Renewable Energy Vol. 2. No. 2. pp. 139-157, 1992 0960 1481/92 $5.00+.00 Printed in Great Britain. Pergamon Press Ltd

CLEAR SKY RADIATION AS A F U N C T I O N OF AL T I TUDE

X. BERGER,* J. BATHIEBO,~" F. KIENOt a n d C. N. AWANOU~ * Laboratoire de Thermodynamique Exp6rimentale, Universit6 de Nice-Sophia Antipolis, C.N.R.S. bf_t.3, rue A. Einstein, Sophia Antipolis 06560 Valbonne, France ; ~ Institut de

Mathbmatiques et de Physique, Universit6 de Ouagadougou, BP 7021 Ouagadougou, Burkina Faso ; ++ Laboratoire de Physique du Rayonnement Facult6 des Sciences et Techniques, Universit+ Nationale

du B6nin, BP 526 Cotonou, B6nin

(Received4 July 1991 ; accepted 14 Au,qust 1991)

Abstract--Using a numerical integration (Lowtran 6 code) stopped at several arbitrary base altitudes, in association with measurements from sounding balloons, clear sky radiation is determined as a function of the altitude parameter. The relations established, and the properties attached, are studied. In application, a natural system to obtain dew in dry climates is explained.

!. INTRODUCTION

In three preceding papers [1-3], clear sky radiation has been studied in its theoretical aspects (influence of atmospheric constituents, spectral radiance, total emissivity), and atmospheric profiles have been shown convenient for its precise formulation at ground level. The data then used came from standard models (McClatchey and Standard U.S.) and sounding bal- loon measurements (three stations of the French meteorological network : Brest, Nimes and Trappes). As all of them corresponded with an altitude of station lower than 200 m above the geoid, the established relations did so.

Nevertheless, many sites around the world are at a higher altitude, and especially a few in countries where the applications in agriculture (radiative cooling, dew condensation), or the consequences upon the climate (local thermal equilibrium) are important : hot dry countries, table-lands and mountains.

So, using supplementary sounding balloon measure- ments from all the world (nebulosity equals 0 octa), and numerical integrations stopped at several arbi- trary base altitudes, the present study led to a for- mulation of the clear sky radiation including the "alti tude" parameter. Finally, the resulting radiative cooling power, increasing when climbing, is discussed, and a natural system to produce dew condensation is presented.

2. RECALL

The celestial vault can be considered as a radiator of which the emission is hemispherical, varying with

zenithal angle and wave number. Sky radiation results from the emission-absorption-reflection by a few gaseous constituents (water vapor, carbon dioxide, ozone and other asymmetrical molecules such as CH 4 and N20), which modulate the transparency of the atmosphere. The contribution of water vapor is the most important (greater than 90%), made of two parts (1): an absorption due to lines in the interval 3 50/tm, and an absorption which can be described as a continuum, overlapping the outside absorption lines.

Consider the theory of molecular band absorption. The extinction of a radiation through an atmospheric layer is proportional to the absorber density, and to the path length ds :

dL~(2, s) = -k(; t , s)L()~, s)p ds (1)

where L is the spectral radiance at the point s ; p is the absorber density ; k()., s) is a coefficient of at tenuation (absorption and scattering). The numerical form of the solution to eq. (1) is (2) :

L~(zo) = e,t~,r(2)L~(tar)z,()0 n

+ Y~ L~(T,)(zi_ ,O0-~,(,~)/T*(;~)) (2) i [

with L~(tar) : spectral blackbody radiance correspond- ing to the target temperature. The observer, at alti- tude z0, is looking at a target, at altitude z,. e,ar(2) : spectral emissivity of the target; L~(Ti): spectral blackbody radiance corresponding to the average temperature 7",. of the ith layer of the atmosphere; ~(2) : spectral transmittance of the atmosphere from

139

140 X BERGER et al.

,°

Zn- 1

Tn Z i

rn Zi_ 1

T i

z 2

z 1

\ \ \ \ \ \ \ E A R T H \ \ \ \ \ \ \ \ \ \ \ z° Fig. 1. Schematic and layered atmosphere for transmission and radiance calculations.

the top of the ith layer to the observer (Fig. 1) ; z*(2) : spectral transmittance of the ith layer accounting only for scattering.

Thus, the spectral radiance at altitude z0 is obtained by a numerical integration from altitude zn to altitude z0, and considering the directional aspect (path length), the various efficient gases and their respective band models [4] of absorption (13 models to set over the whole spectrum from 4 to 40 #m!). The con- centration of ozone (peak at 9.6 ~m) is roughly pre- dominant in the stratosphere (25 km), and it slightly varies with seasons for low latitude sites [5]. The CO2, CH4 and N20 amounts are nearly constant and no significant radiative influences are observed from their variations [6, 7] as they are superimposed on that of water vapor. Finally, the contribution of the other constituents is very small. All this justifies the adop- tion of a single model to compute the radiative influ- ence of the gases different from water vapor.

