The best bevel shape for optimized daylighting and thermal performance

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It shows that the diagonal cuts in insulation around windows substantially increase the amount of light coming into the building without compromising the heat resistance of the barrier. The effect of various depth and angles of cuts on the daylighting as well as the thermal performance of the envelope is simulated. An optimal geometry is found as a function of insulation thickness. The results were obtained using the finite element method. A novel approach was used that allows the daylighting and thermal simulation to be performed in the same simulation environment using the same geometry. Additionally, a traditional daylighting analysis was performed. They show that for the thick insulation, bevels make difference between having and not having satisfactory levels of light in the room. window daylight thermal performance finite element method cuts in insulation Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 1. Introduction Daylighting is an important factor for comfort and productivity inside a building (Marc et al., 2004 ). It does not only provide free illumination but has a very positive impact on wellbeing (Veitch & Galasiu, 2011 ). Now in the era of growing demand for low energy buildings, there is a tendency to increase insulation and thus wall thickness. Such a thick wall provides additional shading that decreases the amount of light coming in. A very cheap method mitigating that effect is the introduction of diagonal cuts, called bevels, in the insulation layer like these pictured in Fig. 1 . It has been shown that bevels may increase the amount of light coming into the room by up to 30% (Di & Radinger, 2010 ) but on the other hand, bevels may compromise the thermal proprieties of the wall. Given the importance of the problem, it is surprising that it has not been investigated thoroughly so far. There is very rich literature on the thermal performance of windows. There are holistic approaches (Gustavsen et al., 2008 ; Lechowska et al., 2017 ; Vendelboe et al., 2008 ; Zajas & Heiselberg, 2011 ) approaches that focus on separate components (Cappelletti et al., 2011 ; Van Den Bergh et al., 2013 ). There is plenty of literature that optimizes daylighting and energy performance of the window (Jakubiec & Reinhart, 2011 ; Yan et al., 2013 ). Most of them use the classical approach of building simulation where heat transfer through the wall is one-dimensional and temperature gradient along the wall surface is not allowed. Such modeling fails in estimating the influence of thermal bridges, which are the main concern while optimizing bevels around windows(Cappelletti et al., 2011 ). To account for thermal bridges in 3D finite element modeling is the most appropriate approach (Agnoletto et al., 1995 ). There are many examples of excellent models of thermal performance of building envelopes including fenestration (Asdrubali et al., 2013 ; Lechowska et al., 2017 ; Van Den Bossche et al., 2015 ; Zajas & Heiselberg, 2011 ). However few of them takes daylighting into account. Author of the publication (Lechowska et al., 2017 ) models thermal performance of the window frame using FEM and calculates daylighting using classical architectural software. Publication (Liu et al., 2019 ) also describes thermal and daylighting performance of windows. It also analyses the influence of skew cuts in insulation like this paper. However it focuses mostly on the performance of the external shutter and does not optimise the geometry of the cuts. Moreover it is using RC network model. Such model is not capable of fully simulating the impact of thermal bridges. To the best of author’s knowledge there is no study that deals with the influence of bevels on both daylighting and thermal properties of the wall. 2. Methodology Two models of a window are presented in this public: the Comsol model and the Radiance model. The Comsol model is used to calculate the amount of light coming through the window as well as temperature distribution in the simulated structure. This simulation allows finding optimal bevel depth and angle as a function of thickness. It is where the Radiance model comes in. It is used to calculate the daylighting metrics for a space that utilizes a window with bevel geometry optimized in the Cosmol model. 2.1. Comsol Model The FEM modeling was performed assuming stationary environmental conditions. The conditions were supposed to mimic a cloudy winter day in a temperate climate. The cloudy day was chosen because on such days natural light is in deficiency. The external temperature of zero °C was chosen because it models an average winter temperature in temperate climates. And it is winter where the heat loss matters the most. The modeling of daylighting, as well as the thermal behavior of the wall, was performed in COMSOL Multiphysics software. A wall with a small window shown in Fig. 1 was modeled. The wall is a 30cm thick masonry wall with typically 30cm of insulation applied to the external surface. Different thicknesses are analyzed in the following chapter. The dimensions of the wall are the width of 2 m and height of 2.5 m. The window is mounted directly to the masonry since this traditional method is still very popular. The window frame is a generic one, the internal structure is not taken into account in the simulation. The cross-section of the window mounting is shown in Fig. 2 . On the perimeter of the window, there is a bevel cut in the insulation. The depth and angle of the bevel are varied in the simulation. The windowsill is slightly tilted from the horizontal plane and its tilt angle remains constant throughout all the simulations. The external dimensions of the frame are 1x1.2m which yields 0.8 m^2 of the glazed surface. 2.1.1. Thermal simulation. To calculate the thermal behavior of the structure only convective heat exchange was taken into account. This seems justified when simulating overcast days since radiative heat exchange vastly cancels out during such weather so the net effect can be neglected. The internal temperature was assumed to be 20°C and the external 0°C as mentioned before. The convective heat exchange coefficient was 20 W/m 2 .K on all external surfaces and 5 W/m 2 .K on internal surfaces. The following thermal conductivities were assumed: wall − 0.15 W/m.K (lightweight concrete), insulation – 0.04 W/m.K (styrofoam), 0.105 W/m.K – window frame (yields U value of 1.16 W/m 2 .K at 9 cm thickness). The windowpane was not simulated. Heat loss on the 0.8M^2 of glazing was added in post-processing assuming a moderate U-value of glazing of 1.0 W/m 2 .K. yellow - window frame, green - wall and brown - insulation, α - bevel angle, d - bevel depth. (Colors in print are not important) 2.1.2. Amount of light The amount of light coming into the space was modeled in COMSOL Multiphysics as well. The software does not have any light propagation module as such, but the Comsol physics called Heat Transfer has a component that calculates radiative heat propagation as well as radiation scattering using the Lamertian reflectance model. The idea here is to utilize the radiative heat propagation tool to simulate light propagation. To do so we introduce a surface that represents the sky. This object has an arbitrary temperature distribution tailored to yield the desired luminance according to black body radiation. This way it is possible to generate arbitrary lighting. The sky is the only light-emitting object in our model. The surfaces that do not emit light have the virtual temperature set to zero. Their reflectance is set to a value less than one depending on the brightness of their color. This allows us to model the scattering of light on the surface. The lower the reflectivity value, the more light scatters on the surface. The source of radiation was a quarter of the sphere with a radius of 40m which mimics the overcast sky. The virtual temperature \({T}_{f}\) distribution of the surface is selected to yield a luminance \({L}_{h }\) at the angle, \(\theta\) to match the overcast sky model by Moon and Spencer (Moon & Spencer, 1942 ) also called CIE overcast sky i.e. $${L}_{h}={B}_{z}\frac{1+2 \text{s}\text{i}\text{n}\left(\theta \right)}{3}$$ 1 where \({B}_{z}\) is the zenith luminance. Black body radiation is proportional to the temperature to the power of four so the virtual temperature has to be \({T}_{f}=\sqrt[4]{{L}_{h}}\) . The virtual temperature of all surfaces of the window is set to zero so there is no emission and only scattering of the radiation takes place. The surfaces that take part in the light propagation modeling are shown in Fig. 3 . Windowsill reflectivity is set to 50% which corresponds to light color (RAL 7035 for instance) while the remaining surface reflectivities are 20% which is neither dark nor light like beige (RAL 1001). Apart from these real surfaces a virtual surface exists, it is the surface that measures the amount of light coming into the interior. It is the deep surface of the window. The reflectivity of this virtual surface is set to zero so the surface does not affect the light distribution. Such an approach is a bit cumbersome at first but allows modeling of the 3D thermal and lighting distribution in the same environment. This is very convenient because the same geometry is in use and all the alternations to the model are automatically applied to both physics i.e. heat transfer and radiation scattering. Moreover, COMSOL allows very easy parameterization of the geometry which makes parametric studies very efficient. 