Window Calculation Module[LINK]

This section describes two potential modeling approaches for Windows. The first (layer by layer) is implemented. The second, simple approach, reuses the layer-by-layer approach but converts an arbitrary window performance into an equivalent single layer.

The primary Window calculation is a layer-by-layer approach where windows are considered to be composed of the following components, only the first of which, glazing, is required to be present:

• Glazing, which consists of one or more plane/parallel glass layers. If there are two or more glass layers, the layers are separated by gaps filled with air or another gas. The glazing optical and thermal calculations are based on algorithms from the WINDOW 4 and WINDOW 5 programs [Arasteh et al., 1989], [Finlayson et al., 1993]. Glazing layers are described using te input object WindowMaterial:Glazing.

• Gap, layers filled with air or another gas that separate glazing layers. Gaps are described using the input object WindowMaterial:Gas.

• Frame, which surrounds the glazing on four sides. Frames are described using the input object WindowProperty:FrameAndDivider.

• Divider, which consists of horizontal and/or vertical elements that divide the glazing into individual lites.

• Shading device, which is a separate layer, such as drapery, roller shade or blind, on the inside or outside of the glazing, whose purpose is to reduce solar gain, reduce heat loss (movable insulation) or control daylight glare. Shading layers are described using “WindowShadingControl” input objects.

In the following, the description of the layer-by-layer glazing algorithms is based on material from Finlayson et al., 1993. The frame and divider thermal model, and the shading device optical and thermal models, are new to EnergyPlus.

A second approch has been developed where windows are modeled in a simplified approach that requires minimal user input that is processed to develop and equivalent layer that then reuses much of the layer-by-model. This “Simple Window Construction: model is described below.

Optical Properties of Glazing[LINK]

The solar radiation transmitted by a system of glass layers and the solar radiation absorbed in each layer depends on the solar transmittance, reflectance and absorptance properties of the individual layers. The absorbed solar radiation enters the glazing heat balance calculation that determines the inside surface temperature and, therefore, the heat gain to the zone from the glazing (see “Window Heat Balance Calculation“). The transmitted solar radiation is absorbed by interior zone surfaces and, therefore, contributes to the zone heat balance. In addition, the visible transmittance of the glazing is an important factor in the calculation of interior daylight illuminance from the glazing.

Glass Layer Properties[LINK]

In EnergyPlus, the optical properties of individual glass layers are given by the following quantities at normal incidence as a function of wavelength:

Transmittance, T

Front reflectance, R${}^f$

Back reflectance, R${}^b$

Here “front” refers to radiation incident on the side of the glass closest to the outside environment, and “back” refers to radiant incident on the side of the glass closest to the inside environment. For glazing in exterior walls, “front” is therefore the side closest to the outside air and “back” is the side closest to the zone air. For glazing in interior (i.e., interzone) walls, “back” is the side closest to the zone in which the wall is defined in and “front” is the side closest to the adjacent zone.

Glass Optical Properties Conversion[LINK]

Conversion from Glass Optical Properties Specified as Index of Refraction and Transmittance at Normal Incidence[LINK]

The optical properties of uncoated glass are sometimes specified by index of refraction, n,* * and transmittance at normal incidence, T.

The following equations show how to convert from this set of values to the transmittance and reflectance values required by Material:WindowGlass. These equations apply only to uncoated glass, and can be used to convert either spectral-average solar properties or spectral-average visible properties (in general, n and T are different for the solar and visible). Note that since the glass is uncoated, the front and back reflectances are the same and equal to the R that is solved for in the following equations.

Given n and T, find R:

$$\begin{array}{c}r={\left(\frac{n-1}{n+1}\right)}^2\\ \tau =\frac{{\left[\right(1-r{)}^4+4r^2{T}^2]}^{1/2}-(1-r{)}^2}{2r^{2}T}\\ R=r+\frac{(1-r{)}^{2}r{\tau }^2}{1-r^2{\tau }^2}\end{array}$$

Example:

$$\begin{array}{c}T=0.86156\\ n=1.526\\ r={\left(\frac{1.526-1}{1.526+1}\right)}^2\\ \tau =0.93974\\ R=0.07846\end{array}$$

Simple Window Model[LINK]

EnergyPlus includes an alternate model that allows users to enter in simplified window performance indices. This model is accessed through the WindowMaterial:SimpleGlazingSystem input object and converts the simple indices into an equivalent single layer window. (In addition a special model is used to determine the angular properties of the system – described below). Once the model generates the properties for the layer, the program reuses the bulk of the layer-by-layer model for subsequent calculations. The properties of the equivalent layer are determined using the step by step method outlined by Arasteh, Kohler, and Griffith (2009). The core equations are documented here. The reference contains additional information.

The simplified window model accepts U and SHGC indices and is useful for several reasons:

1) Sometimes, the only thing that is known about the window are its U and SHGC;

2) Codes, standards, and voluntary programs are developed in these terms;

3) A single-layer calculation is faster than multi-layer calculations.

Note: This use of U and SHGC to describe the thermal properties of windows is only appropriate for specular glazings.

While it is important to include the ability to model windows with only U-value and SHGC, we note that any method to use U and SHGC alone in building simulation software will inherently be approximate. This is due primarily to the following factors:

• SHGC combines directly transmitted solar radiation and radiation absorbed by the glass which flows inward. These have different implications for space heating/cooling. Different windows with the same SHGC often have different ratios of transmitted to absorbed solar radiation.

• SHGC is determined at normal incidence; angular properties of glazings vary with number of layers, tints, coatings. So products which have the same SHGC, can have different angular properties.

• Window U-factors vary with temperatures.

Thus, for modeling specific windows, we recommend using more detailed data than just the U and SHGC, if at all possible.

The simplified window model determines the properties of an equivalent layer in the following steps.

Step 1. Determine glass-to-glass Resistance.[LINK]

Window U-values include interior and exterior surface heat transfer coefficients. The resistance of the bare window product, or glass-to-glass resistance is augmented by these film coefficients so that,

$$\frac{1}{U}=R_{i,w}+R_{o,w}+R_{l,w}$$

Where,

$R_{i,w}$ is the resistance of the interior film coefficient under standard winter conditions in units of m${}^2$·K/W,

$R_{o,w}$ is the resistance of the exterior film coefficient under standard winter conditions in units of m${}^2$·K/W, and

$R_{l,w}$ is the resisance of the bare window under winter conditions (without the film coefficients) in units of m${}^2$·K/W.

