Modern approaches – BSDF and DSHGC

3.2.1. The Klems approach

The modern approaches mainly derived in the recent two decades are based on concepts that date back to earlier work. In his work of 1994 (Klems, 1994b, 1994a), Joe Klems presented a model that allows to describing the solar-optical properties of complex fenestration systems by characterising the layers via matrices that contain spatially averaged but angularly resolved bidirectional reflectance and transmittance values The work of Klems is based on earlier publications of Lawrence Berkely National Laboratory (LBNL), in particular on a paper in 1986 (Papamichael et al., 1986) introducing the matrix concept and a 1988 paper (Papamichael et al., 1988) that proposes a scanning radiometer measurement method to acquire the necessary optical information for the method.
In the Klems matrix approach, the actual spatial (i.e. locational) information is averaged; however, the directional information is kept in an angularly resolved way. For this purpose, Klems introduced an angular segmentation of the hemisphere into 145 patches covering approximately equal solid angles. In order to describe the directional scattering properties of a layer comprehensively, the transmittance or reflectance values for the 145 outgoing directions have to be provided for all of the State-of-the-art analysis RadiCal, D. Rüdisser 16 145 potential incident directions (see Figure 6). Therefore, the entire set comprises 145 x 145 = 21,025 elements. The elements forming the matrices are, of course, not independent, as the fundamental laws of the underlying optical principles have to be obeyed, particularly: positivity, Helmholtz reciprocity and energy conservation. The Klems matrix is, therefore, a discrete representation of a more general concept known as bidirectional reflectance/transmittance distribution functions (BRDF/BTDF) or their combination, referred to as bidirectional scattering distribution functions (BSDF). The BSDF concept and terminology were introduced even earlier, e.g. by Nicodemus (Nicodemus, 1965) for analysis of longwave radiation and more comprehensively in 1980 by Bartell (Bartell, Dereniak and Wolfe, 1981). Klems mainly avoids this terminology but refers briefly to Nicodemus in his first work.

However, in his two important publications in 1994, Klems derives a methodology for calculating solar-optical properties of multi-layer fenestration systems from their individual bi-directional layer properties. Klems begins with non-specular layers and gradually develops the model further to finally include specular layers and consider interreflection between the layers.
Klems notes that his approach is based on two significant approximations:

• 1. Spectrally averaged optical properties:
  The input and output reflectance and transmittance information are valid for integrated wavelength ranges, usually either the visible or global radiation spectrum.
• 2. Spatial averaging:
  While the angular information is tracked, the actual spatial information is lost, as an averaging over all points of a layer is assumed in the model.

Beyond these two approximations raised by Klems, the author of this work believes a third one should be added, although it is related to the second point:

• 3. Infinite extension:
  The layer model intrinsically assumes that all layers are infinitely extended, and any radiation transmitted by one layer will reach the next layer. Therefore, the model cannot cover any boundary effects.

In his work, Klems expressed most concerns regarding the first (spectral) issue and its practical limitations. In contrast, he argues that the second (spatial) issue has limited practical consequences, and he does not raise the third issue. The following section discusses some implications regarding the simplifications contained in the Klems method.
The angular transmittance (BTDF) and reflectance (BRDF) values necessary to define the Klems matrices of a fenestration component can either be determined by measurement or by modelling. In the empirical approach, measurements are mainly performed using spectrophotometers or goniophotometers (Kohler, 2014). While goniophotometers are frequently used to determine optical properties for daylighting applications, their use to measure integrated values for the entire global radiation spectrum is still a matter of research (Curcija et al., 2022; Ward et al., 2022). Since the commonly used silicon sensors cannot capture radiation with wavelengths beyond approx. 1000 nm, a combination with InGaAs or germanium sensors has to be used to extend the measurement capabilities into the near-infrared region. The modelling approaches to determine BSDF matrices are either based on analytical optical calculations, as applied in LBNL Window (LBNL Windows and Daylighting Group, 2022b) or raytracing. In the most commonly used method, however, a combination of measure and modelled data is used: the optical properties of the material’s surfaces are determined empirically and subsequently used in a two- or three-dimensional model to determine the optical properties of the entire system. The reason for that is that, on the one hand, most goniophotometers cannot perform measurements on largely extended objects. On the other hand, the approach allows a more flexible and efficient workflow, as the measured material data can be used to model a variety of different systems.

