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The longwave radiation incident upon inclined surfaces
So/atEnergy,Vol.22,pp.459-d62 © PeqlmonPressLtd..1979. Printedin GreatBritain
003g-092X/"t9/IFa01-O45911t02.00/0
TECHNICAL
NOTE
The iongwave radiation incident upon inclined surfaces R. J. COLE School of Architecture. University of British Columbia, Vancouver, Canada (Received 21 February. 1978: revision accepted 14 August 1978)
I. INTRODUCTION Until approximately 20yr ago. the net Iongwave radiation exchange between building surfaces and their surroundings was considered minimal in comparison to the absorbed solar radiation. Roux[I], and Parmelee and Aubele [2]. however, presented experimental evidence which indicated the fallacy of this supposition. More recent studies comparing the predicted heat flow through elements with measured values[3,4] illustrate that significant errors may be introduced into the calculation of heat flow through building elements if the Iongwave radiation exchange at the outside surface is neglected. Similar arguments can be raised in the evaluation of the performance of solar collector systems. A significant number of relationships are available to predict the atmospheric radiation incident upon a horizontal surface[5] and, as demonstrated by Unsworth and Monteith[6], it is possible to transform these to determine atmospheric radiation incident upon inclined surfaces. However, one of the major limitations has been the determination of the ground component incident upon non-horizontal surfaces. The general assumption that the total Iongwave radiation incident upon vertical surfaces is equal to that of a black-body radiator at screen air temperature. There is no indication as the exact contribution from the ground and the sky vault. This paper presents relationships which describe the Iongwave radiation incident upon inclined surfaces. Using the results of a recent field study[7], in which simultaneous measurements were made of the atmospheric radiation on a horizontal surface and the Iongwave radiation incident upon vertical surfaces, it was possible to ascertain the extent to which the ground radiation differed from that of a black-body radiator. Having established a value of the ground radiation, this could then be combined with the atmospheric component (transformed to the particular tilt angle under consideration) to give the total incident Iongwave radiation.
incident upon a horizontal surface (Wm-2); to is the screen height air temperature (°C), c is the fractional cloud amount. (ii) Vertical surfaces Clear sky. Unsworth and Monteith showed that the normal flux density per unit solid angle, at the zenith angle z is given by R(z)lTr = {a + b . In (u. sec z)}o-T~'
where u is the precipitable water depth of the zenith, a and b are constants having values of 0.70_+0.05 and 0.09_+0.002 respectively during a clear sky. The variation in a and b can be attributed partJy to vertical and partly to horizontal temperature gradients of particulate material (that is, pollution or haze near the ground or water droplets in high cloud). The total atmospheric radiation incident upon a horizontal surface is given by the integration of eqn (4) such that RAo = ~To4(a + b(0.5 + In u)).
R:~, =
crTo4(0.Sa+ b(0.596 + 0,5 In u)).
R/tt. = 0.SRA + 0.~,6b • o'Ta4.
(7)
The value of the atmospheric radiation incident on exposed vertical surfaces during the clear sky condition is given by substituting the value of b (0.09) in eqn (7), such that
RAo = 222 + 4.94t~
(1)
RA¢ = 287 + 6.33to
(2)
RA~, -- 0.5 RAo + 0.0311 o'To~.
