Open Access
Add to BibSonomy
Abstract
Concentrated radiative cooling, an analogous concept of the concentrated solar power technology, has the potential of amplifying both the cooling power and the temperature reduction. However, concentrators have not yet been systematically optimized. Moreover, a widely used theoretical approach to analyze such systems has neglected a fundamental constraint from reciprocity, which can lead to an overestimate of cooling performance and unclarified limits of amplification factors. Here we develop a theoretical framework addressing these shortcomings. Modeling suggests the optimized shape and geometric dimensions of concentrators, as well as the limiting cooling power and temperature reduction. Using an electroplated Al2O3 emitter and an optimized conical concentrator, we experimentally amplify the nighttime radiative cooling by 26%.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The concentrated solar power (CSP) system uses mirrors or lens to concentrate a large area of sunlight onto a receiver to substantially increase its temperature. For example, while unconcentrated sunlight can heat flat plates to 200C to 300C [1,2], receivers under concentrated sunlight can reach 550C to 750C [3–5], which promises greater efficiency and output power. CSP is widely seen as an important component in the quest to power a sustainable future [6–8].
Recently, outer space, which sits at a temperature of -270C , has also been attracting renewed interest for energy harvesting [9–15]. It is basically a “mirror image” resource of the sun from the viewpoint of thermodynamics, which indicates that increasing/decreasing the temperature of the heat source/sink are equally important. Therefore, it is natural to analyze the analogous concentrated radiative cooling technology, which uses mirrors to restrict the solid angle of the radiative cooler to around the zenith direction where the atmosphere is most transparent [16–19].
Motivated by Trombe’s work [20], Smith introduced the concept of amplified radiative cooling, and theoretically predicted the performance of a point emitter equipped with a truncated conical concentrator [16]. Experimentally, Chen et al. applied this idea, as well as sophisticated photonic and thermal designs, to achieve a new record of radiative cooling [17]. Gentle et al. employed a 3D printed parabolic concentrator to amplify the cooling performance of a blackbody emitter [21]. Peoples et al. [19] and Zhou et al. [22] also explored concentrated radiative cooling. The former nested a cooling pipe in a mid-infrared reflecting trough, while the latter mounted a vertically-oriented planar emitter in a V-groove mirror that is also spectrally selective. In both of these works, concentration results in the doubling of the effective radiating area of the emitters [19,22]. In other related works, Zhou et al. designed a tapered waveguide structure to suppress the solar irradiation and the thermal radiation from surroundings, which benefits the daytime cooling although weakens the nighttime cooling due to the reduction of solid angles [23]. Haechler et al. adapted the cooling concentration idea for dew-harvesting enhancement [18].
In spite of the extensive literatures, there are fundamental issues about concentrated radiative cooling that have not been fully addressed. In particular, as we will discuss in detail in this paper, the initial theoretical analysis [16], subsequently taken up in other literature [18,24,25], in fact does not fully take into account the constraint on the performance of these systems as imposed by reciprocity. This result therefore overestimated the theoretical performance of the concentrated radiative cooling system.
This paper is organized as follows. First we highlight the constraint from reciprocity on concentrated radiative cooling. Then we develop a mathematical framework to optimize the geometric dimensions of parabolic and conical concentrators to maximize the nighttime radiative cooling. Fundamental limits are pointed out by combining these concentrators with various emitters under limiting climate conditions. We also quantify the errors of previous approaches that had neglected to take into account this key reciprocity constraint. Finally, experiments with conical concentrators and an electroplated Al2O3 emitter are presented, which validate our theoretical framework and the results from optimization.
2. Theory of concentrated radiative cooling
Figure 1(a)-(b) depicts the geometry of the concentrated radiative cooling system. An emitter is exposed to the sky horizontally. A concentrator with highly reflective inner surfaces is equipped to surround the emitter to restrict its solid angle to around the zenith direction. This system exploits the fact that the atmosphere is more transparent around the zenith direction. Or equivalently, there are more downward radiations from the atmosphere along the directions away from the zenith direction. As a simple picture that explains the operation of the concentrator, the concentrator blocks the downward radiation from the atmosphere away from the normal incidence direction from reaching the emitter. Therefore, the use of the concentrator results in the enhancement of the cooling power of the emitter, as compared to the cooling power of the same emitter in the absence of the concentrator.
Fig. 1. Physical insight of amplifying radiative cooling by a parabolic concentrator. (a) Oblique atmospheric incidence is blocked. (b) Reciprocity requires an additional normal incidence (red), which could not arrive at the emitter if there were no concentrator. (c) Directional mean atmospheric transmittance (${\bar \tau _{8 - 13\mu m}}$; solid line) and emittance (${\bar \varepsilon _{2 - 50\mu m}}$; dashed line). ${\bar \tau _{8 - 13\mu m}}$ (${\bar \varepsilon _{2 - 50\mu m}}$) decreases (increases) with the increase of $\theta $, highlighting the benefit of the concentrator.
