Notes on the radiation spectrum for solar heating and sky cooling
by Stephen Hewitt | Published | Last updated
This article summarises well-known information that could be useful for solar heating.
Figure 1 shows a graph of the spectrum of radiation from the sun, as measured on earth, from a technical report prepared for USA government's Department of Energy.
Figure 2 is a graph from a 2012 science paper  showing a terrestrial solar spectrum from 350nm to 1 μm wavelength, measured in Egypt at several times times of day and looking somewhat different to the corresponding range in Figure 1.
Planck's radiation law
Planck's law describes the power distribution across the spectrum of radiation from a black body at a given absolute temperature. It is an expression for the radiated power at a given wavelength per wavelength interval per unit solid angle per unit surface area of the emitting black body.
Using the symbols defined in Table 1:
Bλ = 2hc2/(λ5(exp(hc/λkT) - 1))
Graphs in this article are all in terms of wavelength, but of course Planck's law could equally be expressed as density per frequency (ν) interval in which case it would be:
Bν = 2hν3/(c2(exp(hν/kT) - 1))
These are density functions, meaning that the value we are actually interested in is obtained by taking the area under the curve between the two wavelengths delimiting the band of interest.
Figure 3 shows plots of this function at three different temperatures. This figure is intended to help visualisation of the range of wavelengths emitted for example by a solar heat receiver at 90°C.
|c||Speed of light||~3.00x108||m/s|
The total power emitted in radiation per unit surface area of a black body is proportional to T4, where T is the absolute temperature of the body.
E' = σT4
where σ is the Stefan-Boltzmann constant ~5.67 x 10-8 W/(m2K4)
Since the Stefan-Boltzmann law gives the total radiation over the whole spectrum, it can be derived by integrating Planck's expression over all wavelengths or frequencies. I have not attempted this but in April 2023 a way to do it was described in a PDF file at https://people.tamu.edu/~kevinkrisciunas/planck.pdf (How to Integrate Planck’s Function, Kevin Krisciunas, SHA256 hash 82ea9dce1c4842e9c6ec3312ea408b048cefac5a43e0693d0931614956da879a.)
The result is the expression above where σ = 2π5k4/(15c2h3)
A real body emits less than the ideal black body:
E' = εσT4
where ε is the emissivity of the body, and 0 < ε< 1
The absorptivity α of a body is the fraction of incident radiation that is absorbed rather than reflected (ignoring transparency for now).
From the literature it seems that to avoid violating the laws of thermodynamics (which are very strict) bodies are required at all times to have ε = α .
However bodies are legally allowed to vary both their emissivity and absorptivity (providing they treat them equally and do not discriminate) with the wavelength of the radiation (and its angle of incidence). Humans have exploited this legal loop-hole by inventing selective coatings, as noted in the next section.
In the case of the solar kettle with the bare copper pipe described in a previous article the Stefan-Boltzmann law allows us to put an upper bound on the amount of radiant heat loss from the pipe because we know that ε < 1. This is the maximum that it might be possible to save if we could stop the pipe radiating at all, for example with a selective coating.
The 22mm pipe is 111cm long, so its total surface area is 0.022 x 1.11 x π
With the pipe at 90°C (387 K) if it were an ideal black body then
radiant heat emitted = σ x 3874 x 0.022 x 1.11 x π
~ 68 W
This is a significant quantity compared with the calculated net heating power of the kettle at 90 °C, as shown in Figure 7 of the earlier article Higher net power from a solar kettle heating water in glass rather than copper . What fraction of this figure is actually radiated depends on the emissivity of the copper pipe. This figure is more than the radiant heat lost to the environment, because some radiant heat will be received back from the environment. Only if the environment was at absolute zero would this much heat be lost by radiation.
The principle of a selective surface is that although emissivity and absorptivity are equal for a given material they can be dependent on wavelength. The idea for solar heating is therefore to find something that has a high emissivity at wavelengths of the main energy in the solar spectrum but low emissivity at the longer wavelengths emitted at the temperature of the heat collector.