(8)
Overcast sky. For the overcast condition Unsworth and Monteith indicate that the value of b will differ from the clear sky value. By assuming that over periods of several weeks the distribution of cloud over the sky is uniform, these workers showed that the mean apparent emissivity of the atmosphere at zenith angle, z, is: E(z. c) = (I - nc)E(z, o)+ n . c
or, more generally, R,~o + (65 + 1.39t.)c
(6)
As a further extension, the value of RA,, can be expressed in terms of the atmospheric radiation incident on a horizontal surface (Ra) and that emitted by a black body radiator at screen air temperature (~rTo4),by combining eqns (5) and (6) such that
(i) Horizontal surfaces In a previous paper, Cole [7] presented relationships giving the incoming atmospheric radiation incident upon a horizontal surface expressed as linear function of screen air temperature and cloud amount, such that:
=
(5)
A similar expression can be established to compute the value of the atmospheric radiation incident upon a vertical surface, such that
2. ATMOSPllEI~CRADIATIONINCIDENTUPON BUILDINGSURFACES
Ra
(4)
(3)
where R,~o is the atmospheric radiation incident upon a horizontal surface during clear sky conditions (Wm-2); RAc is the atmospheric radiation incident upon a horizontal surface during overcast sky conditions (Win-2); RA is the atmospheric radiation
(9)
where, E(z, o) is the apparent emissivity of the cloudless sky at zenith angle, z, and c is the fractional cloudiness. Based upon a long series of measurements in Britain mean values of n for long periods of 3 months were found to range from 0.81 to 0.87, with a mean annual value of 0.84. Therefore, for a
cloudy sky, E(z, c) = a'+ b' In (u - sec z)
459
(10)
460
Technical Note
Where
giving the following results: a'= a +n. c(1-a)
b'= b ( l - n ,
c).
Using these values, Unsworth and Monteith established the general relationship,
Re, = 158+ 2.64to
(clear sky)
(19)
Re, = 155 + 2.38to
(overcast sky)
(20)
/~2 = 162 ÷ 3.15to
(clear sky)
(21)
Re: = 157 + 2.40to
(overcast sky).
(22)
(11)
EA~ =(1 - n . c ) E A o + n " c.
For the present analysis, it was necessary to establish representative instantaneous values of b'. Using the values of atmospheric radiation for clear and overcast conditions as given by eqns (1) and (2) it was possible to establish the dependence of n on screen air temperature. These values are given in Table 1 and can be represented by the linear relationship:
Now, over the temperature range 0-20°C, black-body radiation (~T°4) may be approximated by the linear expression oT~4 = 314 + 5.164to
(23)
and the radiation incident upon a building surfaces inclined at an angle, a, to the horizontal received from the ground is given by:
(12)
n = 0.7067 + 0.00822to.
R a ( a ) = E~n'T/a sin2 (a/2).
(24)
Therefore, the value of the coefficient, b', is given by, b' = b(l - c(0.7067 + 0.00822h))
(13)
where c' is the fractional cloud amount. For the overcast condition, (c = 1), the atmospheric radiation, RA~ = 0.5RAc + 0.346¢To4(0.0264 + 0.00074to).
(14)
Table I. Values of coefficient 'n' as a function of temperature, °C to
f~o
E..+-
n
0 5 10 15 20
0.707 0.725 0.742 0.756 0.768
0.914 0.913 0.946 0.959 0.970
0.706 0.749 0.790 0.831 0.870
3. LONGWAVERADIATION FROM GROUND In addition to the radiation exchange at a horizontal surface, the net radiation balance was measured simultaneously on two exposed vertical building surfaces:
4
(25)
Equation (25), giving the ground radiation incident upon a vertical surface assuming black-body radiation, can now be compared with the measured values for different ground surfaces under clear and overcast skies. Although differences occur, they are within a range of -+ 15 per cent. In view of the experimental error being of a similar order of magnitude to this and the difficulty of being able to characterise the 'ground' radiation (i.e. the extent of the reflected component and its seasonal change), it is felt unnecessary to make any refinements to the black-body assumption. & LONGWAVERADIATIONINCIDENTUPON INCLINED SURFACES The radiation incident upon a surface inclined at an angle, a, to the horizontal (see Fig. i) is a combination of atmospheric and ground radiation, i.e. R(a) = RA(a)+ R o ( a ) .
(26)
It may be shown that the atmospheric radiation incident upon a surface at angle, a, to the horizontal is given by:
Once again linear relationships were between the incident longwave radiation and screen air temperature. Assuming a linear dependence on cloud amount, the relationships describing incident Iongwave radiation for the exposed surfaces are:
Rz = 284 + 5.92to + (20 - 0.20to)C.
I
Ray = ~ aT. = 157 + 2.582t..
Atmospheric component
(i) Facade (l)--Exposed, grass frontage (ii) Facade (2)--Exposed, Tarmac frontage
R~ -- 281 + 5.25t. + (20 + 0.45h)c
Assuming black body radiation, screen air temperature, the ground radiation incident upon a vertical surface is given by:
(15) (16)
RA(a) = ((a' + b'ln u ) K t + bK2) " uTa4.
