KIEAE Journal

Generally, simulation models for horizontal solar radiation are categorized under three headings: 1) regression models such as the Cloud-cover Radiation Model (CRM) 3), the Zhang & Huang Model (ZHM) 4), and the Saudi Arabia Model (SAM) 5), 2) mechanical models such as the Meteorological Radiation Model (MRM) 6) and the Upper –air Humidity Model (UHM) 7), and 3) models using high precision measuring instruments or satellite images 8). As regression models are required to derive regional coefficients from regressions of long-term measured weather data, it cannot be used in simulation programs using only limited site information. Even though mechanical models do not need regional coefficients, simulation process is more complicated and requires extra weather information that the usual weather station does not measure. Models using high precision measuring instruments or satellite data certainly are not suitable for the purpose of this study. Recently the author suggested a hybrid model applying the ZHM for winter season and the CRM for summer season for the sites having critical seasonal weather condition 9).

Thus, the simulation models introduced in this program are using the equations suggested by the most widely approved solar models in the United States, including those of the ASHRAE Hand Book of Fundamentals, Diffie & Beckman, and Kreider & Rabl.

Direct solar radiation on a horizontal surface for a clear day (I_B,H) is determined by beam normal radiation, and the angle between the zenith and the sun. Thus,

Liu & Jordan16) developed an empirical relationship between extraterrestrial radiation and diffuse radiation for clear days. The atmospheric transmittance for diffuse radiation (τ_b) is the ratio of clear sky diffuse radiation to extraterrestrial radiation on a horizontal surface (τ_b = I_B,H / I_O,H) and is calculated by

From the above equations diffuse solar radiation on a horizontal surface on a clear day (I_B,H) can be calculated by

Thus, total solar radiation on a horizontal surface on a clear day (I_G,H) is

The Equation (10) shows that the total solar radiation on a horizontal surface under clear sky can be calculated only by extraterrestrial radiation (I_O,N), zenith angle (θ_z), and atmospheric transmittance for beam radiation (τ_b). However, one can calculate the estimated clear day solar radiation on a horizontal surface if the zenith angle, altitude, and the climate type for the location are given.

Table 6 shows the simulation results of beam, diffuse, and global radiation calculated using the methods of ASHRAE, Duffie & Beckman, and Kreider & Rabl. As we can expect, the maximum radiation on a horizontal surface reaches its maximum point either in May or in June, and its minimum in December.

Total solar radiation incident on a tilted surface can be calculated as the sum of beam radiation, diffuse radiation from the sky, and radiation reflected from various surfaces such as the ground, external shades, and neighboring objects that block the view of the surface. If the view of the surface is blocked from direct sunlight, the total incident beam radiation on a tilted surface (I_B,P) is

where R_b is the tilt factor which is the ratio of the beam radiation component on a tilted surface to that on a horizontal surface, and F_s is the sunlit factor defined by the sunlit area divided by the total surface area. When the incidence angle of direct sunlight (θ_i) is known,

In general, there are two kinds of solar radiation models based upon the method of simulating diffuse solar radiation: an ‘isotropic model’ and an ‘anisotropic model’. In an isotropic sky model, it is assumed that diffuse radiation is isotropic or constant. In other words, the diffuse radiation from the sky is the same regardless of the orientation. However, in an anisotropic sky model, the calculation of the diffuse radiation component on a tilted surface is a little more complicated. Diffuse radiation actually consists of three different components: the ‘isotropic diffuse’ radiation delivered uniformly from the sky hemisphere, the ‘circumsolar diffuse’ radiation scattered around the direct rays of the sun, and the ‘horizon brightening’ which is concentrated near the horizon (Fig. 4).

Fig. 4.  
Beam, isotropic diffuse, circumsolar diffuse, horizon brightening, and ground reflected radiation on a tilted surface

The Liu and Jordan17) model for diffuse radiation is currently the most widely used. In this model, the solar radiation on a tilted surface (I_G,P) is composed of the beam, the isotropic diffuse, and the ground-reflected radiation. Kreider and Rabl used the simple isotropic model, and suggested the equation for the global irradiance on a titled surface as follows:

When vertical surface and the ground in front of the surface is not shaded, the titled surface angle θ_p = 90o and Equation (13) becomes simply

Reindl et al.18) developed an improved anisotropic model called the ‘HDKR model’ by adding a horizon brightening component to the model originally developed by Hay and Davies19) and modified by Klucher20). This model uses an ‘anisotropy index’ to determine the transmittance of the atmosphere for beam radiation. It is assumed that isotropic diffuse radiation from the sky has the same angle as beam radiation. The diffuse radiation on a tilted surface (I_D,P) using the HDKR model is calculated by

where A_i is an anisotropy index and defined by

Perez et al.21) also developed another anisotropic sky model which considers the zenith angle of the direct beam, cloud clearness, air mass, and the brightness coefficient of the sky. Diffuse radiation on a tilted surface is again calculated by

where F₁ and F₂ are circumsolar and horizon brightness coefficients determined by three parameters that describe the sky conditions: the zenith angle, cloud clearness, and brightness. The values of F₁ and F₂ are calculated from the table for brightness coefficients.