Any theoretical analysis, however, must consider all the relevant absorption and emission processes. In Ref. [16], the radiative net flux (with SI units of Wm-2) of a point emitter located at the center of the truncated surface of a conical concentrator is computed as:
We note that there are fundamental issues about this equation. An easy check is to set the atmosphere as a blackbody, i.e. ${\varepsilon _{atm.}}({\lambda ,\theta } )\equiv 1$, which leads to a nonzero $q_{net}^{\prime\prime}$ for an emitter with a concentrator, i.e. ${\theta _c} < \pi /2$, even if ${T_{emitter}} = {T_{ambient}}$. This, of course, violates the 2nd Law of thermodynamics.
Now we examine in more details the issues associated with Eq. (1). The first term of Eq. (1) describes the emission process of the emitter. The second term is intended to describe the absorption process of the emitter. In fact, it considers only the process where a ray, which would have reached the emitter in the absence of the concentrator, is blocked by the concentrator, as shown in Fig. 1(a). However, the presence of the concentrator also results in the process as described by the red arrow as shown in Fig. 1(b), where a ray, which would not have reached the emitter in the absence of the concentrator, is redirected by the concentrator to the emitter. Equation (1) of Ref. [16] ignored this process and therefore results in the overestimate of the cooling power.
We note that the absorption process as described by the red arrow shown in Fig. 1(b) is in fact required by reciprocity. In its reciprocal process (green arrows in Fig. 1(b)), the emitter emits a ray along an oblique angle. The ray is then redirected to the normal direction by the concentrator. To describe the absorption process correctly, one must take into account the suppression of absorption due to the blocking effect as described in Fig. 1(a) for downward radiation at an oblique angle, as well as the enhancement of absorption due to the concentration effect for downward radiation near the normal incidence as described in Fig. 1(b).
Our discussion further implies that if the atmospheric radiation were isotropic, the concentrator could not enhance the radiative cooling, since the effects of suppression and enhancement as discussed above would exactly cancel each other. This conclusion cannot be reached by applying Eq. (1). Fortunately for the development of concentrated radiative cooling systems, the atmospheric emittance is highly angle-dependent. Figure 1(c) shows the algebraic mean of the specular directional atmospheric emittance (transmittance) obtained from MODTRAN [26] and averaged between 2-50 $\mu m$ (8-13 $\mu m$), ${\bar \varepsilon _{2 - 50\mu m}}$ (${\bar \tau _{8 - 13\mu m}}$). Note that 2-50 μm accounts for 97% of the blackbody emissive power at 300 K. Clearly, the oblique incidence (Fig. 1(a)) has higher ${\bar \varepsilon _{2 - 50\mu m}}$ and lower ${\bar \tau _{8 - 13\mu m}}$ than the normal incidence (red in Fig. 1(b)) due to a longer optical path. This directional emittance and transmittance ensure the concentrated radiative cooling.
We now present our theory for the treatment of concentrated radiative cooling. The key here is to compute the atmospheric emission that is absorbed by the emitter. One approach is to consider all the rays along all the downward directions, from the atmosphere to the emitter, and rule out those blocked by the concentrator, and those entered the concentrator but did not arrive at the emitter, since the size of the emitter does not necessarily match that of the bottom aperture of the concentrator (Fig. 2(a)). An equivalent approach, which is easier to implement in actual computation and hence is the one that we will adopt in this paper, is to apply the reciprocity. We consider all the optical paths along which photons are emitted by the emitter and finally escape the concentrator after specular reflections, as shown in Fig. 2(a) for one specific outgoing ray in green. These paths are the only possible paths for the atmospheric photons to reach the emitter, as shown in Fig. 2(a) for the corresponding incoming ray in red, which is shifted slightly for clarity. Here by reciprocity for a point emitter located at the centre of the concentrator we obtain
Fig. 2. Ray tracing scheme to map $({x,y,\theta ,\phi } )$ to a corresponding ${\theta _{atm.}}$ for a conical concentrator. (a) Schematic of a ray emitted from an off-center point $({x,y} )$ of the emitter along a specific direction ($\theta ,\phi $). It finally escapes from the concentrator to the atmosphere along a polar angle, ${\theta _{atm.}}$, which is smaller than $\theta $. Reciprocity requires another ray from the atmosphere follows the same optical path to reach the emitter (red; shifted slightly from the green for clarity). (b) $\theta - {\theta _{atm.}}$ mapping for rays emitted from a specific point, ($\hat x = 0.6,\hat y = 0.6$), along a specific azimuthal angle, $\phi = {45^\circ }$. (c) ${T_{{ {emitter}}}} - {Q_{{ {net}}}}$ relation of a near-ideal emitter. Here ${\ {\varDelta }}T = {T_{ambient}} - {T_{emitter}}$.
