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Tables of solar altitude, azimuth, intensity and heat gain factors for
latitudes from 43 to 55 degrees North
TI{1
N2t-t2
n o . 2 l + 3
e . 2
StDG
J--)
;i, ili,,r;:.ii:l :i,.i
CANADA
DIVISION OI' BUILDING RESEAR.CH
TABLES oF SOLAR ALTITUDE'
AZIMUTH,
INTENSITY AhID
HEAT GAIN F ACTORS FOR LATITUDES FROM 43 to 55
DEGREES NORTH
by
D. G. Stephenson
A T{.fiLYU gD
Technical Paper No, 243
of the
Division of Building Research
Ottawa
A p r i l 1 9 6 7
TABLES OF SOLAR ALTITUDE' AZIMUTH' INTENSITY AND
HEAT GAIN FACTORS FOR LATITUDES FROM 43 tO 55
DEGREES NORTH
b y
Do G. Stephenson
Solar heat usually causes rnuch of the cooling load for
cornrnercial
and residential
buildings.
An accurate estirnate of
cooling load rnust, therefore,
take account of the hourly and seasonal
variations
of solar bearn intensity and of the angle of incidence of the
bearn at the outer surfaces of a building.
The tables in this report
give the solar data needed for designing an air conditioning systern
by either the traditional
hand-calculating
procedures
or digital
corn-puter techniques.
They have been prepared in exactly the sarne way
as the tables in the ASHRAE Handbook of Fundarnentals (196?), but
are for the latitudes of particular
relevance in Canada and for srnall
enough incrernents of latitude that interpolation
is unnecessarye
The
data are in the engineering units cornrnonly used in North America
, 2 ?
(ntu/tt'hr);
and in the rnternational systern (watt/rnetre"), which is
corning into use throughout the world.
Solar Altitude and Azirnuth Angles
The position of the sun in the sky can be expressed most
conveniently by two angular co-ordinatesl
the solar altitude above
the horizon,
and the
a n g l e s a r e r e l a t e d t o
sin Alt
s i n A z
w h e r e
h
L
6
solar azirnuth rneasured frorn South. These
the time,
date and latitude by:
= c o s L c o s 6 c o s h t s i n L s i n 6
= cos 6 sin ly'cos Alt
- 0o 25 fnurnber of rninutes from solar noonl
= latitude
= solar declination
Values of solar altitude and azirnuth angles have been
cornputed for the 21st of. each rnonth and are tabulated in the first
two columns of the tables.
Intensity of Direct Solar Bearn
The intensity of the direct solar bearn, IO*, depends on
of the atrnosphere and the length of the solar bearnls path
The value of IDN
"t
the surface of the earth on a clear
the clarity
through it"
d a y i s w e l l
r e p r e s e n t e d f o r s o l a r a l t i t u d e s g r e a t e r t h a n l 5 d e g r e e s b y
rotti = e/"8/sin
Alt
where A and B are functions of the date that take account of the
seasonal variation of earth-sun distance and the dust, ozone and
water vapour content of the atrnosphet"(t).
values of ro* cornputed
with this forrnula are given in the third colurnn of the tables.
The
values of the A and B parameters
that were used are given in Table I.
They were selected so that the resulting values of IO* would be in
close agreernent with the values given by Threlkeld and Jo"d"rri2) rot
average cloudless days at different tirnes of the year.
For very clear
atrnospheres,
the value of to*
"un
be as rnuch as zo per cent higher
than the values given in these tables.
The local atrnospheric clearness
3
-can be taken into account by rnultiplying
the values of ro*
and the
solar heat gain factors by a clearness Nurnber, which can vary frorn
a b o u t 0 . 8 t o 1 . 2 .
R e f e r e n c e 2 g i v e s c l e a r n e s s n u r n b e r s for the
United States and the rnost southerly parts of Canada1 Further work
i s r e q u i r e d t o p r o v i d e c l e a r n e s s nurnbers for all parts of canada.
u n t i l t h e s e a r e a v a i l a b l e , d e s i g n e r s rnust use their discretion:
a
v a l u e o f 1 . 2 b e i n g a p p r o p r i a t e f o r v e r y clear regions, 1.0 for average
a n d 0 . 8 f o r s r n o g g y i n d u s t r i a l a r e a s .
