. In this work we solve approximately 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 . Our physical model includes the atmosphere, the window , and the backwall that are all in the thermal e quilibrium with each other, and that can exchange energy via radiative heat transfer or convection. Moreover, we use a simplified two -state model for the optical properties of an atmosphere and a window material which assumes two distinct sets of the optic al reflection/absorption/transmission parameters in the visible/near -IR versus mid -IR spectral ranges. Furthermore, w e have distinguished the design goals for the partially transparent windows and radiative coolers and provided optimal choice for the mater ial parameters to reali ze these goals. 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 radiati ve 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 sid e does not exceed that of the atmosphere. We believe that our simple physical models complimented with an in-depth comparative analysis of the radiative windows and coolers can be of interest to a number of scientists and engineers pursuing research in the se disciplines. 1. Introduction Currently, the problem of energy overconsumption presents one of the major challenges for many societies [1]. In particular, energy spending for building cooling, heating and lighting accounts for ~40% of the total energy con sumption in some countries [2, 3]. While several methods to reduce energy use for the cooling of buildings, automobiles, food depots and other structures are known , radiative cooling distinguish es itself by being one of the few temperature reduction method s that require no external energy source s 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 [4]. Standard passive cooling devices use large heat sinks with lower temperature than that of a cooled object to reach large outgoing en ergy fluxes [5]. On a planetary scale, 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 for the planet [4]. 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 depend ing 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 peak in the spectral density covering the wavelength of 8 -15 m [5]. Remarkably, in t he same spectral rage, the Earth’s atmosphere has a transmission window (window of transparency) that allows a considerabl e portion of the mid -IR thermal radiation to leave unimpeded [5,6]. This represents the main mechanism by which terrestrial objects ca n 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 this 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 extending all the way to the near-IR 0.3 -2.5 m [7]. The study of radiative cooling has a long history. It has be en 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 [8, 9]. Temperature reduction of about 5 ℃ was reported by Granqvist in 1981 [10], using a low emittance window surface and night time cooling. This original work was followed by several studies of radiative cooling efficiency under different environmental conditions, such as humidity [11, 12], ambient temperature [1 3], and geographical location [14]. This original work was followed by studies of the day time cooling, which turned to be a much harder problem. Originally, daytime radiative cooling under direct sunlight was achieved by using a radiator that would refle ct most of the sunlight, while radiating efficiently in the mid -IR spectral range [15], 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 design more complicated. This is because the cooler materials have to exhibit widely different thermo -optic properties in the v isible 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 t hermochromic windows [16 -18], p hotonic crystals and metamaterials [19,20] ), as w ell as structures and design optimization (reflecting vs. absorbing structures [21 - 23]), with experimentally demonstrated temperature reduction from 5 ℃ to 42℃ [15, 24, 25] . 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 [12, 26]. Thus, for the night time radiative coolers the predictions range between 15 -42℃ [27 -29], while for the daytime radiativ e coolers the temperature reduction is expected to be only several degrees [15]. An important issue when characterizing radiating coolers is the choice of parameters to characterize the cooler performance. While the cooler itself has a working surface, it s 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 the day and night operation [30]. 