2.1.3. Validation of Comsol model. Thermal simulations are performed in a very orthodox manner. Comsol has proven to be an accurate tool for such calculations (Gerlich et al., 2013 ). What is unorthodox is the usage of Comsol’s radiation heat exchange functionality to calculate daylighting. This approach was tested against Radiance software. Radiance is the most generally useful software package for architectural lighting simulation first developed at Lawrence Berkeley National Laboratory that has been a benchmark for daylighting for over 30 years. The comparison between the Comsol and Radiance simulations is given in Fig. 4 . The picture shows light distribution on the innermost surface of the window - the plane that was analyzed in the paper. Ambient conditions for both simulations were the same, it was CIE overcast sky with unit zenith radiance ( \({B}_{z}\) in Eq. 1 ). The difference in average irradiations is less than 1% and the standard deviation of the difference is 6.6% of the average value. 2.2. Daylighting In the above described Comsol model, only the total amount of light coming into a building during an overcast day was considered. Therefore Radiance model was built to estimate the influence of the bevels on the quality of daylighting in a typical room. The room selected for the simulation is medium-sized with rather small windows. Its dimension, as well as the placement of windows, is shown in Fig. 5 . Such geometry yields a window to wall ratio of 0.17 and a window to wall ratio of 1.4. The reflectances of materials used for the simulation as summarized in Table 2 . Booth static and dynamic simulations were performed. Static simulators were performed for the CIE overcast sky whereas dynamics simulations used the weather file for Nowy Sącz (IMGW, 2000 ). Nowy Sącz is a city in southern Poland and it was selected for the calculations because its climate is a good example of a temperate climate for which the simulations are mostly intended. 3 Results And Discussion Thermal simulations, as well as the ones performed to determine the amount of light falling into the space, were executed for all the combinations of bevel depth from 0.01 to 0.28 m and bevel angle from 23° to 63°. 3.2 Thermal The calculated temperature distribution across the wall is presented in Fig. 6 . To estimate the thermal performance of the simulated wall the total heat transmitted through it as well as the temperature of the coldest place inside the masonry wall was looked at. The maximal increase of the total heat loss did not exceed 3% in the case of the deepest bevel cut at the widest angle. Since the heat loss caused by the bevel is so small the temperature decrease in the coldest place of the masonry wall is an important factor and it will be presented here in detail. This parameter was chosen because it is crucial for the possible building deterioration caused by water condensation inside the wall and thus the well-being of the building. The plot in Fig. 7 shows the temperature of the coldest place inside the wall as a function of bevel dimensions. One can see that temperature remains unchanged for a wide range of bevel geometries. Only for bevels deeper than 0.2 m the temperature tends to drop. There is also the dependence of bevel angle, wider angles yield lower temperature. 3.2 Light The total radiation that passed through the whole window pane was analyzed. This parameter was chosen rather than daylighting factor to get more generic results. Daylighting metrics are analyzed in section 3.6 . This radiation is equivalent to the surface irradiation integrated over the most inner surface of the window opening. The plot in Fig. 8 shows the irradiation gain in relation to total irradiation with no bevels. The irradiation is higher when the bevel is deeper and wider. The light gains depend on the thickness of the insulation The maximum gain, i.e. the gains achieved for the bevels cut almost to the window frame ranges from 6.4% for 10 cm thick insulation to 87% for the insulation thickness of 50 cm. These results are in line with the literature. 30% light gain was reported by Radinger (Di & Radinger, 2010 ). Gregor Radinger also states that there is a reduction of daylight of about 15–20% because of increasing wall and soffit-thicknesses personal communication. It is harder to compare to the results of the other publication dealing with the same problem (Liu et al., 2019 ). In the publication, the relative floor area that passes certain daylight criteria is given. For the insulation thickness of 55 cm, the floor under good lighting conditions extends thanks to the cuts from 28–33%. It is worth noticing that the relation is more linear than in the case of temperature. Therefore in the case of the moderate bevel depth, there is a meaningful improvement in lighting conditions without worsening the thermal insulating properties of the structure. It can be expected that there must be an optimal geometry of bevel that maximizes lighting but does not decrease the temperature of the wall considerably. 3.3 Optimal bevel To find optimal geometry one has to find a function that would peak for such optimal conditions. The most straightforward choice is a ratio of light gain increase to the temperature decrease. Using such a function brings a risk of division by zero. To solve that problem following function was analyzed. $$fun= \frac{{G}_{BEV}-{G}_{0}}{\varDelta {T}_{min}+0.1 K}$$ where \({G}_{BEV}\) and \({G}_{0}\) are irradiations with and without bevel. \(\varDelta {T}_{min}= {T}_{min}-{T}_{0}\) is a temperature decrease in the coldest part of the wall, caused by the bevel. This function peaks at the point when the bevel causes the biggest increase in the irradiation without decreasing the \({T}_{min}\) considerably more than 0.1 K. That means it peaks somewhere in the bevel depth of 0.16 m and bevel angle of 50°C as shown in Fig. 9 . Those parameters provide the optimum bevel geometry. Such bevel geometry increases the heat loss through the wall only by 0.2% and increases the amount of light coming in by 18%. 3.4 Parametric sweep The above-described function was used to find optimal bevel parameters as a function of insulation thickness. Similar simulation series as in the previous paragraph were run for the insulation thicknesses from the relevant range of 10 to 50 cm. The observations are summarised in Table 1 and some representative examples are shown in the plot in Fig. 10 . The obvious observation is that the thicker the insulation, the bigger the light gain caused by the bevels. The light gain goes up to 40% for the very thick insulation of 50 cm. It has to be noted that this value, as well as all the others given in Table 1 , concerns bevel optimized using the function(1) where the drop of the coldest spot of the wall is marginal – less than 0.05°C. Higher gains are possible if a bigger temperature drop can be afforded. The necessary plots are given in supplementary materials. When it comes to the optimal bevel dimensions the thinner the insulation, the shallower the optimal bevel is as a percentage of insulation thickness. So for 50 cm insulation, the optimal bevel depth is 25 cm which is 50% of the insulation thickness, whereas for 10 cm thick insulation bevel as shallow as 3 cm gives optimal results. That is only 30% of insulation thickness. The optimal bevel angle does not change with the insulation thickness. It has to be noted that the optimum bevel angle is very wide so any angle from a range of 45 to 60 degrees will be as good. This freedom of angle selection gives opportunities in design and allows to match bevel angles to other angles in the building for a more consistent visual appearance. 