Question

An evaporator is insulated using glass wool material of \(0.15\ \text{m}\) thickness. The inner most surface and the outer surface of the insulation are at \(700^{\circ}\text{C}\) and \(80^{\circ}\text{C}\), respectively. The mean thermal conductivity of the glass wool under these conditions is \(0.29\ \text{W}\,\text{m}^{-1}\,\text{K}^{-1}\). The rate of heat loss (in W) through \(1.2\ \text{m}^2\) of the evaporator wall surface (rounded off to the nearest integer) is ________________.

Hint:

For flat insulation layers, use \(R = L/(kA)\) and \(\dot Q = \Delta T/R\). With multiple layers, thermal resistances add in series.
Temperature difference is in kelvins or degrees Celsius—use the magnitude; direction is from hot to cold.
Always check dimensional consistency: \(k[\text{W/mK}]\,A[\text{m}^2]\,\Delta T[\text{K}] / L[\text{m}] \Rightarrow \text{W}\).

Answer 1

Correct Answer: 1437

Step 1: For steady 1-D conduction through a plane layer, Fourier’s law gives \[ \dot Q = k\,A\,\frac{\Delta T}{L}. \] Here, \(k=0.29\ \text{W}\,\text{m}^{-1}\,\text{K}^{-1}\), \(A=1.2\ \text{m}^2\), \(L=0.15\ \text{m}\), and \(\Delta T = 700-80=620\ \text{K}\).
Step 2: Substitute and compute: \[ \dot Q = 0.29 \times 1.2 \times \frac{620}{0.15} = 0.348 \times 4133.\overline{3} = 1438.0\ \text{W}\ (\text{approximately}). \] Step 3: Rounding to the nearest integer gives \(\boxed{1438\ \text{W}}\).