Equations for solar heat gain through windows

(1)

Publisher’s version / Version de l'éditeur:

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca.

Questions? Contact the NRC Publications Archive team at

PublicationsArchive-ArchivesPublications@nrc-cnrc.gc.ca. If you wish to email the authors directly, please see the first page of the publication for their contact information.

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

Research Paper (National Research Council of Canada. Division of Building

Research), 1965-06-01

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. https://nrc-publications.canada.ca/eng/copyright

NRC Publications Archive Record / Notice des Archives des publications du CNRC :

https://nrc-publications.canada.ca/eng/view/object/?id=0f701c83-6a5f-406e-811c-d4958a8aadd2 https://publications-cnrc.canada.ca/fra/voir/objet/?id=0f701c83-6a5f-406e-811c-d4958a8aadd2

NRC Publications Archive

Archives des publications du CNRC

This publication could be one of several versions: author’s original, accepted manuscript or the publisher’s version. / La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur.

For the publisher’s version, please access the DOI link below./ Pour consulter la version de l’éditeur, utilisez le lien DOI ci-dessous.

https://doi.org/10.1016/0038-092X(65)90207-0

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

Equations for solar heat gain through windows

(2)

(3)

(4)

..L ... . "

...

. l _ , l . . l l l L

Val. IS, K";uml)ci- 2 , :\lr~.il-.Tur~e, I!i(ib

P ~ i n / < ~ l it! C..X.:I.

Equatioiis for Solar Heat

Gain

Through Wi~idows

1:csearc.h Olficc,r, U r ~ i l t l ~ t ~ g Srrviccs Section, Ilivision of Uniltlitlg Research, S a t ~ o ~ l a l I?cscarcl~ C o u t ~ c i l , C ~ u ~ a t l n

An analysis of solar-radiation records obtained at Searl)oro~tgh, Ontario, indicates that the insola- tion ill Canada call bc significantl? greater than the valucs gitcn by _\loon's1 sta~ldard solar-radia- tion curves. It is possiblc LO represe~it the solar data by a s i ~ l g l e anal3 tical cspression involving an atinospheric evtinctio~l coefficient and ail 31)- parent value of the solar constant. 'l'his c~pression a1lotc.s the calculation of insolation on an? surfacc and the determination of the time :~nc-l date when the ~ n a s i m u m insolation can occur for any sur- face. Sinlple expressions relate the time, date, latitude, building orientation, and the type of' window glass and shading with the solar heat gain through \c-indows. These can be used to program any digital computer to c o n ~ p u t e design values of the instantaileous heat gain through the windows of a building.

S

0 L . m radiatioi~ sliould be considerccl at a \-cry cai,ly stage ill the dcsigii of a building. I t call iiifuciicr

such basic drcisioi~s as: ori(~iitatioii, the amou~it aiicl type of glass t o bc uscd in the various facacles, \vlicthcr it should bc shaded, aricl \vhcthcr tlic huilcliiig will Iiavc to hc air-conditioi~c~l. Idrally, tlic arcliitcct sliould Iial,c> at his disposal comparative valucs of solar licat gaiii for all tlic \\all-n i~iclon. comhiiiatio~is hc is coiisidci~iilg for any particular pi,oj(~ct. Tlic calculatioii of solar heat gaiii call I)(. progiainn~c~d 011 el ell a 11ri.y ii~oclrst

digital computc~.; and oiicc a progl,ain is availablc it is p~.actical to use it for tlic pi.climi11al.y drsign aiialysis as \vcll as for t h r dctailcd dcsig~i of tlic cooli~ig system. This papc.r presents a systcrn of ccluatioiis rclati~ig the solar heat gain thiough miiiclo~vs to tlic, basic data and gives valuc~s for the' data. I t is a relatively simplc task t o organizc tliis iiiformatioli so that it call be used by any available coinputcr. Thc cquatioiis arc scpa- rated irito thrcc catcgorics ~ r l a t i n g to: ( 1 ) Solar irradiation; (2) The angles a t \vhich thc solar bcam strikes ally surfacc, and (3) The t~.a~ismissioii aiid a1)sorptioii factors for r\-iiido\~rs.

