In this work we first solve the radiative heat transfer problem in one dimension to perform a comparative analysis of the time averaged performance of the partially transparent radiative windows and radiative coolers. In doing so w e clearly distinguish the design goals for the partially transparent windows and radiative coolers and provid e optimal choice for the material parameters to realize these goals. Thus, radiative coolers are normally non -transparent in the visible, and the main goal is to design a cooler with the tempera ture of its dark side as low as possible relative to that of the atmosphere. For the radiative windows, however, their surfaces are necessarily partially transparent in the visible. In the cooling mode the main question is rather about the maximal visible light transmission through the window at which the temperature on the window somber side does not exceed that of the atmosphere. We then demonstrate that transmission of the visible light through smart windows can be significantly i ncreased (by as much as a factor of 2) without additional heating of the windows via coupling of the windows to the radiative coolers using transparent cooling liquid that flows inside of the window and radiative cooler structure s. We demonstrate that efficient heat excha nge between radiative coolers and smart windows can be realized using small coolant velocities (sub 1mm/s for ~1m -large windows) or even using a purely passive gravitationally driven coolant flows between a hot smart window and a cold radiative cooler moun ted on top of the window with only a minimal temperature differential (sub -1K) between the two. We believe that our simple models complimented with an in -depth comparative analysis of the standalone and coupled smart w indows and radiative coolers can be of interest to a broad scientific community pursuing research in these disciplines . 1. INTRODUCTION Currently, the problem of energy overconsumption presents one of the major challenges for many societies [1]. In particular, energy s pending for building cooling, heating and lighting accounts for ~40% of the total energy consumption in some countries [1,2]. While several methods to reduce energy use for the cooling of buildings, automobiles, food depots and other structures are known, radiative cooling distinguishes itself by being one of the few temperature reduction methods that require no external energy sources for their operation [1,2]. Design of radiative coolers presents an intriguing multidisciplinary problem at the intersection of physics, material science, optics and engineering. Similarly to other cooling methods, general idea of the radiative cooling is to achieve larger energy outgoing flux than the incoming energy flux [3]. Standard passive cooling devices use large heat s inks with lower temperature than that of a cooled object to reach large outgoing energy fluxes [4]. On a planetary scale, Corresponding author : Maksim Skorobogatiy - maksim.skorobogatiy@polymtl.ca 2 the Earth temperature is ~300K, which is much larger than the temperature of the outer space ~3 K that acts as an infinite heat sink f or the planet [3]. As the outer space is a near perfect vacuum, the main heat loss mechanism for the Earth is not a conduction driven heat transfer but a radiative one. This is due to the fact that objects with temperatures above the absolute zero radiate electromagnetic energy known as thermal radiation or black -body radiation with the spectrum depending on the object temperature and described by the Planck’s law. Thus, objects with temperatures of ~300 K mainly radiate energy in the mid -IR with a broad pe ak in the spectral density covering the wavelength of 8 -15 m [4]. Remarkably, in the same spectral rage, the Earth’s atmosphere has a transmission window (window of transparency) that allows a considerable portion of the mid -IR ther mal radiation to leave unimpeded [4,5]. This represents the main mechanism by which terrestrial objects can dissipate heat into the outer space in the form of electromagnetic waves, which is the basis of radiative cooling. It is pertinent to mention at thi s point that the Earth cooling via radiative heat transfer to the outer space is compensated by heating of the Earth surface via partial absorption of the radiative energy of the Sun that features the highest spectral density in the visible, while extendin g all the way to the near-IR 0.3 -2.5 m [6]. The study of radiative cooling has a long history. It has been well known since the ancient times that using blackened surfaces (radiators) facing a clear night sky could result in sub -ambient temperatures of the radiator, which was even used to make ice [7]. Temperature reduction of about 5℃ was reported by Granqvist in 1981 [8], using a low emittance window surface and night time cooling. This original work was followed by several studie s of radiative cooling efficiency under different environmental conditions, such as humidity [9,10], ambient temperature [11], and geographical location [12]. This original work was followed by studies of the day time cooling, which turned to be a much ha rder problem. Originally, daytime radiative cooling under direct sunlight was achieved by using a radiator that would reflect most of the sunlight, while radiating efficiently in the mid -IR spectral range [13], thus allowing the heat to escape through the atmospheric transparency window. The fact that the radiative energy of the Sun and the thermal radiation of a terrestrial object occupy different and mostly non -overlapping parts of the electromagnetic spectrum make the