S o l a r H e a t G a i n F a c t o r s
The solar heat gain factors are the instantaneous rate of
solar heat transfer through unit area of unshaded double-strength
sheet glass in sorne specific situation.
This is the solar power
transmitted
through the glass plus a fraction of the power absorbed
by the glass.
The rate of heat transfer through other fenestrations
in a sirnilar situation can be found by rnultiplying the tabulated value
of the S" H. G. F. by a constant called the shading coefficient for the
particular
fenestration.
Shading coefficients for rnany of the
corn-rnonly used fenestrations
are given in the ASHRAE Handbook of
Fundarnentals (1967!-, Chapter ?8, as are tables of solar heat gain
f a c t o r s .
E q u a t i o n s a n d D a t a U s e d t o C o r n p u t e S . H . G " F .
The solar radiation falling on a window consists of:
( a ) d i r e c t r a y s f r o r n t h e s u n ;
(b) scattered radiation frorn the sky;
(.)
radiation reflected frorn the ground and surrounding
o b j e c t s .
to the direct rays frorn the sun depends or IDN and the angle of
incidence,
g, between the bearn and a perpendicular to the window
s u r f a c e "
ro
=
I D N . c o s 0
The cosine of the incident angle is related to the solar
altitude and azirnuth and the surface orientation
by:
c o s 0 = cos Alt cos y sin X + sin Alt cos f,
w h e r e
slope angle of the surface
measured frorn horizontal
difference between solar
azirnuth and the azirnuth of a
normal to the surface.
W'hen the surface is horizontalrl
-
0 degrees
t
so that
and for a vertical
surface
so that
cos 0" = sin Alt
t
= 9 0 d e g r e e s
c o s 0,, = cos Alt . cos to
= ,x5 t. .o"i o
J ! 3 O
J
5 '
= .X a. .o"J e
J = O
J
sky or ground
_
t .
- 2 - 5 J
-
e
i 1 o
i + 2
r f Y
bearn radiationo
i . € .
I O
- 0 "
The transrnission
and absorption factors for window
glasses can be approxirnated by a polynornial with cos 0 as the orgur
rnent.
For the direct bearn the factors are:
T r a n s r n i s s i o n 1 ^
' t )
Absorption
aD
and for diffuse radiation frorn the
f d
5
-o d =
* 4 .
_ )
J
j ! "
j + z
values of the coefficients
t. and a. for sorne cornlnon tyPes of
J J
glass have been cornputed frorn the data in Ref , 3 and are given in
Appendix A.
The intensity of the diffuse radiation frorn a cloudless
sky that falls on a horizontal
surface can be approxirnated
by:
r d H = c ' r D N
where c depends on the dust content of the atrnosphere and varies
frorn sumrrrer to winter.
The values of C used for the cornputation
o f t h e s e t a b l e s a r e g i v e n i n T a b l e I .
I f c o s Q
> - 0 . 2
0 . 5 5 + 0 . 4 3 7 c o s e t 0 " 3 1 3 . o " 2 g
o . 4 5 ,
The diffuse radiation
frorn the sky that is incident on a
vertical
surface is the product of Ia"
and Y:.
"r
ernpirical function
o f c o s Q that has been developed by Threlt
" t a ( 4 ) .
Y =
Y =
o t h e r w i s e
The radiation
a v e r t i c a l s u r f a c e i s :
reflected frorn the ground that is incident on
I
t o * ro*
t " *
s h a l t )
where
p_ is the reflectivity
' g
of the ground.
Thus the total diffuse radiation incident on a vertical surface is:
,u = rDN
The solar heat gain factor is given by:
S o H o G . F o = I D ( r o + N i . o D ) * I a ( " a * N . " od)
where N. is the fraction of the absorbed radiation that is transferred
I
to the inside of the building.
These tables have been cornputed using
N . = 0 o 3 .
