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A solar radiation model and energy saving windows in Canada
Solar & Wind Te~hnoloyv Vol. 5, No, 5, pp. 543 548, 1988 Printed in Great Britain.
0741- 983X/88 $3.00 + . 0 0 Pergamon Press pie
A SOLAR R A D I A T I O N M O D E L A N D E N E R G Y SAVING W I N D O W S IN C A N A D A A. S. BARKER, JR Trinity Western University, Langley, B.C., Canada V3A 4R9 and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, U.S.A.
(Received 8 December 1987 ; accepted 25 February 1988) Abstract A model for monthly average solar insolation on a tilted surface of any azimuth is derived which depends only on the average number of bright sunshine hours and other directly known attributes of a site such as latitude and ground albedo. Parameters in the model are derived from data at Canadian stations allowing an assessment of the insolation for north, east, west and south facing windows. By considering windows of various thermal efficiencies and the cost of heat energy, the cost effectiveness of the windows in various northern locations is evaluated. Using present energy costs it is shown that the recently announced super-efficient windows can be cost effective when added as an upgrade over conventional double pane windows and can provide significant energy savings even at far northern latitudes.
1. INTRODUCTION
C a n a d a a n d other high latitude countries [4]. Energy efficient windows, t h o u g h more expensive initially, can be cost effective because of the solar input u n d e r certain conditions. Widespread use o f such windows u n d e r certain conditions can significantly reduce national energy c o n s u m p t i o n . G e o g r a p h i c a n d climatological factors control these conditions in a n essential way. The model introduced here allows assessments o f solar flux based o n a m i n i m u m knowledge of such factors a n d can lead to conclusions o n the value of high efficiency windows. In Section 2 the model is outlined a n d three types o f window defined. Section 3 gives the results o f calculations for six Canadian locations, a n d derives dollar costs associated with the three window types using a s t a n d a r d economic model which includes the cost o f b o r r o w e d money. Section 4 gives the conclusions.
W h e n assessing the heat gain by windows a n d skylights a n d their effect on the winter energy requirements o f a building, a complete description o f the direct a n d indirect (diffuse) radiation flux is needed. Models of m o n t h l y averages of the flux arriving at a horizontal surface have been p r o p o s e d a n d actual m e a s u r e m e n t s of this q u a n t i t y for a network of a b o u t 50 sites in C a n a d a are readily available [1]. The areal coverage of this network is too limited however for direct use in flux calculations. W h e n solar receiving surfaces (e.g. windows) which are n o t horizontal are used, c o r r e s p o n d i n g m e a s u r e m e n t s are usually n o n existent a n d models must be used. There has recently been considerable advance in such modelling to include angular dependence of the diffuse r a d i a t i o n in a realistic manner. We propose such a model to predict m o n t h l y average insolation on a surface of any tilt and azimuth and use it to construct tables of winter heat gains for vertical windows facing north, east (or west) a n d south in C a n a d a . * These tables extend a f u n d a m e n t a l study of winter heat gain t h r o u g h windows carried o u t for the continental U n i t e d States [2]. With the recent a n n o u n c e m e n t of new super efficient windows [3] a n d escalations in the cost o f energy there is a need to reassess the cost effectiveness o f various windows. It is n o t e w o r t h y that, c o n t r a r y to intuition, there is a large potential for energy from the sun in
2. METHODS A precise calculation of the solar flux incident o n a surface of tilt angle T a n d a z i m u t h A depends o n m a n y factors such as angular position of the sun, turbidity, cloud cover, surface albedo, etc. Because of daily a n d even m i n u t e by m i n u t e fluctuations in most o f the i m p o r t a n t variables, various models have been proposed to give average values of the flux. W e follow such an a p p r o a c h here by p r o p o s i n g a m a t h e m a t i c a l expression which gives the energy incident o n a tilted surface over an average day. We take twelve such average days (15 January, 14 February, 15 M a r c h . . . . )
* The present author was a member of the American Institute of Physics team which studied energy conservation and high efficiency windows (see [2]). 543
544 to represent thc pattern over a year and are therefore using m o n t h l y averages. Early models which use monthly averaging have been used quite successfully to predict radiation received on a horizontal window or collector. This is the most c o m m o n measurement made by weather stations. These models always recognize two m a j o r c o n t r i b u t i o n s - - the direct component (all radiation arriving in a cone of solid angle 6.7 x 10 ~ steradian from the direction of the sun), and the diffuse c o m p o n e n t (arriving from the entire hemisphere of sky). Usually they give results for an average day. Since we must calculate the energy entering a window we must explicitly include angle effects. In o u r model we lbllow more recent work [5, 6] which recognizes the significance (on days with low cloudiness) of distinct diffuse c o m p o n e n t s of three kinds. The first is a circumsolar c o m p o n e n t which arrives from the direction of the sun. The second is an isotropic c o m p o n e n t which arrives uniformly from the entire hemisphere of sky, a n d the third is a horizon c o m p o n e n t which arrives from a b a n d near the horizon but which is uniform as a function o f azimuth. Dave has given a theoretical discussion of the origin of these terms [7]. In addition, because o f the frequent presence in C a n a d a of snow cover with its correspondingly high albedo, o u r model must include radiation ,qround-r~[tected into the window and a ,qround r¢flection enhancement of the isotropic diffuse term described above. To evaluate the average daily radiation arriving at a particular surface we start with the following input : latitude of site, date (which gives the sun's declination), sunny day fraction, S, a n d average albedo, R, for the m o n t h . The sunny day fraction is obtained from the hours o f bright sunshine [8] divided by the total hours of sunshine possible at that site. We take albedo to be 0.2 for no snow cover and 0.8 for complete snow cover. Snow cover maps of C a n a d a are readily available for each m o n t h . For an averge sunny day the sky energy incident on a tilted collector is taken to be : E~ = Direct beam + Diffuse = 1353
fexp (--0.28/sin fl) cos Odt
+ 1353 f(0.ZS - 0 . 2 sin fi) exp ( - 0.28/sin fl) • [0.33(1 + c o s T)(1 + R/2) + 0 . 2 5 sin T + 0 . 8 cos 0]dt
(1)
where 1353 W m 2 is used as the power density in the direct beam with no a t m o s p h e r e present. Note that
the square bracket contains the three dilt"t>c t c r n > . isotropic, horizon, and circumsolar multiplied b~ ~ c o m m o n factor related to the brightness of the direct beam. The integrals are to be done from t equals II t~ 24 hours with all terms set to zero whenever the sun's elevation fl is less than zero. 0 is the angle between the sun's beam and the normal to the tilted surface. An~ term in which cos 0 appears must also be set equal to zero when cos 0 is negative (sun behind the surface), It is well k n o w n that on days of low clearness the solar radiation is much more isotropic than predicted by eq. (I). Recently, Massaquoi [9] and others ha'~c stressed the i m p o r t a n c e of the average n u m b e r ,fl hours of bright sun as a p a r a m e t e r in determining thc diffuse c o m p o n e n t . He has proposed a power series dependence of the diffuse radiation on this wmable. We propose a different a p p r o a c h using the concept of two p r o t o t y p e days: average sunny and average cloudy. F o r a 30 day m o n t h , 30S days are average sunny days and 3 0 ( I - S ) days are average cloudy days. F o r an average cloudy day the sky energy on a tilted collector is taken to be isotropic with a strength dependent on the sun's elevation t h r o u g h a factor sin fi: E ~ = 1353 2 (I+cosT)
f 0.25 sin fldt.