Stopping the integration at different arbitrary stan- dard altitudes leads to several determinations of sky radiation at the same time, and in the same weather conditions:

3. THE DATA SET

Two varieties of data were used :

(a) The standard profiles defined by McClatchey with, in addition, the U.S. standard profile (6 models). These standard atmospheres, estab- lished as references, incorporate data from the real atmosphere. But, of course, the real atmo- sphere at a given time and location, differs from these annual (or semi-annual) models con- structed for a single latitude belt (mean values, no day-night effect...). The data taken into account

were temperature, pressure, density, water vapor and ozone mixing ratios, all of them as functions of the altitude (each kilometer). A constant con- centration is adopted for the other gases. Conse- quently, these standard atmospheres correspond with regular profiles, and no problem of inter- polation is attached to them.

(b) The sounding balloon measurements: 73 from Brest, Nimes and Trappes in France, 45 from all the world, mainly in Africa (Table l, Fig. 2). Air and water vapor parameters with altitude are measured and radio-transmitted every 40 s from stratospheric balloons. An analysis of quality of measurements is then made (two indicators : pre- treatment and objective analysis), and also of weather conditions (indicator with five numbers giving five readings only measurements cor- responding with a clear sky were kept : nebulosity equals 0 octa). Each sounding is reduced to a set of values at 13 standard levels by polynomial interpolations (and when possible extrapolations under ground level), and complementary infor- mation was introduced, as standard profiles for CO2, 03 . . . . A few difficulties sprang up when using the mathematical procedure for inter- polation :

(i) inversion phenomenona pointed out with a too high intensity to be realistic ;

(ii) computed profiles rubbing out strong inver- sions ;

(iii) difficulties, and sometimes impossibilities in extrapolating below ground level, because of strong irregularities in the first hundred meters in altitude, mainly at daytime.

Figure 3 shows two profiles. Near ground level, temperatures are subjected to accentuations or inver-

Clear sky radiation as a function of altitude

Table 1. Geographical distribution of the data set

Latitude Longitude Altitude Number of Station C) east C) geoid (m) soundings

( Mid-latitude 0 3 Models ~ Subarctic 0 2

/

l Tropical - 0 1 Brest 48.45 355.59 96 26 Nimes 43.86 4.40 59 41 Trappes 48.76 2.01 168 6 Elbayad 33.66 1.00 1341 2 Tozeur 33.91 8.16 51 4 B~char 31.61 357.77 811 5 Casablanca 33.56 352.34 56 3 In Salah 27.20 2.46 293 7 Tamanrasset 22.78 5.51 1378 3 Nouadhibou 20.93 342.97 3 2 Bamako 12.53 352.05 381 1 Niamey 13.48 2.16 227 7 Nairobi 1.30 36.75 1787 ! Addis Ababa 8.98 38.80 2372 1 Sana'a 15.51 44.18 2205 1 Bangkok 13.73 100.56 3 2 Mendoza - 32.83 291.22 704 5 Khamis -Mushay t 18.30 42.80 2054 1

141

BREST

0

|N-SArAH

MF.NI[~)ZA (ARGENTINE}

Fig. 2. The stations hold for radio-soundings.

~.HAMI.~-M( 15"H AY T

NGKOK -IAn_AND)

142 X. BERGER et al.

o°]

~®~°I '~'*

NIAME 15 04 i 9 8 9 25 H TU

• l x o

i

w

% 2

c , r

-'4o ' 2~o ' o' ' do go ° T AIR (OEGo C)

m : DATA SET , CALCULATED VALUES

-'8o ~ o

x a)

e f x 0

t l . ~ -

!

x

i , , i i t ' - 4 0 - 2 0 0 ~0 o

T DEW (DEGo C)

%

2'0 '

%

x

4'o do ' d0 P ( H P A )

t x

%

I00 i ~lO

IN SALAH 23 02 1988 11 H TU

2-

o .

x l x l ~ .

• (21

& 0 ' 22o ' d 2'o ' X 0 ° 2ao ' T A I R ( [ 3 E G o C )

• , DATA SET = , CALCULATED VALUES

• 0] x o

" i x

t

x

x

i x

' 240 ' 220 ' "~ ' ' ~ ° T DEW (DEG. C)

x

x

x

'X

x |

x

% |

x

x, x m

~ ' 4'o ' do ' eb ' i'~' P ( H P A ) ~10 ~

Fig. 3. Two atmospheric profiles : observed and calculated values.

sions due to heating or cooling of the soil. Dew tem- peratures show irregular profiles, especially in hot dry climates, consequent upon different superposed air masses. Contrarily, atmospheric pressure profiles are regular and identical (they can be superposed), in agreement with the normal exponential decreasing law. They confirm the arrangement of the air masses in altitude submitted to the gravitational effect, and thus, the influence of the altitude relative to geoid, and not to ground.