ICach 01 tlic~sc sul~j(~cts is discussed scpal,atcly bcforc the thrcc arc cornbiiiccl to i~~clicate the ~ilasiinuin v a l ~ ~ c of solar heat gain and \ ~ h c ~ i it call occu~.. Thc gco- 111dric rclatioiisliil~s b(ltiv~('11 i\.a11 oi.iciitatioi1 aiid t h ~

- -

-T h i s ~ a p w i s a c o t ~ t t ~ h u t i o t ~ Irotn I l l ~ I ) ~ v ~ s i o ~ t o f 13u1ltling Itesc~nrch, N : ~ t i o t ~ a l Ilc~scarc~h Cou11c11, C::~~lntl:~, L L I I ~ I is 111111-

11shcd a i t h t h e :IIIII~OV:LI o f t h e I)ir(sc,tor 01 tht' 1)1visiot1. It W ~ L S p ~ e s e ~ ~ t c s c l n t :L Itc~giollnl 'I'cch~~ic:ll Conicxrcl~~cc. ot thcs I ' : I I ~ I ~ I ( Y ~ I . i ~ t g 111stitute of C:I~I:L(~:L :LI 1<i11gsto11, O I I I ; \ I I O , OIL 9 h1:1y 19ti 1,

but 11:~s I I O ~ I ) C ~ % I I ~ I C V I O U S I ~ pr~I)lisI~(~cl.

positioii of the suii arc certaiiily iiot ne\v. Tlicy arc, Iionc\~c>i., cast iii n I'orin that is convci~iciit lor c o i n p ~ ~ t c r

progi~aiiimiiig

.

12asic Equatio~is for Solar Irracliation

Dcsigii valucs oi' the solar ~,adiatioii rcachiiig the suifacc1 of t t ~ c earth shoulcl bc based oil a statistical analysis of solar-radiatioii rcco~~cls. If tlicy are iiot available, thc oiily altcriiativc is to calculate thc valucs tliat \vould obtain lor sorilc assumc~cl atinosphcrc~. Iii 1940 lloolil folloitrccl this latter course aiid published his proposed standard solar-radiation curves for en- giilccriilg use. These data have bccu used siilce that tirile for no st cngincerii~g design calculations iiivolviiig solar ci~crgy. In the rncantinlc, there has becii a great increase in tlic number of stations ~nakirig coiiti~iuous mrasurrmeuts of solar radiation, so that it has become possiblc to chrck 1/Iooii's tlicxorctical staiidaid against ineasurcd valucs.

Tcii y ~ a r s ago I'armc~lrc' puhlishcd results that slioi~~ccl obscrvccl values cscc.cdii~g AIooii's values ~ I J -

about 10 pcrcciit. Iii 19.78 TIirclk(~lc1 aiid .Jordail" inadc a comparison t)ct\vcc~i obscrvccl iralucs of solar radiation

011 clrar clays a ~ i d valucs iiidicatcd by a11 ailalysis simi-

lar to ?\loon's. Tlicy fouiid that thr obsrrvcd ~lalucs at tluxcc wiclrly scparatccl locatioiis in tlic I'iiitcd States mr1.c ill general agrcrmciit with oiic a~iotlicr, hut higlicr

tlia~i hlooii's s t a ~ i d a ~ d . They, tlici~cforc, dcfiiircl a iic~v stanclard. Iii Tlii~clkrld's papc1.s t h r term, clrar clay, is takcii to incall an average cloudless clay rathrr than a dcsigii clay. 111 fact, in l9(53 ThrclkclcIJ rcportccl that on

somc days tlic radiation at 1Iinric~apolis cscc~cclrd his standa1.d by as inucl~ as 20 percent.