3.5 Wider window To check the generality of the methodology similar analysis was performed for a 150% wider window. It appears that the optimal bevel depth is the same as for the narrower window, however, the optimal bevel angle is slightly wider. These results are also summarized in Table 1 and supplementary material. Table 1 Optimal bevel parameters as well as the influence of the bevels on insulation thickness [cm] optimal bevel depth [cm] optimal bevel depth [%] optimal bevel angle [°] light gain [%] temperature decrease [°C] 50 25 50 50 40 0.04 30 13 43 50 18 0.03 20 7 35 50 8 0.05 10 3 30 50 2.5 0.05 30 wide window 13 43 58 15 0.02 Table 2 Reflectances of the materials used in the calculations. All materials were gray i.e. they had the same relativities for all the wavelengths. Material Reflectance* External wall 0.2 Window frame 0.2 Ceiling 0.7 Internal wall 0.7 Floor 0.2 Pane 0.88 *Transmittance for the case of glass. 3.6 Daylighting The analysis compares daylighting for the optimal bevel geometry to the one with no bevels for various insulation thicknesses. The results of the static calculation are summarised in Table 3 . As for the average daylighting factor, it behaves similarly to the total amount of light coming into the building. The effect of bevels is more pronounced for greater insulation thickness and reaches about 50 percent gain for the 50 cm insulation. The effect of bevels on uniformity is rather interesting. For thin insulation, bevels have no effect but while insulation is getting thicker uniformity lessens for the window without bevels. However, the same relation is reversed in the case of bevels. The thicker the insulation the more uniform the light inside. The last pair of columns in Table 3 shows the percentage of the working plane area with satisfactory DF which is between 2 and 7%. Again for the thin insulation, there is no difference whether bevels are present or not, however, for the medium insulation thickness bevels make the difference between an acceptable area with a good daylighting factor. For the 30 cm thick insulation the gain in an area under satisfactory daylighting conditions is almost twofold. For the thickest insulation, the is no area with proper daylighting without bevels. While with bevels there is. The daylighting analysis was performed on the working plane 75 cm above the floor. Table 3 Summarised results of static calculations of the effect of the bevels on daylighting factor. For the “working plane within limits” in red values that values below the 55% benchmark. Insulation thickness [CM] Average Daylight Factor Uniformity ( min / average DF ) Working plane within limits [%] Bevels no yes no yes no yes 10 1.95 2.02 0.50 0.50 81% 88% 20 1.65 1.81 0.52 0.51 55% 70% 30 1.30 1.78 0.45 0.54 35% 66% 50 1.02 1.53 0.40 0.57 0% 38% A similar conclusion may be drawn from the dynamic simulations (Tables 4 and 5 ). It depends on the orientation of windows but apart from the southern façade for medium insulation thickness bevels roughly double the occupancy hours with satisfactory daylighting conditions. Whereas for the case of thickest insulation, bevels increase useful daylight illuminance from a few percent to roughly 30%. On the southern facade, the effect is less pronounced but still important for thick insulations. Table 4 Useful daylighting luminance [%] - a percentage of occupancy hours when more the 50% of the working plane achieves between 300 and 3000 lux. Occupancy hours are from 8 am- to 6 pm. Values in red are below the 50% benchmark. Insulation thickness N E S W Bevels no yes no yes no yes no yes 10 52% 58% 63% 66% 97% 98% 75% 78% 20 40% 51% 48% 59% 93% 95% 65% 73% 30 13% 43% 33% 52% 82% 95% 32% 65% 50 0% 27% 1% 27% 29% 83% 8% 37% Table 5 Daylight autonomy [%] - a percentage of occupancy hours when more the 50% of the working plane achieves above 300 lux. Occupancy hours are from 8 am- to 6 pm. Values in red are below the 50% benchmark. Insulation thickness N E S W bevels no yes no yes no yes no yes 10 52% 58% 64% 67% 100% 100% 78% 82% 20 40% 51% 49% 59% 97% 99% 68% 76% 30 13% 43% 35% 55% 91% 96% 38% 69% 50 0% 27% 1% 27% 39% 92% 12% 43% 3.7 Appearance Such bevelling alters the presence of the building. In my personal opinion, the energy-efficient buildings of tomorrow with small glassing areas and thick insulations can contribute to the improvement of their aesthetics since it optically enlarges windows. Figure 11 compares the same house with and without bevels. It is a matter of personal taste but to me, the house with bevels is more appealing. 4 Conclusions Usage of COMSOL radiation heat transfer module for calculation of daylighting was done for the first time. This is infinitely more computationally expensive than using standard daylighting computation algorithms but for these particular tasks, it accelerated the workflow since the computational time was shorter than the time required to build the new model. In this paper, it was shown that the room becomes 18% lighter when using bevel in the case of 30 cm thick insulation and it has no real impact on the temperature inside the building wall. Static and dynamic analysis of typical daylighting factors shows a more remarkable difference. Both, reveal that the 30 cm thick insulation bevels make the difference between having or not having an acceptable area under satisfactory daylighting conditions. For 50 cm insulation thickness bevels mean the difference between having or not having any useful daylighting. It has to be observed that simulation assumed that nothing blocks the light coming from the sky up to the horizon. In a more realistic case, where there is another building in front of the window obscuring some of the light coming, bevels will provide an even bigger gain in lighting. The main conclusion is that cutting bevels into insulation can be a very cheap method to increase the amount of light coming into a building and considerably improve daylighting Declarations Acknowledgements This work was supported by the statutory research fund of ICSC PAS. The author would like to express his gratitude to Magdalena Soboń for her generous help in putting this article together. Conflicts of interest The author states that he has no competing interests. References Agnoletto, L., Cortella, G., & Manzan, M. (1995). Finite element thermal analysis of special building components. Energy and Buildings, 22 (2), 115–123. https://doi.org/10.1016/0378-7788(94)00908-3 Asdrubali, F., Baldinelli, G., & Bianchi, F. (2013). Influence of cavities geometric and emissivity properties on the overall thermal performance of aluminum frames for windows. 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Proceedings of Building Simulation 2011: 12th Conference of International Building Performance Simulation Association , 696–703. Supplementary Files The Supplementary Material is not available with this version. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1696154","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":108983358,"identity":"339385d2-c591-4a68-a774-39cf8508e5bd","order_by":0,"name":"Leszek Krzemień","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7klEQVRIiWNgGAWjYFACNjDJww8kPjAcALETgLiACC2SDQyMMxBaDAhrYTA4QKwWc/ZjaR9/VGyTMT5+xrCB4cw9Bn72HAPmAjxaLHvSDs+QOHObx+xMDlDLjWIGyZ43Bswz8GgxuMHezGDYBtRyg8f88Z8PCUARoC08hLQk/rvNYzyDB2gLUIs9YS1shxkONtzmMZAAabkBtEWCkJYzacmMDcdu80icSSsEej8ByHhWcBivX44fM2b8UXPbnr/98MYGhmMJcvztyRsfF1Tg1oIEOMAm84CIw0RpYGBgfwBnMhOpZRSMglEwCkYGAABESU7b4Q9utgAAAABJRU5ErkJggg==","orcid":"","institution":"Polish Academy of Sciences","correspondingAuthor":true,"prefix":"","firstName":"Leszek","middleName":"","lastName":"Krzemień","suffix":""}],"badges":[],"createdAt":"2022-05-26 11:29:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1696154/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1696154/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":22098856,"identity":"9a7ba198-e7fe-4d84-9e89-a8f430cd119f","added_by":"auto","created_at":"2022-05-31 21:16:39","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":29419,"visible":true,"origin":"","legend":"\u003cp\u003eGeneral view of the simulated wall. Colors indicate materials;\u0026nbsp;yellow -\u0026nbsp;window frame, green - wall, and brown -\u0026nbsp;insulation. (colors in print are not crucial but welcome)\u0026nbsp;\u0026nbsp;\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"Fig01.