I
The value of pg has been taken as 0o 2, which is an
appropriate
average for ground that has no snow cover, but when the
ground is snow covered the reflection is greatly increasedo
The
presence of snow can be allowed for by increasing the solar heat gain
factors for vertical windows by about 20 pe.r cent for the winter rnonths.
Cooling Load Calculations
The solar heat gain factors rnultiplied by fenestration
area
and shading coefficient
give the instantaneous
rate of heat gain of a
roorn frorn solar energy transfer through the fenestration.
The roorn
rnay also be gaining energy by heat conduction through the fenestration
and through the opaque parts of the wall and roof, and frorn the energy
being supplied to the lights and equiprnent located in the roorrr.
The
surn of all these cornponents is the total instantaneous
heat gain for the
roorrrc
The cooling load, i. e. the rate at which heat rnust be rernoved
frorn the air in the roorn to keep the air ternperattlre at a specified
value, can be quite different frorn the total instantaneous heat gain at
the sarne tirne.
In particular,
the rnaxirnum
cooling load is significantly
srnaller than the rnaxirnurn rate of heat gain if the floor and internal
partitions
have sorrre heat storage capacity.
The sirnplest rnethod of calculating cooling load is to find,
first,
the instantaneous heat gain and then apply a heat storage factor
7
-that has a value less than oneo The heat storage factor depends, on the
h e a t s t o r a g e c a P a c i t y o f t h e s t r u c t u r e o
R e s e a r c h i s u n d e r w a y t o d e t e r
-rnine appropriate
heat storage factors for different types of buildings,
but until these results are available designers must rely on experience"
The solar heat gain factors that were given in the ASHRAE
Guides prior to 1967 were, in general, about z0 per cent smaller than
the values in the 1967 Handbook of Fundarnentals.
There is no doubt
that the new values are more accurate than the previous ones. Cooling
loads were not usually under-estirnated
using the old values because it
has been cornrrron practice to assurne a heat storage factor of one, i. e.
that the cooling load was equal to the instantaneous
rate of heat gain"
This assurnption will give too-large cooling loads when the heat gains
a r e c a l c u l a t e d w i t h t h e s e n e w t a b l e s .
A h e a t s t o r a g e f a c t o r o f 0 . 9 0 i s
applicable to lightweight
structure"(Ul
"
factor 0.25 is reasonable for
the type of buildings where the old Guide data have proved satisfactory(4),
and a factor of 0.60 or less rnight be appropriate for heavy structures
with a lot of solar h""t g"in(6).
Sol-Air
Ternperaturg
The calculation of heat transfer between the outside
environ-rnent and the outer surface of a wall or roof is sirnplified by using the
sol-air ternperature
concept.
The S. A. T. is that ternperature
of the
outdoor air that, in the absence of all radiation exchanges, would give
the sarne rate of heat entry into the surface as would exist with the actual
cornbination
of incident solar radiation,
radiant energy exchange with the
sky and other outdoor surroundings,
and convective
heat exchange with
the outdoor air"
-f-s . A " T . - t
absorptance of surface for solar radiation
emittance of the surface
0
CI
h =
convection
t
alr
AR
total solar radiation incident on the surface = ro * ru
coefficient of heat transfer bv radiation and
at the surface
outdoor air ternpe rature
the difference between the long-rvave radiation incident
on the surface frorn the sky and surroundings and the
radiation ernitted by a block body at outdoor air ternperature.
For horizontal
surfaces that receive long-.wave radiation
only frorn a cloudless sky, an appropriate value of 6R is
7 . 2
about 6O w/rn- (20 Bt{ft"
hr).
For vertical
surfaces it
is cornrnon to assurne that 6R = 0.
The value of I can be approxirnated
frorn the data in these
t a b l e s i v i z
f = 1 . 1 5 S . H . G . F .
REtr'ERENCES
1. stephenson, D.G.,
Equations for solar heat gain through windows.
S o l a r E n e r g y , V o l " I X , N o . 2 , 1 9 6 5 , p . 8 l o N R C 8 5 2 7 .
Zn Threlkeld,
J. L. and R. C. Jordan, Direct solar radiation available
o n c l e a r d a y s .