problem of the radiative cooler desi gn more complicated. This is because the cooler materials have to exhibit widely different thermo -optic properties in the visible and mid -IR spectral ranges. A large number of studies have been conducted to date on a subject of radiative cooler materials (electrochromic, photochromic windows and thermochromic windows [14 -16], photonic crystals and metamaterials [13,17]), as well as structures and design optimization (reflecting vs. absorbing structures [17 - 19]), with experimentally demonstrated temperature reduction from 5℃ to 42℃ [4,13,20]. Furthermore, the question of the fundamental limit in the temperature reduction for radiative coolers was investigated in great details with a consensus that it depends strongly on the environmental conditions [10,21]. Thus, for the night time radiative coolers the predictions range between 15 -42℃ [21 -23], while for the daytime radiative coolers the temperature reduction is expected to be only several degrees [13]. An important issue when characterizing radiating cooler s is the choice of parameters to characterize the cooler performance. While the cooler itself has a working surface, its main function is to cool the air and solid objects behind it. Therefore, while the cooler surface temperature is of importance the more practical parameter is probably the air temperature or the temperature of a solid behind the cooler. Another issue is about on -average versus instantaneous performance of a cooler. While some coolers work mostly during night -time, others are capable of th e day and night operation [24]. They are known as high efficiency radiative coolers or radiative windows if the cooler surface is partially transparent. Additionally, if the window can adjust its radiative properties in response to the changing environment al conditions such as ambient temperature and daylight illumination [25 -27], or if it is capable of both cooling and heating, such windows are frequently referred to as intelligent or smart window [28]. Recently, there has been a strong interest in passiv e and active smart windows that could operate year - long, while providing heating in the cold and cooling in the hot weathers [24,29]. Much research is focusing on developing materials that can simultaneously, while independently manage radiation across sev eral widely 3 spaced spectral ranges covering, for example, visible / near -IR light [12,30,31], or solar / mid -IR light [20]. At the same time, a concept of perfect smart window was proposed to judge the energy efficiency of the existing smart windows [32,33 ]. Thus, the perfect smart window has zero absorptivity for the visible light, near infrared and mid infrared. At the same time, the window features a perfect transmittance in the visible, while the mid -IR light transmittance and reflectance should be eith er (zero, perfect) or (perfect, zero) depending on whether heating or cooling is required. In any case, to switch between the heating and cooling states the intelligent windows should allow large difference in its transmittance/reflectance properties betwe en the two states, while always featuring low absorption of solar radiation [32]. Therefore, much attention was payed to improving the performances of the smart windows through materials research and structural optimization [29,34 -39]. Currently, there th ree main types of smart windows in development that use electrochromic, thermochromic, photochromic materials and combinations of thereof. Electrochromic materials employ reversible redox reaction that affects material’s electronic transitions, and, as a c onsequence, the absorption profile of the solar spectra [28]. Several metal oxides have been reported for applications in smart windows among which the tungsten oxide (WO3) being the most popular one due to material’s fast switching time between opaque and clear sates, as well as high visible light transmission in the clear state [40,41]. Thermochromic smart windows employ materials that exhibit phase transition between the semiconducting monoclinic phase (clear state) and the metallic rutile phase (opaque state) as a function of temperature. Vanadium dioxide (VO2) based materials are the most popular ones to regulate transmission of the visible and near -IR light using the thermochromic effect [41]. At the same time, even in the clear state transmittance of the visible light through this material is relatively low because of the material strong absorption and reflection in short wavelength range [40]. In addition, the VO2 phase transition temperature ~68 ℃ is relatively high for the practical applications, wh ile transitions temperatures in the 20℃~30℃ would be more desired. Finally, photochromic smart window can alter its light transmission properties depending on the intensity of the incident sunlight. Photochromic materials have large working temperature ran ge (from 20℃ to as high as 80℃), and are relatively abundant [40,42]. While much work has been done on theoretical understanding of the functioning of radiative coolers, in the related field of smart windows the performance targets and optimization strate gies are still less understood. As functioning of the smart windows is dominated by the multi -year-long time scale, therefore it is interesting to analyze their averaged -over-time performance rather than an instantaneous response. It is also important to a cknowledge that the function of a window implies nonzero visible light transmission, therefore a tradeoff between the window performance and its esthetic function seems unavoidable. In this respect it