NBSIR 86-3426

Modeling Window Optics for

Building Energy Analysis

George N. Walton

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Building Physics Division

Gaithersburg, MD 20899

July 1986

I

Prepared for; Solar Buildings Technology Division

nffirfi of Solar Heat Technologies

'epartment of Energy

QQ igton, DC 20585

100

.U56

NO.86-3426

1986

NBSIR 86-3426

MODELING WINDOW OPTICS FOR

BUILDING ENERGY ANALYSIS

George N. Walton

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Building Physics Division

Gaithersburg, MD 20899

July 1986

Prepared for: Solar Buildings Technology Division

Office of Solar Heat Technologies

U.S. Department of Energy

Washington, DC 20585

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Director

ABSTRACT

Tliis report discusses modeling the optics of windows for the purposes of

simulating building energy requirements or daylighting availability. The

theory for calculating the optical performance of conventional windows is

reviewed. The simplifications that might commonly be made in creating

computational models are analyzed. Some of the possibilities for more complex

windows are analyzed, and the type of model and data that would be necessary

to simulate such windows in a building energy analysis program are determined.

It is shown that the optical performance of different window types can be

simulated with models which require varying amounts of memory or computing

time. It is recommended that a building energy analysis program have all

models available and use the most efficient for any given window.

iii
SOLAR BUILDINGS RESEARCH AND DEVELOPMENT PROGRAM

CONTEXT STATEMENT

November 21 , 1985

In keeping with the national energy policy goal of fostering an adequate

supply of energy at a reasonable cost, the United States Department of Energy

(DOE) supports a variety of programs to promote a balanced and mixed energy

resource system. The mission of the DOE Solar Buildings Research and Development Program is to support this goal by providing for the development of solar

technology alternatives for the building sector. It is the goal of the

Program to establish a proven technology base to allow industry to develop

solar products and designs for buildings which are economically competitive

and can contribute significantly to building energy supplies nationally.

Toward this end, the program sponsors research activities related to

increasing the efficiency, reducing the cost, and improving the long term

durability of passive and active solar systems for building water and space

heating, cooling, and daylighting applications. These activities are conducted in four major areas: Advanced Passive Solar Materials Research,

Collector Technology Research, Cooling Systems Research, and Systems Analysis

and Applications Research.

Advanced Passive Solar Materials Research - This activity area includes work

on new aperture materials for controlling solar heat gains, and for enhancing

the use of daylight for building interior lighting purposes. It also

encompasses work on low-cost thermal stroage materials that have high thermal

storage capacity and can be integrated with conventional building elements,

and work on materials and methods to transport thermal energy efficiently

between any building exterior surface and the building interior by non

mechanical means.

Collector Technology Research - This activity area encompasses work on advanced low to medium temperature (up to 180°F useful operating temperature)

flat plate collectors for water and space heating applications, and medium to

high temperature (up to 400°F useful operating temperature) evacuated tube/-

concentrating collectors for space heating and cooling applications. The

focus is on design innovations using new materials and fabrication techniques.

Cooling Systems Research - This activity area involves research on high

performance dehumidifiers and chillers that can operate efficiently with the

variable thermal outputs and delivery temperatures associated with solar

collectors. It also includes work on advanced passive cooling techniques.

Systems Analysis and Applications Research - This activity area encompasses

experimental testing, analysis, and evaluation of solar heating, cooling, and

daylighting systems for residential and nonresidential buildings. This

involves system integration studies, the development of design and analysis

tools and the establishment of overall cost, performance, and durability

targets for various technology or system options.

This report is an account of research conducted in the Systems Analysis area concerning the mathematical modeling of window optics for the purposes of

simulating building energy requirements or daylighting availability.

IV
TABLES OF CONTENTS

1 . INTRODUCTION 1

2. OPTICAL THEORY FOR IDEAL PANES OF GLASS 3

2.1 The Air-Glass Interface 3

2.2 Single Glass Pane 5

2.3 Multiple Pane Window 8

2.4 Hemispheric (Diffuse) Performance 11

3. APPROXIMATIONS FOR SIMPLE WINDOWS 13

3.1 Shading Coefficient Procedure 13

3.2 Polarization 16

• » 3.3 Wavelength 19

3.4 Angular Performance 22

3.5 Hemispheric Performance 24

4. APPROXIMATIONS FOR COMPLEX WINDOWS 27

4.1 Diffusing Layers 27

4.2 Slat Shading Devices 29

4.3 Other Devices 31

5. SUMMARY AND CONCLUSIONS 32

REFERENCES 35

APPENDIX: MODEL OF SIMPLE SLAT SHADING DEVICE 36

V
LIST OF TABLES

Table 1.

Table 2.

Table 3.

Wavelength Dependence of Optical Properties

RMS Error of the Polynomial Approximation for Window Optics

Accuracy of Interpolation for Window Optics

20

22

23

VI
LIST OF FIGURES

Figure 1. Reflection and Refraction at an Interface 3

Figure 2. Wavelength Dependence of the Index of Refraction 4

Figure 3. Multiple Reflections in a Pane of Glass 7

Figure 4. Coordinate System for Diffuse Radiation 12

Figure 5. Heat Admission Through Glass 14

Figure 6, Polarization Performance of a Single Pane Window 17

Figure 7. Polarization Performance of a 2-pane Window 17

Figure 8. Polarization Performance of a 3-pane Window 18

Figure 9. Polarization Performance of a 4-pane window 18

Figure 10. Wavelength Performance of Five Common Window Glasses 21

Figure 11. Hemispheric Performance Values by Simpson's Rule 24

Figure 12. Horizontal Slat Shading Device 30

vii
NOMENCLATURE

solar radiation - electromagnetic radiation in the ultraviolet, visible,

and near infrared portions of the spectrun.

light - radiation in the visible portion of the electromagnetic spectrum,

optical interface - boundary between two media where radiation is

reflected and transmitted.

optical layer - a pane of glass or a drape or shading device.

optical system - a window which is composed of one or more optical layers.

Commonly used variables:

a -- absorptance of a layer

A - absorptance of a layer in a system

p - reflectance of an interface

r - reflectance of a layer

R - reflectance of a system

T - transmittance of an interface

t - transmittance of a layer

T - transmittance of a system

6 - angle of incidence

X - wavelength

Subscripts

n — layer number

b - beam

d - diffuse

s - scattered

Superscripts

+ - for radiation going to the right (inside)

- - for radiation going to the left (outside)

Examples

transmittance of layer n for beam radiation toward the inside

transmittance of the system for diffuse radiation to the inside

anb absorptance of layer n for beam radiation to the outside

nb

ns

absorptance of layer n for beam radiation to the outside

when the layer is part of an optical system

transmittance of layer n beam radiation which is incident

from outside and scattered (becomes diffuse) at the inside

viii
1 . INTRODUCTION

Windows are thermally critical elements of the building envelope. They

are difficult to insulate and usually produce much higher conductive gains and

losses than corresponding wall areas. Many can be opened to provide ventilation, but such operable windows frequently become significant sources for

infiltration. The unique characteristic of windows is the transmission of

solar radiation directly into the building. This transmitted radiation

usually produces a desirable energy gain in the winter and an undesirable gain

in the summer. It also provides natural lighting inside the building which

can be used to offset electric lighting requirements. Calculation of solar

heat gain and of daylighting should be closely related since they both concern

radiation from the sun, but there has been no unified approach to the optical

modeling of windows.