.Isid(> fro111 tlic Pact that t h r vai.ious proposed sta~id- arcls appeal t o bc too low for ~ioi~nial clcsigi~ pui,posc.s, they arc iiot xvell suited for iiiclusion ill a computer pi.ogram. A nrn. analysis has bcc~i made, tlic~~cforc, with tllc two-fold objcctivc of dctcrmining design valucs ol' solai ir1.adiatio11 tliat arc applicable for builcling design in Ca~iada and of obtairiing the data in a fo1.111 that is convc~niciit for computer programming.

The solar radiation iiicideiit oil the outside surface of a builclirig coiisists of thrcc co~nporierits: the di~bcct solar l~caiii, I, ; t hc scat t erccl solai radiation co111ing

fi.0111 all parts of the sky, I s ; aiid the solar radiation 1 cflcct cd oiit o t he surfacc by iieiglibouriiig s~ufaccs, I,,

.

Tl~csc compoiiciits will 1)c cliscussed i l l t u r ~ ~ .

12adiation Comiug 1)ircetly from the Slin

Tlic hl(~tcorologica1 Brallcti of tlic Dcpartmciit of Trniispoit, Caiiatla, pul~lislies tlic mcasui~ctl vnluc's of

(5)

TABLE OF NOMENCLATUKE

11 = area of n surface

1 D = direct solar insolation on nny surface

2s = sky-sc:tttered solar i~lsolation on a n y

I R = reflected solar radiatior~ reaching : L I I ~

surface

I = total solar i11~01:~tion 011 ~ L I I Y S I I ~ ~ : L C ~ =

I D

+

I s

S = total solar irr:~diation on nny surface = ID

+

1 s

+

I n

P = reflectivity of IL surface

a further subscript on a n y of the n t ~ o v e (1ua11tit1es identi- fies the surface

a = at,n~ospheric c\tinction coefficient

C,, = polynomial coefficients for absorption and transmission factors of windows

DIVI = direct 11ornla1 i n s o l a t i o ~ ~ a t ground level

F,.., = the geometric view factor between sur-

faces i and j; i.c the fr:ictio~i of (,he radin- ti011 fronl surf:~ce i t h a t falls on j ( 1 , ) ~ ~ ~

F(cos 8 ) = - , a function of incident angle (lJrlor.

I* = apparent solar constant

11i = secant of solar zenith a ~ ~ g l e (approsi- mately equal to a i r mass)

0, P , r = the angles between the solar hear11 2nd a

hol.izonta1 axis pointing south, a horizontal axis pointing west and a vertical asis respect,ively; being the zcnitll angle

I , nr, n = direction cosines of the normal t o a sur-

face referred to t h e samc axis system a s used for solar angles

9 = angle of i~lcitlence of solar \ ) e u ~ n 011 a surface c:os e = 1 c:os a

+

m cos p

+

n cos -j II. = latitude angle

6 = solar declinatioll angle

h = solar hour angle

it z = solar a z i r ~ ~ u t h angle

tl = wall solar : ~ z i ~ n ~ r t h

(0 = profile angle

superscript' = indicates value nt time of maxirnurn insolation

solar radiation fallillg on horizontal surfaces a t about 20 stations across Canada. These values arc the sum of the radiation coming directly from thc sun and thc radiation reaching the surface of the earth after being scattered by the atmosphere. These data alone cannot be used to calculate thc irisolation on si~rfaces that are not horizontal; they must be complenlentcd by a simul- taneous measurement of the direct solar beam. As the direct normal insolation, DiVI, is measured a t only one station in Canada, the Sational Radiation Centrc a t Scarborough, Ontario, orily the Scarborough records are of use in this study. Thcsc direct l~ornlal mclasure-

mcnts n7crc begun in ,June I960 and havc bee11 availablc, sincc that timc (with a gap of a few months in 1961 when the pyrheliometer was damaged by lightnir~g). T h e published values are averages over a one-hour period, ending on the hour in local apparent time. I n the corre- lation that follows it is assumed that thesc hourly averagcs arc the samc as thc instantaneous values a t the mid-point of thc averaging pcriod.