4. SPECTRAL RADIANCE ANALYSIS

Integration of atmospheric radiation from sound- ing balloon measurements and standard models, by using a Lowtran 6 code [4], has led to a determination of the spectral irradiance incident at 6 arbitrarily chosen altitudes: 0, 0.5, 1, 2, 3 and 4 kin. Figure 4 shows the result for mid-latitude summer standard model, and for radio-soundings in Colomb B6char and Bangkok, as examples. A few comments can be expressed.

Clear sky radiation

(a) At low altitudes, warm humid climates lead to a single window, closed by CH4 band at 7.6 ym and CO, band at 14 pro. In the center of the window, at 9.6 ym, is the peak due to 03, also visible at any altitude and for any climate.

(b) A second window emerges when the total water vapor amount above the considered base altitude decreases. But the CO 2 amount remains sufficient to close the window between 14 and 16 #m.

(c) Outside the atmospherical window, the irradiance is that of a black body at an air tem- perature corresponding to the considered base altitude. Nevertheless, one can observe that : (i) For most of the radio-soundings, the

irradiance is lower (black body colder by 1 or 2:C), especially at high base altitude.

(ii) For less than 1/4 of the radio-soundings, the

as a function of altitude 143

irradiance is greater (black body warmer by V'C), especially at low base altitude and for high air temperatures.

Several explanations can be given to these devi- ations.

(i) There may be errors in the integration, in the measurements, or in the interpolations. However, this seems unlikely.

(ii) The high radiative transparency of dry atmo- spheres (high base altitude). The amount of influent gases is then insufficient to close the window.

(iii) Air temperatures in altitude greater than the tem- perature at the considered base, which leads to a transmitted irradiance more intense than that of a black body at the base altitude.

~. Wl(m z ~m sr)

.

oO"

0 C

. .J 7O

%

RADIO SOUNDING

BECHA 2

, ACT (Km) 1.0 Epst, Lon 0°582

8' ' 1'~ ' e '4 ' 3'2 ' 4 ? Lambda

~_ Wl(m 2 #m sr)

00 -

o*- C

,

RADIO SOUNDING

BECHA 2

ALT (Km) 2°0

EpsCLon 0°506

ACr Temp. 6 . 4 0

AV ~4J ~m ' d ' i's ' 2'4 ' ~2 ' go

Lambda

0 c ~o0

7O

CQ"

%,

il(m 2 ~m sr )

#

8

RADIO SOUNDING

BECHA 2

ALT (Km) 3.0 o

Eps&ton 0°433 ~ A L r Temp° 3.10

2.40 ~ _ ~-_

~ m

fe ' ~4 ' 12 ' go °~ Lambda

Wl(m 2 ~m sr)

R/~OIO SOUNDING

BECHA 2

ACT (Km) 4.0

EpsLLon 0°425

/'~ ALr Temp. -5=70

8 16 24 32 40 Lambda

Fig. 4. Atmospheric spectral radiance, and radiance of a black body at the air temperature for the considered altitude.

144 X. BERGER et al.

¢o

o c ~e3 .J qD

~/(m 2 #m st)

S T A N D A R D MODEL

HID-LATITUDE SUMMER

ALT (Km) 0o0 o c ~co / X EpsLLon 0o821 .J

h / ~ ACt Temp= 20.85

,,um I ' l 's ' 2'd ' 3'2 ' 4 b o~

Lambda

~m W/(m & ~m sr)

• STANDARD MOOEL .

co ~ MID-LATI TU~ SUMMER m-

o ~ ~ ALT (Km) 1o0 o c c

I I I & ACr Temp. 16.85 ,o e . t o 0 o . ,

/Jm 8 16 24 32 40 O[

Lambda

°I -o

Y

W/(m 2 ~m sr)

STANDARD MODEL

MID-LATITUDE SUMMER

J

i ACT (Km) 3o0 • Epstton 0o660 ACr Tempo 5°85

°38

~m 8' ' 1'6 ' 2 '4 ' 3 '2 ' 4 '0

Lambda

o

~O-

Q . o c m(.#.

#

Or I

o_ W/(m 2 ~m sr)

STANDARD

M ID-LAT ITUDE S ~ R

ALT (Km) 0:5

EpsLL0n 0.794 ACt Temp: 18:805

/am ' t 's ' 2 '4 ' $ 2 ' 4 'o

Lambda

o W/(m z ~m sr)

STANDARD MOOIEL

NID.-LAT I TUOE SUI~R

ALT ( Km ) ~.O EpsLLon 0°702 ACr Tempo 11;85

~m { 6 2'4 ' %2 4'0

Lambda

W/(m 2 ~m sr)

STANDARO MOOEL

MID-LATITUDE SUPINER

ALT (Kin) 4°0

A~k Atr Tempo-0o15 0°84

#Jm 8' ' t 's ' 2 '4 ' s '2 ' 4'o

Cambda

Fig. 4. (continued)

Clear sky radiation as a function of altitude 145

~(~-

o c . J

(11

t~

o (

.

o c ~ c o

g-

el-

£-

o

x~

y

~t

%

W/(m 2 ,m sr) ~.