Comparing to Eq. (1), the first term on the right-hand side of Eq. (2) is the same. ${\theta _c}$ in the second term of Eq. (1), however, is replaced by $\pi /2$ in Eq. (2). Here ${\theta _{atm.}}(\theta )$ is the direction of the downward radiation that is redirected by the concentrator to be incident on the emitter at an angle $\theta $. For a complete description of the problem, one should also include the third term, the flux emitted by outer space and absorbed by the emitter [27], where ${\tau _{atm.}}({\lambda ,{\theta_{atm.}}(\theta )} )= 1 - {\varepsilon _{atm.}}({\lambda ,{\theta_{atm.}}(\theta )} )$. Numerically, however, this term is negligible because the temperature of the universe, ${T_{univ.}} \approx 3{\textrm{K}}$, is much lower than ${T_{ambient}}$. Therefore, we will follow the convention and neglect this term in all the calculations in this work.
Equation (2) takes advantages of the high-symmetry of the ideal problem, in which a point emitter is located at the center of the truncated surface of a concentrator. To take into account the finite size of the emitter, we compute the radiative net power (with SI units of W) as
Two features are worth noting. First, here we employ an isothermal approximation which neglects the non-uniformity of the temperature (${T_{emitter}}$) across the surface of the emitter. The comparison to an adiabatic approximation is discussed in Note S1 of Supplement 1. Second, here the integrals $\mathop \int \nolimits_{\phi = 0}^{2\pi } d\phi = 2\pi $ and $\mathop \int\!\!\!\int \nolimits_{{A_{emitter}}}{\ {d}{x}{d}}y = {A_{emitter}}$ have already been carried out explicitly for the first term, because ${\varepsilon _{emitter}}$ has azimuthal and translational symmetry. However, we cannot do so for the second and third terms, since now ${\theta _{atm.}}\;$ (Fig. 2(a)), and thus ${\varepsilon _{atm.}}$ and ${\tau _{atm.}}$, depend not only on $\theta $, but also on $\phi $, x, and y. Again, the last term is numerically negligible, which is included here only for the sake of completeness.
We adapt a ray-tracing scheme [28–30] to map $({x,y,\theta ,\phi } )$ to ${\theta _{atm.}}$ for concentrators with various sizes and shapes. Figure 2(b) shows an example of this mapping for a conical concentrator, which has an aspect ratio, $\hat H = H/r = 2$, and a half-angle, $\alpha = {10^\circ }$. Here we arbitrarily pick a point from the emitter, e.g. $({\hat x,\hat y} )= ({0.6,0.6} )$, which are again normalized to r. We fix $\phi = {45^\circ }$ but vary $\theta $. Two features are worth noting. First, ${\theta _{atm.}} \le \theta $, which manifests the function of the concentrator. Second, the first and second discontinuities represent the critical angle transitioning from the 0-reflection to the 1-reflection regime, and that transitioning from the 1-reflection to the 2-reflection regimes, respectively.
Adding the parasitic heat gain, ${h_{parasitic}}{A_{emitter}}({{T_{ambient}} - {T_{emitter}}} )$, where ${h_{{\textrm{parasitic}}}}$ is the parasitic heat transfer coefficient (with SI units of Wm-2K-1), to the right-hand side of Eq. (3), one can either obtain the steady-state temperature of the emitter, ${T_{emitter,s - s}}$, by setting the net cooling power ${Q_{net}} = 0$; or obtain the maximum cooling power ${Q_{net,max.}}$ by setting ${T_{emitter}} = {T_{ambient}}$, which are well-documented in literature [16,31–33].
To illustrate this framework, we choose a conical concentrator with $r = 1\;$m, $H = 2\;$m, and $\alpha = {10^\circ }$, and a near-ideal emitter (inset of Fig. 3(d)) [16,17,31,32] with ${r_{emitter}} = 0.9\;$m. We fix with RH = 40% to obtain a spectral directional ${\varepsilon _{atm.}}$ from MODTRAN [26]. For a vacuum-based thermal design with ${h_{parasitic}} = 0.2\;{\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$[17], we compute the ${Q_{net}}$ vs. ${T_{emitter}}$ relation in Fig. 2(c) for three cases: no concentrator (black), the model neglecting the constraint from the reciprocity (dashed gray; Eq. (S5) of Note S2 of Supplement 1), and the correct model taking into account the reciprocity (blue; Eq. (3)). As expected, ${Q_{net,max.}}$ is amplified from 197 W (black) to 215 W (blue), while ${T_{emitter,s - s}}$ is decreased from -36C (black) to -41C (blue). The dashed gray line, which uses a model that is similar to Eq. (1) that violates reciprocity, overestimates both ${Q_{net,max.}}$ and ${T_{emitter,s - s}}$, as discussed in more details in Note S2 of Supplement 1.