Accuracy is usually the first goal in developing a computer model. This

can include the ability to correctly predict both instantaneous values as well

as long term totals or averages. For windows we are primarily interested in

short term thermal and visual comfort considerations and long term energy

requirements. Since perfect accuracy can never be achieved, the sensitivity

of the model is important. That is, the model should be able to correctly

rank any group of alternatives since selecting the correct alternative is the

primary goal of any design process. This is related to generality, which is

the ability to handle many different conditions. A model which is unable to

simulate one of the alternatives under consideration does not help the

building designer. Similarly, the model must use data that is available to

the designer. Efficiency in terms of execution speed and computer memeory

requirements should also be considered. Decreasing costs of computer memory

and new operating systems make memory a less important consideration than it once was, but speed is always useful in that it allows more alternatives to be

considered by allowing more simulations for a given cost or time.

Several models must be integrated to form an overall model for the

analysis of solar gains and daylighting. First is the model of the angular

distribution of the solar radiation and light that is incident on the window.

This is usually classified as beam radiation which is direct from the sun and

1
diffuse radiation which has been scattered or reflected by the sky, the

ground, or surrounding objects. Another model must describe the distribution

of radiation on the surfaces of the room to determine how much is reflected or absorbed. The optical model of the window establishes the connection between

the outside and the inside. Heat gain calculations must also include a model

of the conduction and convection of heat through the window.

Solar gain models are usually based on the ASHRAE shading coefficient

procedure [1]. This procedure was developed twenty years ago for manual

calculations. Its use for transient analysis of buildings with windows which

are significantly different from the single pane reference window is

questionable (see section 3.1). Some alternatives to shading coefficients

have been included in the major public domain energy analysis programs. DOE-2

[2] includes detailed calculations for a few selected window types. The BLAST

[3] and TARP [4] programs compute the detailed optics of windows consisting

of up to four panes of glass.

Daylighting programs such as SUPERLIGHT [5], CEL-1 [6], and DALITE [7],

include much more detail, particularly in the distribution of light from the

sky and its distribution in the room. However, the detailed sky models apply

only to clear or to completely overcast days. Partly cloudy days must be

approximated. Some of these programs are restricted by lack of generality in

their window models. Their primary drawback, however, is that the

calculations are so detailed and time consuming that they cannot be directly

incorporated into hourly energy analysis programs.

This report is concerned with the modeling of window optics for the

purposes of simulating building energy requirements or daylighting

availability. It reviews the theory for calculating the optical performance

of conventional windows. It analyzes the simplifications that might be made

in creating computational models. It then looks at some of the possibilities

for more complex windows and determines the type of model and data that would

be necessary to simulate such windows in a building energy analysis program.

Such models require more data than is currently available to designers, so the

requirements for additional test data are also discussed.

2
2 . OPTICAL THEORY FOR IDEAL PANES OF GLASS

2,1 The Air-Glass Interface

The optical performance of ordinary window glass is well described by the

classical optical theories [8] of Snell's law, the Fresnel equations, and

Bouger's law. A pane of glass can be considered a partially transparent

dielectric medium which has two smooth parallel glass-to-air interfaces. When

a ray of electromagnetic radiation reaches a smooth interface between two

dielectric media, part of the ray is reflected and part is refracted as shown

in fig 1 . The angle of reflection equals the angle of incidence, 6^. The

angle of refraction, 6^, is given by Snell's law:

sin02/sin62 = vj/v2 (la)

where is the velocity of light in each medium, which is given by v = c//Tji

where c is the velocity of light in a vacuum, e is the medium's dielectric

constant (permittivity), and |x is its magnetic permeability. Defining the

refractive index of a medium to be n = c/v leads to the following version of

Sne 1

1

' s law:

nj^ sinO^ = n2 sin02 (lb)

Incident ray

Reflected

ray

Medium 1 Medium 2

Figure 1 . Reflection and Refraction at an Interface

3
Figure 2. Wavelength Dependence of the Index of Refraction

The refractive index of air is effectively equal to 1 and for glass, since it

is non-magnetic, n = /T. The refractive index of glass is dependent on both

the composition of the glass and the wavelength of the radiation. Fig 2 shows

the relationship of index of refraction to wavelength for several optical

glasses [9]. Trigono.metric identities can convert Snell's law to a relation

of cosines:

COS02 = / 1 - (l-cos^e^) dc)

Electromagnetic theory shows that reflectance and transmittance at an interface depend on the polarization of the incident radiation. The two

components of polarization of interest are pendicular to the plane of

incidence (also called transverse electric or TE) and parallel to the plane of

incidence (also called transverse magnetic or TM) . Unpolarized radiation can

be considered to consist of equal parts of both polarizations. The Fresnel

4
formulae describe the amplitudes of the reflected (gj.) and transmitted (E^)

electric components of the electromagnetic wave relative to the incident (E^)

compone nt.

(Ej./Ej^)<P£ = (n^cosG^ - n2 cos 02 ) / (n^^cosG^ + n2 cos 02 ) (2a)

(Ej./E£)'j'Jj[ = (n2cos02 ” n2^cos02) / (n^cosG^ + n2cos02) (2b)

(E^/E£)j£ = 2n2 cos 0j

/ (n^cosGj^ + n2 cos02 ) (2c)

(E^/Ej^)jjj = 2n2cos02 / (n^cosGj^ + n2cos02) (2d)

However, instead of the ratio of amplitudes, we are more interested in the

energy content of the radiation as given by Poynting's vector. The energy

reflectance, p, is given by p = (E^/g^)^ and the energy transmittance, v, is

given by ^ = (g^./gi)^(n2cos02) / (nicos0i) . If we define

Pi = niCOsGi for TE radiation

and

= cos0£/nj^ for TM radiation

for media, i = 1 and 2. Then for either polarization

2

p = [(Px P2)/^Pi P2^^

and

2

c = (P2 /P 1

) [2px/(Pi+P2)3

(3a)

(3b)

(3c)

(3d)

It is faster to compute the transmittance from the reflectance and the

requirement of conservation of energy: x = 1 - p.

2.2 Single Glass Pane

Glass absorbs radiation passing through it in proportion to the intensity

of the radiation (Bouger's law). The fraction of radiation transmitted, x ^ through a layer of glass is given by the exponential decay formula

X

8

-kx

e (4)

where k is the extinction coefficient of the glass and x is

The value of k is wavelength dependent but independent of

glass). The fraction of radiation absorbed is a = 1 - r

g

g*

of thickness L, the distance the radiation travels through

L/cos0^ ^

the

pol

For

the

path length,

arization (in

a glass pane

glass is I =

A ray incident on a pane of glass will be both reflected and refracted at

5
the front surface. Part of the refracted component will be absorbed in the

glass before reaching the back surface where it is again reflected and

refracted. This continues until all the radiation is either absorbed or escapes from the front or back surfaces of the glass. Fig 3 shows the first

several reflections and refractions of the incident ray. Summation of the

transmitted, reflected, and absorbed components of the incident ray give the

total transmittance, t, reflectance, r, and absorptance, a, of the pane of

glass. These values must be determined separately for both polarization

components, p, T (= 1 - p), T and a (= 1 - x ) are functions of the angle

o o o of incidence and wavelength of the radiation. They are identical at both

air-glass interfaces. The transmittance of the pane is given by

t = (l-p)^x + (i-p)2p2x 3 + (i_p)2p4^ 5 + ...

o o o = (l-p)^Tg (1 + p2xg2 + p4^^4 + ... )

= / (1 _ p^Tg^) (5a)

The reflectance of a pane is given by

r = p + (l-p)^pT ^ + (l-p)2p3t 4 +

= p + (l-p)^pr ^ (1 + p^T ^ + p^T ^ + ...)

= p + x^ p X ^ / (1 - p2 T 2)

g g = P + P t (5b)

The absorptance of a pane is given by

a= [d-p)-(l-p)T^] + [(l-p)pXg-(l-p)pTg2] + [(l-p)p2T^2_(i_pjp2T:^3] + ...