RADIO $ O I . N ~ I N G

BANGK N ~.

ALT (Km) 0°0 EpsCLon 0=905 ~ ALr Temp° 23°80 Dew Tempo 23=00

~ -

Nm

' d ' 1~ ' 2'4 ' 12 ' go ° c L a m b d a

W/(m e um sr) 0.

RADIO SOUNDING

BANGK N mm-

/ ~ ALT (Km) 1o0 o

J ~ Ep,CLon 0°807 ~ / ~ ACt Temp. 21=90

~ = 4 0 a~_

eJ-

/am ' ' 1'6 ' 2'4 12 ' - - 2 o °

Lambda

W/(m 2 ~m s r )

RADIO ~UNDI~

BANGK N

ALT (Km) 3=0 EpsLLon 0°698

~ T e m p o

a' ' I'G ' ~ 4 ' 1 2 ' g0 Lambda

o

W/(m 2 ~m s r )

R A D I O ~NO I NG

BANGK N

ALT (Km) 0°5 EpsCLon 00844 ACr Tempo 24o;~

j ,.,. 16 24 3 2 40

Lambda

W/(m 2 Jum sr)

RADIO SOUNDING

BANGK N

ALT (Km) 2o0 /'~ EpstLon 0=749 I ~ ACt Temp= 15000

8' ' t`6 ' N ' 3'2 ' g0 Lambda

W/(m 2 #m sr)

0o

o

Fig. 4. (continued~

RADIO SOUNDING

BAN~K N

I ALT (Km) 4.0 Epstton 0°655

8 ! G 24 32 40 Lambda

146 X . B E R G E R el al.

A study of these small deviations could be under- taken. It could lead to a reference black body tem- perature more rigorous, and consequently, to another definition ofe.. Such a study has not been done because these deviations are not systematic, and there also are compensation effects in the irregularities of some profiles.

5. THE EMISSIVITY AT EACH ALTITUDE LEVEL

Integrat ing the spectral rad iance leads to the deter-

m i n a t i o n o f the emiss iv i ty . F igure 5 s h o w s the set o f

po in t s e = / ( t d ~ - w ) in func t ion o f several p a r a m e t e r s :

a l t i tude o f the base, h y g r o m e t r y , per iod in the day

(the radio-soundings are usually taken twice a day, at 0 and 12 h UT). The straight lines correspond with the linear regressions (Table 2). A few comments can be expressed :

(i) The night emissivities are greater than the day ones, whatever the altitude. The slopes of the lines are smaller for the first than for the last.

(ii) For identical dew temperatures, day emissi- vities and the geoid level, low hygrometry measure- ments lead to smaller emissivities than high ones :

~, = 0 . 6 9 9 0 + 4 . 4 6 x 10 3/de w

e, = 0 . 7 1 6 6 + 4 . 0 5 X 10 3tale .

hyg < 3 0 % , ( 3 )

hyg > 3 0 % . ( 4 )

§j m- &

d-

g° , J -

tn d-

d

go _3 _ o,)m

&

ALT (Km) = 0.0

-4o '

e, MODEL OF ATMOSPHERE

xa, DAY • HYG < • 30Z

- ~2 ' - ' 24 ' 216 ' - ' 8 ' d ' d 16 T* DEW (CELSIUS DEGREES)

ALT (Km) = 0~-5

z

~_d x A

d o , MODEL OF ATMOSPHERE

xA. DAY • HYG < > ~O%

d" -40 ~ ' 3 2 ' ' 2 4 ' 216 ' ~ ' 0' ' 8' ' IS ' 24 T , DEW ( C E L S I U S OEGREES)

= ALT (Km) = 1=0 x • ~

~ X a ~ x • x x

x • x

-'4Q ' ~ ' "24

e , MODEL OF ATMOSPHERE

x , = DAY • HYG < > $OX

' - - ' is ' ~ a ' o' ' 8' ' 1 ~ ' 2'4 T= DEtJ ( C E L S I U S DEGREES)

~0 d-

z ~- o

d-

ALT (Km} = 2=0

X X ~ - a

(~. MODEL OF ATMOSPHERE

x.= DAY . HYG < > 30%

-40 ' &z ' -'z4 ' -% ' 'B ' d ' d ' I'G ' T. DEW (CELSIUS DEGREES)

ALT (Km) = 3mO

m d-

z~- o _J .

a aa

x xx ~

~ - - X ~ X-XW • e= MODEL OF ATMOSPHERE x X X X ~:

X x-, DAY , HYG < • 30Z x

-4o' '3z ' &4 ' : i s ' -~ ' ~ ' d . - - ¢ ~ ' -4~ T* DEW (CELSIUS DEGREES)

