Chapter 4, Problem 4.51P

An ancient myth describes how a wooden ship was destroyed by soldiers who reflected sunlight from their polished bronze shields onto its hull, setting the ship ablaze. To test the validity of the myth. a group of college students are given mirrors and they reflect sunlight onto a $100\text{mm×100}\text{mm}$ area of a $t=10$ -mm-thick plywood mockup characterized by $k=0.8\text{W/m}\cdot \text{K}\text{.}$ The bottom of the mockup is in water at $T_w=20°\text{C,}$ while the air temperature is $T_{\infty }=25°\text{C}\text{.}$ The surroundings are at $T_{\text{sur}}=23°\text{C}\text{.}$ The wood has an emissivity of $\epsilon =0.90;$ both the front and back surfaces of the plywood are characterized by $h=5{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ The absorbed irradiation from the N students' mirrors is $G_{S,N}=70,000{\text{W/m}}^{\text{2}}$ on the front surface of the mockup.

1. A debate ensues concerning where the beam should be focused, location A or location B. Using a finite difference method with $\Delta x=\Delta y=100\text{mm}$ and treating the wood as a two-dimensional extended surface (Figure 3.17a), enlighten the students as to whether location A or location B will be more effective in igniting the wood by determining the maximum local steady-state temperature.
2. Some students wonder whether the same technique can be used to melt a stainless steel hull. Repeat part (a) considering a stainless steel mockup of the same dimensions with $k=15\text{W/m}\cdot \text{K}$ and $\epsilon =\mathrm{0.2.}$ The value of the absorbed irradiation is the same as in part (a).