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Abstract
Passive radiative cooling has had a renaissance in energy consumption, emission reduction, and environmental protection over the past two decades. Ultimate absorptivity determines the cooler’s performance, so the ideal absorptivity is the target for designing passive radiative coolers. In this paper, we systematically analyzed passive radiative cooling, including angle-dependent and wavelength-dependent thermal radiative power Prad, absorption power from the ambient Patm, their power difference Pdiff, absorption power from the sun Psun and thermally conductive and convection power Pcc. During the analytical process, we show the key factors of cooling and analyze the ideal absorptivity of radiators in four conditions. The analytical progress and results will give a reference to the design of the radiator in the future.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The outer space at a temperature of 3 K is an ultimate heat sink. Passive radiative cooling can cool objects to low temperatures, even below ambient, by radiating excess heat to the outer space through atmospheric transparency windows without any energy input. This plays a vital role in energy saving, emission reduction, and environmental protection due to its passive, effective and renewable properties. Therefore, passive radiative cooling attracts great attention in various applications such as buildings [1–3], temperature-insensitive optoelectronic devices [4–6], textile for body thermal management [7,8], thermal power plants [9–11] and so on.
The applications of passive radiative cooling in daily life can be traced to several centuries ago, but the scientific and systematical investigation was started by Head [12] and Catalanotti [13] in the 1960s. Some researches devoted to investigating the fundamental principles [14–16] such as the selectivity of wavelength, angle dependence, location, air pressure, and so on. Some researches devoted to exploring new materials for passive radiative cooling, such as polymer-based materials [17], inorganic film [18–20], and gas slab [21]. However, earlier works are limited to nighttime cooling because it is difficult to simultaneously realize high reflectivity in the solar spectrum and high emissivity in the atmospheric transparency window.
Benefited from the development of nanophotonics and metamaterials technology, passive radiative cooling has had a renaissance since the Raman group presented a metal-dielectric photonic structure theoretically in 2013 [22]. They introduced seven layers of HfO2 and SiO2 photonic structure experimentally in 2014 [23], which reflects 97% of incident sunlight while emitting strongly and selectively in the atmospheric transparency windows between 8 µm and 13 µm. Now, nanoparticle-based structures [24–27], multilayer film [23,28,29], and patterned surfaces [22,30–33] have been proposed to realize daytime radiative cooling and textile for human thermal management which works at a temperature below ambient, thermophotovoltaics, solar cells and other optoelectronic systems which work at a temperature above ambient. Many groups reviewed the state of the art in passive radiative cooling [3,4,26,34–43] including fundamental physics, materials, structures, applications, and prospects. For solar energy devices such as solar cells, photovoltaic and thermophotovoltaic systems, the radiator should be transparent in the solar spectrum. For other solar-useless devices, such as daytime radiative coolers, textiles, detectors, and so on, the radiator should be a reflecting mirror in the solar spectrum to reduce solar absorption. There are also some researchers using optimization algorithms to optimize radiative coolers [44–46]. However, the investigation on the ideal radiative cooler considering the second atmospheric transparency window (17 µm-24 µm) was rare [22,26,37,47–49]. The transmission of the second atmospheric window (17 µm-24 µm) heavily depends on geographical location and conditions, and we use the atmospheric transmission from Ref. [50] in our work. The atmospheric window between 16 µm and 25 µm might be harnessed for additional radiative cooling [26,51].
Therefore, in this paper, we systematically analyze the ideal absorptivity for different conditions by comparing the radiative cooler worked at different wavelengths and angles. The wavelength result shows that radiation power can be improved by about 30% considering the second transparent atmosphere window is at sub-ambient conditions. Angle analysis shows that the passive cooling radiators emit the heat to outer space most at 45 degrees. The detailed analyses will make people understand more clearly about passive radiative cooling and give a reference to future investigation.
2. Theoretical analysis
A daytime cooler at temperature Tc with angular spectral absorptivity A (λ,θ) is surrounded by air at a temperature of Tatm with angular spectral absorptivity Aatm(λ,θ). Figure 1 shows the schematic and heat transfer model of daytime passive radiative cooling. The net radiative cooling power ${P_{\textrm{net}}}$ of the daytime coolers can be expressed as the following formula:
Fig. 1. Schematic and heat transfer model of daytime passive radiative cooling. Prad is the thermal radiative cooling power by radiating heat energy to outer space at 3 K, Patm is the heating power absorbed from the atmospheric thermal radiation, ${P_{\textrm{sun}}}$ is the heating power absorbed from solar irradiance, and ${P_{\textrm{cc}}}$ is the heating power absorbed through thermal conduction and thermal convection between cooler and ambient.