= (l-p)(l-Tg)(i + pt:^ + p2r^2 + ... )

= “C «g / (1 - p Tg) (5c)

The energy balance (t + r + a = 1) allows any one of the three components to

be computed from the other two.

The extinction coefficient, k, is not commonly reported, however it can

be easily computed from the thickness, refractive index, and the transmittance

at normal incidence (6 = 0) . At normal incidence

2

p = [(nj^ - n2)/(nj + U2

)]

= [(n - l)/(n + 1)]2 (6a)

6
(l-p)^Tjl

(1-p) 2 2 3 P

(1-P I w

Figure 3, Multiple Reflections in a Pane of Glass

7
where n and 1 are the refractive indices of the glass and air, respectively.

Eqn (5a) is a quadratic equation in x ^ whose solution is

X = [ + 4p^t^ - (1-p)^ ] / 2p^t (6b)

O Therefore,

where L is the thickness of the glass.

The equations which were developed above apply to clear and heat

absorbing glass. For these common types of glass, the optical properties of

the pane do not depend on which side of the glass the initial ray is incident.

However, thin film reflective layers do cause directional differences

depending on which side of the glass is coated. Optical theory for thin films

[10, 11] will not be reviewed in this report. The net result of thin film

theory is that the film, or films, which are placed on a pane of glass, can be

considered as a modified air— glass interface with transmission, refelction,

and absorption in the interface. The basic energy conservation laws still

apply for a ray traveling toward the inside or toward the outside of the pane:

*b ^b +®b = and

S ^b *b

where t^ always equals t^j, but r^ need not equal r^j, and a^ need not equal a^.

The + and - directions will be included in the equations that follow to

maintain generality for glass panes with reflective films.

2.3 Multiple Pane Window

The optical properties of a multipane optical system for a given angle of

incidence, wavelength, and polarization are quickly computed from the optical

properties of the individual panes, eqns (5a-c). Instead of summing all the

reflections and refractions of a single ray, a net radiation method is very

effective. Consider the following sketch.

8
1 = n - 1

+ _ + +

(8a) nb M'1 II * ^nb + ^nb * ^nb

(8b) mb ^nb * ^nb ^mb * ^nb

is the sum of all radiation going to the right from layer n. It is

composed of the portion of (the radiation incident on the left side) which

is transmitted through the layer plus the portion of (the radiation

incident on the right side) which is reflected from the layer, and similarly

It is easy to formulate the relationships for a two-layer window as sketched below from the relationships of eqns (8a, b).

Ob

Ob

1a1a y y

e e r

^ ^Ib r <

1 2

'2h

^2b

II 1

O +

^Ib * T+^Ob + Hh * lib (9a)

•+ + + . lb = hb * ^Ob +

^ib * Ilb (9b) + +

lb =^2b

* Ilb + ^2b

* ^2b (9c) + + + 2b

"

^2b

* hh + ^2b

* ^2b (9d)

Equations (9a-d) plus equations defining

matrix form:

1+

qIj and l 2t( can be expressed in

1 0 0 0 0 0

1 M O+

o* + “lb 1 0 -Hh 0 0 Ilb 0

0 1 -^ib 0 0

* lib 0

0 0 “^Jb 1 0 “^2b lib 0

0 0 ‘2b 0 1 -^2b ilb 0

0 0 0 0 0 1

^ ^b - - X2 -

( 10 )

9
To obtain the system properties for light traveling to the right, set ^o

1,0 and X^ to 0.0 and solve the system of equations. This gives Iq^ =1.0 and

^2b “ appropriate values for the other I's, These can be substituted

into the following equations to determine the optical properties of the

two—pane system.

^ ^

^2b f ^Ob ^2b (11a)

K = ^Ob

^Ib = Mb Mb • lib

^2b ~ Mb * ^Ib (lid)

The system properties for light incident from the right, T^^ etc., can be

determined by setting X^ to 0.0 and X2 to 1.0, sol ving for the I values, and

substituting into the following equations.

^ ~^Ob / ^2b “

^Ob (12a)

^ ~

^2b (12b)

^Ib = Mb * ^Ib (12c)

^2b ” Mb Mb * l2b (12d)

This analysis can be extended to any number of layers by increasing the

size of the matrix and solving numerically. Note that the matrix is

diagonally dominant which implies that the simultaneous equations can be

solved iteratively or directly without pivoting. The number of layers in a window should be small enough that solution time is not a significant

consideration for either method. There is a small gain in solution efficiency

by doing LU decomposition [12, sec 2.5] and then solving with the two

different right hand sides. BLAST [3] and TARP [4] use symbolic solutions of

these and similar equations for up to four layers, which is faster, but

greater generality is achieved by using the numeric solution method presented

above.

10
2.4 Hemispheric (Diffuse) Performance

The formulae developed so far give the optical performance of a single or multipane window at a single angle of incidence of the incident radiation. It

is sometimes useful to consider that the radiation is incident at equal

intensity from all angles, that is, the radiation is perfectly diffuse.

Coefficients for optical performance with respect to diffuse radiation are related to the coefficients for beam radiation in the following manner. The

flux, dq, incident at a point due to radiation incident from a small solid

angle, dfl, is given by

dq = I COS0 dQ

= I COS0 (r d0) (r sin0 d^) / r^ (13a)

where I is the intensity of incoming radiation, and the angles .0 and 0 are shown in fig 4. Assuming uniform intensity, the total incident flux is given

by

2n n/2

q = / / I COS0 sin0 d0 d0 = nl (13b)

0=0 0=0

The hemispherical transmittance is obtained by computing the fraction of the

diffusely incident radiation which is transmitted through the window:

2n n/2

f f T(0,0) I COS0 sin0 do d0

Tjj = ,0=0 0=0 (13c)

q

If T is independent of 0, eqn (13c) reduces to

n/2

= 2 f T(0) COS0 sinO d0 (13d)

0=0

Evaluation of this integral is discussed in section 3.5 .
Iqtensity

in direction

Figure 4. Coordinate System for Diffuse Radiation

12
3. APPROXIMATIONS FOR SIMPLE WINDOWS

In the following discussion, 'simple windows' will refer to those whose

optical performance can be computed by the methods presented in the previous

section.

3.1 Shading Coefficient Procedure

Before studying other approximations, it is useful to review the most

common method for computing solar gains through windows, the ASHRAE shading

coefficient method [1]. This method was developed in the early 1960's to

provide a convenient method for manual calculation of solar gains in order to

determine cooling loads for equipment sizing. Figure 5 shows the heat flows

considered in determining the total heat admitted through a pane of glass into

a room. The components of heat admission are

Total heat Sunlight Inward flow Heat flow due to

admission = transmitted + of absorbed + outdoor-indoor

through glass through glass solar radiation air temperature

difference

which, by grouping the radiation terms, becomes

Total heat Solar Conduction

admission = Heat + Heat

through glass Gain Gain

The transmitted radiation causes a heat gain in the room by first being

absorbed by room surfaces and then convected into the room air. There is a delay from the time heat is absorbed until the time it is released by

convection depending on the thermal diffusivity of the surfaces. Some

radiation absorbed in the glass is conducted to the inner surface and

convected into the room air. Since there is usually little mass in the glass

panes, there is negligible delay in this portion of the solar heat gain.

Delay is also ignored in calculating the heat gain due to the outdoor-indoor

temperature difference.

The shading coefficient (SC) of a window's solar heat gain divided by the

(called a solar heat gain factor [SHGF]).

of a window is given by

given window is defined to be the

solar heat gain of a reference glass

Therefore, the total heat admission

13
Reflected

Figure 5. Heat Admission Through Glass

lA

Transmitted
q = sc * SHGF + U (To - Ti) (14)

where U is the overall heat transfer coefficient and (To — Ti) is the outdoorindoor temperature difference.