(2)
To obtain the total energy for an average day we c o m b i n e eqs (1) and (2) and add a g r o u n d reflected energy : E,o, = S E , + ( 1 - S ) E ~ + G R ( I - c o s
T)/2
(3)
where G is the global radiation received by a horizontal surface. G may be taken from m e a s u r e m e n t s for the site or the present theory can be used with T = 0 to evaluate G (G = E,,~, for T = 0). Tables 1 to 6 give the average daily energies for the winter heating m o n t h s at six locations. The constants such as 0.28, 0.25, 0.33, 0.8 in eqs ( 1) and (2) were chosen by studying long term data at various C a n a d i a n sites. F u r t h e r details on the model and a c o m p u t e r p r o g r a m for calculations are available in a separate report [10]. The first test o f the model is to set T = 0, and to calculate the global radiation and diffuse radiation on a horizontal surface for each m o n t h at various C a n a d i a n sites for which measured values are available. Only the sunny day fraction and the latitude of the site are used as input data. This has been done for eight sites. The result is that any m o n t h with average daily G greater t h a n 6 MJ m x day ~ is predicted correctly within 15%, a n d for most m o n t h s within 10%. The percentage error can be larger for m o n t h s with very low G. However, since this energy
545
Solar radiation model and energy saving windows Table 1. Average daily energy incident on a window facing the four cardinal directions for Vancouver, British Columbia (L - 49.25~N)
Table 3. Same as Table 1 for Winnipeg, Manitoba (L = 49.9°N) Month
Month September October November December January February March April May
North
East/West
South
4.4" 3.1 1.9 1.3 1.6 2.6 4.0 5.5 7.0
8. l 5.0 2.5 1.6 2.0 3.8 6.5 9.7 12.7
12.8 9.2 5.0 3.2 3.9 7.3 10.2 12.0 12.5
Net energy gain for various windows for entire heating seasont E (single pane) E (double pane) E ( d o u b l ~ l o w E)
-890* 71 +156
--388 +374 +563
+ 194 +891 +1037
September October November December January February March April May
North
East/West
South
4.4 3.2 2.8 2.8 3.7 6.1 9.2 8.7 7.0
8.2 5.4 3.6 3.4 4.6 8.2 12.6 14.1 13.4
13.2 10.7 6.8 6.3 8.7 14.0 17.8 17.0 13.2
Net energy gain for various windows for entire heating season* E (single pane) E (double pane) E (double--low E)
2277 - 599 -Ill
- 1815 - 188 +265
- 964 + 567 +957
* 9 month heating season with 5920' C days. * Units are MJ m 2 per day or per season as appropriate. "~9 month heating season with 2990"C days. is to be c o m p a r e d with the h e a t loss t h r o u g h the w i n d o w , the larger p e r c e n t a g e e r r o r in a small q u a n tity is n o t significant. F o r m o s t sites tested the difference b e t w e e n the m o d e l a n d the m e a s u r e d global o r diffuse r a d i a t i o n was less t h a n 4-10%-1-0.5 M J m 2 d a y '. T h e n o t a b l e exception was for R e s o l u t e (latitude 74.4 N ) w h e r e the difference was + 15% _+ 1 M J m 2 d a y '. Difficulties with m o d e l l i n g R e s o l u t e d a t a have been n o t e d by R u t h a n d C h a n t [1 I], a n d others. T h e m o r e critical test o f the m o d e l is the anis o t r o p y , i.e. the ability to p r e d i c t the differences in a v e r a g e energy i n c i d e n t o n a n o r t h - f a c i n g w i n d o w Table
2. Same as Table l for Edmonton, Alberta (L = 53.6' N)
Month September October November December January February March April May
North
East/West
South
4.2 2.9 2.5 2.0 2.6 5.0 8.3 9.6 7.0
7.7 5.0 3.2 2.3 3.1 6.4 11.3 15.1 13.6
12.8 I0.6 6.5 4. l 5.7 10.9 16.4 18.7 14.1
c o m p a r e d with a n east- o r s o u t h - f a c i n g w i n d o w . T h e s e crucial differences force r e c o g n i t i o n o f the t h r e e c o m p o n e n t s o f diffuse radiation. T h e relative weights o f these t e r m s in eq. (1) were a d j u s t e d by e x a m i n i n g the w o r k o f Perez et al. [6] b u t the test a d o p t e d was to fit the results o f H o o p e r et al. for T o r o n t o [5]. W e find that o u r m o d e l gives c o r r e c t results within 8 % for s o u t h - f a c i n g a n d east- (or west-)facing w i n d o w s but can be u p to 2 5 % high for n o r t h facing w i n d o w s . Since the total energy is very small for n o r t h - f a c i n g w i n d o w s the a b s o l u t e e r r o r for such a w i n d o w is usually less t h a n 1 M J m 2 day ', w h i c h again is insignificant c o m p a r e d with the total h e a t loss c o n t r i b u t e d . O u r aim is to c o n s t r u c t a simple m o d e l w h i c h predicts total seasonal Table 4. Same as Table 1 for Toronto, Ontario (L - 43.7 N) Month September October November December January February March April May
North
East/West
South
4.7 3.6 2.4 2.0 3.4 3.3 4.5 5.7 7.0
8.9 6.2 3.4 2.7 4.4 5.3 7.5 10.5 12.5
13.2 I 1.2 6.6 5.6 8.0 10.1 11.3 I2.1 I 1.2
Net energy gain for various windows for entire heating season*
Net energy gain for various windows for entire heating season*
E (single pane) E (double pane) E (double--low E)
E (single pane) E (double pane) E ( d o u b l e ~ o w E)
2098 -486 21
-- 1514 +33 +454
* 9 month heating season with 5730'C days.