T r a n s a c t i o n s ,
A S H R A E , V o l . 6 4 , L 9 5 8 , p . 4 5 .
Mitalas,
G"P. and D.G. stephenson, Absorption and transrnission
of therrnal radiation by single and double glazed windowso Decernber
L 9 6 2 ,
3 0 p o I f i g u r e .
N R C 7 L 0 4 .
T h r e l k e l d ,
J . L " ,
S o l a r i r r a d i a t i o n
o f s u r f a c e s o n c l e a r d a y s "
T r a n s a c t i o n s ,
A S H R A E , V o I . 6 9 , 1 9 6 3 , p . 2 4 "
3 "
9
-5. Handbook of air conditioning systern design. McGraw-Hill, !g6s.
6. Stephenson, DoG. and G.P. Mitalae, An analog evaluation of
rnethods for controlling solar heat gain through windows. ASHRAE
Journ4l, VoI. 4, No. 2, February 1962, p. 4l-46. NRC 6560.
Table I
Data for Calculation of Solar Radiation Intensity
Date
J a n 2 l
F e b Z I
M a r 2 1
Apt ?.L
M a y 2 1
June 2l
J u l y 2 l
Aug Zl
S e p t 2 1
O c t Z l
N o v 2 l
D e c 2 l
6
D e g r e e s
- ? 0 " O
- 1 0 . B
0 . 0
X , l " 6
2 0 . A
? 3 . 4 5
z o "
6
r z . 3
0 . 0
- 1 0 . 5
- 1 9 . 8
- 2 3 " 4 5
Btu
?
hr ft'
3 9 0
3 8 5
3 7 6
360
3 5 0
345
344
3 5 1
3 6 5
378
387
39r
z
tn
Iz30
lzl5
1 1 8 6
1 1 3 6
I 1 0 4
I 0 8 8
1 0 8 5
I 1 0 7
r 1 5 1
L T 9 Z
l z z l
I Z 3 3
Air
B
M a s s
0 . I 4 2 .
o . 1 4 4
0 . 1 5 6
0 . 1 8 0
0 . l g 6
0 " 2 0 5
o " 2 0 7
0 . 2 0 1
o . 1 7 7
0 . 1 6 0
0 . 1 4 9
o . . l 4 z
A
w
Dirnensionless
0 . 0 5 8
0 . 0 6 0
0 . 0 7 1
0 " 0 9 7
0 . 1 2 1
o " 1 3 4
0 . 1 3 6
0 . L 2 2
o . o 9 z
0 . 0 7 3
0 " 0 6 3
0 . 0 5 7
I t
-S O L A R P O -S I T I f , | N A N I ) I N I F \ 5 I f V A N O S I ] L A I H [ A T G A I N F A C ' O R 5 F O R , 4 ' D E G . N f J R I H L A T I I U D T D A I E I I I I E S O L A R P ( I S I I I I J A I D I R . N O R I 4 A L A A A L T " A Z I M U I H W A I T S / 5 0 . H . A I O R T H N F - . - - S T J I A R H E A I G A I N F A C I O R S i H A r T S / S Q . i I .E I S I S E S O U T I { S H W E S I N I r H O R .r t t i E
p x
J A N 2 1
8
l 0
l l
t 2
