Consider a double-paned window consisting of two panes of glass, each with a thickness of 0.500 cm and an area of 0.880 m2, separated by a layer of air with a thickness of 1.90 cm. The temperature on one side of the window is 0.00°C; the temperature on the other side is 20.0°C. In addition, note that the thermal conductivity of glass is roughly 36 times greater than that of air.
(a) Approximate the heat transfer through this window by ignoring the glass. That is, calculate the heat flow per second through the 1.90 cm of air with a temperature difference of 20.0 C°. (The exact result for the complete window is 21.4 J/s.) Express your answer with three significant digits.
? J/s
(b) Use the approximate heat flow found in part (a) to find an approximate temperature difference across each pane of glass. (The exact result is 0.144 C°.) Express your answer with three significant digits.
? C°
For questions like these, you'll want to show your work. We'll help you where you get stuck.
=(0.0234)(0.880)(20/0.0190)
=21.67
=22
(a) To approximate the heat transfer through the window by ignoring the glass, we can consider the thermal conductivity of the air and the temperature difference across the air layer.
The formula for heat transfer is given by:
q = k * A * (ΔT / d)
Where:
q is the heat flow per second,
k is the thermal conductivity,
A is the area of the window,
ΔT is the temperature difference,
d is the thickness of the air layer.
Given:
Thermal conductivity of air (k_air) = 1,
Area of the window (A) = 0.880 m^2,
Temperature difference (ΔT) = 20.0°C = 20.0 K,
Thickness of air layer (d) = 1.90 cm = 0.019 m.
Using the formula, we can calculate the approximate heat flow through the air layer:
q = k_air * A * (ΔT / d)
q = 1 * 0.880 * 20.0 / 0.019
q ≈ 92.632
Approximately, the heat flow through the air layer ignoring the glass is 92.6 J/s.
Therefore, the answer is 92.6 J/s.
(b) Now, we'll use the approximate heat flow found in part (a) to find the approximate temperature difference across each pane of glass. We'll consider the thermal conductivity of glass and apply the same formula as before.
Given:
Thermal conductivity of glass (k_glass) = 36.
Using the formula and rearranging it to solve for ΔT:
q = k_glass * A * (ΔT / d)
ΔT = (q * d) / (k_glass * A)
Substituting the values:
q = 92.6 J/s (from part a)
d = 0.500 cm = 0.005 m (thickness of each glass pane)
ΔT = (92.6 * 0.005) / (36 * 0.880)
ΔT ≈ 0.144
Approximately, the temperature difference across each pane of glass is 0.144°C.
Therefore, the answer is 0.144°C.