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A method for comparing the thermal performance of windows
Harrison, S. J.; Barakat, S. A.
https://publications-cnrc.canada.ca/fra/droits
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National Research
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A METHOD FOR COMPARING THE THERMAL PERFORMANCE
OF WINDOWS
by
S.J. Harrison and S.A. Barakat
ANALYZED
Reprinted from
ASHRAE Transactions 1983
Vol. 89, Part 1 A, p. 3
-
11
DBR Paper No. 1 162
Division of Building Research
RESUME
C e t t e comrmnication pr'esente une mSthode s i m p l e d e comparaison d e l a performance thermique d e s v i t r a g e s . I1 e s t p o s s i b l e d ' d t a b l l r une c a r a c t ' e r i s t i q u e u n i v e r s e l l e de rendement pour chaque t y p e d e v i t r a g e e n u t i l i s a n t uniquement s o n c o e f f i c i e n t g l o b a l moyen de t r a n s f e r t d e c h a l e u r e t son c o e f f i c i e n t d'ombre. La d t h o d e permet Sgalement d e c a l c u l e r l e s a p p o r t s n e t s d e c h a l e u r que p e u t p e r m e t t r e un t y p e d e v i t r a g e donn'e. On a pu c o n f i r m e r que les propri'etks thermiques e t o p t i q u e s d e s f e n b t r e s r e s t e n t c o n s t a n t e s e n a n a l y s a n t l e s v a l e u r s d e s a p p o r t s n e t s de c h a l e u r obtenues p a r d e s c a l c u l s d g t a i l l ' e s fond'es s u r d e s donn'ees atmosph'eriques h o r a i r e s r6elles de q u a t r e v i l l e s du Canada. La d t h o d e s e r t egalement 3 pr'evoir l e rendement des f e n s t r e s l o r s q u ' o n u t i l i s e d e s p e r s i e n n e s i s o l a n t e s l a n u i t . - - -
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716 - -No.
2735
A Method for Comparing
the Thermal Performance of Windows
S.J. Harrison
S.A. Barakat
ABSTRACT
!
I A simple method of comparing the thermal performance of glazing systems is presented.
A universal performance characteristic can be established for each glazing system, using only its average over-all heat-transfer coefficient and shading coefficient. The method also permits the calculation of the net heat gain attainable with a given glazing system. The validity of the assumption of constant window thermal and optical properties is confirmed through comparison with values of net heat gain obtained by detailed calculations using actual hourly atmospheric data for four Canadian locations. The method is extended to permit prediction of the
performance of windows using nighttime insulating shutters. An example of its use is given.
INTRODUCTION
It has long been recognized that windows can increase the heating and cooling energy
requirements of buildings. l Recently, however, their otential as suppliers of energy -for the heating of buildings has received increased attention.q Calculated values of net heat gain through windows have shown that even under severe winter conditions double- and triple-glazed windows can supply a positive net heat gain to the indoor space over the heating season.3
The net heat gain through a window is a function of the glazing design ( e . , single, double, etc.), orientation, and the climate to which it is exposed. The selection of a window should therefore account for all these factors. If, however, a window is located for reasons other than maximizing energy gain (i.e., visual communication, lighting, or aesthetics), the glazing system should be designed to minimize energy loss.
A method for selecting windows based on thermal performance would be-of assistance to
building designers who wish to compare glazing options for a specific application. For example, such a method would enable determination of the energy (and ultimately the cost) benefit of reorienting or adding windows or of substituting different windows in a building. The thermal and cost performance of single, double, and triple glazing and of nighttime shutters could also be compared.
The thermal performance of a window is similar to that of a solar collector and should lend itself to the same sort of rating.4 j 5 This paper presents a method of comparing the thermal
performance of glazing systems. A universal performance characteristic is presented for each glazing system, using only its over-all heat-transfer coefficient and shading coefficient. The method also allows the calculation of the net heat gain attainable with a given glazing system. It is assessed through comparison with values of net heat gain obtained by detailed calculations that use actual hourly atmospheric data for four Canadian
location^.^
THERMAL PERFORMANCE OF GLAZING SYSTEMS
The instantaneous rate of energy flow through a glazing system is equal to the difference between the heat gain due to solar radiation and the heat loss by conduction, convection, and
S.J.
