Consider a compound slab consisting of two different material having equal thickness and thermal conductivities `K` and `2K` respectively. The equivalent thermal conductivity of the slab is

Options

A. `sqrt(2K)`
B. `3K`
C. `4/3K`
D. `2/3 K`

Explanation

To find the equivalent thermal conductivity of a compound slab consisting of two different materials with equal thickness and thermal conductivities \( K \) and \( 2K \), we can follow these steps:

### Step 1: Understand the Setup
We have a compound slab made of two materials:
- Material 1 has thermal conductivity \( K \) and thickness \( L \).
- Material 2 has thermal conductivity \( 2K \) and thickness \( L \).

### Step 2: Calculate the Thermal Resistance
The thermal resistance \( R \) for each material can be calculated using the formula:
\[ R = \frac{L}{kA} \]
where:
- \( L \) is the thickness of the material,
- \( k \) is the thermal conductivity,
- \( A \) is the cross-sectional area.

For Material 1 (conductivity \( K \)):
\[ R_1 = \frac{L}{KA} \]

For Material 2 (conductivity \( 2K \)):
\[ R_2 = \frac{L}{2KA} \]

### Step 3: Combine the Resistances
Since the two materials are in series, the total thermal resistance \( R_{total} \) of the slab is the sum of the individual resistances:
\[ R_{total} = R_1 + R_2 = \frac{L}{KA} + \frac{L}{2KA} \]

### Step 4: Simplify the Expression
To combine the resistances, we can find a common denominator:
\[ R_{total} = \frac{L}{KA} + \frac{L}{2KA} = \frac{2L}{2KA} + \frac{L}{2KA} = \frac{2L + L}{2KA} = \frac{3L}{2KA} \]

### Step 5: Calculate the Equivalent Thermal Conductivity
The equivalent thermal conductivity \( K_{eq} \) can be defined using the total resistance:
\[ R_{total} = \frac{L}{K_{eq}A} \]
Setting the two expressions for \( R_{total} \) equal to each other:
\[ \frac{L}{K_{eq}A} = \frac{3L}{2KA} \]

### Step 6: Solve for \( K_{eq} \)
By cross-multiplying and simplifying, we can solve for \( K_{eq} \):
\[ K_{eq} = \frac{2K}{3} \]

### Final Answer
The equivalent thermal conductivity of the compound slab is:
\[ K_{eq} = \frac{2K}{3} \]

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