A 2002 review of selective coatings , although oriented towards coatings that can withstand high temperatures used in solar parabolic trough electricity generation, explains the different kinds and also mentions “black copper” and “black chrome” (see below).
Selective surfaces on copper
For the solar kettle with a copper pipe, it seems from published reports as though a particular kind of copper oxide layer might be effective.
A report with the title “An improved copper oxide selective coating process” “Devices and Services Co” was available in January 2023 at https://devicesandservices.com/TechNotes/TN80-1.pdf (SHA256 hash of this file a2e00f313b328fb788787aed7990930eeafd3e1f120fc95307d7e3983f000753)
This report, dated 1980, says “Copper oxide on copper selective absorbers have been in use for many years, particularly in Australia for hot water heaters”. It mentions immersion of the copper in a hot solution of sodium hydroxide and sodium chlorite to produce a copper oxide layer. One of the cited references for the composition of such a solution is dated 1962.
There are two kinds of copper oxide: CuO “cupric oxide” and Cu2O “cuprous oxide”. The report is not explicit about which one is formed, but a label on one of its figures mentions “the standard CuO coating”.
The report and others use the terms “absorptivity” evidently meaning absorptivity and emissivity at solar wavelengths, and “emissivity” evidently meaning absorptivity and emissivity at the longer wavelengths emitted by the heat collector. For these, it gives figures of 0.90 and 0.15 respectively after the traditional process of forming the copper oxide. It reports corresponding figures of 0.93 and 0.04 for the improved process described.
For comparison, a 2019 article says “State-of-the-art absorptivity of a selective coating has been improved to higher than 0.95, whereas emissivity has been reduced to lower than 0.05.” 
It seems as though the process described with chemicals like sodium hydroxide and sodium chlorite might be within the reach of a home constructor.
In addition to this chemical process it is evident from later publications that copper oxide might also be created on copper by electrodeposition.
A 2014 conference paper reports experiments to do this. Analysing samples electrodeposited under various experimental conditions it reports: “As the oxygen content of our sample in very close enough to the required amount of oxygen to become cupric oxide we can say that it may be cupric oxide.” 
Amongst its other references is a 2006 PhD thesis called “Preparation and characterization of properties of electrodeposited copper oxide films”, which explicitly covers both forms of oxide .
However, with electrodeposition, it might be no harder to deposit the oxide of a different metal onto copper. Two that seem to be mentioned in past literature on selective coatings are “black nickel” and “black chrome”. A 1979 conference proceedings mentions these and other possible selective coatings. .
There are materials such as glass (below) that are transparent at some wavelengths and opaque at others.
Figure 4 shows glass blocking completely radiation with wavelengths above 4.5 μm
Figure 3 illustrates what this might mean for a for a solar heat receiver at 90 °C under a glass cover. The area under the 90°C curve below a wavelength of 4.5 μm is tiny compared with the total area under the curve. It is apparent that only a very small fraction of the total radiation emitted would go directly through the glass.
Unlike glass, polythene and other plastic films tend to transmit longer wavelengths.
This could make them useful as part of a night-time cooling system that radiates heat to the sky. Figure 5 is taken from a scientific paper related to night-time cooling and shows transmittance spectra of various films.
Another idea is a covering on a domestic radiator to reduce heat loss by convection, where only radiative heating is wanted.
In 2022, a supplier of agricultural polytunnels on the web was advertising:
“Thermic films allow short wave infra red from the sun to heat up the soil and contents of the tunnel. When the air temperature drops the hotter objects radiate the heat back again as long wave infra red. What a thermic film does is reflect this heat wavelength back to the ground/contents again, reducing the heat lost through the film. Until about 15 years ago a thermic film would reflect back into the structure about 65% of this heat. With the introduction of SteriLite Premium this was increased to 80-85% reflection. Now with SunMaster SuperThermic this is boosted to a massive 94-95% reflection/heat saving”
Simple Solar Spectral Model for Direct and Diffuse Irradiance on Horizontal and Tilted Planes at the Earth's Surface for Cloudless Atmospheres, R. Bird and C. Riordan, Solar Energy Research Institute, December 1984.