Harrison and S.A. Barakat, Research Officers, Thermal Performance Section, Division of Building Research, National Research Council of Canada, Ottawa, Canada.long-wave radiation. Following the convention of Ref 1,
where
qA = instantaneous rate of heat admission through the glazing, w/m2
F = solar heat-gain coefficient (the dimensionless ratio of the solar heat gain to the incident solar radiation)
It = solar radiation incident on glazing surface, w/m2
U = over-all heat-transfer coefficient of the window, w / m 2 ~ T ,T. = outdoor and indoor air temperatures, respectively, K
0 1
The over-all heat-transfer coefficient, U, takes into account long-wave radiation losses as well as conduction and convection losses due to indoor-outdoor temperature difference. The value of U varies with outdoor air temperature, wind speed, and incident solar radiation.
The solar heat gain coefficient, F, is a unique characteristic for each type of window and represents the sum of the solar radiation transmitted through the glass and the inward flow of solar radiation absorbed by the glazing. It is given by
F = . c + N . a 1 where
T = over-all solar transmittance of window
Ni = inward-flowing fraction of absorbed solar radiation (for example, Ni = U/ho for single glazing)
a = solar absorptance of glazing material h = outside film heat-transfer coefficient
0
For windows with multiple glazings, values of F can be calculated by the method presented in Ref 3. Alternatively, for all glazing systems F can be presented as a fraction of F
single glazing, that is, S
where
SC is the shading coefficient, defined as the ratio of the solar heat gain through a glazing system under a specific set of conditions to the solar gain through a single light of double-strength sheet glass under the same conditions. Values of shading coefficient for a number of glazing systems are given in Ref 1.
From Eq 1 an instantaneous thermal performance factor,
n,
for a window can be defined as,The relation between
,-,
and (Ti - To)/It for any glazing system can be presented graphically as shown in Fig. 1.AVERAGE THERMAL PERFORMANCE
Similar to an analysis presented by Lunde for the thermal efficiency of solar
collector^,^
a relation for the average thermal performance factor of glazing systems can be derived.Consider the instantaneous thermal performance factor of a window represented by Eq
4.
If it is assumed that the shading coefficient, the solar heat gain coefficient for single glazing, and the xindow over-all heat-transfer coefficient are constants over any time period and given by-
SC, F,, and
U,
respectively, then the average heat gain through the window,Q,
over the same time period is given by:where
-
To and
Ti
represent the time-average outdoor and indoor temperatures over the time interval, and H represents the time-average solar radiation incident on the glazing surface.-
For a particular time period,
F
andU
are average values over the period calculated as total solar gainF =
total incident radiation
-
U = total heat loss
average temperature difference times number of hours
The average thermal performance factor,
n ,
of the window during the time period, therefore, is,Eq 6 describes the relation between the average values of the performance factor, incident radiation, outdoor air temperature, and room temperature. It can be graphically represented in the same manner as the instantaneous performance curve (Fig. 1). Note that the slope - of the characteristic line in the figure represents th~oler-all heat-transfer coefficient, U. The optical efficiency of the glazing system
(Co
= SC*Fs? is represented by the condition at zero loss (Ti = To). The value of solar radiation transmitted through the glazing system to the indoor space over the time period, p, is thus given by:The seasonal net heat gain, Q, for a particular combination of outdoor and indoor
conditions and a specific glazing system can be determined analytically from Eq 6 or by using the graphical representation of the thermal performance as in Fig. 1. The graphical approach has the advantage that a number of glazing systems can be compared directly on the same plot.
-
-The linear prescription of in Eq 6 relies on the assumptions that SC, Fs, and
U
are constants for all periods and that the solar gain through the glazing is unaffected by the thermal losses and vice versa. These assumptions are discussed and verified below.WINDOWS WITH NIGHT INSULATING SHUTTERS
Night insulating shutters are often recommended as a means of decreasing the conduction heat loss of windows during nighttime. Since the over-all heat-transfer coefficient,
r,
in Eq 6 is the average value for the window over the period (as defined earlier), it should be possible to represent the case of insulating shutter with a time-weighted-average of U, that is:-
'with shutter ' tl + 'no shutter (24-
ti)u
= 24where
tl is the hours of use of the shutters over a 24-hour period. SENSITIVITY TO VARIATIONS OF PROPERTIES
In the derivation of Eq 5 it was assumed that the optical properties of the glazing, T , U , were constants. The same assumption was also applied to the over-all heat-transfer coefficient
(including the outside film coefficient and the air space resistance). These values, however, do vary with outdoor air temperature, wind speed, and angle of incidence of solar radiation.
