Window Bevel Shape Optimization for Sustainable Daylighting and Thermal Performance in Buildings

Abstract

Thick insulation of external walls is the standard method for passive reduction in heating costs in residential buildings in the northern climate zone. However, increasing the insulation thickness worsens the lighting conditions inside the rooms. This work demonstrates that diagonal cuts in the insulation around windows (bevels) significantly increase the light entering the building without compromising its heat resistance. The optimized window bevel shape is a cost-effective method for improving daylighting in residential buildings. The research employs traditional finite-element modeling (FEM) alongside a novel method that allows for the simultaneous calculation of heat transfer and daylight distribution within the same simulation environment and geometry. The study analyzes the impact of various incision depths and angles on both daylighting and the thermal performance of the building envelope. The results show that the optimal bevel geometry dependent on the insulation thickness without a negative impact on thermal properties may be found. In addition, a traditional daylight analysis shows that for thick insulation, the introduction of bevels makes the difference between satisfactory and inadequate lighting conditions in the room. Moreover, reduced use of insulating material and resulting solar gains may significantly increase the overall sustainability of modern buildings.

1. Introduction

Buildings are the biggest contributors of energy consumption and CO₂ production [1,2,3]. There is a technical possibility to significantly reduce both the energy demand and the carbon footprint of a building in a cold climate by an impressive factor of ten. This substantial reduction in environmental impact can be achieved through extensive insulation of the building envelope [4]. Effective insulation not only involves using thermally resistive wall materials but also sealing the building to minimize energy losses caused by air infiltration. However, this nearly perfect insulation from the exterior may deprive inhabitants of essential natural stimuli, such as the natural day–night cycle. This is why natural daylighting has become increasingly important.

Daylighting is an important factor for comfort and productivity inside a building [5]. It not only provides free illumination but also has a very positive impact on well-being [6]. 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. An unbelievably cheap method mitigating that effect is the introduction of diagonal cuts, called bevels, in the insulation layer like those pictured in Figure 1. It has been shown that bevels may increase the amount of light entering the room by up to 30% [7], 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 yet been investigated thoroughly.

There is a very rich literature on the thermal performance of windows. There are holistic approaches [8,9,10,11] and approaches that focus on separate components [8,12]. There is plenty of literature that optimizes daylighting and energy performance of the window [13,14]. Most studies 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 [15].

A thermal bridge is a specific area in a building’s envelope where heat can flow more easily than in surrounding areas. This phenomenon is often linked to structural elements, such as reinforced-concrete beams, which have low thermal resistance, but thermal bridges can also occur around windows [16]. It is crucial to carefully consider thermal bridges during the design of a building, as they can significantly reduce the overall thermal resistivity of the envelope. Even if the U-value of the envelope appears to be low, thermal bridges introduce additional risks. The lower temperatures created by these bridges can lead to water vapor condensation, which is detrimental to the building. This condensation not only promotes mold growth but also increases the thermal conductivity of the walls, resulting in further condensation and degradation of the building envelope. Typically, energy consumption calculation programs like Energy Plus or WUFI operate on a one-dimensional model of walls, making it impossible to accurately predict the temperature drop caused by thermal bridges. Therefore, in this paper, we chose to utilize FEM simulation software. Instead of factoring in the influence of window bevels on the overall U-value of the envelope, we focus on how these bevels affect the coldest spot in the masonry wall around the window.

To account for thermal bridges in three dimensions, finite-element modeling is the most appropriate approach [17]. There are many examples of excellent models of thermal performance of building envelopes including fenestration [9,11,18,19]. However, few of them consider daylighting. In publication [9], Lechowska et al. model the thermal performance of the window frame using FEM and calculate daylighting using classical architectural software. On the other hand, in publication [20], Liu et al. also describe the thermal and daylighting performance of windows. They also analyze the influence of skew cuts in insulation like this paper. However, they focus mostly on the performance of the external shutter and do not optimize the geometry of the cuts. Moreover, they use the resistance–capacitance (RC) network model—a simplified building energy model, usually applied in fast prediction of building energy consumption [21]. Such a model is not capable of fully simulating the impact of thermal bridges. The influence of bevels on solar gain was actually studied by a very simplified direct beam model, and there was no thermal modeling in the publication [22]. A comprehensive building envelope optimization was presented in [23], but the assumed zero thickness of the walls leaves no place for bevels. Recently, a multi-objective approach similar to the one presented here gained much popularity in all scales from agglomerative [24] to through the whole building [25] to a single window [26], but none of them discussed any skew surfaces near windows. Publication [27] utilizes the concept of diagonal fins as a shading device; however, it does not provide any thermal analysis since such a diagonal fin module does not affect the insulation of the building. To the best knowledge of the authors of this work, there is no study dealing with the influence of bevels on both daylighting and thermal properties of the wall, which is the main novelty of this study.