Use of the SC method is based on an extensive set of SHGF tables for

different latitudes and for different times of the day and the year. In a computer implementation the tables can be replaced by the formulae [1] which

were used to generate them. The primary assumptions in . the SC method are:

(1) the convection coefficients are constant; (2) the relative proportions of

diffuse and beam radiation are set for a clear day; (3) the angular dependence

of the solar heat gain of the window is the same as for the reference glass;

and (4) the proportions of radiation transmitted and absorbed are set for a low absorption glass (t = .86 at normal incidence). Each of these assumptions

causes some loss of accuracy, or more especially sensitivity, for various

windows and conditions. The first two assumptions can be overcome in a computer implementation of the original formulae by using convection

coefficients and beam and diffuse radiation values appropriate for the actual

weather conditions. However, convection coefficients are not well known,

especially for the mixed free and forced convection regime. The last two

assTunptions cannot be improved unless detailed optical calculations are performed. The third assumption can fail completely for some windows (as in

section 4.2). The last assumption can give the wrong portions of radiation

which are absorbed and stored in the room walls or are absorbed and instantly

convected from the glass, thereby giving an incorrect phasing of the heat

gains. This inability to handle dynamic conditions is not surprising since

the SC method was not developed for dynamic energy calculations.

Consideration of the radiation gains into the room indicates that the

following optical performance values are needed for detailed window models:

and Tj for the window and and for each layer (pane) in the window.

In addition to radiation gains, windows also allow radiation losses such as radiation from lights and solar radiation reflected from the surfaces of the room back out the windows. Since such radiation is diffuse, it requires the

and A^jj values. The magnitude of such losses is usually small.

15
3.2 Polarization

Tlie performance of a pane of glass depends on the polarization of the

incident radiation. Fig 6 shows the transmittance, reflectance, and

absorptance of an ordinary pane of glass (t = .86 at normal incidence) for

each polarization and also for the average of the two polarizations as a function of angle of incidence. Note that the polarization dependence is most

significant at about 60® where t^ = ,92 and t^jj = .68 . Beam solar radiation

is usually unpolarized, which is equivalent to consisting of equal parts of

both polarizations. The optical peformance for unpolarized radiation,

therefore, is the average of the peformance at each polarization. Diffuse

radiation from the clear sky is polarized, with the degree of polarization

depending on the direction. Radiation reflected from clouds exhibits little

polarization. Solar radiation can become polarized when it is specularly

reflected from some surfaces, such as the glass of adjacent buildings or water. Building energy analysis programs have not generally considered such

specular reflection energy gains, let alone the more detailed consideration of

polarization effects.

In evaluating the performance of multipane systems eqn (10) should be

evaluated separately for each polarization and the results combined in equal

proportions for unpolarized radiation. Figs 7, 8, and 9 show for 2-, 3-, and

4- pane systems, respectively, the error caused by using the average

properties for each pane instead of the exact procedure. The error in

absorptance is not significant but the error in transmittance increases with

the number of panes and transmittance is always less than the correct value.

In addition, the error does become less significant as more radiation is

absorbed in the panes. The implication of this polarization effect is that

measurements of a single layer should be done for each polarization if that

layer is to be mathematically combined with other layers in computing the

performance of an optical system. Conversely, if data are not available for

both polarizations, errors of the order indicated in figures 7 - 9 can be

expected.

16
TRANSMITTANCE

&

ABSORPTANCE

TRANSMITTANCE

&

ABSORPTANCE

6. Polarization. Performance of a Single Pane Window

Figure 7. Polarization Performance of a 2-pane Window

17
TRANSMITTANCE

&

ABSORPTANCE

0 10 20 30 40 50 60 70 80 90

ANGLE OF INCIDENCE

Figure 8. Polarization performance of a 3-pane Window

Figure 9. Polarization Performance of a 4-pane Window

18
3.3 Wavelength.

A truly detailed • window optical model should consider the wavelength

dependence of optical properties. The formulae for optical properties given

in Section 2 apply to a single wavelength, X,, of radiation. The total

radiation incident on a window is composed of radiation of many wavelengths in

varying proportions. Both the refractive index of glass and the extinction

coefficient, n and k, are wavelength dependent properties. The average

optical performance of a pane of glass is obtained by integrating the

wavelength dependent performance, e.g. transmittance:

where w(X) is a weighting function for different wavelength distributions,

e.g. full solar or visible. It has been common practice to model windows by

using a single average value of refractive index and extinction coefficient.

In order to test this assumption it is necessary to do a detailed

wavelength dependent model of several common window glasses. The wavelength

integration is achieved by rectangular integration using 20 selected

wavelengths from [14] for the solar energy distribution at sea level with air mass 2 (i.e., a solar altitude of 30^) listed in table 1. The spectral

transmittances of five common glasses (3 mm (1/8 inch) regular sheet, 6 mm

(1/4 inch) bronze heat absorbing, 6 mm (1/4 inch) gray heat absorbing, 6 mm

(1/4 inch) green heat absorbing, and 3 mm (1/8 inch) low iron vey clear) were

taken from the extinction coefficients listed in [13]. The refractive index

of typical window glasses is closely represented by the following equation

(15a)

[13]

n 1.5130 - 0.003169X.2 + 0.003962/^^ (15b)

where the wavelength is given in |im

19
Table 1. Wavelength Dependence of Optical Properties

Wavelength index of transmittance

(jxm) refraction clear bronze gray green low Fe

.390 1.539 .888 .463 .400 .683 .908

.444 1.532 .898 .433 .428 .750 .911

.481 1.529 .903 .425 .412 .796 .913

.511 1.527 .906 .441 .383 .794 .914

.543 1.526 .903 .510 .422 .776 .914

.574 1.524 .897 .540 .413 .728 .913

.606 1.523 .888 .546 .382 .668 .911

.639 1.521 .875 .536 .373 .600 .908

.669 1.520 .865 .545 .437 .539 .906

.705 1.519 .853 .564 .567 .488 .903

.745 1.518 .835 .522 .547 .413 .899

.786 1.517 .818 .472 .495 .344 .895

.831 1.517 .799 .429 .444 .287 .891

.877 1.516 .785 .391 .408 .247 .887

.959 1.514 .770 .359 .372 .210 .884

1.026 1.513 .765 .348 .358 .200 .883

1.105 1.512 .764 .344 .346 .198 .883

1.228 1.511 .773 .359 .355 .214 .885

1.497 1.508 .822 .494 .474 .358 .898

1.722 1.505 .846 .574 .544 .453 .904

Integration over the 20 wavelengths gives the detailed performance of the

glass pane, in particular the transmittance at normal incidence, from which it

is possible to generate a wavelength averaging approximation by using an average refractive index of 1.51955 and eqns (6a-c) to determine an average

extinction coefficient. Figure 10 shows the transmittance for the five types

of glass as computed with the wavelength dependent optical properties. The

normal and hemispherical transmittance

s

of the glasses are

clear bronze gray green low Fe

normal .843 .465 .428 .487 .901

hemispherical .781 .402 .367 .429 .842

The hemispherical transmittance based on the wavelength averaging approximation were computed to be

hemispherical .781 .402 .367 .423 .842 error -.03% -.18% .15% -1.41% 0%

Only the performance of the green heat absorbing glass for wavelength

averaging is shown in fig 10. The wavelength averaging approximation for the

20
other glasses do not produce observable differences in the transmittance

curves. Absorptance curves would show even less difference.

This test indicates that there is insignificant error in computing the

optical performance of most common window glasses by using a wavelength

averaging approximation instead of the detailed method, which is about 20

times slower. The least error occurs for glasses which do not have major

wavelength dependencies, such as clear and low iron glasses. Glasses which do

have major wavelength dependencies, such as those using thin reflective films,

are likely to require detailed wavelength modeling.