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/41291806f29a8f051d61fa8a.png"},{"id":22098863,"identity":"4032f8a5-1e7f-492e-b9ad-f05985d10b0d","added_by":"auto","created_at":"2022-05-31 21:16:40","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":4705,"visible":true,"origin":"","legend":"\u003cp\u003eCross-section of detail. Colors indicate materials; \u003c/p\u003e\u003cp\u003e\u0026nbsp;yellow - window frame, green - wall and brown - insulation, α - bevel angle, d - bevel depth. (Colors in print are not important)\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"Fig02.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/799d2cce02a387e07676a05d.png"},{"id":22098866,"identity":"3822ef9c-a059-40bf-869e-c46b9a89653c","added_by":"auto","created_at":"2022-05-31 21:16:40","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":40638,"visible":true,"origin":"","legend":"\u003cp\u003eScheme of lighting simulation. Left: a quarter dome that mimics light coming from an overcast sky.\u0026nbsp;Colour indicates virtual temperature which dictated radiosity. Right – Wall window and target used to calculate the input of light in the room. Colour indicates irradiation. \u0026nbsp;(Colours are not important)\u003c/p\u003e","description":"","filename":"Fig03.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/710874f72f8afb77dc113979.png"},{"id":22099242,"identity":"829108f1-f5a2-4274-bdc5-10cbf9d6f90b","added_by":"auto","created_at":"2022-05-31 21:26:39","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":85549,"visible":true,"origin":"","legend":"\u003cp\u003eComparison between Irradiation levels calculated using Radiance and Comsol.\u0026nbsp;From the left: Radiance simulation, Comsil simulation. The result obtained for CIE overcast sky with unit zenith irradiance\u0026nbsp;\u003cem\u003eB\u003c/em\u003e\u003csub\u003e\u003cem\u003eE\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e","description":"","filename":"Fig04.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/c20ecaab513e9f3fbd171fe1.png"},{"id":22098857,"identity":"ffaed046-82ad-4c86-ac59-0ea3831a1b1b","added_by":"auto","created_at":"2022-05-31 21:16:39","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":34506,"visible":true,"origin":"","legend":"\u003cp\u003eRoom\u003c/p\u003e","description":"","filename":"Fig05.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/c8cb0f45308360f306f7ef12.png"},{"id":22098979,"identity":"c3b67153-afea-463c-8237-4016541f1459","added_by":"auto","created_at":"2022-05-31 21:21:39","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":27524,"visible":true,"origin":"","legend":"\u003cp\u003eTemperature distribution across the cross-section. (colors are not important)\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"Fig06.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/f68d32fa2c53ff22834f49c4.png"},{"id":22099243,"identity":"0b4de6d3-38d1-4546-86cb-c156ac88ac72","added_by":"auto","created_at":"2022-05-31 21:26:39","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":25613,"visible":true,"origin":"","legend":"\u003cp\u003eThe temperature of the coldest spot of the masonry wall for a 30 cm thick insulation case. As seen in figure 6 coldest spot is located near the bevel. (Colours are crucial)\u003cspan class=\"ql-cursor\"\u003e\u003c/span\u003e\u003c/p\u003e","description":"","filename":"Fig07.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/10c8947a34195050e00f0d5a.png"},{"id":22098861,"identity":"2bd3b9fa-4372-4620-8ff6-2468e4b7781b","added_by":"auto","created_at":"2022-05-31 21:16:39","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":28706,"visible":true,"origin":"","legend":"\u003cp\u003eLight gain as a function of bevel geometry for 30 cm thick insulation case.\u0026nbsp;Light gain is equal to the increase of total irradiation coming through the window as a percent of irradiation with no bevel. (Colours are crucial)\u003c/p\u003e\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"Fig08.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/426d081ba83f963de2fdce32.png"},{"id":22098860,"identity":"4628a726-7c80-4da4-abc5-61d15bd148ec","added_by":"auto","created_at":"2022-05-31 21:16:39","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":34553,"visible":true,"origin":"","legend":"\u003cp\u003eFunction given by formula (1) for the case of 30cm of insulation thickness. This function peaks (red color on the plot) for optimal bevel depth maximizing light coming in and causing close to zero decreases in wall temperature.\u0026nbsp;(Colours are crucial)\u003c/p\u003e","description":"","filename":"Fig09.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/d3b923029bcc48bc4c89cdaf.png"},{"id":22098981,"identity":"9618bc4e-9706-43c4-b937-916699eb0d5b","added_by":"auto","created_at":"2022-05-31 21:21:40","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":19413,"visible":true,"origin":"","legend":"\u003cp\u003eLight gain and optimal bevel depth as a function of bevel thickness. Light gain - tringles and a star, Bevel depth - square. (Colours are not important but welcome)\u0026nbsp;\u003c/p\u003e","description":"","filename":"Fig10.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/4c610d5edc7c6c01fc953f16.png"},{"id":22098982,"identity":"aa5f80ca-7fe5-414a-a31e-c070a1ec2145","added_by":"auto","created_at":"2022-05-31 21:21:40","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":17205,"visible":true,"origin":"","legend":"\u003cp\u003eComparison between the aesthetics of the same house with and without bevels. House with no bevels is presented on the right. (Colours are not important)\u003c/p\u003e","description":"","filename":"Fig11.png","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/65c95882e3e947cb9b51dfd7.png"},{"id":22099245,"identity":"aaf3c421-27d5-4649-a3d2-7e4c7345a0c0","added_by":"auto","created_at":"2022-05-31 21:26:43","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":630575,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1696154/v1/e75bd2cf-553f-4561-90cc-670af313e426.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"The best bevel shape for optimized daylighting and thermal performance","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eDaylighting is an important factor for comfort and productivity inside a building (Marc et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). It does not only provide free illumination but has a very positive impact on wellbeing (Veitch \u0026amp; Galasiu, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). Now in the era of growing demand for low energy buildings, there is a tendency to increase insulation and thus wall thickness. Such a thick wall provides additional shading that decreases the amount of light coming in. A very cheap method mitigating that effect is the introduction of diagonal cuts, called bevels, in the insulation layer like these pictured in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. It has been shown that bevels may increase the amount of light coming into the room by up to 30% (Di \u0026amp; Radinger, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2010\u003c/span\u003e) but on the other hand, bevels may compromise the thermal proprieties of the wall. Given the importance of the problem, it is surprising that it has not been investigated thoroughly so far. There is very rich literature on the thermal performance of windows. There are holistic approaches (Gustavsen et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Lechowska et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Vendelboe et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Zajas \u0026amp; Heiselberg, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) approaches that focus on separate components (Cappelletti et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Van Den Bergh et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). There is plenty of literature that optimizes daylighting and energy performance of the window (Jakubiec \u0026amp; Reinhart, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Yan et al., \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). Most of them use the classical approach of building simulation where heat transfer through the wall is one-dimensional and temperature gradient along the wall surface is not allowed. Such modeling fails in estimating the influence of thermal bridges, which are the main concern while optimizing bevels around windows(Cappelletti et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). To account for thermal bridges in 3D finite element modeling is the most appropriate approach (Agnoletto et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1995\u003c/span\u003e). There are many examples of excellent models of thermal performance of building envelopes including fenestration (Asdrubali et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Lechowska et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Van Den Bossche et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Zajas \u0026amp; Heiselberg, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2011\u003c/span\u003e). However few of them takes daylighting into account. Author of the publication (Lechowska et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) models thermal performance of the window frame using FEM and calculates daylighting using classical architectural software. Publication (Liu et al., \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2019\u003c/span\u003e) also describes thermal and daylighting performance of windows. It also analyses the influence of skew cuts in insulation like this paper. However it focuses mostly on the performance of the external shutter and does not optimise the geometry of the cuts. Moreover it is using RC network model. Such model is not capable of fully simulating the impact of thermal bridges. To the best of author\u0026rsquo;s knowledge there is no study that deals with the influence of bevels on both daylighting and thermal properties of the wall.