2. Materials and Methods

Two models of window are presented in this publication: 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 the temperature distribution in the simulated structure. This simulation allows us to find the optimal bevel depth and angle as a function of thickness. This modeling is conducted exclusively under the assumption of an overcast sky, because these are the conditions allowing general conclusions to be drawn. However, the optimal bevel shape should also be tested under more varying lighting conditions. The Radiance model is used to calculate the daylighting metrics for a space that utilizes a window with bevel geometry previously optimized in the COMSOL 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. A cloudy day was chosen because on such days, natural light is deficient. The external temperature of 0 °C was chosen because it models an average winter temperature in temperate climates, and it is in winter that 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 (version 4.3 was used with the heat transfer module [28]). A wall with a small window shown in Figure 1 was modeled.

The wall is a 30 cm thick masonry wall with typically 30 cm of insulation applied to the external surface. Different insulation thicknesses are analyzed in Appendix A. The dimensions of the wall are a width of 2 m and a height of 2.5 m. The window is mounted directly onto 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 mesh was a physics-control tetrahedral with a heat transfer surface-to-surface radiation contributor. It had 9396 elements, excluding the part that mimics the sky.

The cross-section of the window mounting is shown in Figure 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 1 × 1.2 m, which yields 0.8 m² of glazed surface.

2.1.1. Thermal Simulation

To calculate the thermal behavior of the structure, only convective heat exchange was considered. This seems justified for the simulation of overcast days, as the radiative temperature of the sky is about the same as the air temperature [29], and so the radiative heat exchange acts like a small contribution to the conductive heat exchange. The indoor and outdoor temperature was assumed to be 20 °C and 0 °C, respectively. The coefficient for convective heat exchange was 20 Wm⁻²K⁻¹ on all external surfaces and 5 Wm⁻²K⁻¹ on internal surfaces. The following thermal conductivities were assumed: wall—0.15 Wm⁻¹K⁻¹ (lightweight concrete); insulation—0.04 Wm⁻¹K⁻¹ (polystyrene foam); 0.105 Wm⁻¹K⁻¹—window frame (yields a U-value of 1.16 Wm⁻²K⁻¹ at 9 cm thickness). The windowpane was not simulated. The heat loss of the 0.8 m² glazing was added during post-processing, assuming a moderate U-value of the glazing of 1.0 Wm⁻²K⁻¹.

2.1.2. Amount of Light

The amount of light entering the room was also modeled in COMSOL Multiphysics. COMSOL has a module for radiative transfer that contains a ray-tracing algorithm included in COMSOL physics called Heat Transfer. The module has a component that calculates the propagation of radiant heat as well as the scattering of radiation using the Lambertian reflectance model. The main function of this module is to calculate the radiative heat transfer between objects. In this work, however, we use it to calculate daylighting illumination. To do this, the daylighting properties must be replaced by thermal properties so that they fit into the COMSOL module. First of all, there is no light source as such in the radiant heat transfer module. Radiation is emitted from all bodies that have a non-zero temperature according to Plank’s law of black-body radiation, i.e., the radiation intensity is proportional to the fourth power of the temperature. To simulate daylighting illumination in this way, the temperature of all objects except the light sources must be set to zero. Fortunately, COMSOL does allow multiple simulations to be run for one geometry, so we used two independent heat transfer simulations for this model sharing the same geometry. One was the real heat transfer through the envelope described in the previous paragraph. The second was the solution with the parameter called a virtual temperature. This quantity is set to zero for all physical elements of the envelope. The only object with a temperature different from zero is a surface representing the sky. This object has an arbitrary temperature distribution that is tailored to give the desired luminance according to the radiation of the black body. In this way, it is possible to create any arbitrary illumination. The sky is the only light-emitting object in our model, while the virtual temperature is set to zero for all surfaces that do not emit light. Their reflectance is set to a value of less than one depending on their color brightness, which makes it possible to model the light scattering on the surface according to the following relation: the higher the reflectance value, the more light is scattered on the surface.