Values of the net heat gain through single-, double-, and triple-glazing systems for eight glazing orientations and several Canadian locations were analyzed to verify whether the
variability of thermal and optical properties would affect a linear representation of the thermal performance characteristic of a window. The analysis was also intended to establish appropriate average values of the shading coefficients and the over-all heat-transfer
hourly v a r i a t i o n i n t h e g l a z i n g t r a n s m i s s i o n and absorptance, a s well a s t h a t o f t h e o v e r - a l l h e a t - t r a n s f e r c o e f f i c i e n t , U . The c a l c u l a t i o n procedures a r e described i n d e t a i l i n Refs 3 and 8.
-
I n t e g r a t e d monthly v a l u e s , f o r t h e h e a t i n g months, of t h e performance f a c t o r , TI, and (Ti
-
G)/H
f o r t h e t h r e e g l a z i n g t y p e s were obtained and p l o t t e d i n F i g s . 2 , 3, and 4 . Each f i g u r e combines c a l c u l a t e d v a l u e s f o r four l o c a t i o n s (Vancouver, Winnipeg, Ottawa, andFredericton) and e i g h t glazing o r i e n t a t i o n s (north, n o r t h e a s t , e a s t , s o u t h e a s t , south,
southwest, west, and northwest). A l i n e r e p r e s e n t i n g a l e a s t - s q u a r e s f i t t o t h e d a t a p o i n t s i s
a l s o shown. In a d d i t i o n , t h e c a l c u l a t e d thermal performance c h a r a c t e r i s t i c s of a double-glazed window with an i n s u l a t i n g s h u t t e r having a thermal r e s i s t a n c e v a l u e o f 1.06
m2c/w
and closedf o r 8 hours and 12 hours p e r day a r e given i n Figs. 5 and 6, r e s p e c t i v e l y .
A s shown i n Figs. 2 t o 6, t h e d a t a p o i n t s f o r a l l l o c a t i o n s and e i g h t g l a z i n g o r i e n t a t i o n s can be well r e p r e s e n t e d by s t r a i g h t l i n e s . The maximum r m s e r r o r between t h e c a l c u l a t e d v a l u e o f { and t h e s t r a i g h t l i n e f i t , which i s f o ~ t h e s i n g l e - g l a z i n g c a s e , i s l e s s than 0.08. T h i s a l s o i n d i c a t e s t h a t c o n s t a n t v a l u e s of
z,
Fs, and can be used with minimum e r r o r i n Eqs 5 and 6 f o r c a l c u l a t i n g t h e average thermal performance f a c t o r of t h e g l a z i n g o r t h e n e t h e a t flow through it f o r a p e r i o d of time.COMPARISON OF GLAZING SYSTEMS
The c h a r a c t e r i s t i c l i n e s r e p r e s e n t i n g t h e v a r i o u s - - g l a z i n g systems discussed p r e v i o u s l y a r e shown t o g e t h e r i n Fig. 7 . The v a l u e s of c o ( = ~ ~ - ~ s ) and U a r e presented i n Tab. 1, i n a d d i t i o n t o t h e corresponding v a l u e s from Ref. 1. I t should be noted t h a t t h e v a l u e s of Ref 1 can be used t o c o n s t r u c t t h e thermal c h a r a c t e r i s t i c l i n e s of t h e g l a z i n g s o r t o c a l c u l a t e t h e s o l a r h e a t g a i n with no s i g n i f i c a n t l o s s o f accuracy.
Fig. 7 shows c l e a r l y t h e advantage of using windows with a high thermal r e s i s t a n c e value, e s p e c i a l l y i n l o c a t i o n s t h a t experience lbw outdoor a i r temperature and/or low l e v e l s of s o l a r r a d i a t i o n , o r f o r window o r i e n t a t i o n s o t h e r t h a n south.
I t should be p o s s i b l e t o develop s i m i l a r performance c h a r a c t e r i s t i c s f o r improved g l a z i n g systems, such a s honeycomb windows and windows with s e l e c t i v e r e f l e c t i n g f i l m s (e.g., h e a t m i r r o r s ) , provided t h a t v a l u e s of shading c o e f f i c i e n t and o v e r - a l l h e a t - t r a n s f e r c o e f f i c i e n t can be e s t a b l i s h e d by t e s t i n g o r c a l c u l a t i o n .