About 20 of the clearest days in the period between June 1960 and June 1963 \vcl.c sclected for analysis. Thc criteria for selection xrcrc: ( I ) a high value a t mid- day, (2) sylnlnctrical valuc for thc moniing a ~ d after- noon, and (3) low values of the sky radiation as mcas- urcd by a shaded pyranomcter. The days that mere selected represent all seasons, but there arc morc sam- ples per month for summer and autumn than for winter and spring. This may be a result of using only a three- year period or it may indicate that the frequency of clear days is higher in summer and autumn.

For each of the sclccted days thc zenith angle* was calculated for cvcry hour on thc half hour (local apparent time). The logarithm of I>hTI was plotted vs the secant of thc zenith angle (i.c. air mass). T h e curves in Fig. 2 are typical of these plots. The data points fall quitc close t o a straight linc so that D N I can be repre- sented by

where log I* is t h e ordinate of the intersection of the line through the d a t a points with the a s i s d l = 0. Ttic value of a is found from the e x p r e s s i o ~ ~

a = ln(1*/13)/3

where log In is the ortlill:~tc? of the Line t k ~ r o r ~ g k ~ I,he c1;~tn points : ~ t d l = 3.

Moon's standard curve is shon.11 on Fig. 2 for comparison. It is lowcr than ally of the days sclccted for this study.

Tllc values of a vary during thc year, as is shown ill

lpig. 3. 111 winter they arc only half the summer valuc. The lower the value of a the higher D N I is for any spccified air mass; thus, design values of D N I should be based on the minimum values of a that can occur a t any date. X period of observation of only threc years is too short t o establish a minimum for each month on any statistical basis, but the curvc drawn through the points on Fig. 3 is a suggested basis for calculating solar irradiation on very clcar days. It may nced t o be modi- fied slightly when morc data arc availablc.

Short-\Yrave Katliation from the Sky

Thc diffuse or sky-scattercd short-wave radiation iucident on a horizontal si~rfacc was deduccd fro111 the published valuc of D N I and I H a r .

.

The ratio IH/DNI was plotted vs cos 7 _{for all the days iticludcd ill this}

* The angles used in this piLper arc d(,finetl on tho tliagranl in Fig. 1.

(6)

The first terln on the rigl1t side of this ecluatioIl is the Fro. 4-V:~riatiol) of IH/III\'I vs cos 7 .

Vol. 9 , No. 2, 1965 83

component of I, that is due to the direct solar beam, 20

since 7 _{is the incident angle of the solar beam on a}

horizontal surface; and the second term is the sky

conlponent. The records for very clear days show that 15

the sky-scattering is directly proportional to DNI and that the proportionality constant B can be related to

the atnzospheric extirlctioil coefficient a. The values of B 10 B are plotted in Fig. 5 vs the corresponding values of a.

This sinlply indicates that the atmospheric extinction

coefficient increases when there is an increase in scatter- 05

ing by the atmosphere. The spread of the data points in Fig. 5 is partly the result of errors in the graphical

determination of B, hut it also reflects the fact that a o

I I I I x - - 1 7 x

i'

- - x

,/

x x x/x - - x x I I I I

depends on absorption of radiation by C 0 2 aild H:O in 05 10 a 15 20 25

(7)

T A B L E 2-PC)LTSOhII.lL (:OISFI'ICII*:STS IpOli I ~ I ~ S O R I " ~ ' I O N -\ST> TILANShIISSlON I?r\C:'l'Ol:S 01;' \VISDOjjrS

-- - -

I Ordinary \\'indo~r Glass I-Ient-.lbsorbing Pl;ile 1

I

Double

Glnzilig Ordinary \Vindow Glass ! - I-.

i

I o r i ' I ' L I ~ S I ~ ~ S S ~ O I i : l b s o r p l i o ~ ~ I'ransn~ission

1

'I'ransmission Co 0.01154 0.00885 0.01406 -0.00835 0.01407

1

0.00228 -0.00401 C,

I

0 7 i ( i i 4

1

-2.71235 i 4.15058 0 .!)27G(j

1

1 .0li22G

!

