Heat Transfer Mechanisms — Question 2

Question 2

A small thermal bridge can dominate a parallel wall

A plane wall has total area 10.0 m$^2$ and thickness .100 m. Insulation with $k_i=.0400$ W/(m K) occupies 9.90 m$^2$, while metal bridges with $k_m=50.0$ W/(m K) occupy .100 m$^2$. Both paths share a 20.0 K temperature difference. Neglect lateral coupling and surface resistance.

1. Find each heat rate and the fraction carried by the metal.

2. Find the maximum metal area fraction $f$ that would keep total conduction at or below 200 W, with the same total area and thickness.

Question 2 - Solution

A small thermal bridge can dominate a parallel wall

(a) Method and calculation

$$P_i=\frac{.0400(9.90)(20)}{.100}=79.2\,\mathrm W,\quad P_m=\frac{50(.100)(20)}{.100}=1000\,\mathrm W.$$$$P_{\mathrm{total}}=1079.2\,\mathrm W,\quad\frac{P_m}{P_{\mathrm{total}}}=.926612\approx92.66\%.$$

(b) Interpretation and further calculation

$$P=\frac{A\Delta T}{L}[k_i(1-f)+k_mf]=2000[.0400+49.96f].$$$$P\le200\ \Rightarrow\ f\le\frac{.100-.0400}{49.96}=.00120096.$$Only about $.1201\%$ of the area may be metal under these idealized conditions.

Physical check

Area averaging conductivity is valid for this parallel geometry; averaging thermal resistances would describe a different arrangement.