Fig. 3. Optimizing ${{\varDelta }}T = {T_{ambient}} - {T_{emitter,s - s}}$ (upper and middle rows) and $q_{net,max.}^{\prime\prime}$ (lower row) of two concentrators: parabolic (1st column) and conical (2nd and 3rd columns), with three different emitters: ideal (red), near-ideal (blue) and blackbody (black). As the aspect ratio, $\hat H = H/{L_{char.}}$, increases, ${{\varDelta }}T$ and $q_{net,max.}^{\prime\prime}$ first increases and then saturates (left two columns). An optimized ${\alpha _{optimal}}$ is existing for the conical concentrator (right column). In the lower row, $q_{net,max.}^{\prime\prime}$ of the blackbody emitter is exactly the same as that of the ideal emitter (red), and thus is omitted for clarity.
3. Optimization of concentrators
We now discuss the optimization of concentrators. Ideally one prefers a parabolic concentrator, for its ability of redirecting all the rays towards the normal direction (Fig. 1(b)). However, considering the cost and the ease of production in practical applications, a conical concentrator [16–18] might also be considered. Qualitatively, it is obvious that the larger the aspect ratio, $\hat H = H/r$, the stronger the concentrating effect, and thus the better the cooling performance. However, it is not obvious where the point of diminishing returns is for $\hat H$, beyond which further increasing $\hat H$ gives only minimal additional benefit. Another intuition is there exists an optimal angle, $\alpha $, for the conical concentrator, because the concentrating effect diminishes for either $\alpha = 0$ or $\pi /2$. But again it is not obvious what this optimal angle is. At last, with these optimized geometries, what are the fundamental limits of concentrated radiative cooling, and how close is the cooling performance to these limits if one uses a conical concentrator?
Figure 3 addresses these questions quantitatively. Here we analyze three types of emitters, the blackbody, the near-ideal (inset of Fig. 3(d)), and the ideal emitter that optimizes the emissivity to match the spectral directional atmospheric transmittance (Note S3 of Supplement 1) [34,35], equipped with both parabolic and conical concentrators. To highlight some limiting cooling performances, we analyze an ideal scenario (${h_{parasitic}} = 0\;{\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$) under an extremely dry condition (RH=10%), and a practical scenario (${h_{parasitic}} = 8\;{\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$) under an extremely humid condition (RH=100%), both fixed at ${{T_{ambient}} =30}^\circ\textrm{C}$ . Here we assume the surface of all the concentrators has unity reflectivity. We further assume the emitter is a point source overlapping with the center of the bottom aperture, which is also the focal point, of the concentrator, such that one only needs two parameters, H and r, to fully describe a parabolic concentrator (Note S4 of Supplement 1).
The upper row of Fig. 3 shows ${{\varDelta }}T\;({ = {T_{ambient}} - {T_{emitter,s - s}}} )$ of the three emitters (black/blue/red for blackbody/near-ideal/ideal) with the two concentrators (left column for parabolic, and right two columns for conical) with RH = 10% and ${h_{{\textrm{parasitic}}}} = 0\;{\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$. Taking up the questions above: First, we define a critical ${\hat H_{99}}$ as that for which ${{\varDelta }}T$ reaches 99% of its maximal, ${{\varDelta }}{T_{max.}}$. For example, the red line of Fig. 3(a) suggests ${\hat H_{99}} = 3.1$ for the ideal emitter with the parabolic concentrator. Other lines in Fig. 3(a)-(b) indicates similar ${\hat H_{99}}$ ($\le 5.8$) for other emitter-concentrator combinations. Second, Fig. 3(c) shows that the optimal half-angle, ${\alpha _{optimal}}$, is in the range of $20^\circ{-} 25^\circ $ for all the combinations of emitters and concentrators. Third, the combination of the ideal emitter and the parabolic concentrator leads to a limiting temperature reduction, $\varDelta T({\hat H}_{99}) = 138^\circ\textrm{C} $. For all the three emitters, ${{\varDelta }}T({{{\hat H}_{99}},{\alpha_{optimal}}} )$ of the conical concentrator equals to ${{\varDelta }}T({{{\hat H}_{99}}} )$ of the parabolic concentrator.
