heat flux

[logistic_guy]

The thermal conductivity of a sheet of rigid, extruded insulation is reported to be \(\displaystyle k = 0.029 \ \text{W/m} \cdot \text{K}\). The measured temperature difference across a \(\displaystyle 20\)-\(\displaystyle \text{mm}\)-thick sheet of the material is \(\displaystyle T_1 - T_2 = 10^{\circ}\text{C}\).

\(\displaystyle \bold{(a)}\) What is the heat flux through a \(\displaystyle 2 \ \text{m} \times 2 \ \text{m}\) sheet of the insulation?
\(\displaystyle \bold{(b)}\) What is the rate of heat transfer through the sheet of insulation?

 

[logistic_guy]

\(\displaystyle \bold{(a)}\) What is the heat flux through a \(\displaystyle 2 \ \text{m} \times 2 \ \text{m}\) sheet of the insulation?

The heat flux is given by Fourier's law.

\(\displaystyle q''_x = -k\frac{dT}{dx} = k\frac{\Delta T}{L} = 0.029\frac{10}{0.02} = \textcolor{blue}{14.5 \ \text{W/}\text{m}^2}\)

 

[logistic_guy]

\(\displaystyle \bold{(b)}\) What is the rate of heat transfer through the sheet of insulation?

The rate is given by:

\(\displaystyle q_x = q_x''A = 14.5 \times 2 \times 2 = 58 \ \text{W} = \textcolor{blue}{58 \ \text{J/s}}\)