The values for $R_{i,w}$ and $R_{o,w}$ depend on U and are calculated using the following correlations.

$$R_{i,w}=\frac{1}{\left(0.359073Ln\right(U)+6.949915)};forU<5.85$$

$$R_{i,w}=\frac{1}{(1.788041U-2.886625)};forU\ge 5.85$$

$$R_{o,w}=\frac{1}{(0.025342U+29.163853)}$$

So that the glass-to-glass resistance is calculated using:

$$R_{l,w}=\frac{1}{U}-R_{i,w}-R_{o,w}$$

Because the window model in EnergyPlus is for flat geometries, the models are not necessarily applicable to low-performance projecting products, such as skylights with unisulated curbs. The model cannot support glazing systems with a U higher than 7.0 because the thermal resistance of the film coefficients alone can provide this level of performance and none of the various resistances can be negative.

Step 2. Determine Layer Thickness.[LINK]

The thickness of the equivalent layer in units of meters is calculated using,

$$Thickness=\{\begin{array}{cc}0.002& for\frac{1}{R_{l,w}}>7.0\\ 0.05914-\frac{0.00714}{R_{l,w}}& for\frac{1}{R_{l,w}}\le 7.0\end{array}$$

Step 3. Determine Layer Thermal Conductivity[LINK]

The effective thermal conductivity, ${\lambda }_{eff}$, of the equivalent layer is calculated using,

$${\lambda }_{eff}=\frac{Thickness}{R_{l,w}}$$

Step 4. Determine Layer Solar Transmittance[LINK]

The layer’s solar transmittance at normal incidence, $T_{sol}$, is calculated using correlations that are a function of SHGC and U-Factor.

$$T_{sol}=0.939998SHG{C}^2+0.20332SHGC;U>4.5;SHGC<0.7206$$

$$T_{sol}=1.30415SHGC-0.30515;U>4.5;SHGC\ge 0.7206$$

$$T_{sol}=0.41040SHGC;U<3.4;SHGC\le 0.15$$

$$T_{sol}=0.085775SHG{C}^2+0.963954SHGC-0.084958;U<3.4;SHGC>0.15$$

And for U-values between 3.4 and 4.5, the value for $T_{sol}$ is interpolated using results of the equations for both ranges.

Step 5. Determine Layer Solar Reflectance[LINK]

The layer’s solar reflectance is calculated by first determining the inward flowing fraction which requires values for the resistance of the inside and outside film coefficients under summer conditions, $R_{i,s}$ and $R_{o,s}$, respectively. The correlations are:

$$\begin{array}{cc}& R_{i,s}=\frac{1}{\left(29.436546{\left(SHGC-T_{Sol}\right)}^3-21.943415{\left(SHGC-T_{Sol}\right)}^2+9.945872\left(SHGC-T_{Sol}\right)+7.426151\right)};U>4.5\\ & R_{i,s}=\frac{1}{\left(199.8208128{\left(SHGC-T_{Sol}\right)}^3-90.639733{\left(SHGC-T_{Sol}\right)}^2+19.737055\left(SHGC-T_{Sol}\right)+6.766575\right)};U<3.4\\ & R_{o,s}=\frac{1}{\left(2.225824(SHGC-T_{Sol})+20.57708\right)};U>4.5\\ & R_{o,s}=\frac{1}{\left(5.763355(SHGC-T_{Sol})+20.541528\right)};U<3.4\end{array}$$

And for U-values between 3.4 and 4.5, the values are interpolated using results from both sets of equations.

The inward flowing fraction, $Fra{c}_{inward}$, is then calculated using:

$$Fra{c}_{inward}=\frac{\left(R_{o,s}+0.5R_{l,w}\right)}{\left(R_{o,s}+R_{l,w}+R_{i,s}\right)}$$

Then, the solar reflectances of the front face, $R_{s,f}$, and back face, $R_{s,b}$, are calculated using:

$$R_{s,f}=R_{s,b}=1-T_{Sol}-\frac{\left(SHGC-T_{Sol}\right)}{Fra{c}_{inward}}$$

The thermal absorptance, or emittance, is taken as 0.84 for both the front and back and the longwave transmittance is 0.0.

Step 6. Determine Layer Visible Properties[LINK]

The user has the option of entering a value for visible transmittance as one of the simple performance indices. If the user does not enter a value, then the visible properties are the same as the solar properties. If the user does enter a value then layer’s visible transmittance at normal incidence, $T_{Vis}$, is set to that value. The visible light reflectance for the back surface is calculated using:

$$R_{Vis,b}=-0.7409T_{Vis}^3+1.6531T_{Vis}^2-1.2299T_{Vis}+0.4547$$

The visible light reflectance for the front surface is calculated using:

$$R_{Vis,f}=-0.0622T_{Vis}^3+0.4277T_{Vis}^2-0.4169T_{Vis}+0.2399$$

Step 7. Determine Angular Performance[LINK]

The angular properties of windows are important because during energy modeling, the solar incidence angles are usually fairly high. Angles of incidence are defined as angles from the normal direction extending out from the window. The simple glazing system model includes a range of correlations that are selected based on the values for U and SHGC. These were chosen to match the types of windows likely to have such performance levels. The matrix of possible combinations of U and SHGC values have been mapped to set of 28 bins shown in the following figure.

Diagram of Transmittance and Reflectance Correlations Used based on U and SHGC [fig:diagram-of-transmittance-and]

There are ten different correlations, A thru J, for both transmission and reflectance. The correlations are used in various weighting and interpolation schemes according the figure above. The correlations are normalized against the performance at normal incidence. EnergyPlus uses these correlations to store the glazing system’s angular performance at 10 degree increments and interpolates between them during simulations. The model equations use the cosine of the incidence angle, $\cos \left(\phi \right)$, as the independent variable. The correlations have the form:

$$TorR=a\cos (\phi {)}^4+b\cos (\phi {)}^3+c\cos (\phi {)}^2+d\cos \left(\phi \right)+e$$

The coefficient values for a, b, c, d, and e are listed in the following tables for each of the curves.