(27)
Where K~ and K2 are dependent upon a. Again eliminating In u gives RA(a) = RA " KI
+ (K2 -
0.5K,)b' • ~Ta+
= RA" KI + Ksb'" ~To +.
(28) (29)
These relationships (eqns (15) and (16)) give the total longwave radiation incident upon vertical building surfaces with different degrees of exposure. Since the radiation incident upon exposed vertical surfaces (R~ and R2) and the horizontal surface were measured simultaneously, it was possible to determine the ground component incident upon these vertical surfaces by, Re = R~ - RA,..
(17)
Using eqns (8), (14) and (17) the values of the radiation emitted by ground which was incident upon the vertical exposed surfaces (Rat and R<;z) were computed for clear and overcast sky conditions. In order that comparisons could be made with a blackbody radiator these values were correlated against screen air temperature such that, R ~ = I~ + m~to
(18)
Fig. !. Geometric relationships for radiation incident upon inclined surfaces.
Technical Note Where K3 = K 2 - 0.5K~. The values of K~, K~ and K3 for values of slope angle between 0~ and 90" and shown in Table 2. Ground component As discussed in Section 3, the ground component can be adequately computed assuming that the ground radiates as a black-body radiator at screen air temperature. The ground component is thus given by:
R~(a) = c~To4 sin~ (a12).
(30)
Total radiation The total longwave radiation incident upon an inclined surface is given by the addition of eqns (29) and (30). The values of total longwave radiation for both clear and overcast sky conditions as given by these equations are shown in Fig. 2.
5. CONCLUSIONS This paper has presented relationships which describe the Iongwave radiation incident upon inclined external surfaces. The relationships are expressed in terms of cloud cover and screen air temperature are primarily for use in transferring standard weather tapes into input data for mathematical models which predict the performance of solar collector systems. The relationTable 2. Values of K~, K2 and Ks as a function of surface inclination
0 I0 20 30 40 50 60 70 80 %
KI
K2
K3
1.0000 0.9924 0.9698 0.9330 0.8830 0.8214 0.7500 0,6710 0,5868 0.5000
0.5000 0.5184 0.5523 0.5890 0.6214 0.6447 0.6554 0.6514 0.6315 0.5957
0.0000 0.0221 0.0613 0.1225 0,1798 0.2339 0.2803 0.3159 0.3381 0.3457
48o
400
[:
t
~'E:~jO
F'---
. /
0
i
i
I0
20
---~
|
i
i
3O 4O 5O I ncllrlo~oll
Cleor
ships presented here are based on the long term measurement of the radiative beat exchange at building surfaces. The Iongwave radiation incident upon an inclined building surface is given by the sum of the atmospheric (RA(a)) and ground components (Re(a)).
where and
R A ~ = RAKI + b'K3~T~ 4
R~
= ~T~~ sin2 (u/2).
The values of the coefficients are provided in the text. The relationships describing the atmospheric component are transformations of the atmospheric radiation incident upon a horizontal surface. These basic relationships compare favourably with current work derived in other geographical localities (7) and can be considered of universal application. Values of the ground component derived from field measurements were not significantly different from that of a black-body radiator at screen air temperature. Hence the black-body assumption is proposed for the ground component. The Iongwave radiation incident upon inclined building surfaces is almost independent of the inclination angle with overcast skies. However, as would be expected the variation is significant during clear sky conditions. The implications of this can be of considerable significance for those engineers concerned with the design of solar systems or radiative cooling systems. Previously it has been necessary to assume that the collector plate or radiating surface is emitting to the hemispherical sky vault. By virtue of the anisotropic distribution of atmospheric radiation and ground emission, this assumption can lead to errors in excess of 30 W/m 2 during clear sky conditions. NOMENCLATURE a inclination of surface to horizontal c fractional cloud amount EA effective emissivity of atmosphere, i.e. Atmospheric radiation divided by black-body radiation at screen air temperature E(z) effective emissivity of atmosphere at zenith angle, z R~ measured Iongwave radiation incident upon an exposed vertical surface with grass frontage, Win-; R2 measured Iongwave radiation incident upon an exposed vertical surface with tarmac frontage Wm -2 R, Ioogwave radiation incident upon vertical surface, Wm -2 RA~ atmospheric radiation incident upon vertical surface, Wm-2 RA~°~ atmospheric radiation incident upon a surface inclined at an angle, a, to the horizontal, Wm -2 RA atmospheric radiation incident upon an exposed horizontal surface, Wm -2 / ~ , ground radiation incident upon vertical surface, Wm -~" Roqa~ ground radiation incident upon a surface inclined at an angle, a, to horizontal, Wm -2 cr Stefan-Boltzmann's constant, 5.67 × 10-SWm -2 K -4 tA screen height air temperature, °C TA absolute screen height air temperature, K u optical path depth, cm z zenith angle Subscripts o clear sky condition c overcast sky condition.