is interesting to investigate the possibility of improv ing thermo -optical performance of the smart windows by their coupling to the radiative coolers , which to our knowledge has not been studied yet. In this paper we , therefore, consider in more details partially transparent radiative coolers in the context of their application in smart windows. From the onset we distinguish the design goals for the radiative coolers from those of the partially transparent (in the visible) radiat ive windows. Thus, radiative coolers are normally non-transparent in the visible, and the main goal is to design a cooler with the temperature of its dark side as low as possible compared to that of the atmosphere. For the radiative windows, however, their surfaces are necessarily partially transparent in the visible. In the cooling mode, therefore, the main question is about the maximal visible light transmission through the window at which the temperature on the window somber side does not exceed that of the atmosphere. Alternatively, as a measure of the window performance one can use the temperature of the adjacent inner air region or even that of the room wall. Finally , by coupling radiative window to the radiative cooler using active or passive heat e xchange mechanism s (for example, gravitationally driven coolant flow), one can increase transmission of the visible light through the window, while maintaining the same window temperature . In order to model precisely the smart window performance, one has to have a realistic model of the atmospheric optical properties [43], and then solve a full radiative heat transfer problem [44] that includes 4 atmosphere, window and enclosure where the window is installed. While certainly possible, such an overwhelming ap proach will most probably obscure the relatively simple physics behind the problem. Therefore, in this work we strip the radiative heat transfer problem to a bare minimum and confine ourselves to one dimension, while still retaining all the key elements of a problem. Particularly, we consider the atmosphere, the window , the cooler, and the backwall that are all in the thermal equilibrium and that all exchange the energy via radiative heat transfer , convection or coolant flow . Moreover, we use a simplified two-state model for the optical properties of an atmosphere and a window material that assumes two distinct sets of the optical reflection/absorption/transmission parameters in the visible/near -IR and mid-IR spectral ranges. Furthermore, we assume constant temperature across the radiative window/cooler, which is justified for materials with relatively high thermal conductivity. Finally, heat exchange between the window and a radiative cooler is modeled via fixed volumetric rate coolant flow between the internal elements of their structure. We believe that our work captures all the important aspects of the radiative heat transfer in smart windows and allows to make valuable qualitative predictions on the choice of the windows optimal design parameters without resorting to solution of the overcomplicated full radiative transfer problem. 2. SINGLE LAYER MODEL OF THE ATMOSPHERE We start with a single layer model of the atmosphere [ 45] (see Fig. 1) . This model allows to relate together the Earth and the atmosphere average temperatures ( gT and aT ), the atmosphere average emissivity ()a , the planet albedo al (reflectivity) and the total incoming radiative flux from the Sun ()P . This model is a necessary point of departure that allows to define a self -consistent reference (average atmospheric properties versus average Sun radiative flux) for the following models of smart window s. The model assumes equilibrium temperature distribution, and, hence, net zero radiative heat flux through any interface parallel to the Earth. A key element of the model is a recognition that in equilibrium, materi al absorption coefficient equals to the material emission coefficient. Namely, ()a - the emissivity of the atmosphere at frequency  equals to its absorption ()aa as ( ) ( ) 1 ( ) ( )a a a a a r t    = = − − . Here , ()ar is frequency dependent reflection coefficient of the atmosphere by power (mostly due to Rayleigh scattering), ()at is the frequency dependent transmission coefficient of the atmosphere by power given by () ( ) 1 ( ) exp( ( ) )a a a at r L    − − , where th e thickness of the atmosphere is aL and material absorption coefficient of the atmosphere by power is ()a . Finally, w e assume that emissivity of the Earth at any frequency is 1 (blackbody approximation) and define emission spectrum (by power) of a blackbody at temperature T as ( , )ET . Fig. 1. Single layer model of the atmosphere 5 Assuming a constant temperature in the atmosphere aT , we now use the energy conservation principle to write the following equations . At the space edge of the atmosphere we write for the energy fluxes in equilibrium: (1 ( )) ( ) ( ) ( , ) ( ( ) , )a a a a g dd rt d P E T E T         −  =  +    (1) At the Earth level we write for the energy fluxes in equilibrium: ( ) ( ) ( ) ( , ) (1 ( ) ( ) , )a a a a g d t P d E T r E T d          =−  + −    (2) Alternatively, by subtracting (2) from (1) we also get the balance between the absorbed and irradiated energy in the atmosphere: 2 ( ) ( , ) ( ) ( ) ( ) ( ,)a a a ag E T P d a d E ad T           =  +    (3) Now, we introduce several approximations for the values of the various parameters involved in the model (1)-(3) using physical properties of the atmosphere. Fig. 2. Power of the incoming light is mostly concentrated in the visible spectral range, while power of the