Sol-Air Temperature

Sunshine landing on a roof or wall is a heat flow, but a cooling-load calculation is simpler when every heat flow into a surface is driven by one temperature difference. The sol-air temperature turns the sun into a hotter outdoors. It is the outdoor air temperature that would push the same heat into the surface as the real air, the sun and the cold sky do together, so heat can be driven through the wall or roof across the difference between $T_{sa}$ and the temperature on the other side. It is an equivalence worked out from the surface heat balance, not a fitted curve and not a temperature a thermometer reads.

$T_o$ is the shade air temperature, the starting point. $\alpha$ is the solar absorptance, the fraction of the arriving sunlight the surface keeps instead of reflecting: for example 0.9 for a dark roof and 0.3 for a light one. $I$ is the irradiance on the surface plane, for example 800 W/m². The product $\alpha I$ is the sunlight actually absorbed, and it raises the answer. $\Delta Q$ is the extra long-wave heat the surface sheds to the sky, which lowers the answer: zero for a wall, roughly 60 to 70 W/m² for a clear-sky roof. The net heat, $\alpha I - \Delta Q$, is divided by $h_o$, the outside surface heat transfer coefficient (about 17 W/(m²·K) in a 3.4 m/s wind), which turns a heat flux into a temperature rise. A large $h_o$ lets the air carry the heat away easily, so $T_{sa}$ stays close to $T_o$; a small one lets it climb.

Take the sunlit wall: air at 30 °C, $\alpha = 0.6$, $I = 500$ W/m², no sky term and $h_o = 17$ W/(m²·K). The sol-air temperature is 47.647 °C, so for conduction through that wall the outdoors behaves as though it were 47.647 °C, although the shade air is 30 °C. The dark horizontal roof shows the sky term at work: 35 °C air, $\alpha = 0.9$, 800 W/m² and $\Delta Q = 66.3$ W/m² give 73.453 °C.

The result is only as good as its inputs, and the values above are examples, not constants: irradiance depends on where the sun is and what the sky is doing, and $h_o$ on the wind. Use the sky term for surfaces that face the sky, such as a roof, and leave it at zero for a wall. The method is the one in the ASHRAE Handbook: Fundamentals, chapter on nonresidential cooling and heating load calculations.