Normalized Transmittance Correlations for Angular Performance [fig:normalized-transmittance-correlations-for]

Normalized Reflectanct Correlations for Angular Performance [fig:normalized-reflectanct-correlations-for]

Application Issues[LINK]

EnergyPlus’s normal process of running the detailed layer-by-layer model, with the equivalent layer produced by this model, creates reports (sent to the EIO file) of the overall performance indices and the properties of the equivalent layer. Both of these raise issues that may be confusing.

The simplified window model does not reuse all aspects of the detailed layer-by-layer model, in that the angular solar transmission properties use a different model when the simple window model is in effect. If the user takes the material properties of an equivalent glazing layer from the simple window model and then re-enters them into just the detailed model, then the performance will not be the same because of the angular transmission model will have changed. It is not proper use of the model to re-enter the equivalent layer’s properties and expect the exact level of performance.

There may not be exact agreement between the performance indices echoed out and those input in the model. This is expected with the model and the result of a number of factors. For example, although input is allowed to go up to U-7 W/m${}^2$$\cdot$K, the actual outcome is limited to no higher than about 5.8W/m${}^2$$\cdot$K. This is because the thermal resistance to heat transfer at the surfaces is already enough resistance to provide an upper limit to the conductance of a planar surface. Sometimes there is conflict between the SHGC and the U that are not physical and compromises need to be made. In general, the simple window model is intended to generate a physically-reasonable glazing that approximates the input entered as well as possible. But the model is not always be able to do exactly what is specified when the specifications are not physical.

References[LINK]

Arasteh, D., J.C. Kohler, B. Griffith, Modeling Windows in EnergyPlus with Simple Performance Indices. Lawrence Berkeley National Laboratory. In Draft. Available at

Glazing System Properties[LINK]

The optical properties of a glazing system consisting of N glass layers separated by nonabsorbing gas layers (see Figure 4) are determined by solving the following recursion relations for T${}_{i,j}$, the transmittance through layers i to j; R${}^f$${}_{i,j}$ and R${}^b$${}_{i,j}$, the front and back reflectance, respectively, from layers i to j; and A${}_j$, the absorption in layer j. Here layer 1 is the outermost layer and layer N is the innermost layer. These relations account for multiple internal reflections within the glazing system. Each of the variables is a function of wavelength.

$$T_{i,j}=\frac{T_{i,j-1}{T}_{j,j}}{1-R_{j,j}^f{R}_{j-1,i}^b}$$

$$R_{i,j}^f=R_{i,j-1}^f+\frac{T_{i,j-1}^2{R}_{j,j}^f}{1-R_{j,j}^f{R}_{j-1,i}^b}$$

$$R_{j,i}^b=R_{j,j}^b+\frac{T_{j,j}^2{R}_{j-1,i}^b}{1-R_{j-1,i}^b{R}_{j,j}^f}$$

$$A_j^f=\frac{T_{1,j-1}(1-T_{j,j}-R_{j,j}^f)}{1-R_{j,N}^f{R}_{j-1,1}^b}+\frac{T_{1,j}{R}_{j+1,N}^f(1-T_{j,j}-R_{j,j}^b)}{1-R_{j,N}^f{R}_{j-1,1}^b}$$

In Equation [eq:Ajtothefequation], T${}_{i,j}$ = 1 and R${}_{i,j}$ = 0 if i<0 or j>N.

Schematic of transmission, reflection and absorption of solar radiation within a multi-layer glazing system. [fig:schematic-of-transmission-reflection]

As an example, for double glazing (N=2), these equations reduce to:

$$T_{1,2}=\frac{T_{1,1}{T}_{2,2}}{1-R_{2,2}^f{R}_{1,1}^b}$$

$$R_{1,2}^f=R_{1,1}^f+\frac{T_{1,1}^2{R}_{2,2}^f}{1-R_{2,2}^f{R}_{1,1}^b}$$

$$R_{2,1}^b=R_{2,2}^b+\frac{T_{2,2}^2{R}_{1,1}^b}{1-R_{1,1}^b{R}_{2,2}^f}$$

$$A_1^f=(1-T_{1,1}-R_{1,1}^f)+\frac{T_{1,1}{R}_{2,2}^f(1-T_{1,1}-R_{1,1}^b)}{1-R_{2,2}^f{R}_{1,1}^b}$$

$$A_2^f=\frac{T_{1,1}(1-T_{2,2}-R_{2,2}^f)}{1-R_{2,2}^f{R}_{1,1}^b}$$

If the above transmittance and reflectance properties are input as a function of wavelength, EnergyPlus calculates “spectral average” values of the above glazing system properties by integrating over wavelength.

The spectral-average solar property is:

$$P_s=\frac{\int P\left(\lambda \right)E_s\left(\lambda \right)d\lambda }{\int E_s\left(\lambda \right)d\lambda }$$

The spectral-average visible property is:

$$P_v=\frac{\int P\left(\lambda \right)E_s\left(\lambda \right)V\left(\lambda \right)d\lambda }{\int E_s\left(\lambda \right)V\left(\lambda \right)d\lambda }$$

where $E_s\left(\lambda \right)$ is the solar spectral irradiance function and $V\left(\lambda \right)$ is the photopic response function of the eye. The default functions are shown in Table 2 and Table 3. They can be overwritten by user defined solar and/or visible spectrum using the objects Site:SolarAndVisibleSpectrum and Site:SpectrumData. They are expressed as a set of values followed by the corresponding wavelengths for values.

When a choice of Spectral is entered as the optical data type, the correlations to store the glazing system’s angular performance are generated based on angular performance at 10 degree increments. When a choice of SpectralAndAngle is entered as the optical data type, the correlations for the glazing system will be generated using 10 degree increments or more if the SpectralAndAngle properties include data for more angles. For each incident angle, the properties of the SpectralAndAngle layer(s) is calculated by linear interpolation, and then the performance of the entire glazing system is calculated for that angle. The glazing system properties at each angle are used to generate polynomial curve fits with 6 coefficients as a function of cosine of incident angle. The polynomial curves are then used in the simulation to calculate optical properties at each timestep.