irradiated light by the atmosphere and Earth are mostly in the Mid -IR spectral range. First, we note that power of the incoming light from the Sun is mostly concentrated in the visible spectral range, while power of the irradiated light by the atmosphere and Earth are mostly in the mid -IR spectral range, with little overlap between the two (see Fig. 2). Therefore, instead of using frequency -variable transmission and reflection properties of the atmosphere, we rather assume a step -like spectral behavior of its optical properties. Particularly, we assume that in the visible and in the mid -IR spectral ranges, the optical properties of the atmosphere are distinct from each other and frequency invariable as shown in Fig. 3. Fig. 3. Step -like model for the frequency dependent optical properties of the atmosphere (absorption, transmission and ref lection). With thus defined properties of the atmosphere we get the following values for the various integrals : 6 () () () ()44 44( ) ( ) 1 ( ) 1 ; (1 ( )) ( ) 1 (1 ( )) ( , ) ; ( ) ( , ) 1 ( ) ( , ) ; ( ) ( ) 0 ; ( ) ( , )aa a g g a g a g aa g a a a a a gt P d al P al P r P al P r E T d T tdd d d a d aE T T E T T P E T d T                               − = − −  = − −  =  = − = =  =  =       (4) while equations (1), (2) and (3) (only two of the three are independent) become: () () ()44 44 4411 1 2a a a g a a g agal P T T al P T T TT      − = + − − =− + = (5) from which is follows that: () ()4 141 2 2a a gaalTP TT−=− = (6) In order to reproduce with this model the average Earth temperature 288.2gT= , while using for the planet albedo 0.3al= , and the average radiative flux from the Sun incident onto the planet 2342P W m−= , one requires to choose for the atmosphere emissivity 0.78a= . This also resu lts in a somewhat low atmospheric temperature 242.11aTK= ( 30C− ), which is a well know deficiency of a single layer atmospheric model. Note that the quoted value for the average incoming solar radiation takes into accoun t the angle at which the rays strike and that at any one moment half the planet does not receive any solar radiation. It therefore measures only one -fourth of the solar constant, which is an averaged over the year energy flux incident on the Earth as measu red by an orbiting satellite. 3. SINGLE LAYER, PARTIALLY TRANSPARENT RADIATIVE WINDOW We now consider a single layer optically symmetric window placed in the path of a sunlight (see Fig. 5) . The window is assumed to be optically thick (no interference effec ts). We also assume that the temperature across the window is constant and equal to wT , and, therefore, the thermal radiation from the window is the same in both directions. Validity of this approximation is studied in the Supplementary Material A, where we solve a full radiative heat transfer model for a standalone slab and show that tempe rature inside can be considered constant if the slab is made of material with high enough thermal conductivity. Behind the window we place a perfect absorber that we refer to as a wall, and temperature of the wall surface behind the window, which is consid ered to be a blackbody, represented by sT . The wall is assumed to be thermally and radiationally separated from the ground. We, furthermore, assume that there is vacuum between the window and the wall, so we neglect the convection a nd conduction heat transfer in this region. Finally, the Earth atmosphere is characterized by the average temperature and permittivity defined in the previous section. 7 Fig. 5. Model of the optically symmetric single layer window (no convection/conduction between the wall and the window). Spectral dependence of the window material parameters ( ()wr , ()wt , ()wa ) are assumed to be step -like, with two distinct sets of frequency independent parameters defining window properties in the mid -IR and visible ranges as shown in Fig. 4. Fig. 4. Step -like model for the frequency dependent optical properties of the windo w (absorption, transmission and reflection). Spectrum of the incoming radiation from the sun at the ground level by power is given by ()()()gaP t P  = , and average power of the sun at the planet (window) surface is given by () 1 P al P=− . We now use the energy conservation principle to write the following equations . At the window / atmosphere interface we write for the energy fluxes in equilibrium: () (1 ( )) ( ) ( ) ( , ) ( ) ( , ) 1 ( ) ( ) ( , ) g w w a a ww ws dd r P E T t E T T dd rE                −  =  +  − −      (7) At the wall (blackbody) level we write for the energy fluxes in equilibrium: ( ) ( ) ( ) ( , ) (1 ( )) ( , ) ( ) ( ) ( , ) w w w g w a s w a t P d E T r E T d t E T d d               =−  + −  −      (8) Alternatively, by subtracting (8) from (7) we also get the balance between the absorbed and irradiated energy in the window: 2 ( ) ( , ) ( ) ( ) ( ) ( , ) ( ) ( ) ( , ) w s a w w w w a d a d E T P E T d ET aa d                 =  +  +      (9) 8 These equations can be further simp lified when using a step -like model for the material optical properties (only two of the three following equation are independent): 4 4 4 4 4 4 4 4 4(1 ) (1 ) (1 ) 2VIS MIR MIR MIR w w w w s w a a VIS MIR MIR MIR w w w w s w a a MIR VIS MIR MIR w w w w s w a aR P T T T R T T P T R T T T T A P T T                −  =  +  − −  =−  + −  −   =  +  +  (10) Now, using (10), as well as results of the model 1 that relates average atmospheric temperature to the average power flux arriving at the planet surface, we can arrive to the following expressions for the temperatures of the window and the wall: () ()4 4 4 41 22 222MIR VIS VIS w ww MIR w w aa MIR MIR