Examp 1 e
Consider a comparison of g l a z i n g systems l o c a t e d i n t h e n o r t h and south w a l l s of a b u i l d i n g i n Ottawa, Ontario. The mean monthly v a l u e s of t o t a l s o l a r r a d i a t i o n f o r each o r i e n t a t i o n can be obtained from Ref 3 o r 7 . For a heating season assumed t o extend from 1 October t o 30 A p r i l , monthly v a l u e s a r e averaged t o produce a mean v a l u e f o r t h e heating season. Values o f mean d a i l y temperature can be obtained from Ref 9. (Values o f mean d a i l y outdoor temperature can be estimated from degree-day v a l u e s i f t h e a c t u a l v a l u e s a r e n o t a v a i l a b l e . )
The g l a z i n g c h a r a c t e r i s t i c s of Fig 7 a r e reproduced i n Fig. 8 , which a l s o shows t h e v a l u e s
- -
of AT/H f o r t h e south and n o r t h windows i n t h i s example. The r e s u l t i n g v a l u e s o f t h e performance f a c t o r and n e t h e a t g a i n f o r each c a s e a r e given i n Tab. 2 f o r t h e d i f f e r e n t g l a z i n g systems.
A s might be expected, a l l windows f a c i n g n o r t h w i l l experience a n e t h e a t l o s s . On t h e south f a c e , however, only t h e s i n g l e - g l a z e d window w i l l experience a n e t heat l o s s ; a l l t h e o t h e r g l a z i n g systems experience a n e t h e a t gain.
The most e n e r g y - e f f i c i e n t g l a z i n g system i n t h e example i s t h e double-glazed window with s h u t t e r . I t s performance i s n e a r l y matched, however, by t h a t o f t h e t r i p l e - g l a z e d window. The choice of an a p p r o p r i a t e g l a z i n g f o r both o r i e n t a t i o n s must u l t i m a t e l y t a k e i n t o c o n s i d e r a t i o n t h e c a p i t a l c o s t of t h e window a s well a s i t s thermal c h a r a c t e r i s t i c .
I t should be noted t h a t t h e n e t h e a t g a i n c a l c u l a t e d by t h i s method does n o t account f o r t h e dynamic thermal behaviour of t h e b u i l d i n g , t h a t i s , no account i s taken of h e a t s t o r a g e e f f e c t s o r o f h e a t dumping due t o overheating. The n e t h e a t g a i n v a l u e r e p r e s e n t s t h e maximum p o s s i b l e r e d u c t i o n of h e a t i n g demand of a building. I n most i n s t a n c e s t h e u s e f u l s o l a r g a i n depends on such f a c t o r s a s b u i l d i n g h e a t i n g load, thermal stnrage mass, glazed a r e a , and allowable indoor overheating.
A u t i l i z a t i o n f a c t o r , defined a s t h e f r a c t i o n of t h e s o l a r gain t h a t d i r e c t l y c o n t r i b u t e s t o t h e reduction i n heating requirement, can be c a l c u l a t e d f o r combinations of t h e above- mentioned f a c t o r s , a s described i n Ref 10. This, combined with s o l a r gain values a s might be c a l c u l a t e d using t h i s paper, provides a simple procedure f o r c a l c u l a t i n g t h e u s e f u l s o l a r contribution t o t h e heating of houses.
SUMMARY
A simple graphical method f o r comparing t h e thermal performance of glazing systems i s presented. Thermal performance i s represented by j u s t two parameters: t h e o v e r - a l l h e a t - t r a n s f e r
c o e f f i c i e n t and t h e shading c o e f f i c i e n t . A u n i v e r s a l performance c h a r a c t e r i s t i c i s e s t a b l i s h e d f o r each glazing system, independent of geographical l o c a t i o n and window o r i e n t a t i o n . The r e s u l t s i n d i c a t e t h a t t h i s performance c h a r a c t e r i s t i c can be constructed using constant glazing p r o p e r t i e s . The method permits c a l c u l a t i o n of t h e maximum n e t heat gain a t t a i n a b l e with a given glazing system and should be u s e f u l t o building designers i n s e l e c t i n g t h e most e f f i c i e n t and c o s t - e f f e c t i v e glazing system f o r a s p e c i f i c a p p l i c a t i o n .
This paper i s a c o n t r i b u t i o n from t h e Division of Building Research, National Research Council Canada, and i s published with t h e approval of t h e Director of t h e Division.