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 radiati ve properties in response to the changing environmental conditions such as ambient temperature and daylight illumination [31-33], or if it is capable of both cooling and heating , such windows are frequently referred to as intelligent or smart window [34]. Recently, there has been a strong interest in passive and active smart windows that could operate year - long, while providing heating in the cold and cooling in the hot w eathers [30,35]. Much research is focusing on developing materials that can simultaneously, while independently manage radiation across several widely spaced spectral ranges covering, for example, visible / near -IR light [14, 36, 37], or solar / mid -IR lig ht [25]. At the same time , a concept of perfect smart window was proposed to judge the energy efficiency of the existing smart windows [38, 39]. Thus, t he 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 either (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 between the two states, while always featuring low absorpti on of solar radiation [38]. Therefore , much attention was payed to improving the performances of the smart windows through materials research and structural optimization [35, 40 -45]. Currently, there three main types of smart windows in development that use electrochromic, thermochromic , photo chromic materials and combinations of thereof . Electrochromic materials employ reversible redox reaction that affects material’s electronic transition s, and , as a consequence, the absorption profile of the solar spectra [34]. Several metal ox ides have been reported for applications in smart window s among which the tungsten oxide (WO 3) 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 [46,47]. 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 (VO 2) based material s are the mos t popular ones to regulate transmission of the visible and near-IR light using the t hermochromic effect [47]. 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 high reflection [46]. In addition, the VO 2 phase transition temperature ~68 ℃ is relatively high for the practical applications , while transitions temperatures in the 20℃~30℃ would be more desired. Finally, p hotochromic smart windo w can alter its light transmission properties depending on the intensity of the incident s unlight. Photochromic materials have large work ing temperature range (from 20℃ to as high as 80℃), and are relatively abundant [46,50]. While much work has been done on theoretical understanding o f the functioning of radiative coolers, in the related field of smart windows the performance targets and optimization strategies are still less understood . As functioning of the smart windows is d ominated 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 acknowledge that the function of a window implies nonzero visible light t ransmission , therefore a tradeoff between the window performance and its esthetic function seems unavoidable . In this paper we consider in more details partially transparent radiative coolers/heaters in the context of their application in smart windows. Fr om the onset we distinguish the design goals for the radiative coolers from those of the partially transparent (in the visible) radiative 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 atmosphe re. 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 th e 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 adj acent inner air region or even that of the room wall. Finally, one needs to acknowledge that in practic al application, heat convection is always present and its contribution to the smart window performance has to be quantified. Ideally, i n order to model precisely the smart window performance , one has to have a realistic model of the atmospheric optical pro perties [51], and then solve a full radiative heat transfer problem [ 52] that include s atmosphere, window and enclos ure where the window is installed . While certainly possible, such an overwhelming approach will most probably obscure the relatively simple physics behind the problem. Therefore, in this work we strip the radiative heat t ransfer model to a bare minimum and confine ourse lves to one dimen sion, while still retaining all the key elements of a problem . Particularly, we consider the atmosphere, the window and the backwall that are all in the thermal equilibrium and that all exchang e the energy via radiative heat transfer or convection. 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 versus mid-IR spectral ranges . We believe that our work captures all the important aspects of the radiative heat transfer in smart windows and allows to make predictions on the choice of the windows optimal design parameters. 2. Model 1. Single layer model of the atmosphere . We start with a single layer model of the atmosphere [ 53]. This model allow s to relate together the Earth and the atmosphere average temperatures, the atmosphere average emissivity, the planet albe do (reflectivity) and the total incoming radiative flux from the Sun. The model assumes equilibrium temperature distribution, and, henc e, net zero radiative heat flux through any interface parallel to the Earth. A key element of a model is a recognition that in equilibrium, material absorption coefficient equals to the material emission coefficient. This model i s a necessary point of depa rture that allows to define a self-consistent reference ( average atmospher ic properties versus average Sun radiati ve flux ) for the following model s of a smart window. Notation: ( , )ET - emission spectrum (by power) of a blackbody at temperature T (see appendix) ()P - energy flux spectrum of the incoming light from the Sun Atmosphere - aT - temperature of the atmosphere aL - thickness of the atmosphere ()ar - frequency dependent reflection coefficient of the