Composite Slab Equivalent Thermal Conductivity Explained

This question involves understanding how heat transfers through a composite slab made of two different materials. We need to find the overall thermal conductivity when two layers of the same thickness are joined together.

Understanding Thermal Resistance

Heat transfer through a material depends on its thermal conductivity, thickness, and the area through which heat flows. A key concept here is thermal resistance, which measures how much a material opposes heat flow. It's calculated using the formula:

$$ R = \frac{L}{k \cdot A} $$

Where:

• $R$ is the thermal resistance
• $L$ is the thickness of the material layer
• $k$ is the thermal conductivity of the material
• $A$ is the cross-sectional area

When layers of different materials are placed in series (like stacked layers in a slab), their thermal resistances add up to give the total thermal resistance of the composite structure.

Calculating Resistance for the Slab

Let's consider the composite slab:

• It has two layers made of different materials.
• The thermal conductivity of the first layer is $k_1$, and the second is $k_2$.
• Both layers have the same thickness, let's call it $L$.
• The total thickness of the slab is therefore $L + L = 2L$.
• Assume the cross-sectional area is $A$.

The thermal resistance for each layer can be calculated as:

• Resistance of the first layer ($R_1$): $$ R_1 = \frac{L}{k_1 \cdot A} $$
• Resistance of the second layer ($R_2$): $$ R_2 = \frac{L}{k_2 \cdot A} $$

Finding the Total Thermal Resistance

Since the layers are in series, the total thermal resistance ($R_{total}$) is the sum of the individual resistances:

$$ R_{total} = R_1 + R_2 $$

Substituting the formulas for $R_1$ and $R_2$:

$$ R_{total} = \frac{L}{k_1 \cdot A} + \frac{L}{k_2 \cdot A} $$

We can factor out the common term $\frac{L}{A}$:

$$ R_{total} = \frac{L}{A} \left( \frac{1}{k_1} + \frac{1}{k_2} \right) $$

To add the fractions inside the parentheses, we find a common denominator ($k_1 \cdot k_2$):

$$ R_{total} = \frac{L}{A} \left( \frac{k_2}{k_1 \cdot k_2} + \frac{k_1}{k_1 \cdot k_2} \right) $$

$$ R_{total} = \frac{L}{A} \left( \frac{k_1 + k_2}{k_1 \cdot k_2} \right) $$

Determining Equivalent Thermal Conductivity

The equivalent thermal conductivity ($k_{eq}$) is the thermal conductivity of a single layer of material with the same total thickness ($2L$) that would offer the same total thermal resistance ($R_{total}$). The formula for this equivalent resistance is:

$$ R_{total} = \frac{2L}{k_{eq} \cdot A} $$

Now, we equate the two expressions for $R_{total}$:

$$ \frac{2L}{k_{eq} \cdot A} = \frac{L}{A} \left( \frac{k_1 + k_2}{k_1 \cdot k_2} \right) $$

We can cancel out the $\frac{L}{A}$ term from both sides:

$$ \frac{2}{k_{eq}} = \frac{k_1 + k_2}{k_1 \cdot k_2} $$

To find $k_{eq}$, we rearrange the equation:

$$ k_{eq} = 2 \times \frac{k_1 \cdot k_2}{k_1 + k_2} $$

$$ k_{eq} = \frac{2 k_1 k_2}{k_1 + k_2} $$

This formula represents the equivalent thermal conductivity for a composite slab with two layers of equal thickness.