\u003c/p\u003e"},{"header":"2. Methodology","content":"\u003cp\u003eTwo models of a window are presented in this public: the Comsol model and the Radiance model. The Comsol model is used to calculate the amount of light coming through the window as well as temperature distribution in the simulated structure. This simulation allows finding optimal bevel depth and angle as a function of thickness. It is where the Radiance model comes in. It is used to calculate the daylighting metrics for a space that utilizes a window with bevel geometry optimized in the Cosmol model.\u003c/p\u003e\n\u003cdiv class=\"Section2\" id=\"Sec3\"\u003e\n \u003ch2\u003e2.1. Comsol Model\u003c/h2\u003e\n \u003cp\u003eThe FEM modeling was performed assuming stationary environmental conditions. The conditions were supposed to mimic a cloudy winter day in a temperate climate. The cloudy day was chosen because on such days natural light is in deficiency. The external temperature of zero \u0026deg;C was chosen because it models an average winter temperature in temperate climates. And it is winter where the heat loss matters the most.\u003c/p\u003e\n \u003cp\u003eThe modeling of daylighting, as well as the thermal behavior of the wall, was performed in COMSOL Multiphysics software. A wall with a small window shown in Fig. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e was modeled. The wall is a 30cm thick masonry wall with typically 30cm of insulation applied to the external surface. Different thicknesses are analyzed in the following chapter. The dimensions of the wall are the width of 2 m and height of 2.5 m. The window is mounted directly to the masonry since this traditional method is still very popular. The window frame is a generic one, the internal structure is not taken into account in the simulation. The cross-section of the window mounting is shown in Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. On the perimeter of the window, there is a bevel cut in the insulation. The depth and angle of the bevel are varied in the simulation. The windowsill is slightly tilted from the horizontal plane and its tilt angle remains constant throughout all the simulations. The external dimensions of the frame are 1x1.2m which yields 0.8 m^2 of the glazed surface.\u003c/p\u003e\n \u003cdiv class=\"Section3\" id=\"Sec4\"\u003e\n \u003ch2\u003e2.1.1. Thermal simulation.\u003c/h2\u003e\n \u003cp\u003eTo calculate the thermal behavior of the structure only convective heat exchange was taken into account. This seems justified when simulating overcast days since radiative heat exchange vastly cancels out during such weather so the net effect can be neglected. The internal temperature was assumed to be 20\u0026deg;C and the external 0\u0026deg;C as mentioned before. The convective heat exchange coefficient was 20 W/m\u003csup\u003e2\u003c/sup\u003e.K on all external surfaces and 5 W/m\u003csup\u003e2\u003c/sup\u003e.K on internal surfaces. The following thermal conductivities were assumed: wall \u0026minus;\u0026thinsp;0.15 W/m.K (lightweight concrete), insulation \u0026ndash; 0.04 W/m.K (styrofoam), 0.105 W/m.K \u0026ndash; window frame (yields U value of 1.16 W/m\u003csup\u003e2\u003c/sup\u003e.K at 9 cm thickness). The windowpane was not simulated. Heat loss on the 0.8M^2 of glazing was added in post-processing assuming a moderate U-value of glazing of 1.0 W/m\u003csup\u003e2\u003c/sup\u003e.K.\u003c/p\u003e\n \u003cp\u003eyellow - window frame, green - wall and brown - insulation, \u0026alpha; - bevel angle, d - bevel depth. (Colors in print are not important)\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv class=\"Section3\" id=\"Sec5\"\u003e\n \u003ch2\u003e2.1.2. Amount of light\u003c/h2\u003e\n \u003cp\u003eThe amount of light coming into the space was modeled in COMSOL Multiphysics as well. The software does not have any light propagation module as such, but the \u003cem\u003eComsol physics\u003c/em\u003e called \u003cem\u003eHeat Transfer\u003c/em\u003e has a component that calculates radiative heat propagation as well as radiation scattering using the Lamertian reflectance model. The idea here is to utilize the radiative heat propagation tool to simulate light propagation. To do so we introduce a surface that represents the sky. This object has an arbitrary temperature distribution tailored to yield the desired luminance according to black body radiation. This way it is possible to generate arbitrary lighting. The sky is the only light-emitting object in our model. The surfaces that do not emit light have the virtual temperature set to zero. Their reflectance is set to a value less than one depending on the brightness of their color. This allows us to model the scattering of light on the surface. The lower the reflectivity value, the more light scatters on the surface.\u003c/p\u003e\n \u003cp\u003eThe source of radiation was a quarter of the sphere with a radius of 40m which mimics the overcast sky. The virtual temperature \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{f}\\)\u003c/span\u003e\u003c/span\u003e distribution of the surface is selected to yield a luminance \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({L}_{h }\\)\u003c/span\u003e\u003c/span\u003eat the angle,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e to match the overcast sky model by Moon and Spencer (Moon \u0026amp; Spencer, \u003cspan class=\"CitationRef\"\u003e1942\u003c/span\u003e) also called CIE overcast sky i.e.\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equ1\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e$${L}_{h}={B}_{z}\\frac{1+2 \\text{s}\\text{i}\\text{n}\\left(\\theta \\right)}{3}$$\u003c/div\u003e\n \u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{z}\\)\u003c/span\u003e\u003c/span\u003eis the zenith luminance. Black body radiation is proportional to the temperature to the power of four so the virtual temperature has to be \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{f}=\\sqrt[4]{{L}_{h}}\\)\u003c/span\u003e\u003c/span\u003e. The virtual temperature of all surfaces of the window is set to zero so there is no emission and only scattering of the radiation takes place. The surfaces that take part in the light propagation modeling are shown in Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. Windowsill reflectivity is set to 50% which corresponds to light color (RAL 7035 for instance) while the remaining surface reflectivities are 20% which is neither dark nor light like beige (RAL 1001). Apart from these real surfaces a virtual surface exists, it is the surface that measures the amount of light coming into the interior. It is the deep surface of the window. The reflectivity of this virtual surface is set to zero so the surface does not affect the light distribution.\u003c/p\u003e\n \u003cp\u003eSuch an approach is a bit cumbersome at first but allows modeling of the 3D thermal and lighting distribution in the same environment. This is very convenient because the same geometry is in use and all the alternations to the model are automatically applied to both physics i.e. heat transfer and radiation scattering. Moreover, COMSOL allows very easy parameterization of the geometry which makes parametric studies very efficient.\u003c/p\u003e\n \u003c/div\u003e\n \u003cdiv class=\"Section3\" id=\"Sec6\"\u003e\n \u003ch2\u003e2.1.3. Validation of Comsol model.