A partition wall has two layers A and B, in contact, each made of a different material. They have the same thickness but the thermal conductivity of layer A is twice that of layer B. If the steady state temperature difference across the wall is 60 K, then the corresponding difference across the layer A is

Options

A. 10K
B. 20K
C. 30K
D. 40K

Answer

To solve the problem, we need to find the temperature difference across layer A of the partition wall. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the problem
We have a partition wall made of two layers, A and B, with different thermal conductivities. The thermal conductivity of layer A is twice that of layer B. The total temperature difference across the wall is given as 60 K.

### Step 2: Define thermal resistances
The thermal resistance (R) for each layer can be defined as:
- For layer A (thermal conductivity = 2K):
\[ R_A = \frac{L}{2K \cdot A} \]
- For layer B (thermal conductivity = K):
\[ R_B = \frac{L}{K \cdot A} \]

### Step 3: Set up the equation for heat flow
In steady state, the rate of heat flow through both layers is the same. We can express the heat flow (Q) through each layer as:
- For layer A:
\[ Q = \frac{T_1 - T_0}{R_A} = \frac{T_1 - T_0}{\frac{L}{2K \cdot A}} = \frac{2K \cdot A (T_1 - T_0)}{L} \]
- For layer B:
\[ Q = \frac{T_2 - T_1}{R_B} = \frac{T_2 - T_1}{\frac{L}{K \cdot A}} = \frac{K \cdot A (T_2 - T_1)}{L} \]

### Step 4: Relate the temperatures
Given that the total temperature difference across the wall is 60 K, we can express it as:
\[ T_2 - T_0 = 60 \quad \text{(where \(T_2\) is the temperature at the hot side and \(T_0\) is the temperature at the cold side)} \]
Let \(T_1\) be the temperature at the interface between layers A and B.

### Step 5: Set the heat flow equations equal
Since the heat flow through both layers is equal:
\[ \frac{2K \cdot A (T_1 - T_0)}{L} = \frac{K \cdot A (T_2 - T_1)}{L} \]
We can cancel \(K \cdot A\) and \(L\) from both sides:
\[ 2(T_1 - T_0) = (T_2 - T_1) \]

### Step 6: Substitute \(T_2\)
Substituting \(T_2 = T_0 + 60\) into the equation:
\[ 2(T_1 - T_0) = (T_0 + 60 - T_1) \]
Expanding and rearranging gives:
\[ 2T_1 - 2T_0 = T_0 + 60 - T_1 \]
\[ 3T_1 = 3T_0 + 60 \]
\[ T_1 = T_0 + 20 \]

### Step 7: Find the temperature difference across layer A
The temperature difference across layer A is:
\[ \Delta T_A = T_1 - T_0 = 20 \text{ K} \]

### Final Answer
The corresponding temperature difference across layer A is **20 K**.

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