a w w VIS VIS s w w aa MIR MIR a w wTTA T TT T T A TT +  + = + −+ += + −+ (11) The radiative heat transfer model for a single l ayer partially transparen t window can be readily extended to account for the convection heat transfer between the window and the air that surrounds it . The resultant model is virtually identical to the one presented in this section and include s only two new parameters, which are the room temperature ( in the space between the window and the wall), and the heat transfer coefficient h . For completeness , we detail this model and its analysis in Supplementary Material B. The key conclu sion derived from analysis of this model is that convection heat transfer always reduces efficiency of the radiative coolers and window s more so for stronger convectio n rates . Convection contribution becomes important for large enough values of the heat transfer coefficient ()38 (2 )MIR MIR a w w h T T+ , which is equivalent to  6.46 (2 )MIR MIR ww h W K T    + when operating in the vicinity of standard ambient temperatures . The value of the heat transfer coefficient depends stro ngly on the temperature differential, as well as the window size and orientation and is generally in the  1 10 WK− range (see Supplementary Material C for more details) . 3.1 Analysis of the thermal properties of a single layer radiative window Here we present performance analysis of various radiative windows as predicted by the model presented in the previous section . Particularly, we study the trade -off between transmission of the visible light through the radiative window and its temperature, as well as the wall temperature behind the window. 1) A particular solutions of this m odel (Eq. 11) is w s aT T T== when choosing VIS MIR wwTT= , VIS MIR wwA= , for any values of ,MIR MIR wwT , where 10.182a a−==− . Particularly, in the case of highly transmissive windows in the mid -IR 1MIR wT→ , which simultaneously feature low absorption in the visible ( ) 10VIS MIR MIR MIR w w w wA T R  =  =  − − → , window transmission of the visible light can be as high as 18%VIS wT== , while both the room and the wall temperature will not exceed that of the atmosphere. 2) If the window is an efficient absorber or reflector, so that there is no transmission through the window both in the visible and in the mid -IR 0VIS MIR wwTT== , then: ()44 442VIS s w w aa MIR a a wT T A TT= = + − (12) 9 Particularly, if the window absorption in the visible is much smaller that the window absorption in the mid-IR, then the window is an efficient radiative cooler with the window and wall temperatures smaller than that of an atmosphere: 14 0.94VIS MIR w w s w a a aA T T T T =     Alternatively, if the window is a black body or a “balanced” vis/mid -IR absorber, then the window and wall temperature will be those of a ground: 14 2VIS MIR w w s w a gA T T T T=  = = = Finally, if the window is a strong absorber in the visible, but a week absorber in the mid -IR, it becomes an efficient radiative heater: VIS MIR w w s w gA T T T  =  3) What is the smallest wall temperature sT possible? From (11) it follows that: ()14 2when 0 2 0 ; 0min 0.94 1 we must also require 2 1 0 1VIS VIS ww MIR MIR ww VIS VIS ww MIRs a a a VIS w MIR MIR MIR MIR w w w w wTA T TAT TR TT RT R  +=+  → →  +==    → −    + (13) This is the case of an almost completely reflective window in the visible, which at the s ame time has a non-perfect reflectivity in the mid -IR. 4) What is the smallest window temperature wT possible? From (11) it follows that: ()14 1 when 0 0 ; 0 ; 02 we must also requirmin 0.94 1 e 2 ; 10 1MIR VIS VIS w ww MIR VIS w VIS VIS w ww MIR MIR MIR w w w MIR MIR MIRw a a a VIS VIS MIR MIR w w w ww w w MIR wTTA ATTT TAT TTA R RR   +  + =  → → →+  + = += −  →  (14) This is the case of an almost completely reflective window in the visible, which at the same time has a non-perfect reflectivity in the mid -IR. Additionally we have to require that the window absorption in the visible should be much smaller than the window absorption in the mid -IR. 5) What is the maximal transmission of the visible light through the window VIS wT , so that the wall temperature does not exceed the atmospheric temperature a sTT ? From (11) it follows t hat: () ( ) ()4 4 max when 1 ( 0) ; 0 1whe221 re 0.2 11122 1822VIS VIS s w w aa MIR MIR a w w VI VIS MIR MIR MIS RVIS MIR MIR aw ww VIS wwa w w w aw aTT T A TT AT AR TRT     += + −  = = = = = −==−+ − + − −− (15) 10 In order to achieve maximal transmission of the visible light, while still having the wall temperature below that of the atmosphere, we have to demand that window absorption in the visible is zero 0VIS wA→ , while window transmission in th e mid -IR is almost perfect 1MIR wT→ . 6) What is the maximal transmission of the visible light through the window VIS wT , so that the window temperature does not exceed the atmospheric temperature waTT ? From (11) it follows that: () ( ) ()4 4 max 2 0.36 when 1 ( , 0) ; 1 212 1112 0MIR VIS VIS w ww MIR w w aa MIR MIR a w w MIR VIS MIR VIS MIR MIR MIR VIS MIR wMIR VIS aw w w w w MIR aw w w w w w T T RTTA T TT TT T R A A     +  + = + − +  − + − −  + −  = = → → → (16) In order to achieve maximal transmission of the visible light, while still having the window temperature below that of the atmosphere, we have to demand that window transmission in the mid -IR is almost perfect 1MIR wT→ , while window absor ption in the visible is much smaller than that in the mid -IR, and at the same time approaching zero 0VIS wA→ . 