The surfaces taking part in the light propagation modeling are shown in Figure 2. The radiation source was a quarter of a sphere with a radius of 40 m representing an overcast sky. The virtual temperature of the surface distribution was chosen to generate a luminance at a certain angle which matches the overcast sky model [30], often known as the CIE standard overcast sky. This model of the overcast sky assumes that sky brightness does not vary with azimuthal angle. Instead, luminance depends solely on the elevation angle. Such an approach enables us to draw general conclusions from the results, as the geographical orientation of the facade is not a factor. Mathematically, this model is defined in [30] as

$$L_h=B_z{\displaystyle \frac{1+2 \mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right)}{3}},$$

(1)

where $\theta$ is the elevation angle of a sky element above the horizon, and $B_z$ is the zenith luminance in cd/m². Black-body radiation is proportional to the temperature to the power of four, thus the virtual temperature must be expressed as $T_f=\sqrt[4]{L_h}$. The virtual temperature of all window surfaces is set to zero; therefore, there is no emission and only scattering of radiation occurs. Windowsill reflectivity is set to 50%, corresponding to light color (e.g., RAL 7035), while the remaining surface reflectivity is set to 20%, which is neither a dark nor a light color, like beige (RAL 1001). In addition to these real surfaces, the model also includes a virtual surface, the innermost surface of the window that gauges how much light enters the interior. To ensure that the light distribution is unaffected, the reflectance of the virtual surface is set to zero.

Although this procedure may seem cumbersome at first, it allows modeling of both 3D lighting and temperature distribution in the same computational environment. Consequently, this approach is far more practical and effective as the same geometry is in use, and all the alteration to the model are applied automatically to both types of physics used in the model, i.e., heat transfer and light scattering. Furthermore, parametric study is highly effective since the COMSOL interface makes geometry parameterization very simple.

2.1.3. COMSOL Model Validation

Thermal simulations are performed in a very conventional manner, and COMSOL has proven to be a reliable tool for this [31]. In contrast, the use of COMSOL’s radiative heat exchange module for daylighting calculations is unconventional; therefore, this approach was verified with Radiance 5.3 software used as a standard reference. Developed at Lawrence Berkeley National Laboratory, Radiance is the most widely used software tool for simulating architectural lighting and has been a benchmark for daylighting simulations for more than 30 years.

The comparison between our novel approach calculated in COMSOL and simulation of the same geometry with the same conditions using Radiance software is shown in Figure 3. The graph shows light distribution on the innermost surface of the window, i.e., the plane that was analyzed. The same CIE cloudy sky with unit zenith radiance (Equation (1)) served as the ambient conditions for both simulations. The difference in average irradiation is less than 1%, while the standard deviation of the distribution differences is 6.6% of the average value, consequently indicating very good agreement between COMSOL and Radiant calculations.

2.2. Daylighting

The COMSOL model discussed above only considered the total amount of light that enters a building during an overcast day. To calculate the impact of bevels on the daylighting metrics in a typical room, the Radiance model was developed.

The room selected for the simulation is medium-sized with rather small windows. Figure 4 depicts both its size and the arrangement of the windows. With this geometry, the window-to-wall and window head height to room depth ratios are equal to 0.16 and 1.4, respectively. The values of material reflectance used for the simulation are summarized in

Table 1. Reflectance of the materials used in the calculations. All materials were gray, i.e., they had the same reflectance for all wavelengths.

Material	Reflectance
External wall	0.2
Window frame	0.2
Ceiling	0.7
Internal wall	0.7
Floor	0.7
Pane	0.88 *

* Transmittance for the case of glass.

. The geometry for the daylight simulation was created in Google SketchUp 8.0, except for the window frame, which was taken directly from the COMSOL model. The model’s geometry was imported into Blender 2.93 software [32], where the appropriate material was assigned to the surfaces. In the next step, the working plane and the weather .epw file was added. Afterward, the LiVi interface from Vi-suite 0.6 generated the ASCII radiance input files, and these files were edited in a text editor to tailor them to the expected format. Radiance 5.3 [33] was run from the command line with the quality and detail parameters set as ”low”. The resolution was set to 560 × 560, the number of indirect reflections to 3, the ambient value to 0.01, and the ambient divisions to 1024. While dynamic simulations employed the weather of Nowy Sącz [34], static simulations were run for the CIE cloudy sky. The Polish city of Nowy Sącz was chosen for the calculations due to its climate, which is representative of the temperate environment that the simulations are primarily designed for.

3. Results and Discussion

To determine both the amount of light entering the interior and the resulting thermal distribution in relation to the geometry of the bevel, a series of simulations were run for various combinations of bevel depth (from 0.01 to 0.28 m) and bevel angle (from 23° to 63°).