If a glazing layer has optical properties that are roughly constant with wavelength, the wavelength-dependent values of $T_{i,i}$, $R_{i,i}^f$ and $R_{i,i}^b$ in Equations [eq:Tijequation] to [eq:Ajtothefequation] can be replaced with constant values for that layer.

0.0,      9.5,     42.3,    107.8,    181.0,    246.0,    395.3,    390.1,    435.3,    438.9,
 483.7,    520.3,    666.2,    712.5,    720.7,   1013.1,   1158.2,   1184.0,   1071.9,   1302.0,
1526.0,   1599.6,   1581.0,   1628.3,   1539.2,   1548.7,   1586.5,   1484.9,   1572.4,   1550.7,
1561.5,   1501.5,   1395.5,   1485.3,   1434.1,   1419.9,   1392.3,   1130.0,   1316.7,   1010.3,
1043.2,   1211.2,   1193.9,   1175.5,    643.1,   1030.7,   1131.1,   1081.6,    849.2,    785.0,
 916.4,    959.9,    978.9,    933.2,    748.5,    667.5,    690.3,    403.6,    258.3,    313.6,
 526.8,    646.4,    746.8,    690.5,    637.5,    412.6,    108.9,    189.1,    132.2,    339.0,
 460.0,    423.6,    480.5,    413.1,    250.2,     32.5,      1.6,     55.7,    105.1,    105.5,
 182.1,    262.2,    274.2,    275.0,    244.6,    247.4,    228.7,    244.5,    234.8,    220.5,
 171.5,     30.7,      2.0,      1.2,     21.2,     91.1,     26.8,     99.5,     60.4,     89.1,
  82.2,     71.5,     70.2,     62.0,     21.2,     18.5,      3.2                      

0.3000,   0.3050,   0.3100,   0.3150,   0.3200,   0.3250,   0.3300,   0.3350,   0.3400,   0.3450,
0.3500,   0.3600,   0.3700,   0.3800,   0.3900,   0.4000,   0.4100,   0.4200,   0.4300,   0.4400,
0.4500,   0.4600,   0.4700,   0.4800,   0.4900,   0.5000,   0.5100,   0.5200,   0.5300,   0.5400,
0.5500,   0.5700,   0.5900,   0.6100,   0.6300,   0.6500,   0.6700,   0.6900,   0.7100,   0.7180,
0.7244,   0.7400,   0.7525,   0.7575,   0.7625,   0.7675,   0.7800,   0.8000,   0.8160,   0.8237,
0.8315,   0.8400,   0.8600,   0.8800,   0.9050,   0.9150,   0.9250,   0.9300,   0.9370,   0.9480,
0.9650,   0.9800,   0.9935,   1.0400,   1.0700,   1.1000,   1.1200,   1.1300,   1.1370,   1.1610,
1.1800,   1.2000,   1.2350,   1.2900,   1.3200,   1.3500,   1.3950,   1.4425,   1.4625,   1.4770,
1.4970,   1.5200,   1.5390,   1.5580,   1.5780,   1.5920,   1.6100,   1.6300,   1.6460,   1.6780,
1.7400,   1.8000,   1.8600,   1.9200,   1.9600,   1.9850,   2.0050,   2.0350,   2.0650,   2.1000,
2.1480,   2.1980,   2.2700,   2.3600,   2.4500,   2.4940,   2.5370                      

[table:solar-spectral-irradiance-function.]

0.0000,   0.0001,   0.0001,   0.0002,   0.0004,   0.0006,   0.0012,   0.0022,   0.0040,   0.0073,
0.0116,   0.0168,   0.0230,   0.0298,   0.0380,   0.0480,   0.0600,   0.0739,   0.0910,   0.1126,
0.1390,   0.1693,   0.2080,   0.2586,   0.3230,   0.4073,   0.5030,   0.6082,   0.7100,   0.7932,
0.8620,   0.9149,   0.9540,   0.9803,   0.9950,   1.0000,   0.9950,   0.9786,   0.9520,   0.9154,
0.8700,   0.8163,   0.7570,   0.6949,   0.6310,   0.5668,   0.5030,   0.4412,   0.3810,   0.3210,
0.2650,   0.2170,   0.1750,   0.1382,   0.1070,   0.0816,   0.0610,   0.0446,   0.0320,   0.0232,
0.0170,   0.0119,   0.0082,   0.0158,   0.0041,   0.0029,   0.0021,   0.0015,   0.0010,   0.0007,
0.0005,   0.0004,   0.0002,   0.0002,   0.0001,   0.0001,   0.0001,   0.0000,   0.0000,   0.0000,
0.0000                                                                                  

  .380,     .385,     .390,     .395,     .400,     .405,     .410,     .415,     .420,     .425,
  .430,     .435,     .440,     .445,     .450,     .455,     .460,     .465,     .470,     .475,
  .480,     .485,     .490,     .495,     .500,     .505,     .510,     .515,     .520,     .525,
  .530,     .535,     .540,     .545,     .550,     .555,     .560,     .565,     .570,     .575,
  .580,     .585,     .590,     .595,     .600,     .605,     .610,     .615,     .620,     .625,
  .630,     .635,     .640,     .645,     .650,     .655,     .660,     .665,     .670,     .675,
  .680,     .685,     .690,     .695,     .700,     .705,     .710,     .715,     .720,     .725,
  .730,     .735,     .740,     .745,     .750,     .755,     .760,     .765,     .770,     .775,
  .780                                                                                  

[table:photopic-response-function.]

Calculation of Angular Properties[LINK]

Calculation of optical properties is divided into two categories: uncoated glass and coated glass.

Angular Properties for Uncoated Glass[LINK]

The following discussion assumes that optical quantities such as transmissivity, reflectvity, absorptivity, and index of refraction are a function of wavelength, λ. If there are no spectral data the angular dependence is calculated based on the single values for transmittance and reflectance in the visible and solar range. In the visible range an average wavelength of 0.575 microns is used in the calculations. In the solar range an average wavelength of 0.898 microns is used.

