A 1,500 W portable heater is needed to replace the heat lost through a circular window (radius 25 cm) in the wall of a home. The window is made of a single pane of glass 0.30 cm thick. The thermal conductivity for the glass is 0.63 W/(m·K). The temperature inside the house is 20 oC. What is the temperature outside?
Question
A 1,500 W portable heater is needed to replace the heat lost through a circular window (radius 25 cm) in the wall of a home. The window is made of a single pane of glass 0.30 cm thick. The thermal conductivity for the glass is 0.63 W/(m·K). The temperature inside the house is 20 oC. What is the temperature outside?
Solution
To solve this problem, we need to use the formula for heat transfer through conduction, which is:
Q = kA(T1-T2)/d
where: Q is the heat transfer rate (in Watts), k is the thermal conductivity (in W/m·K), A is the area through which heat is transferred (in m²), T1 and T2 are the temperatures on the two sides of the material (in °C), and d is the thickness of the material (in m).
We know Q (1500 W), k (0.63 W/m·K), the radius of the window (0.25 m), and the thickness of the glass (0.003 m). We also know the temperature inside the house (T1 = 20 °C). We need to find the temperature outside the house (T2).
First, let's calculate the area of the window using the formula for the area of a circle, which is πr². The radius of the window is 0.25 m, so the area is:
A = π*(0.25 m)² = 0.196 m²
Now we can rearrange the heat transfer formula to solve for T2:
T2 = T1 - Qd/(kA)
Substituting the known values:
T2 = 20 °C - 1500 W * 0.003 m / (0.63 W/m·K * 0.196 m²) = -15.2 °C
So, the temperature outside the house is approximately -15.2 °C.
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