..
/
461
sky
i
i
i
6O
"tO
8O
9O
Fig. 2. Longwave radiation incident upon an inclined surface during clear and overcast sky conditions.
I~DT[~I[NC~ 1. A. J. A. Roux, The effect of weather conditions on heat transfer through building elements. Build. Res. Cong. D1V. 3, 82-87 (1951). 2. O. V. Parmelee and W. W. Aubele, Radiant energy emission at atmosphere and ground. Trans ASHRAE 511,85-106 (1952). 3. O. Ognnlesi, Solar radiation and thermal gradients in building units. Build. $ci. I, 1-20 (1965).
462
Technical Note
4. B. L. Hoglund, C. P. Mitalas and D. G. Stephenson, Surface Temperatures and heat fluxes for fiat roofs. Build. Sci. 2, 29-36 (1967). 5. R. J. Cole, The longwave radiative environment around buildings. Build. Environ. II(I), 3-13 (1976).
6. M. J. Unsworth and J. L. Monteith, Longwave radiation at the ground (1) Angular distribution of incoming radiation. Quarl. ,l. Roy. Met. Soc. 101, 1-13 (1975). 7. R. J. Cole, The Iongwave radiation incident upon the external sudace of buildings. Build. Serv. Eng. 44, 195-206 (1976).
003g-092X/"t9/IFa01-O45911t02.00/0
TECHNICAL
NOTE
The iongwave radiation incident upon inclined surfaces R. J. COLE School of Architecture. University of British Columbia, Vancouver, Canada (Received 21 February. 1978: revision accepted 14 August 1978)
I. INTRODUCTION Until approximately 20yr ago. the net Iongwave radiation exchange between building surfaces and their surroundings was considered minimal in comparison to the absorbed solar radiation. Roux[I], and Parmelee and Aubele [2]. however, presented experimental evidence which indicated the fallacy of this supposition. More recent studies comparing the predicted heat flow through elements with measured values[3,4] illustrate that significant errors may be introduced into the calculation of heat flow through building elements if the Iongwave radiation exchange at the outside surface is neglected. Similar arguments can be raised in the evaluation of the performance of solar collector systems. A significant number of relationships are available to predict the atmospheric radiation incident upon a horizontal surface[5] and, as demonstrated by Unsworth and Monteith[6], it is possible to transform these to determine atmospheric radiation incident upon inclined surfaces. However, one of the major limitations has been the determination of the ground component incident upon non-horizontal surfaces. The general assumption that the total Iongwave radiation incident upon vertical surfaces is equal to that of a black-body radiator at screen air temperature. There is no indication as the exact contribution from the ground and the sky vault. This paper presents relationships which describe the Iongwave radiation incident upon inclined surfaces. Using the results of a recent field study[7], in which simultaneous measurements were made of the atmospheric radiation on a horizontal surface and the Iongwave radiation incident upon vertical surfaces, it was possible to ascertain the extent to which the ground radiation differed from that of a black-body radiator. Having established a value of the ground radiation, this could then be combined with the atmospheric component (transformed to the particular tilt angle under consideration) to give the total incident Iongwave radiation.
incident upon a horizontal surface (Wm-2); to is the screen height air temperature (°C), c is the fractional cloud amount. (ii) Vertical surfaces Clear sky. Unsworth and Monteith showed that the normal flux density per unit solid angle, at the zenith angle z is given by R(z)lTr = {a + b . In (u. sec z)}o-T~'
where u is the precipitable water depth of the zenith, a and b are constants having values of 0.70_+0.05 and 0.09_+0.002 respectively during a clear sky. The variation in a and b can be attributed partJy to vertical and partly to horizontal temperature gradients of particulate material (that is, pollution or haze near the ground or water droplets in high cloud). The total atmospheric radiation incident upon a horizontal surface is given by the integration of eqn (4) such that RAo = ~To4(a + b(0.5 + In u)).