-809 +659 +812
- 1253 - 177 + 126
669 +342 +600
* 9 month heating season with 3890°C days.
+ 10 +946 + 1153
546
A. S. t]ARKH/.. JR
Table 5. Same as Table I for Montreal, Quebec (L - 45.5 N ) Month
North
East/West
South
4.6 3.4 2.9 3.3 4.1 6.3 8.8 7.5 6.9
8.6 5.7 3.7 4.1) 5. I 8.3 12.2 12.2 12.4
12.9 10.6 6.6 6.9 8.7 13.2 16.6 14.1 11.5
]able 7, Properties of three windows used to calcuiatc nc~ heat gain So]ar
September October November December January February March April May
Net energy gain for various windows for entire heating season* E (single pane) E (double panel E (double--low E)
-1331 - 96 +246
--747 + 423 +721
--42 + 1049 + 1294
* 9 month heating season with 452@~C days. energy flows that are relatively accurate for comparisons amongst various windows. F o r this purpose the model is sufficiently accurate and its simplicity represents an advance over many other procedures. Note that the critical input to this model is S, the fraction of total possible hours in a m o n t h that the sun is shining brightly. These data are available for over 300 stations in C a n a d a [8] and can be estimated with reasonable accuracy at other locations. Equation (3) gives the radiation incident on a window o f any orientation. Following the m e t h o d s o f a previous study [2] we perform calculations on a standard single-glazed window which we take to have a transmission coefficient o f 0.8 for both beam and Table 6. Same as Table 1 for Fort Smith, N.W. Territories (L = 60,0°N) Month September October November December January February March April May June
North
East/West
South
3.5 2.6 1.4 0.7 1.2 3.4 6.9 9,2 7.3 8,7
5,7 3.4 1.5 0.8 1.3 4.2 9, 5 14.7 13.5 14.8
9.5 6.0 2.3 0.9 1.9 7.8 15.0 19.3 15,1 14.6
Net energy gain for various windows for entire heating season* E (single pane) E (double pane) E (double lowE)
-3357 - 1046 -363
2773 - 527 +112
* 10 month heating season with 8050'~C days.
--2214 31 +567
Window type Single pane Double pane 1.2 cm airspace Double pane low emissivity (one interior surface coated)
transmission coefficient*
~,-vatue (W m _~(- L!
0.80
6.4
0.71
2.9
0.65
1.8
* This is an average value which includes effects due to different wavelengths and different angles of incidence of the various solar components of radiation. diffuse irradiance. Therefore, 0.8 Etot gives the energy gained by the building per square metre o f window at a particular site. The window loses energy in winter by convection and radiation. We model this by assuming a U-value for the window and multiplying it by the number o f degree days at the site to give the energy loss per heating season per square metre o f window. C o m b i n i n g 0.8 E~oLs u m m e d over the heating season (usually September through May) minus the heat loss term gives the net energy gain for the heating season. Table 7 gives transmission values and Uvalues used in this study. They are for single pane, double pane (taken to be two panes separated by a sealed 1.2 cm air space) and a new low emissivity double pane window [3]. The low emissivity is obtained by coating one glass surface with a material such as tin oxide and results in a much improved Uvalue for the window at the price o f a somewhat reduced transmission (see Table 7) and a higher initial cost. The figures given in Table 7 are merely typical cases. Qualifications and background information on window performance can be found in the literature [2, 31.
3. R E S U L T S
Tables 1 through 6 give the results o f evaluating the model at six Canadian locations. Five metropolitan areas were chosen along with one far northern location ( F o r t Smith) to test the viability o f high efficiency windows in a location o f low solar elevation and extreme cold. F r o m Table 1 we see that a onesquare-metre single pane window in a north wall in Vancouver supplies 4.4 x 0.8 = 3.52 M J o f energy per day or 3 0 x 3 . 5 2 = 106 MJ for the m o n t h o f September. A south-facing window supplies almost three times as much energy. C o m p a r i s o n with Table 3 shows that in winter (December and January) a north-facing window in Winnipeg supplies twice as
Solar radiation model and energy saving windows much energy as one in Vancouver. This is because o f the brighter skies (i.e. a larger value of S) in Winnipeg. F o r Fort Smith, while the sun only appears briefly in the sky in winter, there are still significant solar c o n t r i b u t i o n s especially for south-facing windows. The tables may be used to find energies received by windows or any o t h e r vertical solar collector. F o r directions between the cardinal directions calculated, linear interpolation as a function of angle m a y be used for rough purposes. Calculations using eq. (3) for intermediate a z i m u t h values show however t h a t such interpolation is only approximate. E q u a t i o n (3) must be used for precise values. We now turn to the calculation of net energy gain by each window over a season. The m e t h o d is illustrated for one case. Consider a north-facing window in Vancouver. S u m m i n g the m o n t h l y c o n t r i b u t i o n s listed under ' N o r t h ' (each m u s t be multiplied by the n u m b e r of days in the m o n t h ) , 954 M J m 2 solar energy is obtained. The net energy gain for the season for a single pane window is then : E, .........(Single Pane, N) - 0.8 x 9 5 4 - 8 6 4 0 0 x 2990 x 6.4 x 10 6 = - 8 9 0 MJ m 2 (Vancouver).
