Temperature distribution and heat radiation of patterned surfaces at short wavelengths

We analyze the equilibrium spatial distribution of surface temperatures of patterned surfaces. The surface is exposed to a constant external heat flux and has a fixed internal temperature that is coupled to the outside heat fluxes by finite heat conductivity across the surface. It is assumed that the temperatures are sufficiently high so that the thermal wavelength (a few microns at room temperature) is short compared to all geometric length scales of the surface patterns. Hence the radiosity method can be employed. A recursive multiple scattering method is developed that enables rapid convergence to equilibrium temperatures. While the temperature distributions show distinct dependence on the detailed surface shapes (cuboids and cylinder are studied), we demonstrate robust universal relations between the mean and the standard deviation of the temperature distributions and quantities that characterize overall geometric features of the surface shape.

Figure 1

Surface patch temperature distribution for models M1, M2, and M3. Colors represent temperature changes from minimum (blue) to maximum (red) temperature. For values see histograms in Figs. 2 to 4 and Table 1.

Figure 2

Histograms for surface patch temperatures of the central unit cell of model M1: (a) temperatures for the three different patch classes vertical ($v$), base ($b$), and top ($t$) for emissivity $\epsilon =0.9$, (b) same as panel (a) for emissivity $\epsilon =0.5$, (c) temperatures for vertical ($v$) patches for emissivity $\epsilon =0.9$, grouped into four different equidistance height classes $H1$ to $H4$ according to their height over the base plane, and panel (d) is the same as panel (c) for emissivity $\epsilon =0.5$.

Figure 3

Histograms for surface patch temperatures as in Fig. 2 for model M2.

Figure 4

Histograms for surface patch temperatures as in Fig. 2 for model M3.

Figure 5

Mean surface temperature (rescaled by the flat surface temperature) as a function of the relative increase $(A_v+A)/A$ in surface area $A=L_x{L}_y$ due to vertical surface patches.

Figure 6

Mean surface temperature (rescaled by the flat surface temperature) as a function of the mean view factor ${\overline{F}}_{\text{b}}\to \text{sky}$ from base surface patches towards the sky.

Figure 7

Standard deviation $\sigma$ of the surface temperature distribution, rescaled by the difference $\overline{T}-T_{\text{flat}}$, as a function of the ratio of vertical surface area $A_v$ and surface area $A_g$ covered by patterns (cuboids, cylinders). Data collapse is observed for different emissivities.