R:~, =
crTo4(0.Sa+ b(0.596 + 0,5 In u)).
R/tt. = 0.SRA + 0.~,6b • o'Ta4.
(7)
The value of the atmospheric radiation incident on exposed vertical surfaces during the clear sky condition is given by substituting the value of b (0.09) in eqn (7), such that
RAo = 222 + 4.94t~
(1)
RA¢ = 287 + 6.33to
(2)
RA~, -- 0.5 RAo + 0.0311 o'To~.
(8)
Overcast sky. For the overcast condition Unsworth and Monteith indicate that the value of b will differ from the clear sky value. By assuming that over periods of several weeks the distribution of cloud over the sky is uniform, these workers showed that the mean apparent emissivity of the atmosphere at zenith angle, z, is: E(z. c) = (I - nc)E(z, o)+ n . c
or, more generally, R,~o + (65 + 1.39t.)c
(6)
As a further extension, the value of RA,, can be expressed in terms of the atmospheric radiation incident on a horizontal surface (Ra) and that emitted by a black body radiator at screen air temperature (~rTo4),by combining eqns (5) and (6) such that
(i) Horizontal surfaces In a previous paper, Cole [7] presented relationships giving the incoming atmospheric radiation incident upon a horizontal surface expressed as linear function of screen air temperature and cloud amount, such that:
=
(5)
A similar expression can be established to compute the value of the atmospheric radiation incident upon a vertical surface, such that
2. ATMOSPllEI~CRADIATIONINCIDENTUPON BUILDINGSURFACES
Ra
(4)
(3)
where R,~o is the atmospheric radiation incident upon a horizontal surface during clear sky conditions (Wm-2); RAc is the atmospheric radiation incident upon a horizontal surface during overcast sky conditions (Win-2); RA is the atmospheric radiation
(9)
where, E(z, o) is the apparent emissivity of the cloudless sky at zenith angle, z, and c is the fractional cloudiness. Based upon a long series of measurements in Britain mean values of n for long periods of 3 months were found to range from 0.81 to 0.87, with a mean annual value of 0.84. Therefore, for a
cloudy sky, E(z, c) = a'+ b' In (u - sec z)
459
(10)
460
Technical Note
Where
giving the following results: a'= a +n. c(1-a)
b'= b ( l - n ,
c).
Using these values, Unsworth and Monteith established the general relationship,
Re, = 158+ 2.64to
(clear sky)
(19)
Re, = 155 + 2.38to
(overcast sky)
(20)
/~2 = 162 ÷ 3.15to
(clear sky)
(21)
Re: = 157 + 2.40to
(overcast sky).
(22)
(11)
EA~ =(1 - n . c ) E A o + n " c.
For the present analysis, it was necessary to establish representative instantaneous values of b'. Using the values of atmospheric radiation for clear and overcast conditions as given by eqns (1) and (2) it was possible to establish the dependence of n on screen air temperature. These values are given in Table 1 and can be represented by the linear relationship:
Now, over the temperature range 0-20°C, black-body radiation (~T°4) may be approximated by the linear expression oT~4 = 314 + 5.164to
(23)
and the radiation incident upon a building surfaces inclined at an angle, a, to the horizontal received from the ground is given by:
(12)
n = 0.7067 + 0.00822to.
R a ( a ) = E~n'T/a sin2 (a/2).
(24)
Therefore, the value of the coefficient, b', is given by, b' = b(l - c(0.7067 + 0.00822h))
(13)
where c' is the fractional cloud amount. For the overcast condition, (c = 1), the atmospheric radiation, RA~ = 0.5RAc + 0.346¢To4(0.0264 + 0.00074to).
