. This method of combining thermal conductivities is essential in understanding how materials interact in terms of heat transfer. If you have any further questions or need clarification on any part of this process, feel free to ask!