3.2.2. The black-box or two-layer model

Independent of the BSDF approach, in 2006 Kuhn (2006) and, more specifically, in 2011 Kuhn, Herkel et al. (2011) propose a significant simplification for the modelling of a shaded glazing system by separating the fenestration system into a shading layer and a glazing system layer:
The proposed model for 𝑔𝑡𝑜𝑡 introduces an additional simplification level and treats the whole glazing unit as one layer and the shading device as another. One advantage is that the angular dependence of the glazing and shading device can be characterized externally with the method that best meets the requirements for the system under consideration. Another advantage is that manufacturers of glazing and shading devices have to guarantee only for the properties of their product, not for the combination. (Kuhn, 2006)
Providing accurate descriptions of the angular-dependent properties of the glazing and shading systems is still necessary for this approach. Kuhn proposes the use of a semi-empirical model suggested by Roos et al. for the glazing layer (2001). The Roos model and underlying models (Montecchi et al., 2002) are, in fact, based on physical optics models and consider Fresnel’s equations and the polarisation state of light. However, finally simplified semi-empirical fitting models are derived. Regarding the angle-dependent optical properties of the shading device layer, Kuhn suggests either (whole system) measurements or the use of 2D raytracing, as demonstrated in his earlier work (Kuhn et al., 2001).
Finally, a virtual two-layer model is used to calculate direct transmittance and solar absorptance in the two virtual layers to derive the secondary heat transfer. For this approach, it is necessary to define thermal boundary conditions, as the relevant thermal transfers leading up to the directional total transmittance values must be considered. For different applications and configurations (e.g. internal or external shading) several equations are provided on how the layerwise models can be State-of-the-art analysis RadiCal, D. Rüdisser 18 integrated to finally derive angle-dependent relations for the total transmittance 𝑔𝑡𝑜𝑡, also referred to as the directional solar heat gain coefficient (DSHGC). The precomputed dataset can then be integrated as ‘black-box’ model into existing dynamic building performance simulation tools. The proposed method allows for a consideration of angular effects and therefore represents a significant improvement over the ISO standard methods. However, the newly introduced simplification made by assuming two independent layers poses a new potential source for inaccuracy. The approach does not sufficiently allow the modelling of specific spectral and angular filtering or conditioning of the solar radiation by the first layer. The angular and spectral profile of the radiation incident to the second layer will be significantly different from the “neutral” radiation profile that is assumed for the calculation of its optical properties. To mitigate some of these issues, analytic relations that mix the layer’s solar and visible optical properties are presented. Furthermore, interreflections between the two layers cannot be modelled accurately, as a simplified model for diffuse radiation is assumed.

3.2.3. BSDF and the two-layer model

Regarding the simplified angular information contained in the Kuhn model, Bueno and others (see Bueno et al., 2015; Bueno et al., 2017) propose an improved approach, as the two-layer model of Kuhn is ‘upgraded to accept BSDF data’. The method described is based on Kuhn’s two-layer model; however, the directional solar heat gain coefficients DSHGC of the glazing unit and the BSDF of the glazing unit and the shading device are considered. The directional information for reflectance, transmittance, absorptance and total transmittance is used in the matrix form proposed by Klems
Regarding modelling the resulting energy flows, the ‘upgraded model’ described by Bueno et al. (2017) still applies the principles as suggested by Kuhn for the two-layer model. In turn, Kuhn’s initial model is based on the simplified approach provided in the standard ISO 52022-1. In the equation proposed for modelling external Venetian blinds, three components are considered:

The optical coefficients (𝜏, 𝛼) as well as the 𝑔 values in equation (3) are functions of the incident light direction; for reasons of clarity, the corresponding arguments were omitted in the above equations. The corresponding directional values are taken from the Klems matrices representing the BSDFs of the shading and glazing unit as well as the DSHGC values of the glazing unit.

device and transmitted to the interior, while the last term represents an additional fractional amount of heat trapped in the glazing unit due to the presence of the shading device. The equation proposed for internal blinds follows a very similar approach but contains additional terms to consider interreflections.