The middle row of Fig. 3 shows ${{\varDelta }}T$ with RH = 100% and ${h_{{\textrm{parasitic}}}} = 8\;{\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$. As compared to the upper row, the trends of the curves are similar, although the magnitudes are significantly reduced. Here three features are worth noting. First, the parabolic concentrator amplifies the ${{\varDelta }}T$ of the ideal emitter from 2.1C to 3.8C, leading to a limiting amplification factor, $\gamma \equiv \frac{{{{\varDelta }}T({with\;concentrator} )}}{{{{\varDelta }}T({no\;concentrator} )}} = 1.81$. Second, the combination of the blackbody emitter and the conical concentrator leads to a ${{\varDelta }}T$ of 2.9C , which is greater than that of the ideal emitter (${{\varDelta }}T$ = 2.1C). This underlines a fact that a concentrator can improve ${{\varDelta }}T$ of an easily-obtained emitter to surpass that of a sophisticated photonic design. Third, as ${h_{{\textrm{parasitic}}}}$ increases from $0\;{\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$ in the upper row to $8\;{\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$ here, the ideal emitter recovers the near-ideal emitter. This emphasizes that, to maximize ${{\varDelta }}T$, it is worth optimizing the emitter to match the spectral directional atmospheric spectrum only in well-insulated systems [17] with small ${h_{{\textrm{parasitic}}}}$ (see Note S3 of Supplement 1 for more discussions).
The lower row of Fig. 3 shows $q_{net,max.}^{\prime\prime}\;({ = {Q_{net,max.}}/{A_{emitter}}} )$ with RH = 10%. Here the maximum concentrated $q_{net,max.}^{\prime\prime}$ is $146\;{\textrm{W}}{{\textrm{m}}^{ - 2}}$. Note here $q_{net,max.}^{\prime\prime}\;$ of the ideal emitter successfully recovers that of the blackbody emitter (omitted from Fig. 3(d)-(f) for clarity), for the latter always gives the maximum $q_{net,max.}^{\prime\prime}$ (Note S3 of Supplement 1) [31].
4. Experimental demonstration
We design a series of experiments to validate our theoretical framework, and to test our optimization results. Here we choose conical concentrators, and a 60-μm-thick Al2O3 emitter electroplated on top of a 300-μm-thick aluminum substrate with a radius of 35 mm.
Figure 4(a) shows the basic experimental setup. To suppress the non-radiative thermal coupling between the emitter and the environment, and to ensure unhindered radiative thermal coupling between the emitter and outer space, we seal the emitter with a polyethylene (PE) film on top and a polystyrene foam on bottom [17,33]. Figure 4(b) shows the spectra of the emitter (blue) and the PE cover (orange), both measured along the normal direction using the Fourier transform infrared spectroscopy (FTIR). The atmospheric transmittance (grey) along the zenith angle, obtained from MODTRAN with Tambient = 10 ℃ and RH = 40%, is also shown for reference.
Fig. 4. Experimental demonstration of amplifying radiative cooling using a conical concentrator. (a) Schematic of the experimental setup (not in scale). (b) Measured spectra of the emitter (blue) and the polyethylene film (orange). (c-d) Experimental procedures and a typical measurement in Nanjing (China) to show the effect of the concentrator.
We first illustrate the amplification of radiative cooling using a conical concentrator with dimensions of r = 60 mm, H = 100 mm, and α = 26°. Figure 4(c)-(d) shows a typical measurement procedure and the corresponding result. We first expose two identical emitters (A & B), which are far away to avoid thermal coupling, to a clear sky to reach steady state (${T_{emitter\;A}} \approx {T_{emitter\;B}} < {T_{ambient}}$). Next we put the conical concentrator on emitter A at ∼ 21:32. ${T_{emitter\;A}}$ (black) rapidly decreases and reaches a new steady state at ${T_{emitter\;A}} < {T_{emitter\;B}}$. To make sure there are no artificial effects, we then remove the concentrator from emitter A at ∼ 21:42, which shows a quick recover of ${T_{emitter\;A}}$ (black) back to ${T_{emitter\;B}}$ (blue). At last, we repeat the same procedure on emitter B. We put the concentrator on emitter B at ∼ 22:02, which again shows a rapid decrease of ${T_{emitter\;B}}$ (blue). Likewise, after removing the concentrator at ∼ 22:20, ${T_{emitter\;B}}$ (blue) rapidly recover back to ${T_{emitter\;A}}$ (black). This experiment clearly verifies the amplifying effect of the concentrator.