ALT (Km) = 4,0

~ X m ~ MODEL OF ATMOSPHERE

X

x A , DAY . HYQ < • 30%

-40 ' ' ~ ' "z4 ' -'16 ' g8 ' d ' d T. DEW (CELSIUS DEGREES)

F i g . 5. Emissivity as a function of the dew temperature at the altitude level, Observations and regression lines. At geoid level, lines corresponding with [2].

w

m ,3-

z C ] ~ o

H

~n

~0 d

§o o ~ t o

to

,3

Clear sky radiation as a function of altitude 147

-'4o

m

ALT (Km) = O.O

d-

go d~

d- (~, MODEL OF ATMOSPHERE

x~, NIGHT, HYG < • 30~

• -'32 ' J24 ' 216 ' " 8 ' d ' 8' ' 1'6 ' 2'4 c]- T , DEW (CELSIUS DEGREES)

ALT (Km) = Do5

/

0, MODEL OF ATMOSPHERE

xa, NIGHT, HYG < • 30%

To DEW (CELSIUS DEGREES)

ALT (Km) = 1.0 • ~ a ~

~ o z. X 0 ~ X J

-'32 '

o . MODEL OF ATMOSPHERE

xa, NIGHT. HYG < > 50%

- ~ 4 ' : I s ' - ~ ' d ' d ' ,~ ' ~ T. DEW (CELSIUS DEGREES)

ALT (Km) = 2 ° 0

X 0

i MODEL DF ATMOSPHERE

x ~ NIGHT* HYQ • > 30%

- - ' 4 o ' - ' s 2 ' - ' 24 ' - ' l ~ ' - ' s ' d ' B' ' 1'6 ' 2'4 To DEW (CELSIUS DEGREES)

ALT (Km) = 3.0 c~

h.

n d

o i MODEL OF ATMOSPHERE

,o. / 30X xa , NIGHT. HYG < • ~ u ~ /

-'40 ' - ~ . ' -~4 ' - ' I S ' -~ d ~ -ss -4e T. DEW (CELSIUS DEGREES)

ALT (Km) = 4.0

x

xa l NIGHT* HYQ < • 3D•

-',m ' - '32 ' -'~4 ' - ' i ~ ' "8 ' o' ' C ' f 6 ' 2'4 To DEW (CELSIUS DEGREES)

Fig. 5. (continued)

Table 2. The sky emissivity obtained by regression (least squares) at each altitude level. Correspondence with Fig. 5

Altitude above geoid Period Emissivity (e,)

0.0 km Day 0 .7087+4 .73 × 10 3 taew Night 0 .7586+4 .28 x l0 ~ td~w

0.5 km Day 0 .7002+4 .68 x 10 3 td,,w Night 0 .7240+3 .85 × 10 3 toew

1.0 km Day 0 .6917+4 .64 × 10 3 td=w Night 0.7065+3.98 x 10 3 tde~

2.0 k m Day 0 .6747+4 .55 × 10 3 tdcw Night 0.6886+4.23 x 10 3 tdc~

3 . 0 k i n Day 0 .6577+4 .46 × 10 3taew Night 0 .6687+4 .19 × 10 3 td~w

4.0 km D a y 0 .6407+4 .37 × 10 3 td~w Night 0.6480+4.15 × 10 3td~ ~

This confirms the theoretical effect previously set up [1, 2]. At higher altitudes, this effect is not visible. The hygrometry effect at geoid level seems more important than predicted. That comes both from the too small number of observations at this level, and from the fact that many of them result from extrapolation below the real ground level of the station (daily profiles in hot dry countries are the most distorted; for night soundings, the errors are minimized). Nevertheless, the lines for night, and for day observations, and for an hygrometry greater than 30%, are in good agreement with refs [1, 2].

As for the site influence, the small number of night profiles outside France does not permit one to confirm the non-influence effect pointed out with the

148 X . B E R G E R e t al.

three French site measurements [2]. For day measure- ments, the only valuable effect is for the 2 3 km alti- tude levels : the emissivity is greater in France than in Africa. The relations drawn in Fig. 6 are :

altitude : 2 km

~ .=0 .67713+5 .237x10 3td,w

e = 0.68929+4.555 x 10 3 t~¢w

altitude : 3 km

e. = 0 .65147+4.664x 10 3ta~w

= 0.66902+4.590 × 1 0 3 t ~

Africa + . . . (5)

France + models (6 )

A f r i c a + . . . (7)

France + models. (8 )

Such a site effect can be explained as follows : in dry tropical climate, the atmospheric profiles are very much disturbed in this ~ 3 km range of altitude. The strong ground heating influences the low atmospheric layers. As for the 2-3 km atmospheric air layers, they are cooled much more than in temperate climate because of the extreme dryness (high radiative trans- parency). At 4 km and above, the air masses are no more under a geographic influence. It could be interesting to build a "dry tropical model of atmo- sphere" to eke out the five McClatchey models, which are related to summer or winter (natural divisions of the year) mid-latitude and sub-arctic climates, and to humid tropical climate (dry and humid are the natural climatic divisions in these countries).

