A sheet of rubber and a sheet of cardboard, each 5 mm thick are pressed together and their outer faces are maintained respectively at 0°C and 18°C. If the thermal conductivities of rubber and cardboard are respectively 0.13 and `0.05 "Wm"^(-1) "K"^(-1)`. find the quantity of heat which flows in one hour across a piece of composite sheer if area is `100 "cm"^(-2)`.

Answer

To solve the problem of finding the quantity of heat that flows across a composite sheet made of rubber and cardboard, we will follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data:**
- Thickness of rubber, \( L_1 = 5 \, \text{mm} = 5 \times 10^{-3} \, \text{m} \)
- Thickness of cardboard, \( L_2 = 5 \, \text{mm} = 5 \times 10^{-3} \, \text{m} \)
- Temperature difference, \( T_1 = 18^\circ C \) (cardboard side) and \( T_2 = 0^\circ C \) (rubber side)
- Thermal conductivity of rubber, \( K_1 = 0.13 \, \text{W/mK} \)
- Thermal conductivity of cardboard, \( K_2 = 0.05 \, \text{W/mK} \)
- Area, \( A = 100 \, \text{cm}^2 = 100 \times 10^{-4} \, \text{m}^2 = 0.01 \, \text{m}^2 \)

2. **Calculate the Thermal Resistance:**
- The thermal resistance \( R \) for each material is given by:
\[ R = \frac{L}{K \cdot A} \]
- For rubber:
\[ R_1 = \frac{L_1}{K_1 \cdot A} = \frac{5 \times 10^{-3}}{0.13 \cdot 0.01} = \frac{5 \times 10^{-3}}{0.0013} \approx 3.846 \, \text{K/W} \]
- For cardboard:
\[ R_2 = \frac{L_2}{K_2 \cdot A} = \frac{5 \times 10^{-3}}{0.05 \cdot 0.01} = \frac{5 \times 10^{-3}}{0.0005} = 10 \, \text{K/W} \]

3. **Calculate Total Thermal Resistance:**
- Since the two materials are in series, the total resistance \( R_{total} \) is:
\[ R_{total} = R_1 + R_2 = 3.846 + 10 = 13.846 \, \text{K/W} \]

4. **Calculate the Heat Current (Q'):**
- The heat current \( Q' \) (in Watts) can be calculated using:
\[ Q' = \frac{T_1 - T_2}{R_{total}} = \frac{18 - 0}{13.846} \approx 1.298 \, \text{W} \]

5. **Calculate the Total Heat Flow in One Hour:**
- To find the total heat \( Q \) that flows in one hour (3600 seconds):
\[ Q = Q' \times t = 1.298 \, \text{W} \times 3600 \, \text{s} \approx 4663 \, \text{J} \]

### Final Answer:
The quantity of heat that flows across the composite sheet in one hour is approximately **4663 Joules**.

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