The angular spectral absorptivity can be regarded as angular spectral emissivity [42]. Therefore, the angular spectral emissivity ɛ(λ,θ) of the daytime cooler at a temperature ${T_c}$ is equal to A(λ,θ).$\; $In Eq. (1), the power Prad(Tc) radiated to outer space can be calculated by:
Thus, Eq. (3) can be expressed by:
There are two main atmospheric transparency windows in 0.3 µm - 40 µm, which distribute in the range 8 µm - 13 µm and 17 µm - 24 µm. Table 1 lists the absorptivity bands of six types of coolers and the radiated and absorbed thermal radiation power values, which are also shown in Fig. 2(a). For these coolers, they have unity absorptivity in the listed bands and zero absorptivity outside the bands. Cooler 1 has unity absorptivity in 8 µm - 13 µm and has zero absorptivity out of the band. Cooler 2 has unity absorptivity in 8 µm - 13 µm and 17 µm - 24 µm and has zero absorptivity at the other wavelengths. Cooler 3 has unity absorptivity in 8 µm - 24 µm and has zero absorptivity at the other wavelengths. Cooler 4 has unity absorptivity in 8 µm - 40 µm, Cooler 5 has unity absorptivity in 4 µm - 40 µm, and Cooler 6 has unity absorptivity in 0.3 µm - 40 µm, while the absorptivity at the other wavelengths are zero.
Fig. 2. (a) Absorptivity of six types of absorbers, labeled as Cooler 1, Cooler 2, Cooler 3, Cooler 4, Cooler 5 and Cooler 6. (b) Thermal radiative cooling power Pλ at ${T_c} = $300 K. (c) Thermal radiative cooling power Pθ at ${T_c} = $300 K. (d) Thermal radiative cooling intensity Prad at different ${T_c}$.
Table 1. Properties of the six types of coolers
Figures 2(b) and 2(c) show Pλ and Pθ of the six types of coolers at ${T_c}\, = $ 300 K, which indicates the radiative intensity increases as the absorptivity band increases. Figure 2(d) shows the radiative intensity Prad of the six types of coolers at different working temperatures, which increases as the working temperature increases. At 300 K, the radiative power Prad of the coolers, labeled as Cooler 1, Cooler 2, Cooler 3, Cooler 4, Cooler 5 and Cooler 6, are 148 W/m2, 227 W/m2, 311 W/m2, 369 W/m2, 432 W/m2, and 433 W/m2, respectively. Cooler 6 has the biggest thermal radiative power and is just 1 W/m2 bigger than Cooler 5, which means the absorptivity of the radiator in 0.3 µm - 4 µm has little effect on Prad(Tc).
Similarly, Patm(Tatm) is calculated by:
Figure 3(a) shows atmospheric transmission at column water vapor 1.0 mm and air mass 1.5. In the atmospheric transparency windows, ɛatm(λ,θ) is near zero, thus ɛ(λ,θ)$\; $can be as big as possible without increasing Patm(Tatm). Figure 3(b) shows the absorptivity of the atmosphere [50] increases in the whole spectra as the incident angle changes from 0 to 89 degrees. When θ increases, ɛatm (λ,θ) in the atmospheric transparency windows increases which means the atmospheric transparency windows disappear gradually. Figure 3(c) shows the wavelength and angle-dependent thermal absorption power Pλθ_atm of Cooler 6 at Tatm = 300 K, which indicates the absorbed power Pλθ_atm localized in 4 µm - 8 µm and 13 µm - 17 µm. Figure 3(d) shows the thermal absorption power Patm of the six coolers increases as the ambient temperature increases. At 300 K, the absorption power Patm of Cooler 1, Cooler 2, Cooler 3, Cooler 4, Cooler 5 and Cooler 6 are 22 W/m2, 64 W/m2, 135 W/m2, 189 W/m2, 240 W/m2, and 240 W/m2, respectively. Cooler 6 has the biggest thermal absorption power and is the same with Cooler 5, which indicates that the absorptivity of the radiator in 0.3 µm - 4 µm has a small impact on Patm.