El §j ~o" LU

El

e.

z ~ - o _J _

`3-

m 3̀-

e. z ` 3 o ._1 .

3̀-

ALT (Km) = O.O

24o ' -'z2 '

`3 e = AFRICA + , FRANCE + MODELS

DAY

'~4 ' - '16 ' -'8 ' d ' d [6 ~ " ~ 4 `3 T. DEW ( C E L S I U S DEGREES)

@ ALT (Km) = 0.5

o , AFRICA += FRANCE + MODELS

DAY

- - - ' 40 ' J32 ' -~4 ' 2)6 ' ~ ' d ' ~ l~ ' ~4 To DEW (CELSIUS DEGREES)

® ALT (Km) = 1=0 ~ " ~

~ 0 @ 0 @ + ¢'

(~ ¢' 4-

® + 0

' 4 o ' - ' 3 2 ' "24 '

E .

¢-.

z ~-

~,~-

=3 ,3-

<~ AFRICA + , FRANCE + RODELS

DAY

- ' I S ' ' 8 ' d ' 8' ' ,'6 ~ `3 T . DEW ( C E L S I U S DEGREES)

ALT (Km) = 2.0

A ~, AFRICA +~ FRANCE + MODELS

DAY

-'4o ' -'32 ' 224 ' -'~s ' -'e ' d ' ~ ' 1~ ' 34 T ° DEW ( C E L S I U S DEGREES)

m

ALT (Km) = 3.0

@ ¢ ' @ z ` 3 '

m(o

o o m

@ @ @ e , AFRICA 3̀" @ + ,~ FRANCE + MODELS

-40 ' "$2 ' "24 ' "IS ' "8 ' d ' 8' T . DEW ( C E L S I U S DEGREES)

ALT (Km) = 4,0

S " ° : : ;;,".'~; • MODELS DAY

2 4 0 ' - '32 ' 22a ' 216 ' ~ ' d ' 8' T° DEW (CELSIUS DEGREES)

Fig. 6. Emissivity as a function of the dew temperature. Lines as in Fig. 5. Effect of the geographical site.

o

~d

:1 3

c;-

z-.

go

~o .

ri-

d

g . J

Clear sky radiation as a function of altitude 149

° d' ALT IKm) = O=O •

j

~o d

g

Ld to

--4o ' - 'zz ' 2 z 4 '

d e, AFRICA A' FRANCE + HODELS

NIGHT

2 i s ' : e ' d ' d ' f s ' ~4 d T. DEW (CELSIUS DEGREES)

ALT (Kin} = 0.5 •0

-'40 '

e, AFRICA A ' FRANCE + MODELS

NIGHT

: z a ' 2z4 ' 21s ' : e ' d ' d ' s~ ' ~a T. DEW (CELSIUS DECREES)

ALT IKm) = 1°0 • • • 0

• • Jl~ 0 §o o~u)

d , AFRICA

&, FRANCE + MODELS NIGHT

-:4o ' 2 z z ' 2 z a ' J I 6 ' ~ ' d ' d I~ ' ~4 d T. DEV (CELSIUS DEGREES)

d ~ i ALT (Km) = 2 * 0

e, AFRICA A' FRANCE + HOOELS

NIGHT

. ~ o ' 23z 2z4 ' 2~6 ' -b ' d ' d ' ~ ' ~4 T. DEW (CELSIUS DEGREES)

ALT (Km) = 5.0

e. . ~ go J .

• L~ o -

~ 6 a

n d- e , AFRICA

It . FRANCE + RODELS ~ _ _ NIGHT / ,~, "~,

240 ' -.'zz ' " 2 4 ' " t s ' - ' s ' d ' B ' " s s ' 2 a ~ ' T , . DEW (CELSIUS DEGREESI

d AL[ (Km) = 4 . 0

o

O • • °

~ ° ~ , AFRICA A, FRANCE + MODELS

NIGHT

-40 ' -'3z ' "z4 ' 2,6 ' 20 ' d ' T. DEW (CELSIUS DEGREES)

Fig. 6. (continued)

6 . E M I S S I V I T Y A S A F U N C T I O N O F T H E

A L T I T U D E

A regression on the calculated (from measure- ments) emissivities was done bo th with day and night subdivisions, and with the al t i tude pa ramete r in addi t ion to the dew temperature. The relat ions d rawn in Fig. 7 are :

night :

e, = 0.75780 - 0 . 0 4 9 4 8 7 alt + 0.0057086 alt 2 ÷ (4.3628

-0 .25422a l t +O .O 5302a l t 2) x 10 3 tdew (9)

day :

~, = 0 . 7 2 5 0 0 - 0.027386 alt ÷ 0.0016595 air 2

+ (3.3700 + 1.07414 alt

- 0 . 2 0 5 8 6 a l t 2) x l0 3 tdew. (10)

This de terminat ion was obta ined with the respect of a cons t ra in t relative to the geoid alt i tude, and justified by the uncertaint ies l inked to measurements at this level (see above) : for n ight observat ions, the relat ion given in Table 2 was adopted, because it was found to be in good agreement with the model buil t for the geoid level (1-2). Fo r day observat ions, an arbi t rary

150 X. BERGER et al.

relation in accordance with the model was adopted (the model, by now, has not to be put beside the question ; measurements at this level, more numerous and precise, should be necessary to have another con- sideration).