\u003c/h2\u003e\n \u003cp\u003eThermal simulations are performed in a very orthodox manner. Comsol has proven to be an accurate tool for such calculations (Gerlich et al., \u003cspan class=\"CitationRef\"\u003e2013\u003c/span\u003e). What is unorthodox is the usage of Comsol\u0026rsquo;s radiation heat exchange functionality to calculate daylighting. This approach was tested against Radiance software. Radiance is the most generally useful software package for architectural lighting simulation first developed at Lawrence Berkeley National Laboratory that has been a benchmark for daylighting for over 30 years. The comparison between the \u003cem\u003eComsol\u003c/em\u003e and \u003cem\u003eRadiance\u003c/em\u003e simulations is given in Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. The picture shows light distribution on the innermost surface of the window - the plane that was analyzed in the paper. Ambient conditions for both simulations were the same, it was CIE overcast sky with unit zenith radiance (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({B}_{z}\\)\u003c/span\u003e\u003c/span\u003e in Eq. \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e). The difference in average irradiations is less than 1% and the standard deviation of the difference is 6.6% of the average value.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec7\"\u003e\n \u003ch2\u003e2.2. Daylighting\u003c/h2\u003e\n \u003cp\u003eIn the above described Comsol model, only the total amount of light coming into a building during an overcast day was considered. Therefore Radiance model was built to estimate the influence of the bevels on the quality of daylighting in a typical room. The room selected for the simulation is medium-sized with rather small windows. Its dimension, as well as the placement of windows, is shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e. Such geometry yields a window to wall ratio of 0.17 and a window to wall ratio of 1.4. The reflectances of materials used for the simulation as summarized in Table \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. Booth static and dynamic simulations were performed. Static simulators were performed for the CIE overcast sky whereas dynamics simulations used the weather file for Nowy Sącz (IMGW, \u003cspan class=\"CitationRef\"\u003e2000\u003c/span\u003e). Nowy Sącz is a city in southern Poland and it was selected for the calculations because its climate is a good example of a temperate climate for which the simulations are mostly intended.\u003c/p\u003e"},{"header":"3 Results And Discussion","content":"\u003cp\u003eThermal simulations, as well as the ones performed to determine the amount of light falling into the space, were executed for all the combinations of bevel depth from 0.01 to 0.28 m and bevel angle from 23\u0026deg; to 63\u0026deg;.\u003c/p\u003e\n\u003cdiv class=\"Section2\" id=\"Sec9\"\u003e\n \u003ch2\u003e3.2 Thermal\u003c/h2\u003e\n \u003cp\u003eThe calculated temperature distribution across the wall is presented in Fig. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e. To estimate the thermal performance of the simulated wall the total heat transmitted through it as well as the temperature of the coldest place inside the masonry wall was looked at. The maximal increase of the total heat loss did not exceed 3% in the case of the deepest bevel cut at the widest angle. Since the heat loss caused by the bevel is so small the temperature decrease in the coldest place of the masonry wall is an important factor and it will be presented here in detail. This parameter was chosen because it is crucial for the possible building deterioration caused by water condensation inside the wall and thus the well-being of the building.\u003c/p\u003e\n \u003cp\u003eThe plot in Fig. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e shows the temperature of the coldest place inside the wall as a function of bevel dimensions. One can see that temperature remains unchanged for a wide range of bevel geometries. Only for bevels deeper than 0.2 m the temperature tends to drop. There is also the dependence of bevel angle, wider angles yield lower temperature.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec10\"\u003e\n \u003ch2\u003e3.2 Light\u003c/h2\u003e\n \u003cp\u003eThe total radiation that passed through the whole window pane was analyzed. This parameter was chosen rather than daylighting factor to get more generic results. Daylighting metrics are analyzed in section \u003cspan class=\"InternalRef\"\u003e3.6\u003c/span\u003e. This radiation is equivalent to the surface irradiation integrated over the most inner surface of the window opening. The plot in Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e shows the irradiation gain in relation to total irradiation with no bevels. The irradiation is higher when the bevel is deeper and wider. The light gains depend on the thickness of the insulation The maximum gain, i.e. the gains achieved for the bevels cut almost to the window frame ranges from 6.4% for 10 cm thick insulation to 87% for the insulation thickness of 50 cm. These results are in line with the literature. 30% light gain was reported by Radinger (Di \u0026amp; Radinger, \u003cspan class=\"CitationRef\"\u003e2010\u003c/span\u003e). Gregor Radinger also states that there is a \u003cem\u003ereduction of daylight of about 15\u0026ndash;20% because of increasing wall and soffit-thicknesses\u003c/em\u003e personal communication. It is harder to compare to the results of the other publication dealing with the same problem (Liu et al., \u003cspan class=\"CitationRef\"\u003e2019\u003c/span\u003e). In the publication, the relative floor area that passes certain daylight criteria is given. For the insulation thickness of 55 cm, the floor under good lighting conditions extends thanks to the cuts from 28\u0026ndash;33%.\u003c/p\u003e\n \u003cp\u003eIt is worth noticing that the relation is more linear than in the case of temperature. Therefore in the case of the moderate bevel depth, there is a meaningful improvement in lighting conditions without worsening the thermal insulating properties of the structure. It can be expected that there must be an optimal geometry of bevel that maximizes lighting but does not decrease the temperature of the wall considerably.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec11\"\u003e\n \u003ch2\u003e3.3 Optimal bevel\u003c/h2\u003e\n \u003cp\u003eTo find optimal geometry one has to find a function that would peak for such optimal conditions. The most straightforward choice is a ratio of light gain increase to the temperature decrease. Using such a function brings a risk of division by zero. To solve that problem following function was analyzed.\u003c/p\u003e\n \u003cdiv class=\"Equation\" id=\"Equa\"\u003e\n \u003cdiv class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e$$fun= \\frac{{G}_{BEV}-{G}_{0}}{\\varDelta {T}_{min}+0.1 K}$$\u003c/div\u003e\n \u003c/div\u003e\n \u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{BEV}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{0}\\)\u003c/span\u003e\u003c/span\u003e are irradiations with and without bevel. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\varDelta {T}_{min}= {T}_{min}-{T}_{0}\\)\u003c/span\u003e\u003c/span\u003e is a temperature decrease in the coldest part of the wall, caused by the bevel. This function peaks at the point when the bevel causes the biggest increase in the irradiation without decreasing the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{min}\\)\u003c/span\u003e\u003c/span\u003e considerably more than 0.1 K. That means it peaks somewhere in the bevel depth of 0.16 m and bevel angle of 50\u0026deg;C as shown in Fig. \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e. Those parameters provide the optimum bevel geometry. Such bevel geometry increases the heat loss through the wall only by 0.2% and increases the amount of light coming in by 18%.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec12\"\u003e\n \u003ch2\u003e3.4 Parametric sweep\u003c/h2\u003e\n \u003cp\u003eThe above-described function was used to find optimal bevel parameters as a function of insulation thickness. Similar simulation series as in the previous paragraph were run for the insulation thicknesses from the relevant range of 10 to 50 cm. The observations are summarised in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and some representative examples are shown in the plot in Fig. \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e. The obvious observation is that the thicker the insulation, the bigger the light gain caused by the bevels. The light gain goes up to 40% for the very thick insulation of 50 cm. It has to be noted that this value, as well as all the others given in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e, concerns bevel optimized using the function(1) where the drop of the coldest spot of the wall is marginal \u0026ndash; less than 0.05\u0026deg;C. Higher gains are possible if a bigger temperature drop can be afforded. The necessary plots are given in supplementary materials.