We next verify our optimization results. We prepare two sets of conical concentrators. In the first set (Fig. 5(a)), we fix the half-angle, α = 26°, and the bottom radius, r = 60 mm, but vary the height, $H$ = 30, 100, and 330 mm (labelled 1, 2, and 3, respectively). In the second set (Fig. 5(a)), we fix $H$ = 100 mm and $r$ = 60 mm, but vary $\alpha $ = 5°, 26°, and 60° (labelled 4, 5, and 6, respectively). We conduct simultaneous measurements on six identical emitters with these two sets of concentrators. Figures 5(c)-(d) show representative experimental results at ${T_{ambient}}$ = 26 ℃ with RH = 62%. Here we also show a control measurement without any concentrator, corresponding to $H$ = 0 mm in Fig. 5(c) or $\alpha $ = 90° in Fig. 5(d). The x- and y-axis error bars are estimated based on the accuracy in measuring the dimensions of concentrators and the datasheet of K-type thermocouples. These measurements are consistent with our mathematical framework (shaded area), which takes into account the finite size of the emitter, the non-unity reflectivity of the concentrator, the optical properties of the PE cover, and estimated range of ${h_{{\textrm{parasitic}}}}$ (Note S5 of Supplement 1).
Fig. 5. Experimental demonstration of our theoretical optimization of conical concentrators. (a-b) Photos of two sets of concentrators. (c-d) Six simultaneous measurements on identical emitters with the two sets of concentrators in (a-b). A control experiment without concentrator is included, corresponding to $H$ = 0 mm in (c) and $\alpha $ = 90° in (d). Shaded areas represent modeling results with a range of ${h_{parasitic}}$ from 2.1 to 3.2 ${\textrm{W}}{{\textrm{m}}^{ - 2}}{{\textrm{K}}^{ - 1}}$ (Note S5 of Supplement 1).
5. Conclusion and discussion
We have developed a mathematical framework to analyze and optimize the concentrated radiative cooling. In particular, we highlighted the importance of carefully considering the constraint from reciprocity, and quantified the error if otherwise. The framework also took into account the finite size of the emitter. Modeling showed that, with optimization, the conical concentrator can perform as good as the ideal parabolic concentrator: the optimized aspect ratio, ${\hat H_{99}}$, should be no less than 5.8; the optimized ha-angle, ${\alpha _{optimal}}$, should be between $20^\circ{-} 25^\circ $. According to these optimizations, we have conducted a series of measurements on an electroplated Al2O3 emitter with two sets of conical concentrators, with the amplification factor, $\gamma $, up to 1.26 under a relatively humid condition (see Note S6 of Supplement 1 for supplemental experiments with pyramidal concentrators).
We conclude by commenting this technology from the perspective of applications, as well as from a fundamental viewpoint. First, this concentrating scheme can be particularly beneficial when a radiative cooler is surrounded by warm and tall objects, e.g. buildings or trees. Second, like the concentrated solar power system, this concentrated cooling technology is also likely to suffer from the extra cost of the mirrors, although here one is relieved from the concern of fatigue of materials at high temperatures. We also note that, instead of simply considering the extra cost of mirrors, a fair comparison should be made between the total cost of the concentrating mirror and an easily-obtained emitter, e.g. a blackbody, and the cost of a sophisticated photonic design, provided that the former performs as good as, if not better than, the latter. Third, we constrain our analysis to nighttime cooling here; for daytime cooling, a solar-absorbing but infrared-transparent cover [14], or a solar-absorbing but infrared-reflective concentrator [23] would be helpful. At last, although our current analysis is constrained from reciprocity, we anticipate that, with the recent development of non-reciprocal photonic designs [36–38], the concentrated cooling performance may be further improved. Comparison to the angularly-restricted photonic design [39,40] (Note S7 of Supplement 1) and maximization of cooling power for a given sky-facing area (Note S8 of Supplement 1) are also discussed.
Funding
National Natural Science Foundation of China (51776038); Innovative and Entrepreneurial Talent Plan (Jiangsu Province, China); Basic Energy Sciences (DE-FG02-07ER46426); Seed Grant from Institutes of Energy and the Environment, the Pennsylvania State University; Postgraduate Research and Practice Innovation Program of Jiangsu (KYCX20_0079); Fundamental Research Funds for the Central Universities (3202002102D).
Acknowledgments
Z. C. thanks Chris Dames for critically reading this manuscript and many constructive comments and suggestions.
Disclosures
The authors declare no competing financial interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. L. Zhao, B. Bhatia, S. Yang, E. Strobach, L. A. Weinstein, T. A. Cooper, G. Chen, and E. N. Wang, “Harnessing Heat Beyond 200 degrees C from Unconcentrated Sunlight with Nonevacuated Transparent Aerogels,” ACS Nano 13(7), 7508–7516 (2019). [CrossRef]
2. D. Real, R. Johnston, J. Lauer, A. Schicho, and N. Hotz, “Novel non-concentrating solar collector for intermediate-temperature energy capture,” Sol. Energy 108, 421–431 (2014). [CrossRef]
3. L. A. Weinstein, J. Loomis, B. Bhatia, D. M. Bierman, E. N. Wang, and G. Chen, “Concentrating Solar Power,” Chem. Rev. 115(23), 12797–12838 (2015). [CrossRef]
4. S. Kuravi, J. Trahan, D. Y. Goswami, M. M. Rahman, and E. K. Stefanakos, “Thermal energy storage technologies and systems for concentrating solar power plants,” Prog. Energ. Combust. 39(4), 285–319 (2013). [CrossRef]
5. “Renewable energy technologies: cost analysis series - concentrating solar power,” International Renewable Energy Agency (IREA) (2012).