Fig. 3. (a) Absorptivity of the atmosphere [50] in the zenith direction in the wavelength range from 0.3 µm to 40 µm. (b) Absorptivity of the atmosphere [50] when the incident angle changes from 0 degrees to 89 degrees. (c) Wavelength and angle-dependent thermal absorption power Pλθ_atm of Cooler 6 at 300 K ambient temperature. (d) Thermal absorption intensity Patm of the six coolers at different ambient temperatures Tatm.
To maximize Pnet, we should maximize ${P_{\textrm{diff}}} = {P_{\textrm{rad}}} - {P_{\textrm{atm}}}$. Figures 2 and 3 exhibit Prad and Patm have similar variation tendencies as the angle, wavelength, and temperature changes. To analyze the limitation of the passive radiative cooling, Fig. 4(a) shows Pdiff_λ of Cooler 6, at different Tc by setting Tatm=300 K. Pdiff_λ increases as Tc increases. Figure 4(b) shows Pdiff_λ at Tc = 260 K, 300 K, and 340 K in Fig. 4(a). Pdiff_λ is near zero before 4 µm when Tc is equal and smaller than 300 K, increases and blue-shift when Tc is bigger than 300 K and increases. Pdiff_λ trends to zero after 30 µm. Pdiff_λ is always positive in the wavelength range between 4 µm and 30 µm when Tc is bigger than 300 K, is positive in the atmospheric windows between 8 µm and 13 µm and between 17 µm and 24 µm when Tc is equal to 300 K and is partial positive in the two atmospheric transparency windows when Tc is smaller than 300 K. As Tc decreases, there is a cutoff temperature where Pdiff_λ is only positive in the wavelength range between 8 µm and 13 µm, and another cutoff temperature where Pdiff_λ is always negative in the whole spectrum. That is if the radiator works at a higher temperature than the air, the radiator's best working wavelength is full-wave band, especially from 4 to infinity. If the radiator works at a lower temperature than the air, the radiator best working wavelength is between 8 µm and 13 µm and between 17 µm and 24 µm. Figure 4(c) shows the wavelength integrated angle-dependent thermal net radiative cooling power Pdiff_θ of Cooler 6 at different ${T_c}\; $when Tatm = 300 K, which shows that the radiative cooler has the biggest Pdiff_θ around 45 degrees. As Tc decreases, Pdiff_θ decreases from positive to negative.
Fig. 4. (a) Pdiff_λ of Cooler 6 at different Tc when Tatm=300 K. (b) The lines in (a) at three different Tc of 260 K, 300 K, and 340 K. (c) Pdiff_θ of Cooler 6 at different ${T_c}\; $when Tatm = 300 K.
Figure 5(a) shows the Pdiff_θ of Cooler 2 has the best performance at Tc = 260 K and Tatm=300 K. Figures 5(a) and 5(d) indicates the ideal absorptivity bands cover the two atmospheric transparency windows between 8 µm and 13 µm and between 17 µm and 24 µm if the radiators work at a lower temperature than air. Figures 5(b) and 5(c) show the Pdiff_θ of Cooler 6 has the biggest value if the radiators work at a higher temperature than air, which means the ideal absorptivity in above-ambient applications should cover all the wavelength when no considering sunlight. Cooler 5 has almost the same value as Cooler 6, which indicates the absorptivity in 0.3 µm - 4 µm has little impact on radiative power. Therefore, if the radiators work at a higher temperature than air, the ideal cooler is Cooler 6, and the sub-ideal is Cooler 5 when no considering sunlight. When the radiator is exposed to the sun, the ideal is Cooler 5 to avoid heating by the sun. Table 2 list Pdiff at different working temperatures and highlight the maximum value with bold.
Fig. 5. (a) At Tc = 260 K and Tatm=300 K, the difference between Pθ and Pθ_atm, labeled as Pdiff_θ. (b) Pdiff_θ at Tc = 300 K and Tatm=300 K. (c) Pdiff_θ at Tc = 340 K and Tatm=300 K. (d) Pdiff as Tc increases when Tatm=300 K.
Table 2. The Pdiff at different Tc when Tatm = 300 K
${P_{\textrm{sun}}}(T )$ is calculated by:
Assuming the absorptivity of radiators in solar spectrum 0.3 µm - 4 µm is 3%, Figs. 8(a) and 8(b) show the net radiative cooling power Pnet as Tc increases with Tatm=300 K and hc is equal to 0 and 6 W/m2/K, respectively. Compared to Fig. 5(d), Fig. 8(a) overall decreases due to the 3% sun radiation absorption, and Fig. 8(b) has sharper lines that indicate the thermally conductive and convection is beneficial to passive cooling in above-ambient applications but is harmful to passive cooling in sub-ambient applications.