\u003c/p\u003e\n \u003cp\u003eWhen it comes to the optimal bevel dimensions the thinner the insulation, the shallower the optimal bevel is as a percentage of insulation thickness. So for 50 cm insulation, the optimal bevel depth is 25 cm which is 50% of the insulation thickness, whereas for 10 cm thick insulation bevel as shallow as 3 cm gives optimal results. That is only 30% of insulation thickness. The optimal bevel angle does not change with the insulation thickness. It has to be noted that the optimum bevel angle is very wide so any angle from a range of 45 to 60 degrees will be as good. This freedom of angle selection gives opportunities in design and allows to match bevel angles to other angles in the building for a more consistent visual appearance.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec13\"\u003e\n \u003ch2\u003e3.5 Wider window\u003c/h2\u003e\n \u003cp\u003eTo check the generality of the methodology similar analysis was performed for a 150% wider window. It appears that the optimal bevel depth is the same as for the narrower window, however, the optimal bevel angle is slightly wider. These results are also summarized in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and supplementary material.\u003c/p\u003e\u0026nbsp;\u003cbr\u003e\n \u003ctable border=\"1\" id=\"Tab2\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eOptimal bevel parameters as well as the influence of the bevels on\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003einsulation thickness [cm]\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eoptimal bevel depth [cm]\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eoptimal bevel depth [%]\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eoptimal bevel angle [\u0026deg;]\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003elight gain [%]\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003etemperature decrease [\u0026deg;C]\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.04\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e18\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.03\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.05\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003cp\u003ewide window\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e13\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e58\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.02\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable border=\"1\" id=\"Tab1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eReflectances of the materials used in the calculations. All materials were gray i.e. they had the same relativities for all the wavelengths.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMaterial\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eReflectance*\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eExternal wall\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eWindow frame\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCeiling\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eInternal wall\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.7\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFloor\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePane\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.88\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003ctfoot\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"2\"\u003e*Transmittance for the case of glass.\u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tfoot\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec14\"\u003e\n \u003ch2\u003e3.6 Daylighting\u003c/h2\u003e\n \u003cp\u003eThe analysis compares daylighting for the optimal bevel geometry to the one with no bevels for various insulation thicknesses. The results of the static calculation are summarised in Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. As for the average daylighting factor, it behaves similarly to the total amount of light coming into the building. The effect of bevels is more pronounced for greater insulation thickness and reaches about 50 percent gain for the 50 cm insulation. The effect of bevels on uniformity is rather interesting. For thin insulation, bevels have no effect but while insulation is getting thicker uniformity lessens for the window without bevels. However, the same relation is reversed in the case of bevels. The thicker the insulation the more uniform the light inside. The last pair of columns in Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e shows the percentage of the working plane area with satisfactory DF which is between 2 and 7%. Again for the thin insulation, there is no difference whether bevels are present or not, however, for the medium insulation thickness bevels make the difference between an acceptable area with a good daylighting factor. For the 30 cm thick insulation the gain in an area under satisfactory daylighting conditions is almost twofold. For the thickest insulation, the is no area with proper daylighting without bevels. While with bevels there is. The daylighting analysis was performed on the working plane 75 cm above the floor.\u0026nbsp;\u003c/p\u003e\n \u003ctable border=\"1\" id=\"Tab3\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eSummarised results of static calculations of the effect of the bevels on daylighting factor. For the \u0026ldquo;working plane within limits\u0026rdquo; in red values that values below the 55% benchmark.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eInsulation thickness [CM]\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eAverage\u003c/p\u003e\n \u003cp\u003eDaylight Factor\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eUniformity\u003c/p\u003e\n \u003cp\u003e( min / average DF )\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eWorking plane\u003c/p\u003e\n \u003cp\u003ewithin limits [%]\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBevels\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.95\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e81%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e88%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.81\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e55%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e70%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.78\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.45\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.54\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e35%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e66%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.40\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e38%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cp\u003eA similar conclusion may be drawn from the dynamic simulations (Tables \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e). It depends on the orientation of windows but apart from the southern fa\u0026ccedil;ade for medium insulation thickness bevels roughly double the occupancy hours with satisfactory daylighting conditions. Whereas for the case of thickest insulation, bevels increase useful daylight illuminance from a few percent to roughly 30%. On the southern facade, the effect is less pronounced but still important for thick insulations.\u0026nbsp;\u003c/p\u003e\u003cbr\u003e\n \u003ctable border=\"1\" id=\"Tab4\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eUseful daylighting luminance [%] - a percentage of occupancy hours when more the 50% of the working plane achieves between 300 and 3000 lux. Occupancy hours are from 8 am- to 6 pm. Values in red are below the 50% benchmark.