0 .:i-l550 0.74050 C?

1

-3.94(i57 -0.62062 -15.062i9 2.15721

/

-,5.50131 - 1 . 1!)008 7 .20350 (23 , 8.57881 . -7 .07320 27 .lB.L02 -S.71420 i 12.15034 i 2 i -20.11763 C., 1 -S.381:15 0 . 7509.5 - 23.8851s !) ,87152 i - 11 . 78092 j -2.OZS7 10 . (j882-1. (I,

1

3.01188

1

- 8 0 2 2 S.O:IlBO -:1.73328 -1 .20070 / 0.72370 - (i .74585 1)iSfusc vnlucs

j

0.0544

1

0.7000

1

0.4641 1 0 i O.O(j00 ! 0.0451

- ~ -p

1

0.6707 X o m s : 1 ) HcnL-:ibsorbi~rg 1)I:~tc h:1s :I t,r:~~rs~~lissiorr for 11orrl1:iI ~ I I C ~ ( ~ C I ~ C C = 50 ~ C ~ C C I I L .

C"

c

c.

c., c,

2) Difl.llsc vnlllcs = 2 -

+

'1

+

2

+

2

+

-

+

-

2 3 4 5 6 7

This iiicthod of arriving a t the sky-scattcrcd short- dcsign valucs of these data for the soulhcrn par1 of wave radiation inciderlt 011 a horizo~ital surfacc iucludcs Canada arc givcu in Figs. 3 and 5 .

tlie radiation fronl all the sky except the vcry small solid angle that comprises the ficld of vic~v of thc pyr- hcliomcter. The for~vard-scattered radiatioil in this region of the sky around the suu is included with thc dircct solar beam. The values for the sky-scattcretl coinpol~ci~t evaluated in this ~ v a y are in general a d . 0 1 cc-

nlent with those mcasured by a shadcd pyranometci.. Thc published valucs of thc sl~adcd pyranomctcr rcsults have bee11 adjusted, ho~vever, to conlpensatc for t11c seg~nent of the sky obscured by thc shade ring. This cor- rection requires so111c assuii~ption about thc variation in intensity of the scathercd radiation ovcr the sky. No such assu~nption is involvcd in the valucs for thc scat- tcred radiation given by t,he method used i11 this analysis.

Thrclkeld4 has shown that t hc difrusc short~vavc radiation fro111 the sky incidcut on a vcrtical surfacc is related to the conconlitant value for a horizontal surfacc and the ariglc of iiicidc~icc of the clirccl bca~ll ou t 1112 vcrtical s~irfacc.

Combinilrg this wit11 l l ~ e obscrvalioir t1i:~L

I.s.~t~,~ = 13 X D.\;r

gives I R , v l r = I)I\~I x 13 x ~ ~ ( c o s e )

wllere P ( c o s 0) is the crrlpirical lullctiol~ giveu ill Tl~relkclcl's paper. I t c:tu be well reprcsel~tetl by tllc fr~llowil~g cspressiolr. ~vheli

:LIICI ~ 1 1 ~ 1 1 cos e < -0.2

I ~ ( c o s ~ ) = 0.45

Tlle ti)lal i ~ r s o l a l , i o ~ ~ , I, 011 a vcrtic:d surfacc is give11 by

I = I*e-a'~"' ~ ( C O S e -1 13 X P ( c o s e l )

I t is shown in the scctio~i dealing ~ v i t h the ec~uations for solar posit ion that the valucs of cos 0 and cos y call be calculated cluitc casily, and that thc irisolatioii on ally ~ ~ r a l l can bc deterniined by tlic above cquatioii when the values of I * , a a i d B are specified. Tentative

Irradiation of' Buildi~lg SurFac:cs

Thc surfacc of a buildiilg can also rccci\lc sigilificailt amounts of solar radiation h y rcflcction honl adjacent surfaces.