atmosphere by power (mostly due to Rayleigh scattering ) ()a - material absorption coefficient of the atmosphere by power ()at - frequency dependent transmission coefficient of the atmosphere by power: () ( ) 1 ( ) exp( ( ) )a a a at r L    − − ()a - emissivity of the atmosphere at frequency equals to its absorption : ( ) ( ) 1 ( ) ( )a a a a a r t    = = − − The Earth - gT - temperature of the Earth ground We assume that e missivity of the Earth at any frequency is 1 (blackbody approximation) Fig. 1 . Single layer model of the atmosphere . Assuming a constant temperature in the atmosphere , we now use the energy conservation principle to write the following equations : At the space edge of the atmosphere we write for the energy flux es in equilibrium: (1 ( )) ( ) ( ) ( , ) ( ( ) , )a a a a g dd rt d P E T E T         −  =  +    (1) At the E arth level we write for the energy flux es 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 atmospher e: 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 vi sible 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: () () () ()4 4 4 4( ) ( ) 1 ( ) 1 (1 ( )) ( ) 1 (1 ( )) ( , ) ( ) ( , ) 1 ( ) ( , ) ( ) ( ) 0 ( ) ( , )a a a g g a g a g a a a a a a a g gd d d d adt P d a P al P r P al lP r E T d T t E T T E T T P E T T ad                               − = − −  = − −  =  = − = = ==         (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 results 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 account 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 measur ed by an orbiting satellite. 3. Model 2. Single layer optically symmetric window, no convection. We no w consider a single layer optically symmetric window placed in the path of a sunlight. The window is assumed to be optically thick (no interference effects ). We also assume that the temperature across the window is constant (see discussion in Appendix A), and, therefore, the thermal radiation from the window is the same in both directions . Behind the window we place a perfect absorber that we refer to as a wall. 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 and conduction heat transfer in this region . Finally, the Eart h atmosphere is characterized by the average temperature and permittivity defined in the previous section . Notation: gaP t P - spectrum of the incoming radiation from the sun at the ground level by power 1 P al P - avera ge power of the sun at the planet (window) surface Window - wT - temperature of the window ()wr - frequency dependent reflection coefficient of the windows by power ()wt - frequency dependent transmission coefficient of the windows by power ()wa - absorption of the atmosphere at frequency equals to its emissivity : ( ) ( ) 1 ( ) ( )w w w w a r t    = = − − Wall - sT - temperature of the wall surface behind the window which is considered to be a blackbody . Spectral dependence of the window material parameters are assumed to be step -like, with two distinct sets of frequency independent parameters defining window proper ties 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 window (absorption, transmission and reflection). Fig. 5. Model of the optically symmetric single layer window (no convection /conduction between the wall and the window ). We now use the energy conservation principle to write the following equations: At the window / atmosphere interface we write for the energy flux es 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 flux es 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) These equation s can be further simplified 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) 3.1 Analysis of the Model 2 1) A particular solutions of Model 1 (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 window s in the mid -IR 1MIR wT→ , which simultaneously feature low a bsorption 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 thro ugh 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) 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 a n almost completely reflective window in the visible, which at the same 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 transmissio n of the visible light through the window VIS wT , so that the wal l temperature does not exceed the atmospheric temperature a sTT . From (11) it follow s that: () ( ) ()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) In order to achieve maximal transmission of the visible light, while still having the wall temperature belo w that of the atmosphere, we have to demand that window absorption in the visible is zero 0VIS wA , while window transmission in the 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, w hile 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 absorption 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 from 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 temperat ure. Intersection of these two regions represent the ra nge 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 attai n 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. 