7) Tradeoff between window transmission and absorption in the visible can be summarized geometrically in Fig. 6. As follows fro m inequalities (15) and (16), shaded regions below the corresponding lines represent the range of values of ,VIS VIS wwTA for which either the wall temperature or the window temperature are smaller than the atmospheric temperature. Intersectio n of these two regions represent the range of values of ,VIS VIS wwTA for which both the wall temperature and the window temperature are smaller than the atmospheric one. Both the window and the wall temperatures attain that of the atmospheric one when VIS MIR wwTT= , VIS MIR wwA= . As clear from the graph, when VIS MIR wwTT and VIS MIR wwA , it is guaranteed that both the window and the wall temperatures will be smaller than the atmospheric temperature. Fig. 6. Tradeoff between window transmission and absorption in the visible (no convection). A region of parameter space that is situated below the two lines define window operation regime for which both the wall and the window temperatures are smaller than that of the atmosphere. 11 4. PARTIALLY TRANSPARENT SMART WINDOW THAT IS THERMALLY COUPLED TO A RADIATIVE COOLER An idea that we explore in this section is to exploit the cooling properties of passive radiative coole rs in order to enhance transmission of the visible light through the smart windows, without increasing the window temperature or the temperature of the wall behind it. This idea can be of practical significance as there were recent reports of successful co oling of water (coolant) flowing through the internal structure of a radiative cooler [4 8]. Particularly, we envision a transparent (in the visible) coolant that passively or actively flows through the window and the radiative cooler inner structure s (see Fig. 7) . The coolant carrie s the h eat from the partially transparent window which is then d issipated into the atmosphere by the radiative cooler . The goal of this section is to quantify the potential improvements in the intensity of transmi tted visible light through the window offered by such a hybrid structure. Fig. 7. Thermally coupled radiative cooler and partially trans parent window via exchange of a cool ing fluid . Thus cooled windows can offer higher transmission intensities of the visible light compared to a standalone window. As established in the previous sections, there is a tradeoff between transmitted intensity of the visible light and the temperature of the wido w and the back wall. For example , we have established that the maximal transmission of the visible light through the window VIS wT , so that the w indow temperature does not exceed the atmospheric temperature a wTT equals to () max 2VIS wT= , where 10.182a a−==− and we have to demand that the window absorption in the visible is zero 0VIS wA→ , while window transmission in the mid -IR is almost perfect 1MIR wT→ . Additionally , we have established that in the case of radiative cooler s, the cooler temperature can be made significantly smaller than that of the atmosphe re ()14min 0.94w a a aT T T=    by requiring that the cooler surface is almost completely reflective in the visible, and that the radiative cooler absorption in the visible is much smaller than the window absorption in the mid -IR. In what follows we explore a scenario where smart window and a radiative cooler are thermally coupled. In this case, radiative cooler is used to reduce the temperature of a smart window, which, in turn, allows to increase the intensity of transmitted visible light through the window, without increasing the window temperature beyond that of the atmosphere. Particularly, w e conside r two channel s, one inside of a smart window and another one inside a radiative cooler that contain a transparent (in the visible coolant ) fluid such as water or oil characterized by the volume heat capacity vC . We suppose that the liquid flows in a closed loop between a smart window and a radiative cooler with a flow speed fv , and that both ch annels have the same crossection area ~channelA L d  , while both the cooler and the window have the same radiative area s 2 .~rad areaAL , where L is a characteristic window/cooler size, and d is the coolant channel size . Then, the heat transfer rate (per unit area) from the window wQ and the cooler cQ can be defined as: 12 ; w c w c QQ TT=  =  (17) where the heat transfer rate per unit temperature is defined as ()2 .areachannel vf radAC v W m KA  =   . Then, using energy conservation of the radiative and thermal fluxes we can get the modified system of equations that describe coupled window/radiative cooler system similar to that presented in the previous sections . Thus, using subscripts c to indicate cooler variables, while using subscript w to indicate window variables (also see Fig. 7) we write: () ()4 4 4 44 44 4 4 44 42 ) 2 (1 )(1 VIS MIR MIR MIR c c sc c a a w c c c VIS MIR MIR MIVIS MIR MIR MIR w w sw w a a c w w w VIS MIR MIR MIR w w a a w w R c c a a c c c scw swA P T T T T T T P T T T A P T T T T T T P T T T R TRT                         +  + +  +  + −  =   +  +  = −   + −  =   +  +  = −         (18) While analytical solution of (18) is possible after linearization of the forth order temperature terms, due to large number of materials coefficients , the resultant expressions are complicated. To demonstrate how coupling of a radiative cooler to a window can result in the window enhanced transmission properties, we first consider a particular case of strongly absorbing in the mid -IR cooler and window materials 1MIR I wMR c== . Moreover, for the radiative cooler we suppose that all the visible radiation is reflected without absorption 0VIS VIS ccTA== . Finally, we suppose