If Llie surfaces rotlcct, tlil'i'usely 1,lle t,ot,al irr:~cliaLio~l, S, f : ~ l l i ~ ~ g on :LIIY s ~ l r f : ~ c c C:LII b e esprcsscd as

As /L ,PI-,* = il?P?+l eLc. tlre :il.e:Ls C : L I ~ Ilc dividetl out; of t,llesc erlual,io~rs Lo give

Sl = I1

+

prFl~.zSa

+

paFl-.as:<

+

. . .

S,y = pLF,y..lS~

+

p?F,v.,?S?

+

p,F,v..:;S:,

+

. . .

+

I, Strictly speaking, this sct of equations should be solved sin~ulta~icously to yicld the valucs of S whcn t h c valucs of I and thc gcomctric vie~v factors are k n o ~ v i ~ . It is sufficiently accuratc for inost practical purposcs, llon-cvci., to use thc valucs of t!~c illsolation I ll~stcacl of thc corrcsponding valucs oC 5' in thc cxprcssions on the right sidc of thesc ccluatioi~s.

Equa~ions f'or Posi~ion of ~ h c Sun

T11c angles used to describe the dircct,ion of t,he solar bean1 rclat ive to a building surfacc are s1101vn in Fig. 1. The basic coordinatc axcs arc lilirs poiiitil~g sout,11, west aiid to t11c xciiith. The incidcut angle, 0 , is the angle _- bctnree~i tlic solar beain alicl a line perpe~idicular to the wall. Thc value of cos I9 call bc d c t e r m i ~ ~ c d cluite readily froni the follonring cxprcssions :

cos 0 = 1 eos a

+

111 cos p

+

rL cos y

Thc dircdion cosiucs of tlic solar bean1 arc furlctior~s of thc lat itude

I),

the hour angle Iz, and the date (i.e. t11c value of the solar declination, 6 ) .

cos 7' = sill $ X sir1 6

+

cos $ X cos 6 X aos h.

c:os p = cos 8 X sill h

cos cY =

+

(1 - (:I)S~ p - C U S S -,.ji

cos a is posit.ive r n h e ~ ~ cos h tall 8/tan $.

(8)

The solar declinatioil varies fro111 +2:3.5 degrees a t the suinnlcr solstice, t h ~ ~ o u g h zcro a t the cquinos, to

-23.3 a t the winter solstice. The values at intcrnlcdiatc

dates arc given in Tablc 1. The solar azinluth angle, Az, is related to the dircctioil cosines of the solar beam by the sinlplc csprc~ssion

the wall solar aziiiiuth, 7, ~vllich is the difFcrei1ce be- t~vcen tlic ~ r a l l aziillutll alld tlic solar azimuth, is given by

ros 7 = cos O/sin 7.

Area of' Glass Receivi~lg Direct Src~~light Tlle area of a nriiido~v that is shaded fro111 direct solar radiation by the n~indow frame or by an external shading fin can be conlputed quite simply. The width of the shadow cast by a projection froill the edge of a ~vindo~v is L X Ian 7, nrhcrc L is the perpeudicular distance froin the plane of the glass to the outermost edge of the projection. Tan 7 is related to the incident and zenith angles by the equatioii

(1

-

cos2 7 - cos?o)5

tan 7 =

cos 0

Whcn cos 0 is negative the surface reccives no direct sunshine. For a windo~v with identical shades a t both sides, the sign of the square root is unimportant, since the change in sign indicates only that the shado~v has changed froin the left to the right side of the ~vindolv.