4. MODEL 3. Single layer optically symmetric window, with convection. In this model we add convection heat transfer between window and air that surrounds it , while ignoring heat conduction through the air . The model is virtually identical to the Model 2, with only two new parameters, which are the room temperature (space between the window and the wall), and the heat transfer coefficient. Value of the heat tr ansfer coefficient is in general temperature dependent and can be significantly different for horizontal or vertical surfaces. In Appendix B, for completeness , we present several well-known models for the heat transfer coefficients at the planar interfaces between gas and solid . Notation: Room – rT - temperature of the air -filled space between window and a blackbody wall. h - temperature dependent heat transfer coefficient that defines convection at the soli d/air surfaces Fig. 7. Model of the optically symmetric single layer window (with convection). We now use the energy conservation principle to write the following equations: At the window / atmosphere interface we write for the energy flux es in equilibrium: () () (1 ( )) ( ) ( ) ( , ) ( ) ( , ) 1 ( ) ( ) ( , ) g w w w s a w w a w a r P E T t E d d d d T r E T h T T              −  =  +  − −   + −    (17) At the window / room interface we write for the energy flux es in equilibrium: () ( ) ( ) ( ) ( , ) (1 ( )) ( , ) ( ) ( ) ( , ) w w w s g w a wa wr t P d E T r E T d d t E T d h T T                =−  + −  −   − −    (18) By subtracting ( 18) from ( 17) we also get the balance between the absorbed and irradiated energy in the window , as well as heat transferred into the window due to convection : ( ) 2 ( ) ( , ) ( ) ( ) ( ) ( , ) ( ) ( ) ( , ) 2w w w a a w s w a wr dd E T P E T E T d h T T T d                   =  +  +   + − −    (19) Finally, in thermal equilibrium, two convection flows inside of the room have to cancel each other: ()()02s r w rsw r h T T T TTTT h− + − = += (19) Assuming step -like spectral properties of the window material optical properties in (17 -19) (see Fig. 4) we can then write the following system of equations (only 2 of them 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 w a VIS MIR MIR MIR w w w w s w a a w r MIR VIS MIR MIR w w w w s w a a w a w rR P T T T R T h T T T P T R T T T h T T T A P T T h T T h T T                −  =  +  − −  + − =−  + −  −  − −  =  +  +  + − + − (21) Now, using results of the M odel 1 that relates atmosph eric temperature to the average light power arriving at the planet surface, we can arrive to the following coupled nonlinear equation s for the temperatures of the window and the wall: () ()3 34 4 4 4321 22 22(2 ) 2 2 (2 2 )2MIR VIS VIS w ww MIR w w aa MIR MIR a w w VIS VIS s w w aMIR w a w s a w MIR MIR MIR a w w a a w w s a MIR MIR a w wa MIR MIR a w w aTTA T TT TT T T T T T h T T T T T T T h TTA TT TT     − − −++ +− ++  + + = + −+ ++ = + −+  (22) These equations can be solved analytically by linearizing the nonlinear terms assuming that the wall and window temperatures are close to the atmospheric temperature 4 ,, 414s w s w a aTT T T , where , , sa sw wT T T . In this case, a system of nonlinear equations (22) becomes linear : ()()3 3232 (2 ) 2(1 4 1 22 42)wMIR MIR w w s w MIR MIR MIR MIR a w w a w a w wMIR VIS VIS w ww M MIR w w aa MIR M IR MIR a w wIR a w w X as aT T T T h T T T T T hTTA T T TTT T T T          ++−+ +  ++ + = − + −+ + ()()2122 sVIS VIS ww aa MIR MIR ww Xs aT TA T T+=−  +−+ (23) and can be solved analytically to give the following expressions: 2 24 (4 )(4 (2 3 )) 4 (2 3 ) (4 )(4 (2 3 )) 2 2 (1 2 ) 2(4 )(4 (2MIR w w w s MIR w w MIR MIR ww a MIR MIR ww MIR w s s w MIR w s MIR MIR ww a MIR MIR ww MIR w w s s MIR s w w r aaTX X X T TT T TX X X T TT T TX X XTT T TT             ++ = + + + + + + − = + + + + + + ++== + + +23 ))MIR MIR ww MIR MIR wwTT+ (24) 4.1 Analysis of MODEL 3 1) A particular solution of Model 2 (Eq. 22) for which convection terms do not contribute is w s wT T T when choosing VIS MIR wwTT , VIS MIR wwA , for any values of ,MIR MIR wwT and h . 2) We now study how addition of convection influences maximal allowed transmission of the visible light VIS wT through the window. As a design condition we demand that either the window temperature or the wall te mperature or the room temperature do not exceed the atmospheric one ,,ws a rTT . From (24), without making any assumption on the strength of a convection coefficient we find that the abovementioned design conditions define the following allowed regions in the S wVIT , VIS wA parameter space (equations in blue in (25)) : ()114 2 1 1 2 2 1 14Window te 42mperature: 0 m4 ax0 MIR MIR MIR MIR MIR VIS VIS MIR w w w w w w w w MIR MIR MIR MIR wwMIR w w w s M w VIR w ww IS ATX X X T TT T T TT A T                  + +  + + + +   + + +                            + +     = () ()02 2 1, 0 - weak convection 114 1where 1,1 , - strong convection1 1 2 2122 42VIS MIR w MIR MIR w w MIR w w MIR MIR w w MIR MIR wMIR MIR MIR MIR ww ww ww T f T T T f TT        = → ++   +   +  + =→   →  +    +  + () 032 1 2 1 1 2 2 1 1 34 4 2 4Wall temp 4 (2 3 erature: 0 max)0 VISMIR MIR MIR MIR VIss S VIS MIR w w w ww MIR MIR MIR w w wwMIR w s VISMR w AIXX T TT T TTTTX A       =           + +  + + +   + + +                   + + −     =           () ()1, 0 - weak convection 1 1 3 24 2where 3 2,1 , - strong convection 31 1 2 22 312 8MIR w MIR MIR w w MIR s w MIR MIR w w MIR MIR w wMIR MIR MIR MIR ww w s w T T f TT f T T        → +++ =→   →   ++  +   +   +         () 022 Room temperature: 0 4max323 1 2 2 1 2 4 1 1 22(1 2 ) 0 24 VISMIR MIR MIR MIR MIR VIS VIS MIR MIR MIRr MIMIR ws R VIMIR MIR S M Rs IR AM ITX TTT TTT A T TX TX      =            + +  + + + +   + + +                       +  =++ + () ()1 1 2 1, 0 - weak convection4where 3, - strong convect4 32 ion1 1 2 43MI MIR MIR r MIR MIRR MIR r T f TfT     ++ →  =→  → ++  +       (25) Tradeoff between window transmission and absorption in the visible can be summarized geometrically in Fig. 8. As follows from inequalities p resented in (25) , regions below the corresponding lines represent the range of values of ,VIS VIS wwTA for which either the window temperature or the wall temperature or the room temperature are smaller th an the atmospheric one. For practical purposes, we are mostly interested in the room temperature and the wall temperature (shaded regions in Fig. 8) . Intersection of the corresponding two regions represent the range of values of ,VIS VIS wwTA for which both the wall temperature and the room temperature are smaller than the atmospher ic one. Both the window, the wall and the room 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 t hat both the window and the room temperatures will be smaller than the atmospheric temperature. Also , from (25) we observe that the maximal allowed v alue of the transmitted light in the visible reduce s somewhat when conve ction is present. The new value represent s a fraction of the original value when no convection is present. Fig. 8. Tradeoff between window transmission and absorption in the visible (with convection) . A region of parameter space which is situated below the two lines define window operation regime for which both the wall and the room temperatures are smaller th an that of the atmosphere. For completeness , a third dotted line is present ed along which the window temperature is equal to that of the atmosphere. 3) We now study how addition of convection influences minimal achievable values of the window, wall and roo m temperatures in the problem of radiative cooling. Particularly, as follows from Model 2, radiative cooling is most efficient when window is almost perfectly reflective in the visible, and when window absorption loss in the visible is much smaller than th at in the mid -IR, which can be summarized as 0, 0,VIS VIS VIS MIR w w w wT A A . In the absence of convection, we then found that 14min min min 0.94w s r a a aT T T T T , or alternatively 1min min min 0.0554w s r a aaaTTT TTT . When convection is present, from analysis of the linearized equations (24) it follows that for the radiative cooling to be most efficient, the same conditions as in the case without convection have to be satisfied 0, 0,VIS VIS VIS MIR w w w wT A A . In this case, 1s w aXX and from the following expressions for the minimal achievable window, wall, and r oom temperatures we conclude that the cooling efficiency degrades (minimal achievable temperatures grow): () ()2 2114 1min41 1 2 34 4 4 1 1 34 1min41 1 2 34 4 4MIR w MIR w a w MIR MIR a ww MIR MIR ww MIR w MIR w a s MIR MIR a ww MIR wT T T TT T T T TT             ++−  =     + + + +         ++−  =     + + + +         ()() 21, 0, weak convection11~ , ,strong convection 4 1 1 24 1min41 1 2 34 4 4a MIR w MIR w MIR w a r MIR MIR a ww MIR MIR wwT T T TT               → −=→   ++  −   =       + + + +           (24) 5. Summary We have considered seve ral simple physical models of the radiative coolers and partially trans parent radiative windows. We have then distinguished the design goals for the se two structures and provided optimal choice for the material parameters to realise these goals . Out findings can be summarized as follows: 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 is 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 o f 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 strong. Partially transparent radiative wind ow 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 window 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 10.18 2a a while the wall, room and window temp eratures are all smaller than the atmospheric on e. If only requiring that the room and window temperatures are s maller than the atmospheric one , then the maximal transmission through the window in the visible can be increased to 40.243 . 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. Effect of convection . Convection contribution becomes important if t he characteristic convection parameter is large 3148 (2 )MIR MIR a w wh TT , which is equivalent to 6.46 (2 )MIR MIR wwWhTK . The value of the heat transfer coefficient depends strongly on the temperature differential, as well as the window size and orien tation and is generally in the 1 10W K range (see Appendix B for more details). 6. Acknowledgements We would like to acknowledge financial support of the l’Institut de l’énergie Trottier and Canada Research Chairs of Prof. Skorobogatiy and Prof. Caloz for this project. 7.