(14)
Table I. Values of coefficient 'n' as a function of temperature, °C to
f~o
E..+-
n
0 5 10 15 20
0.707 0.725 0.742 0.756 0.768
0.914 0.913 0.946 0.959 0.970
0.706 0.749 0.790 0.831 0.870
3. LONGWAVERADIATION FROM GROUND In addition to the radiation exchange at a horizontal surface, the net radiation balance was measured simultaneously on two exposed vertical building surfaces:
4
(25)
Equation (25), giving the ground radiation incident upon a vertical surface assuming black-body radiation, can now be compared with the measured values for different ground surfaces under clear and overcast skies. Although differences occur, they are within a range of -+ 15 per cent. In view of the experimental error being of a similar order of magnitude to this and the difficulty of being able to characterise the 'ground' radiation (i.e. the extent of the reflected component and its seasonal change), it is felt unnecessary to make any refinements to the black-body assumption. & LONGWAVERADIATIONINCIDENTUPON INCLINED SURFACES The radiation incident upon a surface inclined at an angle, a, to the horizontal (see Fig. i) is a combination of atmospheric and ground radiation, i.e. R(a) = RA(a)+ R o ( a ) .
(26)
It may be shown that the atmospheric radiation incident upon a surface at angle, a, to the horizontal is given by:
Once again linear relationships were between the incident longwave radiation and screen air temperature. Assuming a linear dependence on cloud amount, the relationships describing incident Iongwave radiation for the exposed surfaces are:
Rz = 284 + 5.92to + (20 - 0.20to)C.
I
Ray = ~ aT. = 157 + 2.582t..
Atmospheric component
(i) Facade (l)--Exposed, grass frontage (ii) Facade (2)--Exposed, Tarmac frontage
R~ -- 281 + 5.25t. + (20 + 0.45h)c
Assuming black body radiation, screen air temperature, the ground radiation incident upon a vertical surface is given by:
(15) (16)
RA(a) = ((a' + b'ln u ) K t + bK2) " uTa4.
(27)
Where K~ and K2 are dependent upon a. Again eliminating In u gives RA(a) = RA " KI
+ (K2 -
0.5K,)b' • ~Ta+
= RA" KI + Ksb'" ~To +.
(28) (29)
These relationships (eqns (15) and (16)) give the total longwave radiation incident upon vertical building surfaces with different degrees of exposure. Since the radiation incident upon exposed vertical surfaces (R~ and R2) and the horizontal surface were measured simultaneously, it was possible to determine the ground component incident upon these vertical surfaces by, Re = R~ - RA,..
(17)
Using eqns (8), (14) and (17) the values of the radiation emitted by ground which was incident upon the vertical exposed surfaces (Rat and R<;z) were computed for clear and overcast sky conditions. In order that comparisons could be made with a blackbody radiator these values were correlated against screen air temperature such that, R ~ = I~ + m~to
(18)
Fig. !. Geometric relationships for radiation incident upon inclined surfaces.
Technical Note Where K3 = K 2 - 0.5K~. The values of K~, K~ and K3 for values of slope angle between 0~ and 90" and shown in Table 2. Ground component As discussed in Section 3, the ground component can be adequately computed assuming that the ground radiates as a black-body radiator at screen air temperature. The ground component is thus given by:
R~(a) = c~To4 sin~ (a12).
(30)
Total radiation The total longwave radiation incident upon an inclined surface is given by the addition of eqns (29) and (30). The values of total longwave radiation for both clear and overcast sky conditions as given by these equations are shown in Fig. 2.