The previous study ascertained that the method developed by Liu and Jordan is less accurate than the anisotropic models22). The model developed by Perez et al. is also difficult to apply in a computer program, because it requires values that weather stations do not usually measure. Thus, the anisotropic model developed by Reindl et al. (i.e., HDKR model) is the most suitable for the calculation of diffuse radiation on a vertical surface.

The ground-reflected solar radiation on a tilted surface (I_R,P) is determined by the total radiation on the horizontal surface (I_G,H), the view factor of the surface to the ground (F_grd), and the ground reflectance (ρ_g). For a surface directly in the front of a collector extending in all directions, the view factor to the ground is (1 - cosβ/2.

Generally, in sky radiation model the reflected radiation from the surrounding objects is ignored, because it is much smaller than the reflected radiation from the ground. However, in the case when a large portion of the window surface is blocked by external surfaces, the reflected radiation from these surfaces should be considered. The total reflected radiation from the surrounding surfaces (I_R,i) can be calculated by the multiple of the solar radiation incident (I_i) on the ith surface, the diffuse reflectance(ρ_i) of that surface, and the view factor(F_i) from the ith surface to the analyzed tilted surface.

Finally, the total solar radiation on a tilted surface (I_G,P) is calculated as the sum of these components. Thus, the total solar radiation on a tilted surface is

For the computerized algorithm of the total solar radiation incident on a tilted surface, Equation (19) was most suitable. In this equation, it is critically important to calculate the incident angles and view factors between glazing and shading objects.

Table 7 shows the simulation results tested for solar radiation on a vertical wall facing 15oSE. It clearly shows the rapid decrease of direct radiation during the summer season (e.g., the value in June 21 is just about 10 % of that in December 21 in Duffie & Beckman model), and almost constant values of diffuse radiation throughout the whole year. However, the reflected radiation increases during the summer season.

When the reflection and absorption losses through the glazing are considered, the transmittance (τ) of a single-glazed window can be calculated by applying ray-tracing methods to Fresnel’s equation and Snell’s equation. The total reflection (γ) of unpolarized radiation was calculated using perpendicular (γ⊥ ) and parallel (γ∥ ) components of unpolarized radiation as follows

where I_i and i_r are incident and reflected solar radiation, and θ₁ and θ₂ are incidence and refraction angles respectively. The total transmittance (τ) of single-pane glazing can be calculated applying the ray-tracing method to Fresnel’s equations as follow

where τ⊥ and τ∥ are the perpendicular and parallel components of transmittance of the glazing and τα is the transmittance of glazing when only absorption losses have been considered. The value of τα is defined as,

where K = the extinction coefficient

   L = the thickness of the glazing

The total amount of the transmitted solar radiation through window glazing can be calculated from the above equations. However, some portion of the transmitted solar radiation will be absorbed by the inside surface and some will be reflected back to the glazing. Thus, for the more accurate estimation of heat gain from the transmitted solar radiation, we need to calculate how much of the transmitted solar radiation is actually absorbed by the inside surfaces. The total solar radiation transmitted through glazing and absorbed by the inside surfaces can be calculated using the ‘transmittance-absorptance product (τα)’ as follows

Applying the same ray-tracing method here the (τα) can be calculated by

where ρ_d is the reflectance of the glazing for the diffuse radiation from the inside surface and α is the absorptance of the inside surface. This solar absorptance is dependent on the angle of incidence of the radiation striking the surface, and can be calculated as a function of the normal incidence for a flat black surface as follows

where α_n is the normal incidence for a flat black surface and θ is the incidence angle on the surface.

Fig 5 shows the solar test bench, which was used for the validation of the transmitted solar radiation calculation as well as the glazing transmittance test. Besides this solar test bench, a physical model was constructed and used to collect the real measured transmitted solar radiation though window glazing. For more detailed description of physical model and validation of the simulation model, please refer to the previous research22).