The spectral data include the transmittance, T, and the reflectance, R. For uncoated glass the reflectance is the same for the front and back surfaces. For angle of incidence, $\varphi$ , the transmittance and reflectance are related to the transmissivity, $\tau$, and reflectivity, $\rho$, by the following relationships:

$$T\left(\varphi \right)=\frac{\tau {\left(\varphi \right)}^2{e}^{-\alpha d/\cos {\varphi }^{\prime }}}{1-\rho {\left(\varphi \right)}^2{e}^{-2\alpha d/\cos {\varphi }^{\prime }}}$$

$$R\left(\varphi \right)=\rho \left(\varphi \right)\left(1+T\left(\varphi \right)e^{-\alpha d/\cos {\varphi }^{\prime }}\right)$$

The spectral reflectivity is calculated from Fresnel’s equation assuming unpolarized incident radiation:

$$\rho \left(\varphi \right)=\frac{1}{2}\left({\left(\frac{n\cos \varphi -\cos {\varphi }^{\prime }}{n\cos \varphi +\cos {\varphi }^{\prime }}\right)}^2+{\left(\frac{n\cos {\varphi }^{\prime }-\cos \varphi }{n\cos {\varphi }^{\prime }+\cos \varphi }\right)}^2\right)$$

The spectral transmittivity is given by:

$$\tau \left(\varphi \right)=1-\rho \left(\varphi \right)$$

The spectral absorption coefficient is defined as:

$$\alpha =\frac{4\pi \kappa }{\lambda }$$

where $\kappa$ is the dimensionless spectrally-dependent extinction coefficient and $\lambda$ is the wavelength expressed in the same units as the sample thickness.

Solving Equation [eq:RhoofPhi] at normal incidence gives:

$$n=\frac{1+\sqrt{\rho \left(0\right)}}{1-\sqrt{\rho \left(0\right)}}$$

Evaluating Equation [eq:RofPhiEquation] at normal incidence gives the following expression for $\kappa$:

$$\kappa =-\frac{\lambda }{4\pi d}\ln \frac{R\left(0\right)-\rho \left(0\right)}{\rho \left(0\right)T\left(0\right)}$$

Eliminating the exponential in Equations [eq:TofPhiEquation] and [eq:RofPhiEquation] gives the reflectivity at normal incidence:

$$\rho \left(0\right)=\frac{\beta -\sqrt{{\beta }^2-4(2-R(0\left)\right)R\left(0\right)}}{2(2-R(0\left)\right)}$$

where

$$\beta =T(0{)}^2-R(0{)}^2+2R\left(0\right)+1$$

The value for the reflectivity, $\rho$(0), from Equation [eq:Rho0Equation] is substituted into Equations [eq:NAsFunctionOfRhoEquation] and [eq:KappaAsFunctionOfLambdaRTEquation]. The result from Equation [eq:KappaAsFunctionOfLambdaRTEquation] is used to calculate the absorption coefficient in Equation [eq:AlphaAsFunctionOfKappaLambda]. The index of refraction is used to calculate the reflectivity in Equation [eq:RhoofPhi] which is then used to calculate the transmittivity in Equation [eq:TauofPhiEquation]. The reflectivity, transmissivity and absorption coefficient are then substituted into Equations [eq:TofPhiEquation] and [eq:RofPhiEquation] to obtain the angular values of the reflectance and transmittance.

Angular Properties for Coated Glass[LINK]

A regression fit is used to calculate the angular properties of coated glass from properties at normal incidence. If the transmittance of the coated glass is > 0.645, the angular dependence of uncoated clear glass is used. If the transmittance of the coated glass is $\le$ 0.645, the angular dependence of uncoated bronze glass is used. The values for the angular functions for the transmittance and reflectance of both clear glass $({\overline{\tau }}_{clr},{\overline{\rho }}_{clr})$ and bronze glass $({\overline{\tau }}_{bnz},{\overline{\rho }}_{bnz})$ are determined from a fourth-order polynomial regression:

$$\overline{\tau }\left(\varphi \right)={\overline{\tau }}_0+{\overline{\tau }}_1\cos \left(\varphi \right)+{\overline{\tau }}_2{\cos }^2\left(\varphi \right)+{\overline{\tau }}_3{\cos }^3\left(\varphi \right)+{\overline{\tau }}_4{\cos }^4\left(\varphi \right)$$

and

$$\overline{\rho }\left(\varphi \right)={\overline{\rho }}_0+{\overline{\rho }}_1\cos \left(\varphi \right)+{\overline{\rho }}_2{\cos }^2\left(\varphi \right)+{\overline{\rho }}_3{\cos }^3\left(\varphi \right)+{\overline{\rho }}_4{\cos }^4\left(\varphi \right)-\overline{\tau }\left(\varphi \right)$$

The polynomial coefficients are given in Table 4.

These factors are used as follows to calculate the angular transmittance and reflectance:

For T(0) > 0.645:

$$T\left(\varphi \right)=T\left(0\right){\overline{\tau }}_{clr}\left(\varphi \right)$$

$$R\left(\varphi \right)=R\left(0\right)(1-{\overline{\rho }}_{clr}(\varphi \left)\right)+{\overline{\rho }}_{clr}\left(\varphi \right)$$

For T(0) $\le$ 0.645:

$$T\left(\varphi \right)=T\left(0\right){\overline{\tau }}_{bnz}\left(\varphi \right)$$

$$R\left(\varphi \right)=R\left(0\right)(1-{\overline{\rho }}_{bnz}(\varphi \left)\right)+{\overline{\rho }}_{bnz}\left(\varphi \right)$$

Angular Properties for Simple Glazing Systems[LINK]

When the glazing system is modeled using the simplified method, an alternate method is used to determine the angular properties. The equation for solar transmittance as a function of incidence angle, $T\left(\varphi \right)$, is:

$$T\left(\varphi \right)=T\left(\varphi =0\right)\cos \left(\varphi \right)\left(1+(0.768+0.817SHG{C}^4){\sin }^3\left(\varphi \right)\right)$$

where,

$T\left(\varphi =0\right)$ is the normal incidence solar transmittance, $T_{Sol}$.