(4)
The c o n s t a n t 86400 is the n u m b e r of seconds in one day, 2990 is the n u m b e r o f degree days and 6.4 is the U-value which controls the heat loss, 10-6 converts Joules to Megajoules. Such a window in a north-facing wall gives a net energy loss which must be m a d e up by energy supplied by the heating system (or possibly by south-facing windows in the same room). In contrast, we note from Table 1 that the same window in a south-facing wall gives a net energy gain of 194 M J m 2 over the heating season. This latter window saves fuel a n d therefore provides a value to off'set its cost. Passing to the new low emissivity type window we see t h a t it provides a net energy gain when used in any o r i e n t a t i o n in Vancouver. Season energy totals shown in the lower part of each table m u s t be used with caution. Note t h a t they result from the difference o f two large values, one o f which may be in error by _+ 15%. Small values such as - 2 1 M J m 2 in Table 2 m a y have a n uncertainty of ± 150 MJ m 2 in absolute terms. Recall that this arises from our use o f the model for the solar c o n t r i b u t i o n for this location. The numerical value is nonetheless useful for c o m p a r i s o n with o t h e r window types a n d other directions. The uncertainty has little effect on these comparisons. * The usual practice of computing present worth of future cash flows is used. We assume a lifetime of 20 years for the window and follow the methods of Ref. [2] Appendix B.
547
We may now draw some preliminary conclusions a b o u t the energy savings for various windows a n d their cost effectiveness. T o quantify the model we assume t h a t building heat is supplied by an oil furnace. Oil is t a k e n to cost 30 cents per litre a n d furnace efficiency is t a k e n as 70%. With these assumptions, heat energy costs 1.1 cent per MJ. F r o m Table 1 we note that for a south-facing single pane window of one square metre there is an energy gain w o r t h 194 x 0.011 = $2.13 per season. Switching to a double pane window increases this to $9.80. W i t h this incremental a n n u a l saving o f fuel oil w o r t h $7.67, one can a d o p t an economic model to calculate the allowable initial cost to upgrade from single pane to double pane.* A s s u m i n g m o n e y can be b o r r o w e d at 12% interest rate, the cost to a n n u a l savings ratio must be less t h a n 13.7. T h a t is, the upgrade to a double pane window m u s t cost less t h a n 13.7 x 7.67 = $105 per square metre. The incremental seasonal savings on going from double pane to a low emissivity window in this location is $1.60. The low emissivity coating process must therefore cost less t h a n 13.7 x 1.60 = $21.90 per square metre, at the retail level over the cost of conventional double pane windows, to be viable. F o r north-facing windows in Vancouver, b o t h single a n d double pane units are net energy losers. F r o m a strict economic view there should be no such windows. If such windows are needed for o t h e r than economic reasons, it can still be cost effective to upgrade from single pane to double pane or low emissivity windows. A r g u m e n t s similar to those above show that a double pane window should cost less t h a n $123 more than single pane per square metre to be cost effective a n d the upgrade from double pane to low emissivity window m u s t cost less t h a n $34.25 per square metre (retail). This latter n u m b e r doubles to $70.00 per square metre for a north-facing window in E d m o n t o n a n d rises to $103.00 per square metre for a north-facing window in F o r t Smith. M a n y other c o m p a r i s o n s can be m a d e using the data in the tables. 4. CONCLUSIONS A simple model for solar insolation has been developed which requires only the m o n t h l y h o u r s o f bright sunshine, latitude a n d g r o u n d albedo as input parameters. Various constants o f the model have been a d o p t e d which yield reasonable values for radiation o n a tilted surface of any o r i e n t a t i o n in C a n a d a , a n d which include circumsolar a n d bright horizon comp o n e n t s as well as an isotropic c o m p o n e n t for diffuse radiation. The model has been evaluated for six sites
548
A. S. BARKER, ,IR
in C a n a d a and then combined with performance data for three types o f windows. The calculations show that at every site considerable energy can be gained from the solar flux, even for north-facing windows. A s s u m i n g that the windows are not blocked or shaded by adjacent structures and that reflective blinds or shades are not used to prevent capture o f the solar input, dollar savings can be calculated. The results show that at every location and for any window orientation the upgrade from single pane to double pane windows results in net savings if it can be done for less than $100 per square metre o f window. The savings are extremely large for a northern location such as F o r t Smith. The viability o f one type o f the new super-window has been considered also. If this window can be manufactured and installed at an incremental cost o f less than $21.90 per square metre (over standard doublepane windows), it is cost effective for a south-facing window in Vancouver. Even at a considerably higher price it is cost effective as an upgrade at other locations. We can c o m m e n t on the significance o f the super-window in Table 7 to the national energy picture. Approximately 0 . 5 x l0 TM J is consumed annually in C a n a d a for residential space heating. Using the concept o f an averge single family residence defined in a previous study [2], and assuming that only half o f the window area on average will be free from shading and left clear o f curtains, we estimate an 8% saving in space heating energy for an average home with single pane converted to double pane windows. A 2.5% saving can be achieved for the upgrade from double pane to the type o f low emissivity double pane window discussed above. It appears that such window
improvement is now cost effective in new cons[lUCtion. Refitting old windows could become cost effective if d e m a n d incrcascs and manufacturers begin t~ market suitable units. Finally, in spite o f the harsh northern winter climate o f Canada, significant national energy savings can be achieved through the use o f new types o f architectural window. REFERENCES
1. Canadian Climate Normals, Vol. 1, Solar Radiation. Canadian Climate Center, Downsview, Ontario (1982). 2. Efficient use of energy, Am. Inst. o[" Phys. Con/~ Proc. No. 25, pp. 245 304. A.I.P., New York (1975). 3. V. E. Gilmore, Superwindows. Popular Science, pp. 76 1i5 (March 1986). 4. P. W. Suckling, Residential solar space-heating potential across the United States and Southern Canada. Canadian Geographer 26, 158 165 (1982). 5. F. C. Hooper, A. P. Brunger and C. S. Chan, A clear sky model of diffuse sky radiance, in Solar Engbwerin.q, Proc. A S M E Solar Energy Division Fourth Annual Conl l. pp. 90 95. ASME, New York (1982). 6. R. Perez, R. Stewart, C. Arbogast, R. Seals and J. Scott, An anisotropic hourly diffuse radiation model for sloping surfaces. Sol. Energy 36, 481 497 (1986). 7. J. V. Dave, Validity of the isotropic-distribution approximation in solar energy estimations. Sol. Energy 19, 331 333 (1977). 8. Canadian Climate Normals, Vol. 7, Bright Sunshine. Canadian Climate Center, Downsview, Ontario (1982). 9. J. G. M. Massaquoi, Predicting diffuse radiation where only data on sunshine duration is available. Sol. Wind Technol. 4, 205 210 (1987). 10. A. S. Barker Jr, A simple computer model for solar radiation on a tilted surface in Canada, unpublished (available from the author). 11. D. W. Ruth and R,. E. Chant, The relationship of diffuse radiation to total radiation in Canada. Sol. Energy 18, 153 154 (1976).
