heat radiating upward from roof http://people.csail.mit.edu/jaffer/SimRoof/ThermalOptics

SimRoof: Thermal-Infrared Optics

Although electromagnetic radiation in the 5 μm to 100 μm (infrared) band obeys the laws of physics governing all bands, radiation interacting with matter at the same temperature which emits it has unique constraints. These constraints derive primarily from the the Second Law of Thermodynamics and its corollary, Kirchhoff's Law of Thermal Radiation.

Blackbody Radiation

Spectral density of blackbody radiation at temperatures from 273K to 310K. Planck's Blackbody Formula gives the maximum spectral density of power radiated by a blackbody, "a perfect radiator which absorbs all radiation incident upon it". To the right is a graph of the Planck formula for temperatures spanning 98% of the weather experienced by the 19 tropical locations in the TMY3 data-set.
temperaturetotal radiated power
310 K=37°C=98°F524 W/m2
300 K=27°C=80°F459 W/m2
290 K=17°C=62°F401 W/m2
[a wavenumber (waves per centimeter) scale is used for the abscissa so that equal areas represent equal powers.]

Materials in the Thermal-Infrared

Materials can transmit, reflect (or scatter), or absorb and emit electromagnetic radiation. Most materials absorb and emit room-temperature thermal-infrared radiation.

Materials which transmit most room-temperature thermal-infrared radiation are rare: anhydrous salt crystals and unweathered polyethylene. Even a desert atmosphere obstructs so much thermal-infrared that the cooling powers given above cannot be approached at sea-level.

Electrically conductive metals reflect infrared radiation. Building materials which reflect most thermal-infrared are termed low-emissivity, and are valued for retaining interior heat or rejecting exterior heat.

Semiconductors can be conditionally reflective. If photons with higher than the band-gap energy impinge on a semiconductor, they will be absorbed and generate carriers, thereby causing the (lower-energy) infrared radiation to be reflected. This is the case for all but the highest bandgap semiconductors.

Granular Materials

The preceding discussion applied to pure materials. The bulk material refractive-index won't necessarily hold for the material in a granulated form where the granules are smaller than the wavelength under consideration.

In particular, conductive metals in granular form act as high-IR dielectrics, as detailed in Granular Metal Films. For thermal-infrared applications, granular-metal composites may be the only practical source of high refractive-index dielectrics.

From Optics for Passive Radiative Cooling

The Second Law of thermodynamics dictates that no net radiative energy transfers between two objects at the same temperature.

Thus the blackbody radiative flux through an aperture can be no greater than that produced by a blackbody at the same temperature having that aperture as its surface.

Therefore, blackbody radiation cannot be concentrated by any arrangement of mirrors. The rate of cooling is limited by the smallest aperture through which the blackbody radiation all flows.

Kirchhoff's Law of Thermal Radiation

The Second Law of Thermodynamics dictates that no net radiative energy transfers between two passive objects at the same temperature. Now consider the case of two objects at the same temperature which are separated by a (wavelength-selective) optical bandpass filter.
This system is still passive, so no net radiative energy should transfer between the objects. But the bandpass filter only transmits the radiation within its pass-band. Thus it is also the case that no net radiative energy in any band transfers between two passive objects at the same temperature.

Now suppose that the filter passes radiation only in the 5 μm to 10 μm band, the object on the right is a perfect blackbody, and the object on the left emits only 35% of the (perfect) blackbody radiation in the 5 μm to 10 μm band. The blackbody on the right will emit 100% of the blackbody radiation in that band. If the object on the left absorbs more than 35% of of the impinging radiation in the 5 μm to 10 μm band, then there would be a net transfer of blackbody radiation to the body on the left, which would contradict the finding above. Similarly, if the object on the left absorbs less than 35% of of the impinging radiation in the 5 μm to 10 μm band, then there would be a net transfer of blackbody radiation to the body on the right, which would again contradict the finding above. Thus, in every band, a body must emit the same proportion of thermal radiation as it absorbs.

Infrared Scattering

In the description of the ATRAN program A New Software Tool for Computing the Earth's Atmospheric Transmittance of Near-Infrared and Far-Infrared Radiation [51], S.D. Lord makes no mention of clouds or scattering. The application of ATRAN data by astronomers is probably only for clear viewing conditions.

It seems reasonable that neither Rayleigh scattering nor Mie scattering would be involved in clear (infrared) skies because the air molecules are so much smaller than the thermal-infrared wavelengths.

Infrared Scattering by Water Clouds

Clouds have mean droplet sizes ranging from 6 μm to 14 μm, with the largest sizes occurring "over remote tropical oceans"[53], and the smaller sizes occurring over land, particularly over polluted areas. Infrared scattering in the infrared atmospheric window between 8 μm and 14 μm by water droplets is particularly important to address.

First consider small droplets. The refractive-indexes of water and ice have significant k>0 through their thermal ranges (4.5 μm to 40 μm):
spectral refractive-index of water spectral refractive-index of ice
Waves which are larger than droplets will induce electric dipoles in those droplets. This, and the droplets comprising a low volume fraction of clouds, indicates that Maxwell Garnett Theory should apply. The transmission through a 250 m thick cloud holding 1 g/m3 of water in small droplets is nearly null [a wavenumber scale is used for the abscissa so that equal areas represent equal energies]:
infrared transmission through water cloud infrared transmission through ice cloud
1 g/m3 of water (ice, on right) in a 250 m thick cloud corresponds to 0.25 mm of precipitable-water, a nearly unnoticeable amount if it fell as rain. The transmission through this equivalent 0.25 mm film of water (ice) is the lighter trace in the graph. [The refractive index of ice varies with temperature; but only at wavelengths longer than those of interest.]

Imagine breaking this 0.25 mm=250 μm film into 250 μm diameter droplets distributed uniformly-randomly in a 250 m thick cloud. This cloud has the same precipitable-water as the earlier hypothesized cloud, but in much larger droplets. In the process of scattering infrared light, the droplets will have roughly the same attenuation as the 0.25 mm film. Smaller droplets, such as those occuring in clouds, should have transmission between the two curves, both of which are negligible.

The change in direction of infrared photons by scattering affords opportunities for more scattering and hence even more attenuation.

Because of water's and ice's positive k, scattering by droplets absorbs infrared radiation, making water and ice clouds effective blackbodies for thermal radiation.

In Atmospheric radiation near the surface of the ground: A summary for engineers [52], R. Bliss simply treats clouds as infrared blackbody radiators.

Additional support is found in "Analysis of Visible and Infrared Cirrus Cloud Optical Properties Using High Spectral Remote Sensing,"[54]:

IR Scattering from Mie Model

The infrared scattering cross-section for molecules is much smaller than the absorption cross-section; and extinction in the IR is dominated by absorption such that scattering can be neglected. However, ice crystals and water droplets found in clouds are of similar magnitude or greater in size than wavelengths associated with IR radiation. Although extinction by absorption continues to dominate, scattering of upwelling terrestrial and atmospheric emission from below the cloud will provide a small contribution to the downwelling measured radiance...

Copyright © 2010 Aubrey Jaffer

Radiative Cooling in Hot Humid Climates
agj @ alum.mit.edu
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