The equation for solar reflectance as a function of incidence angle, $R\left(\varphi \right)$, is:

$$R\left(\varphi \right)=\frac{R\left(\varphi =0\right)\left(f_1\left(\varphi \right)+f_2\left(\varphi \right)\sqrt{SHGC}\right)}{R_{fit,o}}$$

where,

$$f_1\left(\varphi \right)=\left(\left(\left(2.403\cos \left(\varphi \right)-6.192\right)\cos \left(\varphi \right)+5.625\right)\cos \left(\varphi \right)-2.095\right)\cos \left(\varphi \right)+1$$

$$f_2\left(\varphi \right)=\left(\left(\left(-1.188\cos \left(\varphi \right)+2.022\right)\cos \left(\varphi \right)+0.137\right)\cos \left(\varphi \right)-1.720\right)\cos \left(\varphi \right)$$

$$R_{fit,o}=0.7413-\left(0.7396\sqrt{SHGC}\right)$$

Calculation of Hemispherical Values[LINK]

The hemispherical value of a property is determined from the following integral:

$$P_{hemispherical}=2{\int }_0^{\frac{\pi }{2}}P\left(\varphi \right)\cos \left(\varphi \right)\sin \left(\varphi \right)d\varphi$$

The integral is evaluated by Simpson’s rule for property values at angles of incidence from 0 to 90 degrees in 10-degree increments.

Optical Properties of Window Shading Devices[LINK]

Shading devices affect the system transmittance and glass layer absorptance for short-wave radiation and for long-wave (thermal) radiation. The effect depends on the shade position (interior, exterior or between-glass), its transmittance, and the amount of inter-reflection between the shading device and the glazing. Also of interest is the amount of radiation absorbed by the shading device.

In EnergyPlus, shading devices are divided into four categories, “shades,” “blinds,” “screens,” and “switchable glazing.” “Shades” are assumed to be perfect diffusers. This means that direct radiation incident on the shade is reflected and transmitted as hemispherically uniform diffuse radiation: there is no direct component of transmitted radiation. It is also assumed that the transmittance, ${\tau }_{sh}$, reflectance, ${\rho }_{sh}$, and absorptance, ${\alpha }_{sh}$, are the same for the front and back of the shade and are independent of angle of incidence. Many types of drapery and pull-down roller devices are close to being perfect diffusers and can be categorized as “shades.”

“Blinds” in EnergyPlus are slat-type devices such as venetian blinds. Unlike shades, the optical properties of blinds are strongly dependent on angle of incidence. Also, depending on slat angle and the profile angle of incident direct radiation, some of the direct radiation may pass between the slats, giving a direct component of transmitted radiation.

“Screens” are debris or insect protection devices made up of metallic or non-metallic materials. Screens may also be used as shading devices for large glazing areas where excessive solar gain is an issue. The EnergyPlus window screen model assumes the screen is composed of intersecting orthogonally-crossed cylinders, with the surface of the cylinders assumed to be diffusely reflecting. Screens may only be used on the exterior surface of a window construction. As with blinds, the optical properties affecting the direct component of transmitted radiation are dependent on the angle of incident direct radiation.

With “Switchable glazing,” shading is achieved making the glazing more absorbing or more reflecting, usually by an electrical or chemical mechanism. An example is electrochromic glazing where the application of an electrical voltage or current causes the glazing to switch from light to dark.

Shades and blinds can be either fixed or moveable. If moveable, they can be deployed according to a schedule or according to a trigger variable, such as solar radiation incident on the window. Screens can be either fixed or moveable according to a schedule.

Shades[LINK]

Shade/Glazing System Properties for Short-Wave Radiation[LINK]

Short-wave radiation includes:

• Beam solar radiation from the sun and diffuse solar radiation from the sky and ground incident on the outside of the window,

• Beam and/or diffuse radiation reflected from exterior obstructions or the building itself,

• Solar radiation reflected from the inside zone surfaces and incident as diffuse radiation on the inside of the window,

• Beam solar radiation from one exterior window incident on the inside of another window in the same zone, and

• Short-wave radiation from electric lights incident as diffuse radiation on the inside of the window.

Exterior Shade

For an exterior shade we have the following expressions for the system transmittance, the effective system glass layer absorptance, and the system shade absorptance, taking inter-reflection between shade and glazing into account. Here, “system” refers to the combination of glazing and shade. The system properties are given in terms of the isolated shade properties (i.e., shade properties in the absence of the glazing) and the isolated glazing properties (i.e., glazing properties in the absence of the shade).

$$T_{sys}\left(\varphi \right)=T_{1,N}^{dif}\frac{{\tau }_{sh}}{1-R_f^{dif}{\rho }_{sh}}$$

$$T_{sys}^{dif}=T_{1,N}^{dif}\frac{{\tau }_{sh}}{1-R_f^{dif}{\rho }_{sh}}$$

$$A_{j,f}^{sys}\left(\varphi \right)=A_{j,f}^{dif}\frac{{\tau }_{sh}}{1-R_f{\rho }_{sh}},j=1toN$$

$$A_{j,f}^{dif,sys}=A_{j,f}^{dif}\frac{{\tau }_{sh}}{1-R_f{\rho }_{sh}},j=1toN$$

$$A_{j,b}^{dif,sys}=A_{j,b}^{dif}\frac{T_{1,N}^{dif}{\rho }_{sh}}{1-R_f{\rho }_{sh}},j=1toN$$

$${\alpha }_{sh}^{sys}={\alpha }_{sh}\left(1+\frac{{\tau }_{sh}{R}_f}{1-R_f{\rho }_{sh}}\right)$$