3.1. Thermal Performance

To evaluate the thermal performance of the simulated wall, the total heat transferred as well as the temperature of the coldest spot inside the masonry wall were analyzed (Figure 5a). Temperature distribution calculated across the wall is presented in Figure 5a. In the case of the deepest bevel cut and the widest angle (upper right corner of Figure 5a), the maximum increase in total heat loss does not exceed 3%. This means that in a wide range of bevel dimensions, they do not increase heat loss on a practical level. On the other hand, the drop in temperature at the coldest point of the masonry, which is caused by such modification in the insulation layer, is also an important factor, as it is crucial for possible damage to the building due to water condensation in the masonry and thus for the overall condition of the building. Therefore, temperature distribution relations are presented below in detail.

The plot in Figure 5b presents the temperature of the coldest point inside the wall (black rectangle in Figure 5a) as a function of bevel dimensions. It is evident that the temperature is practically unaffected by a variety of bevel shapes. The temperature drops only for bevels deeper than 0.2 m. Additionally, there is the bevel angle dependence, where larger angles result in lower temperatures.

3.2. Light Performance

To get more generic results, the total amount of light passing through the whole windowpane was selected instead of daylighting metrics like daylighting factor (DF), which is analyzed in Section 3.4. This parameter corresponds to the surface irradiation integrated over the innermost surface of the window opening. The graph in Figure 6 shows the illuminance gains in relation to the total illuminance without bevels calculated for a 30 cm thick insulation layer. As can be clearly seen, the illuminance is higher when the bevel is deeper and wider. At the same time, 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 a 10 cm thick insulation to 87% for the insulation thickness of 50 cm (Appendix A). These results are consistent with previous findings present in the literature, e.g., a 30% light gain was reported in ref. [7]. The same author also states that a reduction in daylight of about 15–20% due to increasing wall thickness can be observed.

It is important to note that daylighting depends more linearly on the bevel depth than temperature does (Figure 5b). Hence, with a moderate bevel depth, there is a noticeable enhancement in lighting conditions while maintaining the thermal insulating properties of the structure. It can be assumed that there exists an ideal bevel shape that maximizes lighting without significantly reducing the temperature of the wall. A detailed analysis will be presented in the following section.

3.3. Optimal Bevel

To find the optimum bevel geometry, one needs to find a function that would peak at such optimal conditions. The most straightforward selection is a ratio of an increase in light gain to a decrease in temperature. Using such a function creates a risk of division by zero. To solve this problem, the following function

$$fun={\displaystyle \frac{G_{BEV}-G_0}{∆T+0.1 K}}$$

(2)

was analyzed, where $G_{BEV}$ and $G_0$ are irradiations with and without bevels. The temperature decrease in the coldest point of the wall caused by the bevel $∆T$ is defined as the difference between the lowest wall temperature with ($T_{min}$) and without ($T_0$) bevels. This function peaks at the point where the bevel causes the greatest increase in the irradiation without reducing $T_{min}$ considerably (more than 0.1 K). This means that it reaches the maximum somewhere around 0.16 m and 50 °C for the bevel depth and angle, respectively, as shown in Figure 7. Those values of the parameters provide the optimum bevel geometry, which in turn increases the heat loss only by 0.2% and increases the amount of light entering by 18%.

Formula (2) was used to find optimum bevel parameters as a function of insulation thickness. A similar series of simulations as presented in the previous chapter was run for the insulation thicknesses in realistic range between 10 and 50 cm. The results are summarized in

Table 2. Optimal bevel parameter and its influence on light and thermal performance.

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 (150% wider window)	13	43	50 (58)	18 (15)	0.03 (0.02)
20	7	35	50	8	0.05
10	3	30	50	2.5	0.05

, and some representative examples are shown in Figure 8. The obvious observation is that the thicker the insulation, the bigger the light gains caused by bevels. The light gain increases up to 40% for a very thick insulation of 50 cm. It should be noted that this value, along with all others collected in Table 2, relates to a bevel optimized with Function (2), where the temperature decrease of the wall’s coldest spot is very small, i.e., lower than 0.05 °C. It is possible to obtain greater light gains if bigger temperature drops are allowed. The necessary plots are given in Appendix A.

When it comes to the optimum bevel dimensions, the thinner the insulation, the shallower the optimal bevel (expressed as a percentage of insulation thickness). Therefore, in the case of 50 cm thick insulation, the optimum bevel depth is 25 cm, which is 50% of the insulation thickness. On the other hand, for 10 cm thick insulation, bevels as shallow as 3 cm give optimal results. Three centimeters is only 30% of the insulation thickness. The optimal bevel angle does not depend on the insulation thickness. It should be noted that the optimum bevel angle range is very wide, i.e., any angle ranging from 45 to 60 degrees gives the same good result. This flexibility in selection of angles opens various design possibilities and enables one to match the bevel angle to other façade’s angles for a more unified and visually consistent appearance of the building.