0 10 20 30 40 50 60 70 80 90

INCIDENT ANGLE

Figure 10. Wavelength Performance of 5 Common Window Glasses

21
3.4 Angular Performance

Although the optical theory for simple windows is straightforward and

easy to implement for computation, it can become time consuming if the optical

performance of every window in a building is to be evaluated hourly for a one-year simulation as is commonly done for energy analysis. In such a case

it is important to save computation costs by using an approximation to the

detailed theory. The most common such approximation is the polynomial

expression for transmittance as a function of the cosine of the angle of

incidence which has the form

T(cosO) = c^ cos6 + C 2 cos^® "* cos^Q •••

. cos^e

1 i=l

and similarly for absorptance. This form is particularly convenient because

cosO (rather than 9) is directly calculable for the sun's rays incident on a surface with any orientation; transmittance and absorptance are both zero at

cosO = 0; and the expression is quickly evaluated by the use of Horner's rule:

T(cosO) = Cj^(cos0 + C2 (cos 9 + c^(cos9 + ... ))) (16b)

The coefficients, c^^ are computed by a least squares fit to the exact optical

performance. High precision calculations are recommended because the

Table 2. RMS Error of the Polynomial Approximation for Window Optics

Avg. N = 2 N = 3 N = 4 N = 5 N = 6

T .784 .03859 .00738 .00727 .00429 .00228

A1 .071 .01909 .01168 .00777 .00540 .00388

T .655 .02077 .02117 .01038 .00257 .00061

A1 .078 .02574 .01576 .00984 .00636 .00432

A2 .058 .00670 .00240 .00131 .00083 .00048

T .559 .02843 .02624 .00931 .00139 .00142

A1 .082 .02734 .01637 .00978 .00614 .00417

A2 .064 .01107 .00473 .00209 .00091 .00039

A3 .048 .00243 .00054 .00031 .00018 .00018

T .482 .03390 .02797 .00793 .00346 .00225

A1 .084 .02786 .01648 .00960 .00596 .00411

A2 .067 .01245 .00525 .00120 .00067 .00028

A3 .053 .00482 .00085 .00040 .00040 .00039

A4 .041 .00161 .00162 .00060 .00015 .00015

22
simultaneous equations for the fit are relatively ill-conditioned

Table 2 shows the accuracy for different numbers of coefficients in eqn

(16a) for single through quadruple pane windows, each pane having a normal

transmittance of .86 and a refractive index of 1.52 . The root mean square

(rms) error is reported for the system transmittance and the absorptance in

each pane. 'Avg. ' refers to the hemispherically averaged performance values.

Sixteen equally spaced values of cos0 were used to determine the curve fit,

and the rms error is based on calculations at every one degree from 0^ to 89^.

An alternative to the fitted polynomial is the interpolating polynomial

[12, sec 3.4], formed by interpolation (e.g. Lagrangian) using two or more

points. Table 3 shows the accuracy of interpolation using different numbers

of points selected from values computed at 10° intervals. In particular, N =

.2 refers to linear interpolation which compares favorably with the 4th order

polynomial curve fit. It is also possible to do interpolation with unevenly

spaced angles for higher accuracy by using closer spacing where performance is

changing rapidly with angle of incidence. Interpolation is not as fast as eqn

(16b) because of the time required to determine the interpolation points.

Maximum error in both methods occurs at high angles of incidence where other

factors, such as small window setbacks and the actual thickness of the window,

make the idealized theory inexact so computed errors at high 6 may not be

meaningful

.

Table 3. Accuracy of Interpolation for Window Optics.

Avg. N = 2 N = 3 N = 4 N = 5 N = 6

T .784 .00691 .00533 .00535 .00469 .00399

A1 .071 .01000 .00790 .00695 .00628 .00582

T .655 .01415 .01162 .00923 .00704 .00566

A1 .078 .00466 .00302 .00232 .00182 .00153

A2 .058 .00231 .00141 .00113 .00097 .00087

T .559 .00973 .00614 .00309 .00117 .00102

A1 .081 .00371 .00236 .00198 .00182 .00181

A2 .064 .00206 .00108 .00095 .00094 .00098

A3 .048 .00051 .00019 .00016 .00016 .00017

T .482 .00799 .00422 .00132 .00196 .00245

A1 .084 .00363 .00228 .00195 .00185 .00187

A2 .067 .00196 .00125 .00131 .00078 .00079

A3 .053 .00078 .00068 .00076 .00078 .00079

A4 .041 .00052 .00037 .00022 .00014 .00013

23
In most cases th.e polynomial expression is the preferred method for

representing window optical performance because of its speed, however some windows may be better modeled by interpolation when performance changes

rapidly with respect to the angle.

3.5 Hemispheric Performance

There are several methods for computing the hemispheric performance, eqn

(13d), for simple windows. One method ia s Simpson's rule numeric integration

[12, sec 4.10] which requires evaluation of the window optical performance at

n+1 equally spaced angles between 0° and 90°, where n is an even number.

Figure 11 shows the computed hemispheric performance (T, R, and A for both

polarizations) for values of n up to 20. There is no significant improvement

in accuracy beyond n = 8 (values every 11.25°). Computed hemispheric values

given in previous sections are based on n = 10.

1.0

0.8

UJO

z

<

QC

o

c 0.6

UJ

Q.

<

i

0.4

UJX

Q.<0

UJX

0.2

—1

1 A

1

A

1 T"

A A

[—

A

1

A

1

A

—1

A A

— A

—

A A A A A A A A

A Transmittance

O Reflectance Both polarizations —

Absorptance -

O O O O O O O O o

O

0 0 0 § § 0 0

? 1 1 1 1 1 1 _J

0 2 4 6 8 10 12 14 1 6 18 2C

NUMBER OF DIVISIONS

Figure 11. Hemispheric Performance Values by Simpson's Rule

24
A disadvantage of using Simpson's rule is that the window optical

performance must be computed at evenly spaced angles which are not necessarily

the angles convenient for angular performance fitting as described in the

previous section. Performance fitting using evenly spaced interpolation

values can provide the values necessary for a Simpson's rule integration.

Unevenly spaced interpolation values and values taken at regular intervals of

cos6 do not. The unevenly spaced interpolation values can be used for a trapezoidal integration [ 12 , sec 4.8] which has the same accuracy as shown in

Table 3 for linear interpolation (N = 2) since trapezoidal integration is an application of linear interpolation.

By substituting the polynomial expression, eqn (16a), into eqn (13d) a very simple expression for hemispheric performance in terms of the polynomial

coefficients is obtained:

This expression gives the hemispheric performance much more accurately than

the rms errors reported in the angular comparisons reported in Table 2. For

example, for the single pane window with N = 2, eqn (17) gives T^ = 0.789 and

= 0,072, and for N = 3 it gives T^ = 0.784 and = 0.070 versus values of

0,784 and 0,071, respectively, obtained by Simpson's rule.

Given the simplicity and the accuracy, eqn (17) should be used to

evaluate hemispheric performance whenever the polynomial expression is being

used to represent angular performance. The appropriate numeric integration

should be used when angular performance is approximated by interpolation.

It may be noted that basic energy conservation applies to the hemispheric

performance values just as it does to the angule dependent values in eqns

N

i=l

(17)

(7a, b)

and

(18a)

^d ^d ^d = (18b)

However, this relationship cannot be used to develop an accurate method for

computing the hemispheric performance of a multipane window from the hemispheric performance of its individual panes as done in eqns (8) through (12)
for individual rays, Tliere is an error because the portions of diffuse

radiation which are transmitted through or reflected from a pane are no longer

perfectly diffuse because of the angular performance characteristics of glass,

i.e., more diffuse radiation is transmitted at low 6 and more is reflected at

high 6. The underprediction of transmission in a multipane system introduced

by this approximation is about the same as the error caused by the

polarization approximation discussed previously.