6. R. Pitz-Paal, “Concentrating solar power: Still small but learning fast,” Nat. Energy 2(7), 17095 (2017). [CrossRef]
7. S. Pfenninger, P. Gauche, J. Lilliestam, K. Damerau, F. Wagner, and A. Patt, “Potential for concentrating solar power to provide baseload and dispatchable power,” Nat. Clim. Change 4(8), 689–692 (2014). [CrossRef]
8. X. H. Xu, K. Vignarooban, B. Xu, K. Hsu, and A. M. Kannan, “Prospects and problems of concentrating solar power technologies for power generation in the desert regions,” Renew. Sust. Energ. Rev 53, 1106–1131 (2016). [CrossRef]
9. S. J. Byrnes, R. Blanchard, and F. Capasso, “Harvesting renewable energy from Earth's mid-infrared emissions,” Proc. Natl. Acad. Sci. U. S. A. 111(11), 3927–3932 (2014). [CrossRef]
10. S. Buddhiraju, P. Santhanam, and S. Fan, “Thermodynamic limits of energy harvesting from outgoing thermal radiation,” Proc. Natl. Acad. Sci. U. S. A. 115(16), E3609–E3615 (2018). [CrossRef]
11. S. H. Fan, “Thermal Photonics and Energy Applications,” Joule 1(2), 264–273 (2017). [CrossRef]
12. X. B. Yin, R. G. Yang, G. Tan, and S. H. Fan, “Terrestrial radiative cooling: Using the cold universe as a renewable and sustainable energy source,” Science 370(6518), 786–791 (2020). [CrossRef]
13. A. P. Raman, W. Li, and S. H. Fan, “Generating Light from Darkness,” Joule 3(11), 2679–2686 (2019). [CrossRef]
14. Z. Chen, L. X. Zhu, W. Li, and S. H. Fan, “Simultaneously and Synergistically Harvest Energy from the Sun and Outer Space,” Joule 3(1), 101–110 (2019). [CrossRef]
15. W. Li, S. Buddhiraju, and S. Fan, “Thermodynamic limits for simultaneous energy harvesting from the hot sun and cold outer space,” Light Sci. Appl. 9(1), 68 (2020). [CrossRef]
16. G. B. Smith, “Amplified radiative cooling via optimised combinations of aperture geometry and spectral emittance profiles of surfaces and the atmosphere,” Sol. Energ. Mat. Sol. C. 93(9), 1696–1701 (2009). [CrossRef]
17. Z. Chen, L. Zhu, A. Raman, and S. Fan, “Radiative cooling to deep sub-freezing temperatures through a 24-h day-night cycle,” Nat. Commun. 7(1), 13729 (2016). [CrossRef]
18. I. Haechler, H. Park, G. Schnoering, T. Gulich, M. Rohner, A. Tripathy, A. Milionis, T. M. Schutzius, and D. Poulikakos, “Exploiting radiative cooling for uninterrupted 24-hour water harvesting from the atmosphere,” Sci. Adv. 7(26), eabf3978 (2021). [CrossRef]
19. J. Peoples, Y.-W. Hung, X. Li, D. Gallagher, N. Fruehe, J. B. Anil Yuksel, T. Horton, and X. Ruan, “Concentrated Radiative Cooling,” arXiv: 2010.02426v1 (2020).
20. F. Trombe, “Perspectives sur l’utilisation des rayonnements solaires et terrestres dans certaines regions du monde,” Rev. Gen. Therm. 6(70), 1285–1314 (1967).
21. A. Gentle, A. Nuhoglu, M. Arnold, and G. Smith, “3D printable optical structures for sub-ambient sky cooling,” in SPIE 10369, Thermal Radiation Management for Energy Applications, (San Diego, California, United States, 6 September 2017), paper 103690B.