Fig. 6. Solar spectrum of AM1.5 [53] in 0.3 µm - 4 µm. The part in 2.5 µm - 4 µm is enlarged in the inset.
Fig. 7. Thermal conductive and convection power as the temperature difference ${T_{\textrm{atm}}} - {T_\textrm{c}}$.
Fig. 8. (a) In the daytime, net radiative cooling power Pnet as Tc increases when Tatm=300 K, assuming the absorptivity of radiators in solar spectrum 0.3 µm - 4 µm is 3% and hc = 0. (b) In the daytime, net radiative cooling power ${P_{\textrm{net}}}$ as Tc increases when Tatm=300 K, assuming the absorptivity of radiators in solar spectrum 0.3 µm - 4 µm is 3% and hc = 6 W/m2/K. (c) The ideal absorptivity in above-ambient and sub-ambient applications with or without exposure to sunlight.
3. Ideal absorptivity discussion
3.1 Above-ambient passive radiative cooling
For the terrestrial cooler, radiative cooling can reduce a system’s temperature without energy input. When Tc is bigger than Tatm, Pdiff_λ is always positive in the whole spectrum, as seen in Figs. 4(a) and 4(b). Cooler 6 has the biggest Pdiff and Cooler 1 has the smallest Pdiff, as shown in Fig. 5(d). Therefore, for the nighttime or the condition unexposed to sunlight, the ideal absorptivity spectrum for above-ambient cooling is Cooler 6, which has unity absorptivity in the whole spectrum, as shown in Fig. 8(c) in red circles.
For the daytime cooling, the intensity of sunlight is about 1000 W/m2. To avoid heating by the sun, the coolers should have high reflectivity before 4 µm and have high emissivity after 4 µm especially in 4 µm - 40 µm, simultaneously. The ideal absorptivity of above-ambient coolers with exposure to sunlight is shown in Fig. 8(c) in blue circles.
3.2 Sub-ambient passive radiative cooling
When Tc is smaller than Tatm, Pdiff_λ is partially positive in atmospheric transparency windows and negative at the other wavelengths, as shown in Figs. 4(a) and 4(b). If daytime radiator works at a lower temperature than air, the daytime cooler can radiate heat energy to air through the two atmospheric windows 8 µm - 13 µm and 17 µm - 24 µm. As shown in Table 2, when Tc = 280 K and Tatm = 300 K, Cooler 2 has the biggest Pdiff which shows 30% radiative power improvement against Cooler 1. The ideal cooler for the daytime cooling in sub-ambient applications is Cooler 2, that is the ideal absorptivity should have high emissivity in the atmospheric windows between 8 µm and 13 µm and between 17 µm and 24 µm, and have high reflectivity at other wavelengths, simultaneously, as shown in Fig. 8(c) in green squares.
For the nighttime or the condition unexposed to sunlight, the ideal absorptivity spectrum for sub-ambient passive radiative cooling is also similar to the daytime, except the absorptivity before 4 µm has little impact on cooling, which can be any value between 0 and 1. The ideal absorptivity is shown in Fig. 8(c) in the black circles. The black line before 4 µm is not shown in Fig. 8(c), which means the absorptivity before 4 µm can be any value between 0 and 1 because it has little impact on cooling.
4. Conclusion
In conclusion, through systematically analyzing the ideal cooler, we find the key indicator of a radiator is the equilibrium temperature or the temperature difference between radiator and ambient. The key design criteria is the ideal absorptivity which is impacted by the working temperature below or above ambient, the air condition and the possible exposure to sunlight. We give the ideal absorptivity in Fig. 8(c) for above-ambient or sub-ambient with or without exposure to sunlight. The atmospheric transparency windows in 17 µm - 24 µm are beneficial to sub-ambient, which can improve the cooling power by about 30%. Angle analysis shows that the passive cooling radiators should be angle-independent with the maximum cooling power at 45 degrees. The detailed analyses will make people understand more clearly about passive radiative cooling and give a reference to the future investigation.
Funding
National Natural Science Foundation of China (61504078); China Postdoctoral Science Foundation (2015M571545).
Acknowledgments
The authors thank Jun Yin for technical supporting.
Disclosures
The authors declare no conflicts of interest.
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