There is another assumption implicit in using T^ and values to compute

the performance of a window for non-direct incident radiation. This is the

assumption that non-direct radiation around buildings is perfectly diffuse,

which is almost never completely true although it is approached under some conditions. A detailed model of the non-direct radiation incident upon

windows would have to include the non-uniform distribution of radiation from

the sky under different weather conditions and the reflection of solar

radiation from the ground and other objects seen from the window. This would

be a complex calculation whose cost must be considered in relation to the

additional accuracy and sensitivity which could be gained. The historic trend

of every decreasing computation cost would indicate that even if such a calculation is not practical now, it may be in the future.

26
4. APPROXIMATIONS FOR COMPLEX WINDOWS

In the following discussion complex windows will refer to those whose

optical performance cannot be computed by the methods presented in section 2.

4.1 Diffusing Layers

One of the most common window treatments is the drapery or curtain.

Draperies can transmit from almost none to over one half of the incident

radiation depending upon fabric color, thickness, and closeness of the weave. There is a fundamental difference from the optical performance of glass: some

of the beam radiation transmitted and reflected by the drapery is diffusely

scattered as radiation is reflected from individual fibers. If the thermal

model of the room requires information on how much solar radiation strikes

each surface, then it is important to know the amounts of radiation

transmitted as beam and as diffuse radiation.

For the purposes of developing an optical model, a drapery will be

considered an optical layer which interacts with a ray of radiation by

transmitting part as beam radiation, transmitting another part as scattered

radiation, reflecting part as beam radiation, reflecting another part as scattered radiation, and absorbing the remainder. This five-way distribution

of the incident energy at layer n is expressed by

^nb ^ns ^nb ^ns ®^nb ~ 1.0 (19a)

^nb ^ns ^nb ^ns ^nb ~ 1.0 (19b)

By assuming that the scattered radiation is perfectly diffuse. the diffuse

performance values and relationships of a layer can also be used:

4.'^ + +• - * ^nd ^nd ^nd “ (20a)

^nd ^nd *nd =1*0 (20b)

This model applies to other features besides draperies, such as a rough

surfaced pane of glass intended specifically to diffuse the transmitted

radiation. In most cases it will be necessary to determine the components of

the optical performance in eqn (19) by measurements.

The optical relationships for a diffusing layer can be expressed ii

27
manner similar to that developed in section 2.3 . Consider the sketch below.

^mb + ^ib * ^nb (21a)

^md “

lib * ^ns + 4b * 4s + ^nd * * ^id (21b)

^mb “

lib « ^nb + ^mb

* ^ib (21c)

Imb * + 4b * r + ns 4d * ^nd ^ ^nd * ^nd (21d)

This general layer model shows the values which most be measured to fully

determine optical performance of a diffusing layer: the transmittance and

reflectance for both beam and diffuse radiation for both directions of

incidence. The general layer model also leads to a set of simultaneous

equations which can be used to determine the optical performance of a system

of layers:

1 0 0 0 0 0 0 0 0 0 0 0

^Sb 1

-

0 1 0 0 0 0 0 0 0 0 0 0 4d Yi +^ib 0 1 0 0 0 -

'Ib 0 0 0 0 0 ^Ob 0 + -'Id 0 1 0 0 -

'is -'id 0 0 0 0 4d 0

-‘lb 0 0 0 1 0 -

'Ib 0 0 0 0 0 4b 0

-‘Is “4d 0 0 0 1 -'Is -'Id 0 0 0 0 4d 0

0 0 0 0 ”4b 0 1 0 0 0 -'2b 0

* iib = 0

0 0 0 0 -‘L

+ -'2d 0 1 0 0 -'2s -'2d ^Id 0

0 0 0 0 -‘2b 0 0 0 1 0 -'2b 0 4b 0

0 0 0 0 -‘L '2d 0 0 0 1 -'2s -'2d ^2d

! 0

0 0 0 0 0 0 0 0 0 0 1 0

^2b ^2

. 0 0 0 0 0 0 0 0 0 0 0 1 L 4d J Y2 .

28
Tlie system optical performance values for beam radiation traveling to the

right and to the left are obtained by setting and X2 , respectively to 1.0

and setting remaining elements of the right hand side of (22) to zero and

solving the system of equations. If, in addition to the assumption of a perfectly diffusing layer, it can be assumed that the diffuse performance

values are accurate for nearly diffuse radiation, then the system of equations

(22) is accurate for a limited number of multilayer systems. These systems

are g-s, s-g, s-g-s, and g-s-g where g represents a glass pane and s represents one or more scattering layers. This limitation occurs because

pefectly diffuse radiation ceases to be perfectly diffuse after transmission

through or reflection from a glass pane. The limitation on the number of

glass panes can be overcome by treating a group of panes as a single pane when

solving (22) and then determining the absorption values of the individual

panes in the group. The system optical performance values for energy

calculations must include the system inward scattering transmittance, T^^

addition to the values for a simple window, T^^

^Ib* ^2b» ®tc.

One should avoid the temptation to compute performance for diffuse

radiation by setting or Y2 to one, because of the diffuse radiation problem

for glass panes, and should use integration of the beam radiation values as described in section 3.5 .

4.2 Slat Shading Devices

Another common window feature is the slat shading device, which may

consist of vertical slats or of horizontal slats, either large (e.g. Venetian

blinds) or small (some types of sun screen). Figure 12 shows the general

configuration and angles used to define a horizontal slat shading device. The

regular geometry of such a device allows calculation of its optical

performance. See Appendix A for an ideal (thin diffusely reflecting slats and

no supporting structure) slat shading device. Some of the incident beam

radiation is diffusely transmitted through and reflected from a slat shader so many of the ideas from the previous section are applicable. The critical new feature of a slat device is that its optical performance is dependent on the

profile angle, Q in fig 11, and not the angle of incidence, 6, or the azimuth

angle, d. At some profile angles there is no transmittance of beam radiation

through the slats except as scattered radiation. The optical performance of

29
the slat device can be described by functions (t^^ tg, tg, rg, rg) of the

profile angle from -90® to +90®. (note: rj = = q) The difficulty comes when the slat device layer is combined with more conventional layers whose

performance depend on the angle of incidence. The performance of such a system of layers is dependent on both Q and d.

Figure 12. Horizontal Slat Shading Device

30
Th.e most general method of fast calculation for energy analysis is a two—variable interpolation. This solution has a high storage cost. For

example, establishing interpolation points at every 10® in Q from -80® to +80®

and every 10® in 0 from 0 to 80® (values at 90® can be assumed) requires a 17

by 9 array of values for each optical component, T^^ x^, ..•» needed. An even larger data structure would be required for a window with both horizontal

and vertical slat layers. The calculation of hemispherical performance values

must be based on numerical evaluation of eqn (13c).

4.3 Other Devices and Models

One of the important assumptions which has been made in developing the

formulae for multilayer systems is that edge effects are unimportant. That

is, the total thickness of the window is very small compared to its width and

height. The error caused by this assumption has not been analyzed.

Fortunately, transmittance is small at high angles of incidence where edge

effects become important, and shadow casting features of the architecture

often shade the window at high 6. Such edge effects play an important part in

the performance of a common residential window type — the colonial window

consisting of many small panes separated by mullions. The performance

characteristics of such a window should probably be determined by experimental

measurements. The data structure developed for the window with a slat shading

device can also handle such edge effects.