0741- 983X/88 $3.00 + . 0 0 Pergamon Press pie
A SOLAR R A D I A T I O N M O D E L A N D E N E R G Y SAVING W I N D O W S IN C A N A D A A. S. BARKER, JR Trinity Western University, Langley, B.C., Canada V3A 4R9 and Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853-2501, U.S.A.
(Received 8 December 1987 ; accepted 25 February 1988) Abstract A model for monthly average solar insolation on a tilted surface of any azimuth is derived which depends only on the average number of bright sunshine hours and other directly known attributes of a site such as latitude and ground albedo. Parameters in the model are derived from data at Canadian stations allowing an assessment of the insolation for north, east, west and south facing windows. By considering windows of various thermal efficiencies and the cost of heat energy, the cost effectiveness of the windows in various northern locations is evaluated. Using present energy costs it is shown that the recently announced super-efficient windows can be cost effective when added as an upgrade over conventional double pane windows and can provide significant energy savings even at far northern latitudes.
1. INTRODUCTION
C a n a d a a n d other high latitude countries [4]. Energy efficient windows, t h o u g h more expensive initially, can be cost effective because of the solar input u n d e r certain conditions. Widespread use o f such windows u n d e r certain conditions can significantly reduce national energy c o n s u m p t i o n . G e o g r a p h i c a n d climatological factors control these conditions in a n essential way. The model introduced here allows assessments o f solar flux based o n a m i n i m u m knowledge of such factors a n d can lead to conclusions o n the value of high efficiency windows. In Section 2 the model is outlined a n d three types o f window defined. Section 3 gives the results o f calculations for six Canadian locations, a n d derives dollar costs associated with the three window types using a s t a n d a r d economic model which includes the cost o f b o r r o w e d money. Section 4 gives the conclusions.
W h e n assessing the heat gain by windows a n d skylights a n d their effect on the winter energy requirements o f a building, a complete description o f the direct a n d indirect (diffuse) radiation flux is needed. Models of m o n t h l y averages of the flux arriving at a horizontal surface have been p r o p o s e d a n d actual m e a s u r e m e n t s of this q u a n t i t y for a network of a b o u t 50 sites in C a n a d a are readily available [1]. The areal coverage of this network is too limited however for direct use in flux calculations. W h e n solar receiving surfaces (e.g. windows) which are n o t horizontal are used, c o r r e s p o n d i n g m e a s u r e m e n t s are usually n o n existent a n d models must be used. There has recently been considerable advance in such modelling to include angular dependence of the diffuse r a d i a t i o n in a realistic manner. We propose such a model to predict m o n t h l y average insolation on a surface of any tilt and azimuth and use it to construct tables of winter heat gains for vertical windows facing north, east (or west) a n d south in C a n a d a . * These tables extend a f u n d a m e n t a l study of winter heat gain t h r o u g h windows carried o u t for the continental U n i t e d States [2]. With the recent a n n o u n c e m e n t of new super efficient windows [3] a n d escalations in the cost o f energy there is a need to reassess the cost effectiveness o f various windows. It is n o t e w o r t h y that, c o n t r a r y to intuition, there is a large potential for energy from the sun in
2. METHODS A precise calculation of the solar flux incident o n a surface of tilt angle T a n d a z i m u t h A depends o n m a n y factors such as angular position of the sun, turbidity, cloud cover, surface albedo, etc. Because of daily a n d even m i n u t e by m i n u t e fluctuations in most o f the i m p o r t a n t variables, various models have been proposed to give average values of the flux. W e follow such an a p p r o a c h here by p r o p o s i n g a m a t h e m a t i c a l expression which gives the energy incident o n a tilted surface over an average day. We take twelve such average days (15 January, 14 February, 15 M a r c h . . . . )
* The present author was a member of the American Institute of Physics team which studied energy conservation and high efficiency windows (see [2]). 543
544 to represent thc pattern over a year and are therefore using m o n t h l y averages. Early models which use monthly averaging have been used quite successfully to predict radiation received on a horizontal window or collector. This is the most c o m m o n measurement made by weather stations. These models always recognize two m a j o r c o n t r i b u t i o n s - - the direct component (all radiation arriving in a cone of solid angle 6.7 x 10 ~ steradian from the direction of the sun), and the diffuse c o m p o n e n t (arriving from the entire hemisphere of sky). Usually they give results for an average day. Since we must calculate the energy entering a window we must explicitly include angle effects. In o u r model we lbllow more recent work [5, 6] which recognizes the significance (on days with low cloudiness) of distinct diffuse c o m p o n e n t s of three kinds. The first is a circumsolar c o m p o n e n t which arrives from the direction of the sun. The second is an isotropic c o m p o n e n t which arrives uniformly from the entire hemisphere of sky, a n d the third is a horizon c o m p o n e n t which arrives from a b a n d near the horizon but which is uniform as a function o f azimuth. Dave has given a theoretical discussion of the origin of these terms [7]. In addition, because o f the frequent presence in C a n a d a of snow cover with its correspondingly high albedo, o u r model must include radiation ,qround-r~[tected into the window and a ,qround r¢flection enhancement of the isotropic diffuse term described above. To evaluate the average daily radiation arriving at a particular surface we start with the following input : latitude of site, date (which gives the sun's declination), sunny day fraction, S, a n d average albedo, R, for the m o n t h . The sunny day fraction is obtained from the hours o f bright sunshine [8] divided by the total hours of sunshine possible at that site. We take albedo to be 0.2 for no snow cover and 0.8 for complete snow cover. Snow cover maps of C a n a d a are readily available for each m o n t h . For an averge sunny day the sky energy incident on a tilted collector is taken to be : E~ = Direct beam + Diffuse = 1353
fexp (--0.28/sin fl) cos Odt
+ 1353 f(0.ZS - 0 . 2 sin fi) exp ( - 0.28/sin fl) • [0.33(1 + c o s T)(1 + R/2) + 0 . 2 5 sin T + 0 . 8 cos 0]dt
(1)
where 1353 W m 2 is used as the power density in the direct beam with no a t m o s p h e r e present. Note that
the square bracket contains the three dilt"t>c t c r n > . isotropic, horizon, and circumsolar multiplied b~ ~ c o m m o n factor related to the brightness of the direct beam. The integrals are to be done from t equals II t~ 24 hours with all terms set to zero whenever the sun's elevation fl is less than zero. 0 is the angle between the sun's beam and the normal to the tilted surface. An~ term in which cos 0 appears must also be set equal to zero when cos 0 is negative (sun behind the surface), It is well k n o w n that on days of low clearness the solar radiation is much more isotropic than predicted by eq. (I). Recently, Massaquoi [9] and others ha'~c stressed the i m p o r t a n c e of the average n u m b e r ,fl hours of bright sun as a p a r a m e t e r in determining thc diffuse c o m p o n e n t . He has proposed a power series dependence of the diffuse radiation on this wmable. We propose a different a p p r o a c h using the concept of two p r o t o t y p e days: average sunny and average cloudy. F o r a 30 day m o n t h , 30S days are average sunny days and 3 0 ( I - S ) days are average cloudy days. F o r an average cloudy day the sky energy on a tilted collector is taken to be isotropic with a strength dependent on the sun's elevation t h r o u g h a factor sin fi: E ~ = 1353 2 (I+cosT)
f 0.25 sin fldt.
