Worked example — Thermal blanket versus window

Problem

Two “walls” separate a room at $20\ ^{\circ}$C from the outside at $0\ ^{\circ}$C. The first is a glass pane of area $A_v=2\ \mathrm{m}^2$, thickness $d_v=4\ \mathrm{mm}$, conductivity $k_v=1{,}0\ \mathrm{W/(m\cdot K)}$. The second is a glass-wool blanket of the same area, thickness $d_c=8\ \mathrm{cm}$, $k_c=0{,}04\ \mathrm{W/(m\cdot K)}$. How much heat escapes by conduction in one hour, in the two cases?

Solution

Setup. The heat flow by conduction is $\dot Q=\dfrac{k\,A\,\Delta T}{d}$, with $\Delta T=20\ \mathrm{K}$. Glass: $\dot Q_v=\frac{1{,}0\cdot 2\cdot 20}{0{,}004}=10000\ \mathrm{W}$ Thermal blanket: $\dot Q_c=\frac{0{,}04\cdot 2\cdot 20}{0{,}08}=20\ \mathrm{W}$ The ratio is $500$ to $1$. In one hour the window loses $\dot Q_v\cdot 3600\approx 36\ \mathrm{MJ}$, the blanket only $72\ \mathrm{kJ}$. $\ev{\dot Q_v : \dot Q_c\approx 500:1}$ This is why double glazing (an air gap of ~1 cm, $k\approx 0{,}025$) is so effective: the real insulator is still air, not glass (Bloomfield 2016) Riferimenti bibliografici.

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Topics: Termologia Concepts: Trasmissione del calore