A more complicated form of the slat shading device has specularly

reflecting surfaces on the upper sides of the slats. This has been proposed

to reflect light upward to the ceiling for daylighting purposes. Simplified

modeling would require additions to the data structure: both the fraction of

beam light transmitted upward and the angle at which it is transmitted (if the

light rays are being followed in detail).

An even more complicated window could be developed by the use of a holographic coating. This could allow incident radiation to be transmitted

and scattered in arbirtary directions. A very general data structure could be

required. One structure, continuing the description of radiation as beam and

diffuse components, would give at every incidence direction (Q, t)

,

the inward

scattering transmittance, inward beam transmittance, T^, and the new

31
direction q£ beam. Even more general would be a data structure

giving the portion of tbe incoming ray scatted in every direction £qj.

every incoming direction (2, o). This is the 'bi-directional transmittance'.

Values of 2 and o every 10® would require an array of 17 by 17 by 17 by 17 ( = 83521 ) elements, which is probably impractical for any building energy or daylighting analysis.

Another advanced window type would change its optical properties as a function of temperature or some other, perhaps electric, input. This should

use the data structures developed above, except that the window would be

described by multiple structures, one for each condition the window can have.

The building analysis program would have to select which data structure to use depending on the conditions that control the window. This model also applies

to some very conventional control mechanisms — human intervention such as changing the position of drapes and blinds.

5. SUMMARY AND CONCLUSIONS

This report has discussed the modeling of window optics for the purposes

of simulating building energy requirements or daylighting availability. The

theory for calculating the optical performance of conventional windows was reviewed. This included reflection and refraction at an air-to-glass

interface, multiple reflections in a pane of glass, the net radiation method

for computing the optical performance of a multipane window, and hemispheric

performance. Special reflective films were not discussed.

The simplifications that might commonly be made in creating computational

models were analyzed. Use of the ASHRAE shading coefficient procedure for

transient analysis of buildings with windows which are significantly different

for the single pane reference window is questionable. Windows with heat

absorbing glass, reflective films, or shading slats are difficult to model

with shading coefficients. Accurate optical analysis requires treating

typical unpolarized radiation as consisting of equal parts of TE and T.M polarized radiation. Failure to do this causes an error in transmittance

which increases with the number of panes. The error does become less

significant as more radiation is absorbed in the panes. It was shown that

32
evaluating the optical performance of most common window glasses by using a wavelength averaging approximation instead of the detailed method is

satisfactory. The least error occurs for glasses which do not have major

wavelength dependencies, such as clear and most heat absorbing glasses.

Glasses which do have major wavelength dependencies, such as those using thin

reflective films, are likely to require detailed wavelength modeling.

Consideration of the radiation heat gains into the room indicates that

the following optical performance values are needed for detailed window

models: Transmittance into the room for beam and diffuse radiation (T^ and

absorptance of each pane and for inward beam and diffuse

radiation, and window transmittance and layer absorptances (T^ and A~^) for

outward diffuse radiation. Evaluation of daylighting performance requires the

^ and T^ values evaluated for the visible portion of the available solar

radiation. In long term energy analysis simulations, it is desirable to

quickly evaluate the performance for beam radiation as a function of angle of

incidence. In most cases a polynomial expression is the preferred method

because of its speed, however some windows may be better modeled by

interpolation when performance changes considerably over small angles. The

hemispheric performance (for diffuse radiation) can be quickly and accurately

evaluated from the coefficients of the polynomial expression for beam

performance. Other numeric integration methods may be used when angular

performance is approximated by interpolation.

Some of the possibilites for more complex windows were analyzed, and the

type of model and data that would be necessary to simulate such windows in a building energy analysis program were determined. The common drapery was modeled as a radiation scattering layer, and a theoretical method was presented for determining its optical performance in a multipane window. The

window optical performance values for energy calculations must include the

system inward scattering transmittance, T^^ in addition to the values for a simple window. The slat style shading device introduces a major complication

because optical performance is dependent on two angles, profile and azimuth,

so the data structure for quick calculation must reflect this fact. This is

best handled by a two-variable interpolation. This methodology is applicable

to other and more complex window types.

33
It was shown that the optics of different window types can be simulated

with models which require varying amounts of memory or computing time. It is

possible to develop a very general modeling method applicable to nearly all

windows, but such a method would be inefficient in either computation time or data storage for the majority of window types. It is recommended that a building energy analysis program have all models available and use the most

efficient for any given window.

Although theoretical approaches are possible for developing modeling data

for most window configurations, the data should also be obtainable from

experimental measurements. The measurements could also indicate the validity

of some of the assumptions that must be made to obtain theoretical models.

The applicability of available measurement techniques should be studied.

There is also a need for a sensitivity analysis to determine the impact of

different optical simulation methods on the computed loads and daylighting

availability in buildings. This can be done for typical buildings and window

types, but there is a limit to the usefulness of such analysis because

simulation has its greatest benefit in the atypical cases. Similarly, the

detailed analysis program will be used in developing simpler design tools

capable of handling common window types, but the detailed program is necessary

for predicting the performance of unusual windows.

34
REFERENCES

[1] ASHRAE Handbook - 1985 Fundamentals , ch 27.

[2] York, D, and Cappiello, C. C, (ed.), DOE-2 Engineers Manual, Los

Alamos National Laboratory report LA-8520-M (also LBL-11353), 1981.

[3] Hittle, D. C. , 'The Building Loads Analysis and System Thermodynamics

(BLAST) program. Version 2.0: User Manual', CERL Construction

Engineering Research Laboratory (CERL) report E-153, 1979.

[4] Walton, G. N. , 'Thermal Analysis Research Program Reference Manual',

National Bureau of Standards (U.S.) report NBSIR 83-2655, 1983.

[5] Modest, M. , 'Daylighting Calculations for Non-rectangular Interior Spaces

with Shading Devices', Illumination Engineering Society Annual Technical

Conference, Toronto, Canada Aug. 9-13, 1981, and Lawrence Berkeley

Laboratory report LBL-12599, June 1981.

[6] Brackett, W. , 'CEL-1 Lighting Computer Program - User's Guide, CR

81.026', Naval Civil Engineering Laboratory, 1981.

[7] Gillette, G. , 'A Daylighting Model for Building Energy Simulation',

National Bureau of Standards (U.S.) BSS 152, 1983.

[8] Born, M. , and Wolf, E. , Principles of Optics , 4th edition, Pergamon

Press, 1970.

[9] Morey, G. W. , Propeties of Glass . Reinhold Publishing Corp., 1938.

[10] Heavens, 0. S., Optical Properties of Thin Solid Films , Academic Press,

1955, pp 51-58.

[11] Rubin, M. , 'Solar Optical Properties of Windows', Energy Research , vol 6,

1982, pp 123-133.

[12] Gerald, C. F. , and Wheatley, P. 0., Applied Numerical Analysis , Third

Edition, Addison-Wesley, Reading MA, 1984.

[13] 'Standard Test Method for Solar Energy Transmittance and Reflectance

(Terrestial) of Sheet Materials', ASTM E424-71.

[14] Rubin, M. , 'Optical Properties of Soda Lime Silica Glasses', Solar Energy

Materials , Vol. 12 (1985), pp275-288.

35
APPENDIX A: SIMPLE SLAT SHADING DEVICE MODEL

The following model describes an idealized slat shading device which is

composed of thin flat slats which are T wide, with spacing S between slats,

and with a tilt angle p from the vertical as shown in figure Al. Fig A1 shows

three angles, and a^, which define four regions which must be

considered separately.

Region 1 (0 < a < as in fig A2): There is no direct transmission through

the shade. The lower slat is partially irradiated.

Region 2 < a < 02 as in fig A3): There is direct transmission through the

shade. The lower slat is partially irradiated.

Region 3 (02 < o < Oj as in fig A4): There is direct transmission through the

shade. The bottom side of the upper slat is partially irradiated.