(2)
To obtain the total energy for an average day we c o m b i n e eqs (1) and (2) and add a g r o u n d reflected energy : E,o, = S E , + ( 1 - S ) E ~ + G R ( I - c o s
T)/2
(3)
where G is the global radiation received by a horizontal surface. G may be taken from m e a s u r e m e n t s for the site or the present theory can be used with T = 0 to evaluate G (G = E,,~, for T = 0). Tables 1 to 6 give the average daily energies for the winter heating m o n t h s at six locations. The constants such as 0.28, 0.25, 0.33, 0.8 in eqs ( 1) and (2) were chosen by studying long term data at various C a n a d i a n sites. F u r t h e r details on the model and a c o m p u t e r p r o g r a m for calculations are available in a separate report [10]. The first test o f the model is to set T = 0, and to calculate the global radiation and diffuse radiation on a horizontal surface for each m o n t h at various C a n a d i a n sites for which measured values are available. Only the sunny day fraction and the latitude of the site are used as input data. This has been done for eight sites. The result is that any m o n t h with average daily G greater t h a n 6 MJ m x day ~ is predicted correctly within 15%, a n d for most m o n t h s within 10%. The percentage error can be larger for m o n t h s with very low G. However, since this energy
545
Solar radiation model and energy saving windows Table 1. Average daily energy incident on a window facing the four cardinal directions for Vancouver, British Columbia (L - 49.25~N)
Table 3. Same as Table 1 for Winnipeg, Manitoba (L = 49.9°N) Month
Month September October November December January February March April May
North
East/West
South
4.4" 3.1 1.9 1.3 1.6 2.6 4.0 5.5 7.0
8. l 5.0 2.5 1.6 2.0 3.8 6.5 9.7 12.7
12.8 9.2 5.0 3.2 3.9 7.3 10.2 12.0 12.5
Net energy gain for various windows for entire heating seasont E (single pane) E (double pane) E ( d o u b l ~ l o w E)
-890* 71 +156
--388 +374 +563
+ 194 +891 +1037
September October November December January February March April May
North
East/West
South
4.4 3.2 2.8 2.8 3.7 6.1 9.2 8.7 7.0
8.2 5.4 3.6 3.4 4.6 8.2 12.6 14.1 13.4
13.2 10.7 6.8 6.3 8.7 14.0 17.8 17.0 13.2
Net energy gain for various windows for entire heating season* E (single pane) E (double pane) E (double--low E)
2277 - 599 -Ill
- 1815 - 188 +265
- 964 + 567 +957
* 9 month heating season with 5920' C days. * Units are MJ m 2 per day or per season as appropriate. "~9 month heating season with 2990"C days. is to be c o m p a r e d with the h e a t loss t h r o u g h the w i n d o w , the larger p e r c e n t a g e e r r o r in a small q u a n tity is n o t significant. F o r m o s t sites tested the difference b e t w e e n the m o d e l a n d the m e a s u r e d global o r diffuse r a d i a t i o n was less t h a n 4-10%-1-0.5 M J m 2 d a y '. T h e n o t a b l e exception was for R e s o l u t e (latitude 74.4 N ) w h e r e the difference was + 15% _+ 1 M J m 2 d a y '. Difficulties with m o d e l l i n g R e s o l u t e d a t a have been n o t e d by R u t h a n d C h a n t [1 I], a n d others. T h e m o r e critical test o f the m o d e l is the anis o t r o p y , i.e. the ability to p r e d i c t the differences in a v e r a g e energy i n c i d e n t o n a n o r t h - f a c i n g w i n d o w Table
2. Same as Table l for Edmonton, Alberta (L = 53.6' N)
Month September October November December January February March April May
North
East/West
South
4.2 2.9 2.5 2.0 2.6 5.0 8.3 9.6 7.0
7.7 5.0 3.2 2.3 3.1 6.4 11.3 15.1 13.6
12.8 I0.6 6.5 4. l 5.7 10.9 16.4 18.7 14.1
c o m p a r e d with a n east- o r s o u t h - f a c i n g w i n d o w . T h e s e crucial differences force r e c o g n i t i o n o f the t h r e e c o m p o n e n t s o f diffuse radiation. T h e relative weights o f these t e r m s in eq. (1) were a d j u s t e d by e x a m i n i n g the w o r k o f Perez et al. [6] b u t the test a d o p t e d was to fit the results o f H o o p e r et al. for T o r o n t o [5]. W e find that o u r m o d e l gives c o r r e c t results within 8 % for s o u t h - f a c i n g a n d east- (or west-)facing w i n d o w s but can be u p to 2 5 % high for n o r t h facing w i n d o w s . Since the total energy is very small for n o r t h - f a c i n g w i n d o w s the a b s o l u t e e r r o r for such a w i n d o w is usually less t h a n 1 M J m 2 day ', w h i c h again is insignificant c o m p a r e d with the total h e a t loss c o n t r i b u t e d . O u r aim is to c o n s t r u c t a simple m o d e l w h i c h predicts total seasonal Table 4. Same as Table 1 for Toronto, Ontario (L - 43.7 N) Month September October November December January February March April May
North
East/West
South
4.7 3.6 2.4 2.0 3.4 3.3 4.5 5.7 7.0
8.9 6.2 3.4 2.7 4.4 5.3 7.5 10.5 12.5
13.2 I 1.2 6.6 5.6 8.0 10.1 11.3 I2.1 I 1.2
Net energy gain for various windows for entire heating season*
Net energy gain for various windows for entire heating season*
E (single pane) E (double pane) E (double--low E)
E (single pane) E (double pane) E ( d o u b l e ~ o w E)
2098 -486 21
-- 1514 +33 +454
* 9 month heating season with 5730'C days.