Fig. 5.  
Configuration of solar test bench (a) multi-pyranometer array with an artificial horizon, (b) Eppley normal incidence pyrheliometer, (c) horizontal solar transmittance test box, (d) shadow band, (e) Eppley shadow band pyranometer, (f) Eppley precision spectral pyranometer, (g) test stand for calibrating pyranometers

To calculate very accurately the transmitted and absorbed solar radiation through glazing, the shading analysis must be included in the computerized calculation process.

Several methods have been developed for the calculation of the sunlit and shaded area of a window. One of the simplest methods is an algorithm with ‘discrete element analysis with grids’ (Fig. 6). This method was first developed by Groth and Lokmanheim and used in the earlier version of DOE-223) and BLAST24). In this method, the receiving surface is divided into a two dimensional grid. The center point of each element is then tested by a shading projection algorithm to determine whether it is in sunlit or shaded areas. The sum of the sunlit or shaded grid elements is then used to obtain the sunlit and shaded fraction, respectively. Unfortunately, this method can require excessive processing time for higher resolution. Furthermore, it can only solve rectangular plane surfaces, which means that non-rectangular surfaces must be represented with combinations of rectangular planes.

Fig. 6.  
Calculation of the shaded area using discrete grid elements

In the recent version of DOE-2, an ‘improved bar-method’ (Fig. 7) was developed that uses bars instead of grids for a discrete element analysis. This method increased both the speed and accuracy of the calculation. However, the processing speed is still dependent on the desired accuracy and the method still has similar geometrical limitations as the grid method, although not as restrictive.

The most recent version of BLAST uses a ‘convex polygon clipping algorithm with homogeneous coordinates’ (Fig. 8) developed by Walton25). In Walton’s method, the n dimensional Cartesian coordinates are transformed into n+1 dimensional homogeneous coordinates. For example, a point given by two dimensional coordinates (x, y) is represented by three dimensional coordinates (h_x , h_y , h), where h is an arbitrary number. After the coordinate transformation, the algorithm finds the vertices of one polygon within the other and visa versa. Then, the intersecting points of the boundary of both polygons are determined. These overlapped vertices and intersecting points are then transformed again into Cartesian coordinates, ordered clockwise and the area is computed. Though this method is much faster than the discrete element analysis method, it is restricted to the use of convex polygons.

Fig. 8.  
Calculation of the shaded area using convex polygon clipping with homogeneous coordinates

The extraterrestrial radiation and beam normal radiation on a clear day were estimated pretty accurately from most solar models. Even though the intensity of direct solar radiation is determined by seasonal variations and atmosphere conditions such as water vapor and dust in the air, the simulation results were pretty close to the real measured data.

The intensities of diffuse radiation either on a horizontal surface or a tilted surface were almost the same throughout the whole year. However, as the tilt factor Rb decreases rapidly during winter, the direct and reflected solar radiation also decreased in the same way. In this case ASHRAE clear sky model produces greater errors than we had expected and showed some limitations in being applied in a computer program.

The strength and incident angle of beam radiation on a transparent surface determine the intensity of transmitted and absorbed solar radiation through window glazing. When the window is partly blocked with shading devices or external walls, the estimation of transmitted solar radiation becomes more complicated, because it involves three dimensional shading analysis. Even though the convex polygon clipping method was faster than other models, it was restricted to certain shapes of the shaded area.

For the validation process in the development of the program in this research, two different types of accuracy tests were conducted (i.e., the comparative test and the empirical validation test). The comparative test was conducted by comparing the simulated results against the results of the DOE-2 program. For the empirical validation test, a specially designed physical test box was constructed and tested under various conditions. The environment of the physical model was restricted to focus on the effects of solar gain through window glazing.

A series of comparison tests of the simulation results from these models against solar data measured on the solar test bench as well as simulated results from other simulation programs demonstrated that Duffie & Beckman’s method introducing the HDKR anisotropic model has provided the most reliable simulation results. The simulation results were pretty close to the real measured data when the intensity of solar radiation was relatively high. However, as the solar intensity becomes lower in winter, the differences of estimation results produced more errors.

The main aim of this study is to devise an effective algorithm for complicated and delicate problems which can be calculated by step by step series of calculations the result of which can correct previous mistakes in our assumptions. This study shows how important experimental and theoretical validation is at each stage of the calculation process. This is especially true when dealing with complicated matters in our research such as the transmitted solar radiation through multi-pane window systems. Scrutinizing advantages and limitations of simulation models at each stage could improve the mathematical precision and practical usability of the computer program appling the algorithm that we had proposed.