Problem — Single-pane window

Problem

A single-pane window has area $A = 2$ m², thickness $d = 4$ mm and conductivity $k_{\text{glass}} = 1.0$ W/(m·K). The room inside is at $T_1 = 20\ ^{\circ}$C, the outside at $T_2 = -5\ ^{\circ}$C. Calculate the heat flow in steady state, assuming perfect thermal contact between the glass and the air.

Solution

Fourier’s law. In steady state the flow through a flat slab is $\dot Q = k \, A \, \frac{\Delta T}{d}$ Substituting the data. The temperature gap is $\Delta T = 20 - (-5) = 25$ K and the thickness $d = 4$ mm $= 0.004$ m: $\dot Q = 1.0 \cdot 2 \cdot \frac{25}{0.004} = \frac{50}{0.004} = 1.25 \cdot 10^{4}\ \mathrm{W}$ The idealised value is huge: in reality the flow is much smaller, because the thin layers of air in contact with the glass (which we neglect here) dominate the thermal resistance. That’s precisely why a single pane insulates so poorly. $\ev{\dot Q \approx 1.25 \cdot 10^{4}\ \mathrm{W}}$

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Topics: Termologia Concepts: Trasmissione del calore