The proposed relations link the six arbitrarily chosen altitudes, and show a very good agreement with the observations and models. Figure 8 shows the altitude effect, Fig. 9 the day-night effect.

7. A P P L I C A T I O N : D E W C O N D E N S A T I O N W I T H A

" T E M P O R A L C O N V E C T I V E " FOCUSING SYSTEM

In a first step, the possibilities in condensing the atmospheric water vapor were studied in their general aspect: on one hand, the psychrometric diagram

expands when arising in altitude because of the atmo- spheric pressure decrease. From this point, the possi- bilities in condensing are reduced, but slightly. On the other hand, the above formulas lead to an increased possibility in condensing (the curve t~y < t~w is moved towards the low hygrometries). Even if both the local air temperature and hygrometry decrease when the altitude is increasing, the final result is a greater possibility at 1 km than at geoid level (Fig. 10).

In a second step, the conception criteria for a con- densing system were considered : in a previous paper [8], the interest for selective coatings has been dis- cussed : an emissivity near 1 in the window, and low outside, does not exist in practice; moreover, the investment cost and the maintenance are not negli-

~o- ul

d-

_1 .

ix z d- q .

ALT (Km) = O.O

e, MODEL OF ATMOSPHERE

xa, DAY • HYG < > 30%

24o ' -'3z ' "zd ' -'12 ' -'e ' d ' ~ ' ~'G ' z'd r. DE~/ (CELSIUS DEGREES)

ALT (Km) = 1.0 _.. aaa a a~1-

x x XX • x

0, MODEL OF ATMOSPHERE

xa , DAY . HYG < > 30%

-~o ' g 3 z ' ~2~ ' 2 1 2 ' ~ ' d ' d ' 1~ ' ~4 T, DEW (CELSIUS DEGREES)

ALT (Km) = 3*0

z ~

to

~o _]

oom

rx ~

x x ~i( w x xx

x • in x x x x

x e,, MODELMODEL OFOF ATMOSPHERE o

X xA, DAY • HYG < • 301 x

T. DEV (CELSIUS DEGREES)

ALT (Km) = 0.5

x a ~

o, MODEL OF ATMOSPHERE

xm., DAY . HYD < > 30%

- '40' -'3z ' ~z4 ' - ' t2 ' -~ ' d ' d ' tb ' ~d T, DEW (CELSIUS DEGREES)

ALT (Km) = 2=0

x e l MODEL OF ATMOSPHERE

xa J DAY • HYG < > 30%

-'4o ' "s2 ' -'z4 ' "t2 ' -~ ' d ' 9' ' t'2 ' 34 T. DEW (CELSIUS DEGREES)

ALT (Km) = 4,0

X e, MODEL OF ATMOSPHERE

xA, DAY . HYG < > 30%

To DEW (CELSIUS DECREES)

Fig . 7. E m i s s i v i t y as a f u n c t i o n o f t h e d e w t e m p e r a t u r e : r e g r e s s i o n , b y a l e a s t - s q u a r e s m e t h o d , w i t h all t h e o b s e r v a t i o n s in t h e s a m e t i m e . C o n s t r a i n t for g e o i d a l t i t u d e a n d d a y o b s e r v a t i o n s .

C l e a r s k y r a d i a t i o n a s a f u n c t i o n o f a l t i t u d e 151

t-. §o

J

J

^ L T (Km) = 0.0

g- ~ -

e. MODEL OF ATMOSPHERE

xa. NIGHT, HYG < > 30%

~ o ' -'~o ' - ' 2 4 ' - ' t~ ' - '8 ' d ' B' ' t'~ ' z',~ To DEW ( C E L S I U S D E G R E E S )

ALT (Km) = I,0 & ~ . . . . ~

S x X x

m

zo

d .