22. L. Zhou, H. M. Song, N. Zhang, J. Rada, M. Singer, H. F. Zhang, B. S. Ooi, Z. F. Yu, and Q. Q. Gan, “Hybrid concentrated radiative cooling and solar heating in a single system,” Cell Rep. Phys. Sci. 2(2), 100338 (2021). [CrossRef]
23. Y. Zhou, H. M. Song, J. W. Liang, M. Singer, M. Zhou, E. Stegenburgs, N. Zhang, C. Xu, T. Ng, Z. F. Yu, B. Ooi, and Q. Q. Gan, “A polydimethylsiloxane-coated metal structure for all-day radiative cooling,” Nat. Sustain. 2(8), 718–724 (2019). [CrossRef]
24. X. Lu, P. Xu, H. L. Wang, T. Yang, and J. Hou, “Cooling potential and applications prospects of passive radiative cooling in buildings: The current state-of-the-art,” Renew. Sust. Energ. Rev. 65, 1079–1097 (2016). [CrossRef]
25. A. R. Gentle and G. B. Smith, “Angular selectivity: impact on optimised coatings for night sky radiative cooling,” in Proceeding of SPIE (2009), paper 74040J.
26. S. S. Shen, A. Berk, P. E. Lewis, G. P. Anderson, P. K. Acharya, L. S. Bernstein, L. Muratov, J. Lee, M. Fox, S. M. Adler-Golden, J. J. H. Chetwynd, M. L. Hoke, R. B. Lockwood, J. A. Gardner, T. W. Cooley, C. C. Borel, P. E. Lewis, and E. P. Shettle, “MODTRAN5: 2006 update,” in Algorithms and Technologies for Multispectral, Hyperspectral, and Ultraspectral Imagery XII (2006).
27. Here we model the atmosphere as a partly transmitting and partly absorbing layer, and further treat outer space as a blackbody with εuniv. (λ,θatm. (θ)) ≡ 1, because the effective surface area of this huge “outer space enclosure” is much larger than that of the earth.
28. J. Miller, W. Jang, and C. Dames, “Thermal Rectification by Ballistic Phonons in Asymmetric Nanostructures,” in Summer Heat Transfer Conference, (ASME, San Francisco, California USA, 2009), paper HT2009-8848.
29. J. P. Miller, “Thermal Rectification by Ballistic Phonons in Asymmetric Nanostructures,” Master thesis (adviser: Chris Dames) (University of California, Riverside, 2009).
30. J. Lee, W. Lee, G. Wehmeyer, S. Dhuey, D. L. Olynick, S. Cabrini, C. Dames, J. J. Urban, and P. Yang, “Investigation of phonon coherence and backscattering using silicon nanomeshes,” Nat. Commun. 8(1), 14054 (2017). [CrossRef]
31. P. Berdahl, M. Martin, and F. Sakkal, “Thermal Performance Of Radiative Cooling Panels,” Int. J. Heat Mass Tran. 26(6), 871–880 (1983). [CrossRef]
32. E. Rephaeli, A. Raman, and S. Fan, “Ultrabroadband photonic structures to achieve high-performance daytime radiative cooling,” Nano Lett. 13(4), 1457–1461 (2013). [CrossRef]
33. M. Dong, N. Chen, X. Zhao, S. Fan, and Z. Chen, “Nighttime radiative cooling in hot and humid climates,” Opt. Express 27(22), 31587–31598 (2019). [CrossRef]
34. L. Fan, W. Li, W. Jin, M. Orenstein, and S. Fan, “Maximal nighttime electrical power generation via optimal radiative cooling,” Opt. Express 28(17), 25460–25470 (2020). [CrossRef]
35. S. Jeon and J. Shin, “Ideal spectral emissivity for radiative cooling of earthbound objects,” Sci. Rep. 10(1), 13038 (2020). [CrossRef]
36. L. X. Zhu and S. H. Fan, “Near-complete violation of detailed balance in thermal radiation,” Phys. Rev. B 90(22), 220301 (2014). [CrossRef]
37. B. Zhao, C. Guo, C. A. C. Garcia, P. Narang, and S. Fan, “Axion-Field-Enabled Nonreciprocal Thermal Radiation in Weyl Semimetals,” Nano Lett. 20(3), 1923–1927 (2020). [CrossRef]
38. S. Pajovic, Y. Tsurimaki, X. Qian, and G. Chen, “Intrinsic nonreciprocal reflection and violation of Kirchhoff's law of radiation in planar type-I magnetic Weyl semimetal surfaces,” Phys. Rev. B 102(16), 165417 (2020). [CrossRef]
39. S. Jeon and J. Shin, “Directional radiation for optimal radiative cooling,” Opt. Express 29(6), 8376–8386 (2021). [CrossRef]
40. Y. R. Qu, M. Y. Pan, and M. Qiu, “Directional and Spectral Control of Thermal Emission and Its Application in Radiative Cooling and Infrared Light Sources,” Phys. Rev. Applied 13(6), 064052 (2020). [CrossRef]