-809 +659 +812
- 1253 - 177 + 126
669 +342 +600
* 9 month heating season with 3890°C days.
+ 10 +946 + 1153
546
A. S. t]ARKH/.. JR
Table 5. Same as Table I for Montreal, Quebec (L - 45.5 N ) Month
North
East/West
South
4.6 3.4 2.9 3.3 4.1 6.3 8.8 7.5 6.9
8.6 5.7 3.7 4.1) 5. I 8.3 12.2 12.2 12.4
12.9 10.6 6.6 6.9 8.7 13.2 16.6 14.1 11.5
]able 7, Properties of three windows used to calcuiatc nc~ heat gain So]ar
September October November December January February March April May
Net energy gain for various windows for entire heating season* E (single pane) E (double panel E (double--low E)
-1331 - 96 +246
--747 + 423 +721
--42 + 1049 + 1294
* 9 month heating season with 452@~C days. energy flows that are relatively accurate for comparisons amongst various windows. F o r this purpose the model is sufficiently accurate and its simplicity represents an advance over many other procedures. Note that the critical input to this model is S, the fraction of total possible hours in a m o n t h that the sun is shining brightly. These data are available for over 300 stations in C a n a d a [8] and can be estimated with reasonable accuracy at other locations. Equation (3) gives the radiation incident on a window o f any orientation. Following the m e t h o d s o f a previous study [2] we perform calculations on a standard single-glazed window which we take to have a transmission coefficient o f 0.8 for both beam and Table 6. Same as Table 1 for Fort Smith, N.W. Territories (L = 60,0°N) Month September October November December January February March April May June
North
East/West
South
3.5 2.6 1.4 0.7 1.2 3.4 6.9 9,2 7.3 8,7
5,7 3.4 1.5 0.8 1.3 4.2 9, 5 14.7 13.5 14.8
9.5 6.0 2.3 0.9 1.9 7.8 15.0 19.3 15,1 14.6
Net energy gain for various windows for entire heating season* E (single pane) E (double pane) E (double lowE)
-3357 - 1046 -363
2773 - 527 +112
* 10 month heating season with 8050'~C days.
--2214 31 +567
Window type Single pane Double pane 1.2 cm airspace Double pane low emissivity (one interior surface coated)
transmission coefficient*
~,-vatue (W m _~(- L!
0.80
6.4
0.71
2.9
0.65
1.8
* This is an average value which includes effects due to different wavelengths and different angles of incidence of the various solar components of radiation. diffuse irradiance. Therefore, 0.8 Etot gives the energy gained by the building per square metre o f window at a particular site. The window loses energy in winter by convection and radiation. We model this by assuming a U-value for the window and multiplying it by the number o f degree days at the site to give the energy loss per heating season per square metre o f window. C o m b i n i n g 0.8 E~oLs u m m e d over the heating season (usually September through May) minus the heat loss term gives the net energy gain for the heating season. Table 7 gives transmission values and Uvalues used in this study. They are for single pane, double pane (taken to be two panes separated by a sealed 1.2 cm air space) and a new low emissivity double pane window [3]. The low emissivity is obtained by coating one glass surface with a material such as tin oxide and results in a much improved Uvalue for the window at the price o f a somewhat reduced transmission (see Table 7) and a higher initial cost. The figures given in Table 7 are merely typical cases. Qualifications and background information on window performance can be found in the literature [2, 31.
3. R E S U L T S
Tables 1 through 6 give the results o f evaluating the model at six Canadian locations. Five metropolitan areas were chosen along with one far northern location ( F o r t Smith) to test the viability o f high efficiency windows in a location o f low solar elevation and extreme cold. F r o m Table 1 we see that a onesquare-metre single pane window in a north wall in Vancouver supplies 4.4 x 0.8 = 3.52 M J o f energy per day or 3 0 x 3 . 5 2 = 106 MJ for the m o n t h o f September. A south-facing window supplies almost three times as much energy. C o m p a r i s o n with Table 3 shows that in winter (December and January) a north-facing window in Winnipeg supplies twice as
Solar radiation model and energy saving windows much energy as one in Vancouver. This is because o f the brighter skies (i.e. a larger value of S) in Winnipeg. F o r Fort Smith, while the sun only appears briefly in the sky in winter, there are still significant solar c o n t r i b u t i o n s especially for south-facing windows. The tables may be used to find energies received by windows or any o t h e r vertical solar collector. F o r directions between the cardinal directions calculated, linear interpolation as a function of angle m a y be used for rough purposes. Calculations using eq. (3) for intermediate a z i m u t h values show however t h a t such interpolation is only approximate. E q u a t i o n (3) must be used for precise values. We now turn to the calculation of net energy gain by each window over a season. The m e t h o d is illustrated for one case. Consider a north-facing window in Vancouver. S u m m i n g the m o n t h l y c o n t r i b u t i o n s listed under ' N o r t h ' (each m u s t be multiplied by the n u m b e r of days in the m o n t h ) , 954 M J m 2 solar energy is obtained. The net energy gain for the season for a single pane window is then : E, .........(Single Pane, N) - 0.8 x 9 5 4 - 8 6 4 0 0 x 2990 x 6.4 x 10 6 = - 8 9 0 MJ m 2 (Vancouver).
(4)
The c o n s t a n t 86400 is the n u m b e r of seconds in one day, 2990 is the n u m b e r o f degree days and 6.4 is the U-value which controls the heat loss, 10-6 converts Joules to Megajoules. Such a window in a north-facing wall gives a net energy loss which must be m a d e up by energy supplied by the heating system (or possibly by south-facing windows in the same room). In contrast, we note from Table 1 that the same window in a south-facing wall gives a net energy gain of 194 M J m 2 over the heating season. This latter window saves fuel a n d therefore provides a value to off'set its cost. Passing to the new low emissivity type window we see t h a t it provides a net energy gain when used in any o r i e n t a t i o n in Vancouver. Season energy totals shown in the lower part of each table m u s t be used with caution. Note t h a t they result from the difference o f two large values, one o f which may be in error by _+ 15%. Small values such as - 2 1 M J m 2 in Table 2 m a y have a n uncertainty of ± 150 MJ m 2 in absolute terms. Recall that this arises from our use o f the model for the solar c o n t r i b u t i o n for this location. The numerical value is nonetheless useful for c o m p a r i s o n with o t h e r window types a n d other directions. The uncertainty has little effect on these comparisons. * The usual practice of computing present worth of future cash flows is used. We assume a lifetime of 20 years for the window and follow the methods of Ref. [2] Appendix B.