Interior Shade

The system properties when an interior shade is in place are the following:

$$T_{sys}\left(\varphi \right)=T_{1,N}\left(\varphi \right)\frac{{\tau }_{sh}}{1-R_b^{dif}{\rho }_{sh}}$$

$$T_{sys}^{dif}=T_{1,N}^{dif}\frac{{\tau }_{sh}}{1-R_b^{dif}{\rho }_{sh}}$$

$$A_{j,f}^{sys}\left(\varphi \right)=A_{j,f}\left(\varphi \right)+T_{1,N}\left(\varphi \right)\frac{{\rho }_{sh}}{1-R_b^{dif}{\rho }_{sh}}{A}_{j,b}^{dif},j=1toN$$

$$A_{j,f}^{dif,sys}=A_{j,f}^{dif}+T_{1,N}^{dif}\frac{{\rho }_{sh}}{1-R_b^{dif}{\rho }_{sh}}{A}_{j,b}^{dif},j=1toN$$

$$A_{j,b}^{dif,sys}=\frac{{\tau }_{sh}}{1-R_b^{dif}{\rho }_{sh}}{A}_{j,b}^{dif},j=1toN$$

$${\alpha }_{sh}^{sys}\left(\varphi \right)=T_{1,N}\left(\varphi \right)\frac{{\alpha }_{sh}}{1-R_b^{dif}{\rho }_{sh}}$$

$${\alpha }_{sh}^{dif,sys}=T_{1,N}^{dif}\frac{{\alpha }_{sh}}{1-R_b^{dif}{\rho }_{sh}}$$

Long-Wave Radiation Properties of Window Shades[LINK]

Long-wave radiation includes:

• Thermal radiation from the sky, ground and exterior obstructions incident on the outside of the window,

• Thermal radiation from other room surfaces incident on the inside of the window, and

• Thermal radiation from internal sources, such as equipment and electric lights, incident on the inside of the window.

The program calculates how much long-wave radiation is absorbed by the shade and by the adjacent glass surface. The system emissivity (thermal absorptance) for an interior or exterior shade, taking into account reflection of long-wave radiation between the glass and shade, is given by:

$${\epsilon }_{sh}^{lw,sys}={\epsilon }_{sh}^{lw}\left(1+\frac{{\tau }_{sh}^{lw}{\rho }_{gl}^{lw}}{1-{\rho }_{sh}^{lw}{\rho }_{gl}^{lw}}\right)$$

where ${\rho }_{gl}^{lw}$ is the long-wave reflectance of the outermost glass surface for an exterior shade or the innermost glass surface for an interior shade, and it is assumed that the long-wave transmittance of the glass is zero.

The innermost (for interior shade) or outermost (for exterior shade) glass surface emissivity when the shade is present is:

$${\epsilon }_{gl}^{lw,sys}={\epsilon }_{gl}^{lw}\frac{{\tau }_{sh}^{lw}}{1-{\rho }_{sh}^{lw}{\rho }_{gl}^{lw}}$$

Switchable Glazing[LINK]

For switchable glazing, such as electrochromics, the solar and visible optical properties of the glazing can switch from a light state to a dark state. The switching factor, f${}_{switch}$, determines what state the glazing is in. An optical property, p, such as transmittance or glass layer absorptance, for this state is given by:

$$p=(1-f_{switch})p_{light}+f_{switch}{p}_{dark}$$

where

p${}_{light}$ is the property value for the unswitched, or light state, and p${}_{dark}$ is the property value for the fully switched, or dark state.

The value of the switching factor in a particular time step depends on what type of switching control has been specified: “schedule,” “trigger,” or “daylighting.” If “schedule,” f${}_{switch}$ = schedule value, which can be 0 or 1.

Thermochromic Windows[LINK]

Thermochromic (TC) materials have active, reversible optical properties that vary with temperature. Thermochromic windows are adaptive window systems for incorporation into building envelopes. Thermochromic windows respond by absorbing sunlight and turning the sunlight energy into heat. As the thermochromic film warms it changes its light transmission level from less absorbing to more absorbing. The more sunlight it absorbs the lower the light level going through it. Figure 5 shows the variations of window properties with the temperature of the thermochromic glazing layer. By using the suns own energy the window adapts based solely on the directness and amount of sunlight. Thermochromic materials will normally reduce optical transparency by absorption and/or reflection, and are specular (maintaining vision).

Variations of Window Properties with the Temperature of the Thermochromic Glazing Layer [fig:variations-of-window-properties-with]

On cloudy days the window is at full transmission and letting in diffuse daylighting. On sunny days the window maximizes diffuse daylighting and tints based on the angle of the sun relative to the window. For a south facing window (northern hemisphere) the daylight early and late in the day is maximized and the direct sun at mid day is minimized.

The active thermochromic material can be embodied within a laminate layer or a surface film. The overall optical state of the window at a given time is a function primarily of:

• thermochromic material properties

• solar energy incident on the window

• construction of the window system that incorporates the thermochromic layer

• environmental conditions (interior, exterior, air temperature, wind, etc).

The tinted film, in combination with a heat reflecting, low-e layer allows the window to reject most of the absorbed radiation thus reducing undesirable heat load in a building. In the absence of direct sunlight the window cools and clears and again allows lower intensity diffuse radiation into a building. TC windows can be designed in several ways (Figure 6), with the most common being a triple pane windows with the TC glass layer in the middle a double pane windows with the TC layer on the inner surface of the outer pane or for sloped glazing a double pane with the laminate layer on the inner pane with a low-e layer toward the interior. The TC glass layer has variable optical properties depending on its temperature, with a lower temperature at which the optical change is initiated, and an upper temperature at which a minimum transmittance is reached. TC windows act as passive solar shading devices without the need for sensors, controls and power supplies but their optical performance is dependent on varying solar and other environmental conditions at the location of the window.

Configurations of Thermochromic Windows [fig:configurations-of-thermochromic-windows]

EnergyPlus describes a thermochromic window with a Construction object which references a special layer defined with a WindowMaterial:GlazingGroup:Thermochromic object. The WindowMaterial:GlazingGroup:Thermochromic object further references a series of WindowMaterial:Glazing objects corresponding to each specification temperature of the TC layer. During EnergyPlus run time, a series of TC windows corresponding to each specification temperature is created once. At the beginning of a particular time step calculations, the temperature of the TC glass layer from the previous time step is used to look up the most closed specification temperature whose corresponding TC window construction will be used for the current time step calculations. The current time step calculated temperature of the TC glass layer can be different from the previous time step, but no iterations are done in the current time step for the new TC glass layer temperature. This is an approximation that considers the reaction time of the TC glass layer can be close to EnergyPlus simulation time step say 10 to 15 minutes.