REFERENCES
1. ASHRAE Handbook of Fundamentals
-
1977 Volume, Chapter 26 (New York: ASHRAE, 1977).2. G.P. Mitalas, "Net Annual Heat Loss Factor Method f o r Estimating Heat Requirements of
Buildings (BRN 117, Div. of Building Research, National Research Council of Canada, 1976).
3. S.A. Barakat, "Solar Heat Gains through Windows i n Canada" (DBR Paper No. 944, Div. of Building R e s e a r c ~ a t i o n a l Research Council of Canada, 1980).
4 . ASHRAE Standard 93-77, "Methods of Testing t o Determine t h e Thermal Performance of S o l a r
collector^^^ (New York: ASHRAE, 1978).
5. A.C. Lindsay, "Performance of S e l e c t i v e R e f l e c t i v e Glazing Components," Proceedings of t h e 1980 Annual Meeting of t h e American Section of t h e I n t e r n a t i o n a l Solar Energy Society
(Phoenix, 1980).
6. P.J. Lunde, ltPrediction of Average Collector Efficiency from Climatic Data," Solar Energy 19 (1977). pp. 685-689.
7. J . E . Hay, "An Analysis of Solar Radiation Data f o r Selected Locations i n Canada," Climatological Studies No. 32 (Atmospheric Environment Service, Toronto, 1977).
8. S.A. Barakat, A Fortran I V Program t o Calculate Net Heat Gains through Windows, DBR Computer Program No. 47 (Div. of Building Research, National Research Council of Canada, 1980). 9. Canadian Normals, Vol. 1--SI, Temperatures 1941-70 (Environment Canada, Atmospheric
Environment
,
1975).10. S.A. Barakat and D.M. Sander, l l U t i l i z a t i o n of Solar Gain through Windows f o r Heating of Houses" (BRN 184, Div. of Building Research, National Research Council of Canada, 1982).
TABLE 1
Glazing Characteristics
Glazing Type Fig. 7 Ref 1
-
-
SC U, (w/k*m2) 'lo SC U, (w/k*m2)
Single glass 0.83 1 5.63 0.87 1 6.25
Insulating glass: double
1.27 -cm air space 0.73 0.88 2.85 0.77, 0.89 2.78 Insulating glass: triple
1.27 -cm air space 0.67 0.81 1.89 0.7 0.80 1.75 Double with R1.06 shutter
closed 12 h- per day 0.73 0.88 1.86 0.77 0.89
-
TABLE 2
Net Heat Gain
for
Windows in OttawaVertical S Window Vertical N Window
Single-glazed -0.25 -29.6 -1.77 -84.8 Double-glazed 0.185 21.9 -0.58 -28 Triple-glazed 0.31 36.3 -0.20 -9.8 Double-glazed with 12-h shutter closure (R=l. 06 m2*c/w) 0.38 44.7 -0.12 -6.
UI
F-"
~ . ~ t \
( T i-
T o l / I , ( O R17,
-
T o l l i i , S L O P E a - U\
F i g u r e 1 . G l a z i n g t h e r m a l c h a r a c t e r i s t i c F i g u r e 3 . D o u b l e - g l a z i n g t h e r m a l c h a r a c t e r i s t i c F i g u r e 4 . T r i p l e - g l a z i n g t h e r m a l c h a r a c t e r i s t i cD O U B L E G L A Z I N G
2*.,
N O S H U T T E R %,'.
' b - 4 . 0' i
'*. '*., D O U B L E GLAZING>*-., N O S H U T T E R %. - 4 . 0'.,
'.
'\ F i g u r e 5 . T h e r m a l c h a r a c t e r i s t i c o f d o u b l e F i g u r e 6 . T h e r m a l c h a r a c t e r i s t i c of d o u b l e g l a z i n g w i t h 1 2 - h n i g h t s h u t t e . g l a z i n g w i t h 8-h n i g h t s h u t t e r ( R = 1 . 0 6 m2-
"c/w) ( R = 1.06 m 2 - ' ~ / ~ ) IF -1.0-
IF- 1 . 0-
'-DOUBLE GLAZ ING '- DOUBLE GLAZING
- 2 . 0
-
'\
S l NGLE GLAZ l NG SINGLE GLAZING
- 3 . 0
-
F i g u r e 7 . C o m p a r i s o n o f t h e r m a l F i g u r e 8 . E x a m p l e f o r w i n d o w s i n O t t a w a c h a r a c t e r i s t i c s o f d i f f e r e n t
DISCUSSION
J . E . R i z z u t o , New York S t a t e E.R.D.A., Albany: Is r a d i a t i o n t o and from t h e window i g n o r e d o r i n c l u d e d i n t h e U v a l u e ?