The relevant solar angle for a horizontal projection froin the lop of a window is the profile angle p. It is related to the incident and zenith angles by the sinzplc

t a n rp = cos ?/cos 0

The height of thc shadon- cast by a horizonlal projcc- tioii extending a distance I, fro111 the plane of the window is L X tan p. The sunlit area of a windolv is, therefore, (hcight - L tan p) X (wicltll - L tan 7). Translr~ission and Absorption Factors for Glass

Thc absorptivity and transn~issivity of a miiido~v depend 011 the Lypc and thickness of the glass; on

~vhcther it is single or clouble glazed; on the angle t h a t the incident bean1 of light makes with the 11ormal to t,hc surface; aiid on the degree of polarizat.ion of the inci- clent beam. Values of absorptivity aiid transmissivit;y have been compu(,cd and tabulated as fuiictioiis of incident angle for single- and double-glazed ~vindoms by llitalas and S t e p h e n ~ o n . ~ These data are fine for hand comput,ation, but are not well suited for inclusioll in a computer program. Ipor this purpose a polynonlial ill cos 0 has been fit'tcd t'o 1,hc data,

6

i.e. ~ ' ( 8 ) =

c

c,,

x eosfh e

71=O

TAI3LIC l-S01~.lI< II15CLIX.lTIOS 1.S (',lLL~:SD.+lK DATE

-

Dcclinat~on (d c g ~ c e \ )

I

Dates (from 1964 \aut!cal Almanac)

23.5

1

21 J u n r 20 24 July 20 M:iy 15 12 August 1 M a y 10

I

2 i A u g u s t 16 April 5 10 Scptcrnhcr 2 April 0

1

21 Scptcrnhcr 21 March -5 (i Octobcr 8 Marc11 - 10 19 0ctobc.r 24 Fchru:~r>- - 15 I :$ Novcmbcr (3 l ~ c l ~ r u i ~ r y - 20

1

22 N o v c ~ n h e r 22 Jariu:~ry -23.5 23 Ilcccnibrr I

TTalucs of the cocfIicien(s for these polynonlials \\rill be published shortly by the Xational Research Council. Table 2 presents the coeficients for a single sheet of ordinary niindolv glass, a single sheet of heat absorbing glass (that transmits about 50 percent a t norinal inci- dence) and a double window wit 11 both panes of ordinary glass.

Cosine 0 has been used as the argunlent for thcsc polynonlials for two reasons: il is evaluated directly fro111 the direction cosines of the solar bean1 and the liornld to the surface, so that no inverse trigononletric function needs to be evaluated; and the integration t obtain the values for diffusely inciderit radiation is par- ticularly sinlple to evaluate:

T i 2

CII

Y7,,ff,,,. = ?'(el sin 28 08 = 2 -

n=o n

+

2

The values for diffusc radiation, obtaincd from the coeficicnts in this way, arc includcd in Tablc 2. 3Iasimunl Insolation o n a Vertical Surface

The incident angle of the solar heam on any vertical surface can bc expressed as

cos 0 = cos 7 X sin r

Thus, the direct insolatioll on a vertical surface is

This is illaxinluill ~ v l ~ c n 7 is zcro aiid -y satisfies the relationship

The values of the angles that obtain when the insolation is maximuin are designated by a priille superscript. Values of 7' corresponding to the values of a that occur on very clear days a t Scarborough are shon~ii ill Fig. 6. T h e range is only fro111 60 degrees in suiumer to about 67 degrees in wint cr.

The tinle and date nrhcn the illasiilluill illsolation can occur are functions of latitude and orient ation.

cos 6' = sin i l z w , ~ ~ X sin

Conlbinirig this wit 11 the equat,ion for cos -y gives

eos ; ' = sir1 $ s i l ~ 6'

+

cos $(cos' 6' - cos? P')h

~vhich can t)c solved for the declination 6'. T h c c o r ~ c - sponding hour angle is given by sin h' = cos @'/cos 6'. Vol. 9, No. 2 , 1965

(9)