5. CONCLUSIONS This paper has presented relationships which describe the Iongwave radiation incident upon inclined external surfaces. The relationships are expressed in terms of cloud cover and screen air temperature are primarily for use in transferring standard weather tapes into input data for mathematical models which predict the performance of solar collector systems. The relationTable 2. Values of K~, K2 and Ks as a function of surface inclination
0 I0 20 30 40 50 60 70 80 %
KI
K2
K3
1.0000 0.9924 0.9698 0.9330 0.8830 0.8214 0.7500 0,6710 0,5868 0.5000
0.5000 0.5184 0.5523 0.5890 0.6214 0.6447 0.6554 0.6514 0.6315 0.5957
0.0000 0.0221 0.0613 0.1225 0,1798 0.2339 0.2803 0.3159 0.3381 0.3457
48o
400
[:
t
~'E:~jO
F'---
. /
0
i
i
I0
20
---~
|
i
i
3O 4O 5O I ncllrlo~oll
Cleor
ships presented here are based on the long term measurement of the radiative beat exchange at building surfaces. The Iongwave radiation incident upon an inclined building surface is given by the sum of the atmospheric (RA(a)) and ground components (Re(a)).
where and
R A ~ = RAKI + b'K3~T~ 4
R~
= ~T~~ sin2 (u/2).
The values of the coefficients are provided in the text. The relationships describing the atmospheric component are transformations of the atmospheric radiation incident upon a horizontal surface. These basic relationships compare favourably with current work derived in other geographical localities (7) and can be considered of universal application. Values of the ground component derived from field measurements were not significantly different from that of a black-body radiator at screen air temperature. Hence the black-body assumption is proposed for the ground component. The Iongwave radiation incident upon inclined building surfaces is almost independent of the inclination angle with overcast skies. However, as would be expected the variation is significant during clear sky conditions. The implications of this can be of considerable significance for those engineers concerned with the design of solar systems or radiative cooling systems. Previously it has been necessary to assume that the collector plate or radiating surface is emitting to the hemispherical sky vault. By virtue of the anisotropic distribution of atmospheric radiation and ground emission, this assumption can lead to errors in excess of 30 W/m 2 during clear sky conditions. NOMENCLATURE a inclination of surface to horizontal c fractional cloud amount EA effective emissivity of atmosphere, i.e. Atmospheric radiation divided by black-body radiation at screen air temperature E(z) effective emissivity of atmosphere at zenith angle, z R~ measured Iongwave radiation incident upon an exposed vertical surface with grass frontage, Win-; R2 measured Iongwave radiation incident upon an exposed vertical surface with tarmac frontage Wm -2 R, Ioogwave radiation incident upon vertical surface, Wm -2 RA~ atmospheric radiation incident upon vertical surface, Wm-2 RA~°~ atmospheric radiation incident upon a surface inclined at an angle, a, to the horizontal, Wm -2 RA atmospheric radiation incident upon an exposed horizontal surface, Wm -2 / ~ , ground radiation incident upon vertical surface, Wm -~" Roqa~ ground radiation incident upon a surface inclined at an angle, a, to horizontal, Wm -2 cr Stefan-Boltzmann's constant, 5.67 × 10-SWm -2 K -4 tA screen height air temperature, °C TA absolute screen height air temperature, K u optical path depth, cm z zenith angle Subscripts o clear sky condition c overcast sky condition.
..
/
461
sky
i
i
i
6O
"tO
8O
9O
Fig. 2. Longwave radiation incident upon an inclined surface during clear and overcast sky conditions.
I~DT[~I[NC~ 1. A. J. A. Roux, The effect of weather conditions on heat transfer through building elements. Build. Res. Cong. D1V. 3, 82-87 (1951). 2. O. V. Parmelee and W. W. Aubele, Radiant energy emission at atmosphere and ground. Trans ASHRAE 511,85-106 (1952). 3. O. Ognnlesi, Solar radiation and thermal gradients in building units. Build. $ci. I, 1-20 (1965).
462
Technical Note
4. B. L. Hoglund, C. P. Mitalas and D. G. Stephenson, Surface Temperatures and heat fluxes for fiat roofs. Build. Sci. 2, 29-36 (1967). 5. R. J. Cole, The longwave radiative environment around buildings. Build. Environ. II(I), 3-13 (1976).
6. M. J. Unsworth and J. L. Monteith, Longwave radiation at the ground (1) Angular distribution of incoming radiation. Quarl. ,l. Roy. Met. Soc. 101, 1-13 (1975). 7. R. J. Cole, The Iongwave radiation incident upon the external sudace of buildings. Build. Serv. Eng. 44, 195-206 (1976).