In March 2023 this was at https://www.nrel.gov/docs/legosti/old/2436.pdf as a PDF with SHA256 6eb32071407ab66743b334f5b29f3d162c4a9409252c923ccf9188ebb69eeef2
Studying the effect of spectral variations intensity of the incident solar radiation on the Si solar cells performance, A. E. Ghitas, Article in NRIAG Journal of Astronomy and Geophysics, December 2012
Aging Study of Plastics to Be Used as Radiative Cooling Wind-Shields for Night-Time Radiative Cooling—Polypropylene as an Alternative to Polyethylene, Ingrid Martorell, Jaume Camarasa, Roger Vilà , Cristian Solé and Albert Castel, Energies, 2022.
In April 2023, this was available at https://www.mdpi.com/1996-1073/15/22/8340 as a PDF with SHA256 hash 4ce786851814b233a3579b0923e2bf96a19a116f42efeadcce63ad6f4b3aa95a
Optical properties of soda lime silica glasses, M. Rubin, Solar Energy Materials, Volume 12, Issue 4, September–October 1985
The graph shown here was extracted from a version dated 1984 which in April 2023 was at https://escholarship.org/content/qt8vm0x7jd/qt8vm0x7jd.pdf?t=p0wsns with SHA256 hash aaf2d166a3cdef8bf7622431349853af32d24d1269902a98b3c240cdade71943 linked from https://escholarship.org/uc/item/8vm0x7jd while the journal article was embargoed behind publisher Elsevier's paywall.
Novel Methods to Harness Solar Radiation for Advanced Energy Applications Yin Gao,Ziman Wang, Ding Ding, Wenjia Li, Yaoguang Ma, Yong Hao, and Hang Zhang, ES Energy and Environment, 2019
In February 2023 this was available as a PDF file with SHA256 hash 31d9d1be42285e0b58a5ce86480997d6d00bbed389cf26ecc02c0bef6b4e5751 at https://www.espublisher.com/uploads/article_pdf/esee8c328.pdf
Electrodeposition and Characterization of Copper Oxide Thin Films for Solar cell Applications, A.S.M. Sayem Rahman et al, Procedia Engineering 105 (2015)
In April 2023 this was available as a PDF file with SHA256 hash 678acf9c6c9c71dc1bfed56c365a61cae169259f2294960c42d869a4f0544cf3 at https://core.ac.uk/download/pdf/82680242.pdf
Preparation and characterization of properties of electrodeposited copper oxide films, Longcheng Wang, PhD thesis, University of Texas, 2006.
In April 2023 this was available as a PDF file with SHA256 hash 1fb0c190b22afb14c305397ac7d1149b9881aab947d0e9fe45ab947cb964387d at https://rc.library.uta.edu/uta-ir/bitstream/handle/10106/668/umi-uta-1592.pdf
Proceedings Second Annual Conference on Absorber Surfaces for Solar Receivers, USA government's Department of Energy, 24,25 January 1979
In February 2023 this was available as a PDF file with SHA256 hash 1569745fd99b0282645806bc70a4dc49c2c7880f0ef8c4ffd5c69a5db2e6ca93 at https://www.nrel.gov/docs/legosti/old/182.pdf
Review of Mid- to High Temperature Solar Selective Absorber Materials, C.E. Kennedy, National Renewable Energy Laboratory, USA, July 2002
In April 2023 this was available as a PDF file with SHA256 hash 429634fcdd591ff3024bc37a2e0334afbf02e3e59620b29f24f180229c0c3a10 at https://digital.library.unt.edu/ark:/67531/metadc1394743/m2/1/high_res_d/15000706.pdf