that window material does not absorb visible light 0VIS wA= . In this case one can demonstrate that sc cTT= , while equations (18) can be simplified as follows: 4 444 4 4 3 4 4 4 32 c w c aVIS w c w wa a a a a a VIS sw c w w a a a aa a a a a aT T T PTT T T T T T T T PTTT T T T TT TTT TT   = + −  =  + + −    =   + +   −      (19) Linearization of Eq. (20) around aT by using ()44 4 34a a a T T T T T T = +  +  we can transform (19) into : 3 34 431 4 4 4 1 2 4 4 1 44VIS w a c w w a a a a a VIS sw a c w w a a a a a c a w c a a a aT T T T T TT T T PTT T T T T T T T PTT T T T T T               − −=− + −   =  − + −    −=  − + −           (20) Finally, r emembering expression for ()42aa PT= −  from a single layer model of the atmosphere, equation (20) can be rewritten in the following form: 13 1 31 3 1 312114 8 2 1211412314 8 2 82VIS c a a w aVIS w a a w aa VIS sw a a aa awTTTT TTTT TTTT             − − − −− =− + −− =− +  + +   −− =− +  + +   − +            (21) Thus, in the case of no heat exchange between radiative cooler and a window 0= we retr ieve results of Section 2 , while in the case of efficient heat exchange between the radiative cooler and a window 321aT we find that both the window and the wall (behind the window) temperatures have decreased, while the temperature of the cooler surface has inc reased no heat exchange between window and cool er 0 strong heat1 2 1 2214 8 4 8 1 2 1 2434 8 8 44 1VIS VIS w a a w a a ww aa VIS VIS s c a c aaw a a sw a a ww aa w aTTTTTT TTTTTT T TTT TT               =− − − −=− +   =− +   − − − −=− +   =− +  = −=→ − 3 exchange between window and cooler 2aT (22) From expressions (22) it also follows that maximal visible transmission through the window can be increase d in the case of a coupled radiative cooler / window system, while still maintaining the window and the wall (behind the window) temperatures below that of the atmosphere. For example, in the case of a standalone window , from (22) it follows that both the w all and the window temperatures are smaller than the atmosphere ,0w swTT as long as 0.092VIS wT . At the same time, for the thermally coupled window and radiative cooler, it follows from (2 2) that both the wall and the wind ow temperatures are smaller than that of the atmosphere as long as 20.123VIS wT , which is a significant improvement over the uncoupled case. 4.1 Enhancement of the visible transmission of a smart window coupled to a radiative cooler as a function of the heat exchange efficiency between the two. General case Finally, in the general case of nonlinear equations (18) , analytical solution can still be found in the limit of strong heat exchange ()34MIR wMR c aIT  (this condition can be derived in a general case by linea rizing nonlinear terms in (18)) . Indeed, in that case the temperature of the window w ill be equal to the temperature of the radiative cooler and to that of the cooler fluid f w c T T T= and the system equations (18) can be simplif ied by adding the first and the third equations to give: ()() ()4 44 4 4 444 4 4(1 ) (1 )2VIS MIR MIR c c c sc VIS MIR MIR MIR c c a a cVIS MIR MIR w w w sw VIS MIR MIR MIR cMIR MIR a a w c f f fw w a a w w s scwP T T AT AT T P T T T P TT TR T TT R              +  +  + +  + + +  = − =+ +  +  = −    (23) 14 Remembering expression for ()42aa PT= −  from a single layer model of the atmosphere, system of equations (23) can then be solved analytically to give : () ()4 4 4 4 03 4 441 limit of strong heat exchange : no heat exchang1 21 0 e:22 2 MIR VIS VIS c cMIR VIS VIS w ww MIR w w MIR Ma IR a w w VIS ww wc MIR c c MIR MIR ac aa ac aaTTA T TTTTA T TT TA T TT        → =→  +  ++  +  + + == + − + = − + + =+ ()4 4 01 22 21 ;MIR VIS w MIR w MIR MIRMIR VIS VIS c cc MIR c c aa MIR MIR a c c wwTTA T TTT T =+  + = + −++  + (24) From (24) it follows that the temperature of a smart window in the limit of strong heat exchange with a radiative cooler will always be smaller than t hat of a standalone window 0 wwTT→ = as long as the temperature of a standalone radiative cooler is smaller than that of a standalone window 0 0 c w T T = = . 4.2 Potential realizations of the strong heat exchange regime between sma rt windows and radiative coolers Here , we conside r several practical examples of the forced and the gravitationally driven heat exchange s between a smart window and a radiative cooler of characteristic size ~1 Lm . We suppose that the heat transfer between the two is realized via exchange of a coolant that flows across the edge of the window and into the cooler (see Fig. 7). Assuming that the coolant fluid of volumetric heat capacity ()63~ 4 10 vC J m K (water ) flows through a channel of size ~ 3 d mm (and width ~ L ), then the coolant flow speed necessary to realize the regime of strong heat exchange is given by: 3 3 .area44 ~ 0.5 channel a v f v f a f rad vAT d L mmC v C v T vA L C d s=   =     (25) We note that thus found coolant speed is quite modest and can be easily realized via passive gravitationally driven liquid flow. Indee d, by mounting a radiative cooler of temperature cT on top of a smart window of temperature wT , due to coolant density dependence on temperature, this arrangement will produce a pressure differential across the window/cooler assembly: ()() ( )()() ~w af cc w P gL T TT T T gL     = −   − (26) where the cooler fluid thermal expansion coefficient is 4~ 2 10f− (water), coolant density at the ambient temperature is ()33~ 10 aT kg m (water), and a freefall acceleration is 29.8 g m s . This pressure differential then drives an upward flow of the hot coolant from a smart window and a downward flow of the cold coolant