3.2.4. LBNL Window and Radiance

In the currently most advanced and probably most commonly used approach for modelling the energetic performance of complex fenestration systems, the free tools LBNL Window (LBNL Windows and Daylighting Group, 2022b) and Radiance (McNeil et al., 2013) are used simultaneously. LBNL Window allows a quite accurate modelling of glazing systems comprising coated and/or uncoated glass panes. For normal or near-normal incidence, the results of the calculations can be considered very accurate, as LBNL Window uses the measured product data contained in the extensive IGDB – International Glazing Data Base (LBNL Windows and Daylighting Group, 2021) maintained by the same workgroup.
However, regarding the important directional dependencies of the optical coefficients for the glass panes, LBNL Window makes use of simplified models that consider unpolarised radiation only. There are currently two different angular models implemented in the Window software (Curcija et al., 2018). The approach for uncoated glass utilizes the empirical transmittance and reflectance spectra, measured at normal incidence only, to estimate a spectrally resolved refractive index in the first step. Subsequently, this quantity is used in Fresnel’s equations to derive the angular reflectance and transmittance of the glass. For coated glass panes, the angular properties are determined based on a regression fit. For the fitting process, two sets of fitting parameters are provided. A threshold value of 0.645 for the transmittance is used to determine if the coefficients for clear glass or bronze glass are used in the polynomial regression formula. The same approach for calculating angular properties is applied in the building performance simulation tool EnergyPlus (U.S. Dept. of Energy, 2022). Regarding spectral resolution, LBNL Window allows various approaches. The spectral resolution of the calculation is mainly determined by the specular information available for the components. Systems containing exclusively glass layers usually allow spectrally resolved computation. If a shading layer is present, the calculations are primarily performed for integrated spectral ranges (e.g. the solar or visible range).
Shading devices are integrated into the calculation process in the form of shading layers. LBNL Window allows the following options (Mitchel et al., 2019):

• Homogeneous diffusing shade
• Perforated screen
• Woven shade
• Venetian blind, horizontal
• Venetian blind, vertical
• Shade with XML data
• THERM file

The software performs calculations based on analytical or geometrical methods for the first five

options. The materials used for the shades are generally considered to be diffusely reflecting (Lambertian surfaces). For the simulation of Venetian blinds, LBNL Window has implemented a method that complies with the standard ISO 15099. However, a set of options and additional features are available that are beyond the scope of the standard and should be able to provide higher accuracy. For example, the geometry of the slats can be assumed as non-flat by providing a curvature radius, or the simplified modelling of diffuse radiation can be overruled by selecting a directional diffuse calculation method.
options. The materials used for the shades are generally considered to be diffusely reflecting (Lambertian surfaces). For the simulation of Venetian blinds, LBNL Window has implemented a method that complies with the standard ISO 15099. However, a set of options and additional features are available that are beyond the scope of the standard and should be able to provide higher accuracy. For example, the geometry of the slats can be assumed as non-flat by providing a curvature radius, or the simplified modelling of diffuse radiation can be overruled by selecting a directional diffuse calculation method.
In order to model more complex shading devices or to model shading devices in a more detailed way, Window allows the integration of externally computed BSDF data. The options Shade with XML data and THERM file are available for this purpose. The latter option allows the definition of the geometry and material using LBNL’s THERM software (LBNL Windows and Daylighting Group, 2022a).
In this approach, the actual BSDF information is not generated by LBNL Window, but with the external tool genBSDF (McNeil, 2015). This program is a three-dimensional raytracing representing a spin-off of the powerful open-source daylighting tool Radiance (Radiance Community, 2022). An overview of the workflow is depicted in Figure 8. The illustration is taken from the technical documentation for LBNL Window (Curcija et al., 2018).
The genBSDF tool offers no graphical user interface. It is a command line tool; the necessary input and output are provided as scripts and reports.
Since Radiance is a rendering software, it follows an RGB approach to achieve realistic visual representations. To still use the raytracer for energetic purposes, it is commonly recommended to apply Radiance material definitions of type plastic or translucent plastic and use identical values for the red, green and blue reflectance (or transmittance) values. The constant reflectance values provided should represent the measured spectrally integrated reflectance for either the solar or the visible band. The resulting BSDF files are then valid for the provided spectral range but do not contain any spectrally resolved information. This can be problematic for modelling systems that contain spectrally selective layers, as will be pointed out in the next section.