Askiitians Tutor Team
Last Activity: 5 Months ago
To solve the problem of finding the temperature difference across layer A of the wall, we need to apply the concept of thermal resistance and the relationship between thermal conductivity and temperature difference. Given that the thermal conductivity of layer A is three times that of layer B, we can set up a relationship based on the heat transfer through both layers.
Understanding Thermal Conductivity and Resistance
Thermal conductivity (k) is a measure of a material's ability to conduct heat. When two layers of different materials are in contact, the heat transfer through each layer can be described using the formula:
Q = (k * A * ΔT) / d
Where:
- Q is the heat transfer rate.
- A is the area through which heat is being transferred.
- ΔT is the temperature difference across the layer.
- d is the thickness of the layer.
In steady-state conditions, the heat transfer rate (Q) through both layers must be equal. Therefore, we can express the heat transfer through layers A and B as follows:
Setting Up the Equations
Let’s denote:
- k_B = thermal conductivity of layer B
- k_A = 3 * k_B (since A's conductivity is three times that of B)
- ΔT_A = temperature difference across layer A
- ΔT_B = temperature difference across layer B
According to the problem, the total temperature difference across the wall is:
ΔT_A + ΔT_B = 36 °C
Using the heat transfer equations for both layers, we have:
For layer A: Q = (k_A * A * ΔT_A) / d
For layer B: Q = (k_B * A * ΔT_B) / d
Since Q is the same for both layers, we can equate them:
(k_A * ΔT_A) = (k_B * ΔT_B)
Substituting k_A = 3 * k_B into the equation gives us:
(3 * k_B * ΔT_A) = (k_B * ΔT_B)
Solving for Temperature Differences
We can simplify this equation by dividing both sides by k_B (assuming k_B is not zero):
3 * ΔT_A = ΔT_B
Now, we can express ΔT_B in terms of ΔT_A:
ΔT_B = 3 * ΔT_A
Substituting this back into the total temperature difference equation:
ΔT_A + 3 * ΔT_A = 36 °C
4 * ΔT_A = 36 °C
Now, divide both sides by 4:
ΔT_A = 36 °C / 4 = 9 °C
Final Result
The temperature difference across layer A is therefore 9 °C. This means that while the total temperature difference across the wall is 36 °C, layer A experiences a smaller portion of that difference due to its higher thermal conductivity compared to layer B.