l o . I 8 1 0 2 6 5 ! 2 0 l 8 l 4 s o 6 6 2 4 1 1 1 1 2 1 4 0 6 5 9 1 8 4 6 9 0 1 6 4 6 0 5 6 t 7 9 9 1 2 9 0 2 6 9 6 2 4 6 1 t l t ? 4 6 5 4 5 6 t 7 l 1 2 9 4 8 6 l 6 8 7 l 4 2 F E B 2 1 1 I l o I t t 2 6 . 1 t 4 . 6 2 1 . ? 2 5 . 5 2 r . o J . t | 1 . 4 2 2 . ? l 2 9 . 6 ) 4 . 5 J 6 . 2 1 0 . 9 ? . 1 . 4 r l . l - r 9 . 3 4 1 - O f . 9 t 8 . 8 2 9 . 1 4 0 . 1 4 9 . 3 5 6 . 0 5 8 . 6 t . 2 1 1 . 5 2 . . . t ) 5 . 2 4 6 . O 5 5 . 9 6 1 . I 6 t . o 4 t . 4 ) j 9 7 0 1 l 2t 1 1 6 1 t 1 4 8 5 1 o . 0 8 9 9 H A L F O A Y T O T A L S l l l l 2 7 4 t 2 t 2 1 3 5 t 4 6 4 6 2 5 4 2 5 q 5 4 3 3 6 I 6 0 5 6 1 6 4 t 2 l f 2 t 7 l 9 3 6 4 0 1 5 4 6 5 t 5 6 1 9 4 4 l 1 4 2 6 l 6 2 6 A , l l B 6 4 8 . 7 B8r' l o ? 8 3 1 ) q . \ 9 0 7 4 0 . 2 2 l . c 0 . 02 t . o
o . 0
1 0 4 . 9 9 5 . 3 8 5 . 1 1 0 7 . 6 I 7 6 9 2 6 9 5 0 9 5 1 3 2 9 4 8 6 l 6 8 7 6 2 4 4 2 4 4 7 f l 9 l 3 4 4 l 6 4 5 6 6 8 2 9 4 t 0 l l l l 4 5 8 I t 4 6 0 ? 9 9 4 l 0 6 l l l l 2 q 5 4 46 1 . 0
6 5 1
5 4 2 9 t a f 2 5 l5 9 1 2 A 1 2 9 5 6 5 1 6 2 4 9 9 5 l 4 1 A 1 7 B 6 5 I t ? 7 z t l t 6 9 7 f 4 5 a 4 0 1 6 5 9 7 r 1 7 5 4 7 l A 5 ? 3 1 6 0 ? 5 1 6 8 t I 4 9 5 4 t t ? 0 2 4 ( r r t 6 6 , 2 4 t 4 l 6 3 0 7 5 0 4 2 4 6 6 4 1 5 1 a 5 5 6 4 9 4 2 4 2 6 4 1 6 5 1 2 . 6 5 9 1 4 8 0 6 8 0 6 8 0 2 6 1 9 ) 4 9 f 2 2 9 t 1 2 1 t o t t 8 0 t ? t 6 6 3 5 4 6 0 5 2 4 5 f o { 6 0 1 t 4 6 6 6 6 3 5 6 6 0 2 9 t 8 2 4 t 7 6 1 2 4 1 9 t 0 0 2 5 3 5 t 4 5 0 1 1 8 , I I I t t 2 5 2 9 ) 7 2 1 0 5 7 t 2 t 5 l 6 8 s 6 5 8 3 0 l 0 I I 4 1 0 2 4 0 t 4 t 4 6 6 4 4 2 1 6 4 6 0 6 9 4 5 t 6 I 0 0 7 9 6 l B 5 1 1 1 9 5 9 4 4 6 5 5 4 0 1 0 2 I I 0 2 5 4 4 4 1 t 1 6 L 4 4 1 2 4 2 8 8 4 O 2 2 8 8 t 2 l 5 z e o a 1 2 7 2 6 \ 7 1 2 4 ) 9 9 9 5 0 I 2 ) a 4 t 4 l 6 6 0 4 0 1 6 9 6 6 6 8 0 4 9 9 9 5 A q 6 0 5 5 1 t t 6 t 9 9 4 5 5 4 9 6 2 5 5 I 1 4 2 5 2 t 9 8 a 2 t 1 3 6 1 2 9 2 5 4 1 4 7 2 5 4 ) t 2 l 2 7 3 2 t l ? B 6 5 1 l 8 l , 2 2 4 5 A 2 1 2 3 6 1 6 6 4 8 4 1 4 6 ? 