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

Explanation

To solve the problem of finding the equivalent thermal conductivity of a compound slab consisting of two materials with thermal conductivities \( K \) and \( 2K \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Properties of the Slabs**:
- Let the thickness of each slab be \( l \).
- The thermal conductivity of the first slab is \( K \).
- The thermal conductivity of the second slab is \( 2K \).

2. **Calculate the Thermal Resistance of Each Slab**:
- The thermal resistance \( R \) for a slab is given by the formula:
\[ R = \frac{l}{K \cdot A} \]
- For the first slab (conductivity \( K \)):
\[ R_1 = \frac{l}{K \cdot A} \]
- For the second slab (conductivity \( 2K \)):
\[ R_2 = \frac{l}{2K \cdot A} \]

3. **Combine the Thermal Resistances**:
- Since the slabs are in series, the total thermal resistance \( R_{total} \) is the sum of the individual resistances:
\[ R_{total} = R_1 + R_2 = \frac{l}{K \cdot A} + \frac{l}{2K \cdot A} \]

4. **Find a Common Denominator**:
- To combine the fractions, find a common denominator:
\[ R_{total} = \frac{2l}{2K \cdot A} + \frac{l}{2K \cdot A} = \frac{2l + l}{2K \cdot A} = \frac{3l}{2K \cdot A} \]

5. **Relate Total Resistance to Equivalent Conductivity**:
- The equivalent thermal resistance can also be expressed in terms of the equivalent thermal conductivity \( K_{eq} \):
\[ R_{total} = \frac{l}{K_{eq} \cdot A} \]
- Setting the two expressions for \( R_{total} \) equal gives:
\[ \frac{3l}{2K \cdot A} = \frac{l}{K_{eq} \cdot A} \]

6. **Solve for the Equivalent Thermal Conductivity**:
- Cancel \( l \) and \( A \) from both sides:
\[ \frac{3}{2K} = \frac{1}{K_{eq}} \]
- Taking the reciprocal gives:
\[ K_{eq} = \frac{2K}{3} \]

### Final Result:
The equivalent thermal conductivity of the compound slab is:
\[ K_{eq} = \frac{4}{3} K \]