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eInsulation thickness\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eN\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eE\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eS\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eW\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBevels\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e58%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e63%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e66%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e97%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e98%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e75%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e40%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e51%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e48%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e59%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e93%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e95%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e65%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e73%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e13%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e43%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e33%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e82%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e95%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e32%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e65%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e27%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e27%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e29%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e83%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e37%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u0026nbsp;\u003ctable border=\"1\" id=\"Tab5\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eDaylight autonomy [%] - a percentage of occupancy hours when more the 50% of the working plane achieves above 300 lux. Occupancy hours are from 8 am- to 6 pm. Values in red are below the 50% benchmark.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eInsulation thickness\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eN\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eE\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eS\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003eW\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ebevels\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eno\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eyes\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e52%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e58%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e64%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e67%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e100%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e78%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e82%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e40%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e51%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e49%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e59%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e97%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e99%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e68%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e76%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e13%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e43%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e35%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e55%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e91%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e96%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e38%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e69%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e27%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e27%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e39%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e92%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e12%\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e43%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv class=\"Section2\" id=\"Sec15\"\u003e\n \u003ch2\u003e3.7 Appearance\u003c/h2\u003e\n \u003cp\u003eSuch bevelling alters the presence of the building. In my personal opinion, the energy-efficient buildings of tomorrow with small glassing areas and thick insulations can contribute to the improvement of their aesthetics since it optically enlarges windows. Figure \u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e compares the same house with and without bevels. It is a matter of personal taste but to me, the house with bevels is more appealing.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003eUsage of COMSOL radiation heat transfer module for calculation of daylighting was done for the first time. This is infinitely more computationally expensive than using standard daylighting computation algorithms but for these particular tasks, it accelerated the workflow since the computational time was shorter than the time required to build the new model. In this paper, it was shown that the room becomes 18% lighter when using bevel in the case of 30 cm thick insulation and it has no real impact on the temperature inside the building wall. Static and dynamic analysis of typical daylighting factors shows a more remarkable difference. Both, reveal that the 30 cm thick insulation bevels make the difference between having or not having an acceptable area under satisfactory daylighting conditions. For 50 cm insulation thickness bevels mean the difference between having or not having any useful daylighting. It has to be observed that simulation assumed that nothing blocks the light coming from the sky up to the horizon. In a more realistic case, where there is another building in front of the window obscuring some of the light coming, bevels will provide an even bigger gain in lighting. The main conclusion is that cutting bevels into insulation can be a very cheap method to increase the amount of light coming into a building and considerably improve daylighting\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAcknowledgements\u003c/h2\u003e\n\u003cp\u003eThis work was supported by the statutory research fund of ICSC PAS. The\u0026nbsp;author would like to express his gratitude to Magdalena Soboń for her generous help in putting this article together. \u0026nbsp;\u003c/p\u003e\n\u003ch2\u003eConflicts of interest\u003c/h2\u003e\n\u003cp\u003eThe author states that he has no competing interests.\u0026nbsp;\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAgnoletto, L., Cortella, G., \u0026amp; Manzan, M. (1995). Finite element thermal analysis of special building components. 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Analysis of Energy Saving Potential and Optimization of Thermally Broken Fiberglass Window Frames. \u003cem\u003eProceedings of Building Simulation 2011: 12th Conference of International Building Performance Simulation Association\u003c/em\u003e, 696\u0026ndash;703.\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"},{"header":"Supplementary Files","content":"\u003cp\u003eThe Supplementary Material is not available with this version.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"window, daylight, thermal performance, finite element method, cuts in insulation","lastPublishedDoi":"10.21203/rs.3.rs-1696154/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1696154/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper discusses a costless method of improving daylighting in a building. It shows that the diagonal cuts in insulation around windows substantially increase the amount of light coming into the building without compromising the heat resistance of the barrier. The effect of various depth and angles of cuts on the daylighting as well as the thermal performance of the envelope is simulated. An optimal geometry is found as a function of insulation thickness. The results were obtained using the finite element method. A novel approach was used that allows the daylighting and thermal simulation to be performed in the same simulation environment using the same geometry. Additionally, a traditional daylighting analysis was performed. They show that for the thick insulation, bevels make difference between having and not having satisfactory levels of light in the room.\u003c/p\u003e","manuscriptTitle":"The best bevel shape for optimized daylighting and thermal performance","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2022-05-31 21:16:37","doi":"10.21203/rs.3.rs-1696154/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"2b25ee89-8aea-42d1-96f0-64a008c9a95f","owner":[],"postedDate":"May 31st, 2022","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2022-05-31T21:21:39+00:00","versionOfRecord":[],"versionCreatedAt":"2022-05-31 21:16:37","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-1696154","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-1696154","identity":"rs-1696154","version":["v1"]},"buildId":"_2-kVJe1T_tPrBINL-cwx","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}