The analysis presented in this paper is limited to the flat plane cuts in isolation. This selection was motivated by the simplicity of making such cuts. It is possible that curved shapes would improve the daylighting even more, but the shape would have to be calculated for all the window sides, so the analysis would lose its universality.

For a window that is 150% wider, we conducted a similar analysis to check if the presented methodology is applicable. We found that the best bevel depth remains the same as for the narrower window, but the optimal bevel angle is slightly wider. This example is also detailed in Table 2, and more results are presented in Appendix A.

3.4. Daylighting

In the previous section, we covered the COMSOL combined lighting and thermal analysis results. In contrast, in this section, we discuss an extended daylighting analysis using Radiance software. We focus on daylighting metrics for the optimal bevel geometry compared to the case without bevels for various insulation thicknesses. The results of the static calculations are summarized in

Table 3. Summarized results of static calculations of the influence of bevels’ presence on daylighting factor (DF). In the last column, values in italics indicate those below 55% of the benchmark.

Insulation Thickness (cm)	Average Daylight Factor (DF)		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	4.53	0.40	0.57	0	38

. As for the average daylighting factor (DF), it behaves similarly to the total amount of light entering the building. The effect of bevels is more pronounced for thicker insulation and reaches about 50% gain for the 50 cm thick insulation. The influence of bevels’ presence on uniformity is not very distinct. For thin insulation, bevels have no effect, but with increasing insulation thickness, the uniformity decreases for the window without bevels. However, this relation is reversed when bevels are present. The thicker the insulation, the more uniform the light inside. The last column in Table 3 shows the percentage of the working plane area with satisfactory DF. The daylighting analysis was performed on the working plane located 75 cm above the floor level. The working plane covered the whole area of the room. As in the case of the previous analysis, for 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 and unactable area with a good DF.. For the 30 cm thick insulation, introducing bevels doubles the area under satisfactory daylighting. As a curiosity, for very thick insulation, there is no area with proper daylighting without bevels, while with bevels present, such area accounts for 38%.

The dynamic simulations summarized in

Table 4. Useful daylighting illuminance (%)—a percentage of occupancy hours when more than 50% of the working plane achieves between 300 and 3000 lux. Occupancy hours are from 8 am to 6 pm. Values in italics are below the 50% benchmark.

Insulation Thickness (cm)	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
60	0	27	1	27	29	83	8	37

lead to a similar conclusion. The impact depends on the window orientation, but for medium insulation thickness, window bevels approximately double the hours of occupancy with satisfactory daylighting conditions, known as useful daylight illuminance (UDI). In the case of the thickest insulation, bevels increase the UDI from a few percent to roughly 30%. This effect is less significant on the southern façade, but it is still important for buildings with thick insulation.

Regarding the comparison of results obtained in this work with some others existing in the literature, it was very difficult to find identical calculations. Reference [20] provides similar daylighting metrics but for the internal bevels on roof windows. Its author states that the bevels increased a sufficiently illuminated working plane by 5 percent, from 28 to 33%. Our results, on the other hand, prove that the working plane within limits increases from 0 to 38% for a similar insulation thickness. This difference shows that bevels around wall windows are far more beneficial than those around roof windows.

4. Conclusions

Our publication details an unconventional but successful application of COMSOL Multiphysics to calculate the daylighting and thermal performance of a building envelope. The use of finite-element method (FEM) in this context may appear unusual initially, given its higher computational cost compared to standard daylighting computational algorithms. However, in this specific task, it significantly accelerated the entire workflow by enabling precise 3D temperature distribution calculations and an effective parametric study within the same program. As a result, the total computational time was significantly shorter than the time required to create a new model every time a new geometry was introduced

In this work, we show 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 presence of 30 cm thick insulation bevels doubles the well-illuminated working plane. In the case of 50 cm insulation thickness, the presence of bevels means the difference between having and not having any useful daylighting. Moreover, they have a positive impact on light uniformity. The main conclusion is that cutting bevels into insulation can be a very cheap method to increase the amount of light entering the building and considerably improve daylighting. Additionally, beveling can contribute to the improvement of building aesthetics, since it optically enlarges the windows. This aspect is especially important nowadays in the case of energy-efficient buildings with thick insulation and small glazing areas. On the other hand, reduced use of insulating material together with resulting solar gains may significantly increase the overall sustainability of modern buildings.