from a radiative cooler. For the coolant flow confined to a narrow channel of size d , and assuming that the channel width is comparable to the channel length ~ L one can then define the channel hydrolic resist ance as 312cRd= , where the coolant viscosity is 310 Pa s− (water) . The h ydrolic resistance can then be used to relate the pressure differential across the channel to the volume flow rate 15 through the channel 3 V t m s , or equivalently to the gravitationally driven heat transfer rate (per unit area, per unit temperature) through the channel as follows: ()()3 .. ~12a v f vv grav c rac d surface rad surface cwT C g CC V P V P d t R A tTATRL   =  = =   − (27) Finally, in order to reali ze the strong heat exchange regime between a smart window and a radiative cooler using only gravitationally driven flow we must demand that: ()()3 3 3484 ~ 0.35 a grav awatwerc a v fT LTKT C g dTT− (28) As follows from Eq. (28) efficient heat exchange between a smart window and a radiative cooler is possible to realize by simply mounting one on top of the other and using a completely passive gravitationally driven flow, as long as temperature differential between the cooler and a window is larger than a fraction of one degree. Considering that standard radiative coolers can achieve temperatures that are tens of degrees below that of the ambient, then even under the daylight illumination conditions, we believe that proposed gravitationally driven heat exchange mechanism is viable . SUMMARY We have considered several simple physical models of the radiative coolers and partially transparent radiative windows. We have then distinguished the design goals for these two structures and provided optimal choice for the material parameters to reali ze these goals. Finall y, we studied a thermally coupled window / radiative cooler system and showed how transmission of the visible light through the window can be enhanced without raising the window temperature above that of the ambient. Out findings can be summarized as follo ws: Radiative cooling problem. Radiative coolers are normally non -transparent, and the main question is how to design a cooler so that the temperature on its dark side is as low as possible compared to that of the atmosphere. This is achieved when window i s almost perfectly reflective in the visible 1VIS wR→ (alternatively 0, 0VIS VIS wwTA→→ ), and when window absorption loss in the visible is much smaller than that in the mid -IR VIS MIR wwA . Then, in the absence of convection, and for any choice of the material optical parameters in the mid-IR ( ,MIR MIR wwT ) ()()14min min 0.94w s a a aT T T T  = =    . In the presence of convection, radiative cooling efficiency reduces potentially to zero when convection is s trong. Partially transparent radiative window problem. For the radiative windows, their surfaces are partially transparent, and the main question is rather about the maximal visible light transmission through the window at which the temperature on the wind ow somber side does not exceed that of the atmosphere. This is achieved when window transmission in the mid -IR is almost perfect 1MIR wT→ (alternatively 0, 0MIR MIR wwR→→ ), while window absorption in the visible is small 0VIS wA→ and at the same time much smaller than that in the mid - IR VIS MIR wwA . Then, in the absence of convection, the maximal allowed transmission through the window in the visible is 10.182VIS a w aT−= = =− while the wall, room and window temperatures are all smaller than the atmospheric one. If only requiring that the room and window temperatures are smaller than the atmospheric one, then the maximal transmission through the window in the visible can be inc reased to 40.243VIS wT==  . In the presence of convection, maximal allowed window transmission in the visible reduces somewhat (at most as a certain multiplicative factor) even in the presence of strong convection. 16 Effect of convection. Convectio n contribution becomes important for high enough values of the heat transfer coefficient 38 (2 )MIR MIR a w w h T T + , which for the standard ambient temperatures is equivalent to ()212 (2 )MIR MIR ww h W m K T   + . The value of the heat transfer coefficient depends strongly on the temperature differential between the window and the ambient temperatures , as well as the window size and orientation and is generally in the ()21 10 W m K− range for the non -forced convection (see Supplementa ry Material C for more details). Finally, by thermally coupling smart windows to radiative coolers using passive or active flows of a fluid coolant placed inside of the window and cooler structures, one can significantly increase transmission of the visibl e light through the window, while keeping the window temperature below that of the ambient. This enhancement is a function of the heat transfer rate per unit temperature , and the largest enhancement is achieved in the limit of strong heat exchange between a window and a cooler 34MIR awMIR c T    , which for the standard ambient temperatures is equivalent to ()26MIR cMIR w W m K   . Furthermore, strong heat exchange regime betwee n radiative coolers and smart windows can be realized with small coolant velocities (sub 1mm/s for ~1m-large windows) or even using a purely passive gravitationally driven coolant flows between a hot smart window and a cold radiative cooler mounted on top with minimal temperature differential (sub -1K) between the two . ACKNOWLEDGMENT We would like to acknowledge financial support of the l’Institut de l’énergie Trottier and Canada Research Chair of Prof. Skorobogatiy for this project . 17