Region 4 (a^ < a < n as in fig A5): There is no direct transmission through

the shade. The bottom side of the upper slat is partially irradiated.

Consider only p < n/2, since other tilt angles can be considered by

changing the origin for measuring angles. Then the lengths and angles in fig

Al are given by:

R = / t2 + s2 - 2TScosp

Q = y T^ + + 2TScosp

cosuj^ = (r2 + $2 - t2)/2RS

COS012 = —cosp

cosa^ = -(q2 + S2 - t2 )/ 2QS

The non—zero view factors between the four surfaces (including the open ends:

1 and 4) are given by:

Fi2 = (S+T-R)/2S = (S+T-Q)/2S F

;^4 = (R+Q-2T)/2S

F

21 = (S+T-R)/2T F23 = (R+Q-2S)/2T F24 = (S+T-Q)/2T

•'31 * F24 ^32 = ^23 F 34 " *^21 ^41 “ ’'14 ^42 = *^13 F43 = Fi 2

36
Figure A2 shows the condition for a in region 1. U is the length of the

bottom slat which is in direct sun. The lengths in fig A2 are given by:

U = Ssina/sin(a+P

)

V = Ssin^/sin(a+P

W = T - U

X = y + 2SWcosp

The additional view factors are given by:

la = (S+U-V)/2S Fib = (T+V-R-U)/2S

'3 a = (GH-V-S-X)/2T F3^ = (R+X-S-V)/2T

’4a = (T+X-Q-W)/2S F4b = (S+W-X)/2S

'ai = (S+U-V)/2U Fa3 = ((i+V-S-X)/2U Fa4 = (T+X-Q-W)/2U

= (T+V-R-U)/2W F^3 = (R+X-S-V)/2W Fb4 = (S+W-X)/2W

Figure A3 shows the condition for a in region 2. X is the length of the open

side which is in direct sun. X is given by:

X = Tsin(a+P) /sina

Figure A4 shows the condition for a in region 3 where X is given by:

X = -Tsin(a+P) /sina

Figure A5 shows the condition for a in region 4 where direct transmission is

again blocked. U is the length of the top slat which is in direct sun. The

lengths in fig A5 are given by:

U = -Ssina/sin(a+P)

V = Usinp/sina

W = T - U

X = y S'^ + W'^ - 2SWcosp ?2 _i_

The additional view factors are given by:

la = (S+U-V)/2S Fib = (T+V-Q-U)/2S

3a = (R+V-S-X)/2T ^3b =,(CH-X-S-V)/2T

4a = (T+X-R-W)/2S P4b = (S+W-X)/2S

ai = (S+U-V)/2U Fa3 = (R+V-S-X)/2U Fa4 = (T+X-R-W)/2U

|,1 = (T+V-Q-U)/2W F^3 = (OfX-S-V)/2W Fb4 = (S+W-X)/2W

In all four regions the angle of incidence of

(measured in the same plane as p and a) is given by:

COS0 = I sin(a+P

)

I

the ray on the slat

37
Calculation of the optical performance of the slat shading device can be

accomplished by considering the parallelogram formed by surfaces 1 through 4

to be an enclosure and following the general enclosure theory presented by

McCabe [Al] . In that theory all surfaces reflect diffusely, and any surface

may be transparent. Since surface 1 and 4 of the slat shader are imaginary,

they have have transmittances of 1 and reflectances of 0. The enclosure

equations use the following additional quantities:

J = radiosity = flux of radiant energy leaving a surface diffusely,

G diffuse irradiance,

= diffuse irradiance on the outside of the surface,

H = beam irradiance,

= reflectance for diffuse radiation,

= reflectance for beam radiation, and

= transmittance for diffuse radiation.

Conservation of energy implies the following relationships:

= Pd.j®: * Pb.jHj j op*?”'

= Pa + “^A j transparent

J “»JJ “»JJ

and

Since is a function of the J of all other surfaces in the enclosure, it

must be computed by a set of simultaneous equations expressed in matrix for

as

^“^dl^ll "PdlFl2 "Pdl^ia “PdlFl4 Jl Pbl^l‘*^dlGl°

”^d2^21 ^~Pd2^22 ~Pd2^23 "Pd2^24

« J2 Pb2^2'*"^d2^2

”^d3^31 ”Pd3^32 l“Pd3^33 “Pd3p34 J 3 Pb3H3'*’'^d3G3°

"Pd4^41 “Pd4p42 "Pd4p43 l“Pd4^44 J4 . Pb4^4'*^d4G4°

In this set of equations is always 0, and the optical performance of the

shade is determined by setting different values for the right hand side (RUS)

of the equations. The LHS matrix is diagonally dominant. When the incident

ray is in regions 2 and 3, the direct transmittance is given by t^ (S-X)/S

and, of course, in regions 1 and 4 t^ =0. In addition, the set of

simultaneous equations must be expanded to include subsurfaces a and b in

38
these regions.

A critical aspect of the slat shader is that some of the beam radiation

is scattered as it is reflected from the slats. This scattered radiation may

be directly transmitted or reflected from the device, or it may be indirectly

transmitted or reflected after further diffuse reflections between the slats.

The scattering transmittance, T^, is obtained by setting the appropriate H

component to cos0 and all other components in the RES to 0, computing the ^ and then computing which is the scattering transmittance. The scattering

reflectance is given by jhe optical performance for radiation

incident from the opposite side, evaluated at the same

time as x'*‘(7r-a) and p'*’(n-a) because of geometric similarity with the relative

s s positions of surfaces 2 and 3 reversed.

The transmittance of the shader for diffuse radiation, is obtained by

setting Gj^° to 1 and all other components in the RHS to 0, computing the ^ and again computing G^^ diffuse reflectance is given by G^. The reverse

diffuse transmittance, v~, is obtained by setting G,° to 1 and solving for G.

d 4 1

•

Another major difference between the slat shader and the

glass is that the optical performance of the shader is not

function of incidence angle. This has a significant inpact on performance can be represented in a simple manner.

common pane of

symmetric as a how the optical

[Al] McCabe, M. E. , and Van Migom, M. , Application of Transparent Enclosure

Theory to Solar Energy Absorption by Cylindrical Tubes in Sunspaces',

ASME Journal of Solar Energy Engineering, Vol 107, Feb 1985.

39
Figure A3

40
U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date 'BIBLIOGRAPHIC DATA

SHEET (See instructions)

REPORT NO.

NBSIR-86/3426 AUGUST 1986

4. TITLE and subtitle

Modeling Window Optics for Building

Energy Analysis

5. AUTHOR(S)

George N. Walton

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No,

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Store, ZIP)

Solar Buildings Technology Division

Office of Solar Heat Technology

U.S. Department of Energy

Washington, DC 20585

10. supplementary notes

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here)

This report discusses modeling the optics of windows for the purposes of simulating

building energy requirements or daylighting availability. The theory for calculating

the optical performance of conventional windows is reviewed. The simplifications

that might commonly be made in creating computational models are analyzed. Some

of the possibilities for more complex windows are analyzed, and the type of model

and data that would be necessary to simulate such windows in a building energy

analysis program are determined. It is shown that the optical performance of

different window types can be simulated with models which require varying amounts

of memory or computing time. It is recommended that a building energy analysis

progam have all models available and use the most efficient for any given window.

12. KEY WORDS (Six to twelve entries; alphabetical order; caoitalize only proper names; and separate key

daylighting; fenestration, optics; solar heat gam; thermal modeling

words by semicolon s i

13. AVAILABILITY 14. NO. OF

Unlimited

PRINTED PAGES

1 1 For Official Distribution. Do Not Release to NTIS

'

' 20402.

„ . 15. Price

I^^Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $9.95

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