6 2 6 8 0 5 2 ? l o t a 2 6 0 8 5 5 9 1 9 0 9 1 4 5 9 5 2 1 2 9 t I I 3 2 5 4 i t t t 6 5 | 4 5 L 2 1 2 8 2 1 9 I 2 a 2 - f l 7 l ? - a 7 t t 2 4 9 6 5 6 2 8 1 1 6 6 1 9 l 8 6 0 t 4 3 t 5 5 4 9 6 7 6 5 1 9 t 4 2 7 0 6 1 6 6 t 1 2 1 9 8 7 4 6 f 6 1 0 4 0 2 1 0 6 2 5 t 5 l 7 4 A 2 I 8 2 I | 7 3 6 1 5 0 9 1 6 l 2 9 4 8 l l 1 4 L 6 1 t 6 7 1 4 4 I 3 6 q 6 1 2 5 6 1 2 6 ? : ' 2 . 1 t 5 0 6 0 1 7 l 7 4 ( J 9 7 0 4 6 0 7 0 9 5 4 5 9 9 2 4 0 6 2 1 6 ! t O 2 6 ) 9 1 4 6 7 6 5 9 4 6 f 2 4 1 2 r r i l g 2 2 t t 7 a 2 8 l 7 1 2 2 l 5 1 0 5 6 0 2 _ 7 0 3 l 5 4 8 1 2 9 4 7 1 5 3 t 2 7 7 5 0 6 2 1 1 2 6 2 1 6 6 1 5 7 2 0 : r 3 0 1 9 5 l l 7 5 2 5 3 1 t ? 8 0 1 0 2 5 2 4 1 4 7 i 6 2 5 | 3 o s I 7 5 1 2 -4 -4 0 6 q 6 4 6 5 t 8 1 6 6 1 2 6 6 q 6 1 6 0 l 8 l 6 7 U 1 5 0 1 6 A 6 ? 5 5 1 f A 5 5 5 t t 2 6 6 2 6 3 e 2 4 2 C 8 - | 9 l t 3 2 1 2 4 0 E 4 9 2 5 2 1 I t 5 4 6 7 5 248 4 2 2 5 5 t l 63 lr 6 6 4 I 2 2 6 6 3 9 1 . 9 J 3 8 0 5 4 2 6 6 5 1 4 1 ? 6 9 I 29..9 1l o ?
6
2 8 1
5
4 5 8
4
6 0 7
1
1 2 2
2
7 9 4
I
8 1 8
t 2
1 t l 1
1 7
7
1 3 9
6
3 l l
,
4 A t
4
6 2 6
t
1 7 1
2
9 0 6
I
8 2 4
t 2
, 5 9 1
4 7
l l l
6
z g t
5
4 5 ?
4
6 0 1
3
7 1 5
2
7 8 6
I
8 0 s
1 2
t159
4 2 6
1 9 5
5
1 7 6
4
5 3 t
1
6 5 2
2
7 2 1
I
f s t
t 2
2904
7 1
5
2 4 0
4
4 0 8
a
5a6
2
6 1 5
I
6 4 2
t 2
? 1 9 2
I 4 ? I 2 5 4 3 2 I 2 6 4 3 2 I f t 2 9 4 8 6 l ? l 2 4 2 , 2 4 4 1 6 4 f 7 8 5 8 7 t 4 2 4 5 6 6 a 2 9 4 l 0 t 1 0 4 4 5 6 I 3 4 6 0 1 9 9 4 l 0 6 l t l l l 5 5 4 1 9 4 l 6 6 8'r 9 9 l t o I t 7 l l 9 5 8 4 3 6 6 2 8 2 9 1 I O 9 l l 5 l l 8 5 5 8 t 8 4 9 7 0 8 7 9 9 l 0 6 l 0 9 4 8 1 2 5 5 0 6 8 8 0 8 8 9 l t 5 8 9 4 l 8 4 9 9 I l 0 I l ? t 2 9 5 8 7 2_ 1 6 6 2 a 2 9 1 1 0 9 I l 5 l 2 ? 5 4 , I I 8 4 9 7 0 8 7 9 9 1 0 6 I l ? 4 8 6 5 0 6 8 8 0 8 8 q 7 3 6 0 l B . o o , o 9 4 1 9 5 1 H A L F O A Y I O I A L S 2r 4 a R 2 l
I
I
l 0
t l
l 2
A P A 2 1
6
I 9l 0
l l
t ?
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