A carpenter builds a solid wood door with dimensions 2m x 0.95m x 5cm. Its thermal conductivity is k= 0.120 w/mk . The air films on the inner and outer surface of the door have the same combined thermal resistance as an additional 1.8cm thickness of solid wood. The inside air temperature is 20°C and the outside air temperature is 8°C. What is the rate of heat flow through the door?

To find the rate of heat flow through the door, we will use the concept of thermal resistance and the formula for heat transfer through a composite wall. The rate of heat flow (Q) through a composite wall is given by:

\[
Q = \frac{\Delta T}{R_{\text{total}}}
\]

where:
- \( \Delta T \) is the temperature difference across the door.
- \( R_{\text{total}} \) is the total thermal resistance of the door.

### Step 1: Calculate Thermal Resistance of the Door

The total thermal resistance \( R_{\text{total}} \) consists of:
1. The thermal resistance of the solid wood door.
2. The thermal resistance of the equivalent additional thickness due to the air films.

The thermal resistance \( R \) for a layer is calculated using the formula:

\[
R = \frac{L}{kA}
\]

where:
- \( L \) is the thickness of the material (in meters).
- \( k \) is the thermal conductivity of the material (in W/m·K).
- \( A \) is the surface area of the door (in square meters).

#### a) Thermal Resistance of the Solid Wood Door

Given:
- The thickness of the door, \( L = 5 \) cm = 0.05 m
- The thermal conductivity, \( k = 0.120 \) W/m·K
- The dimensions of the door: 2 m x 0.95 m, so the surface area \( A = 2 \times 0.95 = 1.9 \) m²

Now, let's calculate the thermal resistance of the door:

\[
R_{\text{wood}} = \frac{L_{\text{wood}}}{k \cdot A} = \frac{0.05}{0.120 \times 1.9}
\]

#### b) Thermal Resistance of the Air Films

The equivalent thermal resistance of the air films is given as being the same as an additional 1.8 cm (0.018 m) thickness of solid wood. We calculate this resistance similarly:

\[
R_{\text{air}} = \frac{L_{\text{air}}}{k \cdot A} = \frac{0.018}{0.120 \times 1.9}
\]

#### c) Total Thermal Resistance

The total thermal resistance is the sum of the two resistances:

\[
R_{\text{total}} = R_{\text{wood}} + R_{\text{air}}
\]

### Step 2: Calculate the Rate of Heat Flow

The temperature difference between the inside and outside air is:

\[
\Delta T = 20^\circ \text{C} - 8^\circ \text{C} = 12^\circ \text{C}
\]

Now, we can find the rate of heat flow:

\[
Q = \frac{\Delta T}{R_{\text{total}}}
\]

Let's compute these values step-by-step.Here are the calculated values:

1. **Thermal resistance of the wood door**:  
   \( R_{\text{wood}} = 0.219 \) \( \text{m}^2 \cdot \text{K/W} \)

2. **Thermal resistance of the air films**:  
   \( R_{\text{air}} = 0.079 \) \( \text{m}^2 \cdot \text{K/W} \)

3. **Total thermal resistance**:  
   \( R_{\text{total}} = 0.298 \) \( \text{m}^2 \cdot \text{K/W} \)

4. **Rate of heat flow through the door**:  
   \( Q = 40.24 \) W

### Conclusion:
The rate of heat flow through the door is approximately **40.24 watts**.

Would you like a deeper explanation or have any questions? Here are some related questions:

1. How would the rate of heat flow change if the door was thicker?
2. What would happen if the thermal conductivity of the material was different?
3. How can the thermal resistance of air films affect insulation?
4. How does this heat transfer relate to energy consumption in a household?
5. What materials could provide better insulation than wood?

**Tip:** Understanding thermal resistance helps in choosing materials for efficient thermal insulation!

Learn how to calculate the rate of heat flow through a solid wood door with detailed step-by-step solutions. This problem involves heat transfer principles, thermal resistance calculations, and composite wall analysis. Suitable for advanced learners and professionals.

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