Blinds[LINK]

Window blinds in EnergyPlus are defined as a series of equidistant slats that are oriented horizontally or vertically. All of the slats are assumed to have the same optical properties. The overall optical properties of the blind are determined by the slat geometry (width, separation and angle) and the slat optical properties (front-side and back-side transmittance and reflectance). Blind properties for direct radiation are also sensitive to the “profile angle,” which is the angle of incidence in a plane that is perpendicular to the window plane and to the direction of the slats. The blind optical model in EnergyPlus is based on Simmler, Fischer and Winkelmann, 1996; however, that document has numerous typographical errors and should be used with caution.

The following assumptions are made in calculating the blind optical properties:

• The slats are flat.

• The spectral dependence of inter-reflections between slats and glazing is ignored; spectral-average slat optical properties are used.

• The slats are perfect diffusers. They have a perfectly matte finish so that reflection from a slat is isotropic (hemispherically uniform) and independent of angle of incidence, i.e., the reflection has no specular component. This also means that absorption by the slats is hemispherically uniform with no incidence angle dependence. If the transmittance of a slat is non-zero, the transmitted radiation is isotropic and the transmittance is independent of angle of incidence.

• Inter-reflection between the blind and wall elements near the periphery of the blind is ignored.

• If the slats have holes through which support strings pass, the holes and strings are ignored. Any other structures that support or move the slats are ignored.

Slat Optical Properties[LINK]

The slat optical properties used by EnergyPlus are shown in the following table.

It is assumed that there is no direct-to-direct transmission or reflection, so that ${\tau }_{dir,dir}=0$, ${\rho }_{dir,dir}^f=0$, and ${\rho }_{dir,dir}^b=0$. It is further assumed that the slats are perfect diffusers, so that ${\tau }_{dir,dif}$, ${\rho }_{dir,dif}^f$ and ${\rho }_{dir,dif}^b$ are independent of angle of incidence. Until the EnergyPlus model is improved to take into account the angle-of-incidence dependence of slat transmission and reflection, it is assumed that ${\tau }_{dir,dif}$ = ${\tau }_{dif,dif}$, ${\rho }_{dir,dif}^f$ = ${\rho }_{dif,dif}^f$, and ${\rho }_{dir,dif}^b$ = ${\rho }_{dif,dif}^b$.

Direct Transmittance of Blind[LINK]

The direct-to-direct and direct-to-diffuse transmittance of a blind is calculated using the slat geometry shown in Figure 7(a), which shows the side view of one of the cells of the blind. For the case shown, each slat is divided into two segments, so that the cell is bounded by a total of six segments, denoted by s${}_1$ through s${}_6$ (note in the following that s${}_i$ refers to both segment i and the length of segment i).The lengths of s${}_1$ and s${}_2$ are equal to the slat separation, h, which is the distance between adjacent slat faces. s${}_3$ and s${}_4$ are the segments illuminated by direct radiation. In the case shown in Figure 7(a) the cell receives radiation by reflection of the direct radiation incident on s${}_4$ and, if the slats have non-zero transmittance, by transmission through s${}_3$, which is illuminated from above.

The goal of the blind direct transmission calculation is to determine the direct and diffuse radiation leaving the cell through s${}_2$ for unit direct radiation entering the cell through s${}_1$.

(a) Side view of a cell formed by adjacent slats showing how the cell is divided into segments, $s_i$, for the calculation of direct solar transmittance; (b) side view of a cell showing case where some of the direct solar passes between adjacent slats without touching either of them. In this figure ${\varphi }_s$ is the profile angle and ${\varphi }_b$ is the slat angle. [fig:a-side-view-of-a-cell-formed-by-adjacent]

Direct-to-Direct Blind Transmittance[LINK]

Figure 7(b) shows the case where some of the direct radiation passes through the cell without hitting the slats. From the geometry in this figure we see that

$${\tau }_{bl,f}^{dir,dir}=1-\frac{|w|}{h},|w|\le h$$

where

$$w=s\frac{\cos ({\phi }_b-{\phi }_s)}{\cos {\phi }_s}$$

Note that we are assuming that the slat thickness is zero. A correction for non-zero slat thickness is described later.

Direct-to-Diffuse Blind Transmittance, Reflectance and Absorptance[LINK]

The direct-to-diffuse and transmittance and reflectance of the blind are calculated using a radiosity method that involves the following three vector quantities:

J${}_i$ = the radiosity of segment s${}_i$, i.e., the total radiant flux into the cell from s${}_i$

G${}_i$ = the irradiance on the cell side of s${}_i$

Q${}_i$ = the source flux from the cell side of s${}_i$

Based on these definitions we have the following equations that relate J, G and Q for the different segments:

$$\begin{array}{cc}{J}_1& =Q_1\\ J_2& =Q_2\\ J_3& =Q_3+{\rho }_{dif,dif}^b{G}_3+{\tau }_{dif,dif}{G}_4\\ J_4& =Q_4+{\tau }_{dif,dif}{G}_3+{\rho }_{dif,dif}^f{G}_4\\ J_5& =Q_5+{\rho }_{dif,dif}^b{G}_5+{\tau }_{dif,dif}{G}_6\\ J_6& =Q_6+{\tau }_{dif,dif}{G}_5+{\rho }_{dif,dif}^f{G}_6\end{array}$$

In addition we have the following equation relating G and J:

$$G_i=\sum _{j=1}^6{J}_j{F}_{ji},i=1,6$$

where $F_{ji}$ is the view factor between $s_j$ and $s_i$, i.e., $F_{ji}$ is the fraction of radiation leaving $s_j$ that is intercepted by $s_i$.

Using $J_1=Q_1=0$ and $J_2=Q_2=0$ and combining the above equations gives the following equation set relating J and Q:

$$J_3-{\rho }_{dif,dif}^b\sum _{j=3}^6{J}_j{F}_{j3}-{\tau }_{dif,dif}\sum _{j=3}^6{J}_j{F}_{j4}=Q_3$$

$$J_4-{\tau }_{dif,dif}\sum _{j=3}^6{J}_j{F}_{j3}-{\rho }_{dif,dif}^f\sum _{j=3}^6{J}_j{F}_{j4}=Q_4$$

$$J_5-{\rho }_{dif,dif}^b\sum _{j=3}^6{J}_j{F}_{j5}-{\tau }_{dif,dif}\sum _{j=3}^6{J}_j{F}_{j6}=Q_5$$