547
We may now draw some preliminary conclusions a b o u t the energy savings for various windows a n d their cost effectiveness. T o quantify the model we assume t h a t building heat is supplied by an oil furnace. Oil is t a k e n to cost 30 cents per litre a n d furnace efficiency is t a k e n as 70%. With these assumptions, heat energy costs 1.1 cent per MJ. F r o m Table 1 we note that for a south-facing single pane window of one square metre there is an energy gain w o r t h 194 x 0.011 = $2.13 per season. Switching to a double pane window increases this to $9.80. W i t h this incremental a n n u a l saving o f fuel oil w o r t h $7.67, one can a d o p t an economic model to calculate the allowable initial cost to upgrade from single pane to double pane.* A s s u m i n g m o n e y can be b o r r o w e d at 12% interest rate, the cost to a n n u a l savings ratio must be less t h a n 13.7. T h a t is, the upgrade to a double pane window m u s t cost less t h a n 13.7 x 7.67 = $105 per square metre. The incremental seasonal savings on going from double pane to a low emissivity window in this location is $1.60. The low emissivity coating process must therefore cost less t h a n 13.7 x 1.60 = $21.90 per square metre, at the retail level over the cost of conventional double pane windows, to be viable. F o r north-facing windows in Vancouver, b o t h single a n d double pane units are net energy losers. F r o m a strict economic view there should be no such windows. If such windows are needed for o t h e r than economic reasons, it can still be cost effective to upgrade from single pane to double pane or low emissivity windows. A r g u m e n t s similar to those above show that a double pane window should cost less t h a n $123 more than single pane per square metre to be cost effective a n d the upgrade from double pane to low emissivity window m u s t cost less t h a n $34.25 per square metre (retail). This latter n u m b e r doubles to $70.00 per square metre for a north-facing window in E d m o n t o n a n d rises to $103.00 per square metre for a north-facing window in F o r t Smith. M a n y other c o m p a r i s o n s can be m a d e using the data in the tables. 4. CONCLUSIONS A simple model for solar insolation has been developed which requires only the m o n t h l y h o u r s o f bright sunshine, latitude a n d g r o u n d albedo as input parameters. Various constants o f the model have been a d o p t e d which yield reasonable values for radiation o n a tilted surface of any o r i e n t a t i o n in C a n a d a , a n d which include circumsolar a n d bright horizon comp o n e n t s as well as an isotropic c o m p o n e n t for diffuse radiation. The model has been evaluated for six sites
548
A. S. BARKER, ,IR
in C a n a d a and then combined with performance data for three types o f windows. The calculations show that at every site considerable energy can be gained from the solar flux, even for north-facing windows. A s s u m i n g that the windows are not blocked or shaded by adjacent structures and that reflective blinds or shades are not used to prevent capture o f the solar input, dollar savings can be calculated. The results show that at every location and for any window orientation the upgrade from single pane to double pane windows results in net savings if it can be done for less than $100 per square metre o f window. The savings are extremely large for a northern location such as F o r t Smith. The viability o f one type o f the new super-window has been considered also. If this window can be manufactured and installed at an incremental cost o f less than $21.90 per square metre (over standard doublepane windows), it is cost effective for a south-facing window in Vancouver. Even at a considerably higher price it is cost effective as an upgrade at other locations. We can c o m m e n t on the significance o f the super-window in Table 7 to the national energy picture. Approximately 0 . 5 x l0 TM J is consumed annually in C a n a d a for residential space heating. Using the concept o f an averge single family residence defined in a previous study [2], and assuming that only half o f the window area on average will be free from shading and left clear o f curtains, we estimate an 8% saving in space heating energy for an average home with single pane converted to double pane windows. A 2.5% saving can be achieved for the upgrade from double pane to the type o f low emissivity double pane window discussed above. It appears that such window
improvement is now cost effective in new cons[lUCtion. Refitting old windows could become cost effective if d e m a n d incrcascs and manufacturers begin t~ market suitable units. Finally, in spite o f the harsh northern winter climate o f Canada, significant national energy savings can be achieved through the use o f new types o f architectural window. REFERENCES
1. Canadian Climate Normals, Vol. 1, Solar Radiation. Canadian Climate Center, Downsview, Ontario (1982). 2. Efficient use of energy, Am. Inst. o[" Phys. Con/~ Proc. No. 25, pp. 245 304. A.I.P., New York (1975). 3. V. E. Gilmore, Superwindows. Popular Science, pp. 76 1i5 (March 1986). 4. P. W. Suckling, Residential solar space-heating potential across the United States and Southern Canada. Canadian Geographer 26, 158 165 (1982). 5. F. C. Hooper, A. P. Brunger and C. S. Chan, A clear sky model of diffuse sky radiance, in Solar Engbwerin.q, Proc. A S M E Solar Energy Division Fourth Annual Conl l. pp. 90 95. ASME, New York (1982). 6. R. Perez, R. Stewart, C. Arbogast, R. Seals and J. Scott, An anisotropic hourly diffuse radiation model for sloping surfaces. Sol. Energy 36, 481 497 (1986). 7. J. V. Dave, Validity of the isotropic-distribution approximation in solar energy estimations. Sol. Energy 19, 331 333 (1977). 8. Canadian Climate Normals, Vol. 7, Bright Sunshine. Canadian Climate Center, Downsview, Ontario (1982). 9. J. G. M. Massaquoi, Predicting diffuse radiation where only data on sunshine duration is available. Sol. Wind Technol. 4, 205 210 (1987). 10. A. S. Barker Jr, A simple computer model for solar radiation on a tilted surface in Canada, unpublished (available from the author). 11. D. W. Ruth and R,. E. Chant, The relationship of diffuse radiation to total radiation in Canada. Sol. Energy 18, 153 154 (1976).