S . J . H a r r i s o n : I n c a l c u l a t i n g t h e U v a l u e o f t h e window, t h e t e m p e r a t u r e o f e a c h g l a z i n g is
c a l c u l a t e d e v e r y h o u r . T h i s c a l c u l a t i o n t a k e s a c c o u n t o f t h e v a r i a t i o n i n f i l m c o e f f i c i e n t , and t h e s o l a r r a d i a t i o n a b s o r b e d by t h e g l a s s . The g l a s s t e m p e r a t u r e is t h e n used t o d e t e r m i n e t h e a i r s p a c e r e s i s t a n c e , u t i l i z i n g a n o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t a c c o u n t i n g f o r
r a d i a t i o n and c o n v e c t i o n h e a t t r a n s f e r . The d e t a i l s o f t h e c a l c u l a t i o n s a r e p r e s e n t e d i n R e f s . 3 and 8 l i s t e d i n t h e p a p e r .
P . S c h a t k u n , New York S t a t e Div. o f Housing, New York: What was t h e t h i c k n e s s o f g l a s s u s e d i n t h e s t u d y f o r s i n g l e - , d o u b l e - , a n d t r i p l e - g l a z i n g ?
H a r r i s o n : G l a s s t h i c k n e s s was: s i n g l e 0.125 i n . (0.3175 cm) d o u b l e 0 . 2 5 i n . ( 0 . 6 3 5 cm) b o t h t r i p l e t - 1 / 8 - & i n . (0.3175
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0.635)S c h a t k u n : What t h i c k n e s s o f a i r s p a c e was u t i l i z e d between t h e d o u b l e and t r i p l e g l a z i n g ? H a r r i s o n : An a i r s p a c e o f i n . ( 1 . 2 7 cm) was g i v e n i n Tab. 1 o f t h e p a p e r .
S c h a t k u n : What t i m e o f y e a r , i . e . , h e a t i n g o r c o o l i n g s e a s o n s , were u t i l i z e d i n d e v e l o p i n g t h e h e a t t r a n s f e r f a c t o r s ?
H a r r i s o n : The h e a t i n g s e a s o n was d e f i n e d a s O c t . 1st t h r o u g h Apr. 30.
S c h a t k u n : Is t h e r e a follow-up t h e r o e t i c a l o r p r a c t i c a l e x p e r i m e n t b e i n g c o n s i d e r e d t o d e t e r - mine t h e e f f e c t o f d i f f e r e n t r e f l e c t i v e f i l m c o a t i n g s f o r d o u b l e and t r i p l e t h i c k n e s s g l a z i n g f o r h e a t i n g o r c o o l i n g a p p l i c a t i o n s ?
H a r r i s o n : A t h e o r e t i c a l s t u d y i s underway t o c a l c u l a t e t h e p e r f o r m a n c e o f r e f l e c t i v e f i l m c o a t i n g s and o t h e r window improvements f o r h e a t i n g a p p l i c a t i o n s . T h i s s t u d y w i l l a l s o be accompanied i n t h e f u t u r e by kindow t e s t i n g t o d e t e r m i n e t h e i r c h a r a c t e r i s t i c s .
It s h o u l d be n o t e d t h a t t h e v a l u e s u s e d f o r t h e window s p e c i f i c a t i o n s were c h o s e n t o c o r - r e s p o n d w i t h t h o s e p r e s e n t e d i n Ref. 1. The r e s u l t s p r e s e n t e d a r e i n t e n d e d t o i l l u s t r a t e and c o n f i r m t h e s i m p l e r e p r e s e n t a t i o n o f window performance. I n f a c t , any window may be r e p r e s e n t e d and t h e a n a l y s i s performed i f v a l u e s o f SC and U a r e known.
T h i s paper, w h i l e being d i s t r i b u t e d i n r e p r i n t form by t h e Division of Building Research, remains t h e copyright of t h e o r i g i n a l publisher. It should n o t be reproduced i n whole o r i n p a r t without t h e permission of t h e publisher.
A l i s t of a l l p u b l i c a t i o n s a v a i l a b l e from t h e Division may be obtained by w r i t i n g t o t h e P u b l i c a t i o n s S e c t i o n , D i v i s i o n of B u i l d i n g R e s e a r c h , N a t i o n a l R e s e a r c h C o u n c i l of Canada, Ottawa, O n t a r i o ,