~o-

e~ MODEL OF ATMOSPHERE

xA i NIGHT, HYG < > 30%

~ o ' - '32 ' " 2 4 ' - ' t s ' " e ' o' ' e' ' t'G ' d4 ':; To DEW (CELSIUS DEGREES)

ALT (Km) = 3,0

X ~ ^

e , MODEL OF ATMOSPHERE

, ,~,~ , _~8o

co ,5-

~o J . O3¢=

m~

z

T . DEW (CELSIUS DEGREES)

ALT (Km) = 0,5

e, MODEL OF ATMOSPHERE

xa , NIGHT, HY6 < • 30%

-~0 ' J3z ' - '24 ' L i e ' -'8 ' d ' d ' ~ ' ~4 T. DEW (CELSIUS DEGREES)

ALT (Km) = 0.0

-

o= MODEL OF ATMOSPHERE

xA. NIGHT, HYR < > 30%

~ 4 o ' : 3 z ' - ~ d ' 4 6 ' ~e ' d ' d ' ~ ' ~4 T o D E W ( C E L S I U S D E G R E E S )

ALT (Km) = 4.O

~ A T M O ~ P H E R E

xA . NIGHT, HYQ < > 30%

- ~ o ' L 3 z ' - ~ 4 ' - % ' - b ' d ' d To DEW ( C E L S I U S D E G R E E S )

Fig. 7. (continued)

i] ^LI [ TUOE (Kml

O.O 0.5

Z 1.0

] " °

B o 1 6 2 4 T, DEW (CELSIUS DEGREES)

:1 ^LIlTUOE lKml

0°0 0,5

2.0 Z ~ 4.0 d

-'4o ' 2~2 ' -'24 ' -'~G ' -'9 ' d ' d ' t'6 ' 2'. T° DEW (CELSIUS DEGREES)

F i g . 8 . E f f e c t o f t h e a l t i t u d e o n t h e s k y e m i s s i v i t y .

152

go

txl

g _1

d

d

d.

e. z d.

~o

d

X. BERGER e t a l .

-'40 '

o~ ALT (Km) = 0 . 0 d"

j

m d-

g -

laJ d-

NIGHT

DAY

- ~ z ' - '24 ' 2 i s ' 2 e ' d ' d ' =~ ' T= DEW ( C E L S I U S DEGREES)

ALT ( K m ~

-'3,?. ' .-.~.4 '

g o

J _

d-

r, zd- o

&

ALT (Km) = 3.0

NIGHT

DAY

216 ' Ja ' d ' d ' i~ T . DEW (CELSIUS DEGREES}

DAY

--~o ' "z~' ' --'z4 ' .-'IS ' -'8 ' d ' d T= DEW ( C E L S I U S DEGREES)

ALT (Km) = 0.5

NIGHT

DAY

~ o ' -~ z ' Jz4 ' JiG ' -'8 ' d ' d ' ~ ' T. DEW ( C E L S I U S DEGREES)

ALT (Km) = 2.0

NIGHT

OAY

-%o ' 2zz ' -%4 ' 2LG ' ee ' d ' d ' L% ' T . DEW ( C E L S I U S DEGREES)

ALT (Km) = 4.0

T° DEW (CELSIUS DEGREES)

Fig. 9. Day night effect on clear sky emissivity.

gible. As for anti-convective films, their transparency is no more than 80% in the infrared spectrum; their maintenance (dust, senescence) can be neglected ; they also constitute a barrier which prevents the necessary air renewal (for dew condensation) near the cold radiator. All this leads to the preference for black paints and natural systems.

When considering the radiative refreshment of a radiator looking at the sky, and the minimum limit to reach condensation (/sky < 10ew), the convective heat- ing (and also the conductive back heating from ground, but it is here neglected!) by the environmental air is the strong exchange against which all the efforts have to be made. Indeed, such a simple system leads to the following equation of equilibrium :

a(Tb 4 - T~ky) = 2.8(Ta~r-- Tb),.25

where Tb is the radiator temperature, 2.8 (Tai r - - Tb) 025

the lowest convective heat transfer coefficient (natural convection). Thus, in a dry climate, it is a minimum 10°C below the air temperature which has to be obtained for the radiator temperature (Fig. 10). Else- where, the air refreshed by the radiator is heavier than the environmental air, and thus, lies down on the ground if the radiator is flat, and at a higher level. If the radiator is at ground level, then, the natural ventilation takes this refreshed air away. The renewal avoids the radiator being cooled.

A border reduces the hemispherical solid view angle of sky. Of course, the emissivity is the lowest in the 0 zenithal direction, and draws near 1 in the horizontal one. But, in the first case, the solid angle 2~z sin 0 d O is the smallest, and the greatest in the last. So, it is not convenient to cut too high a horizontal band. The

Clear sky radiation as a function of altitude 153

PSYCHROMETRIC DIAGRAM ~ - c l ~ / / / ~ i

Al l i l tK le : 0 km ( 1013 mb) tsky = "~6

\ =

2O

,0-~ / ? ,, / / / _ _ . _ , M . / , ~ / 1~ )

o c , , % %

" ~ ~ ~ . J < ~ . ~ ",

- 5 - 3 --1 1 ,3 5 7 9 I1 13 15 17 19 :21 23 25 2 7 29 ,.31 3.3 3 5 37 39 o l r L ~ m p e r o t u r e ( C )

PSYCHROMETRIC DIAGRAM 3o

Al t i tude = I km (913 mb) 26

24

22

20

la l r - t b ~ S "C 16

E 12 o ° I 0

• 8 ~

4 Io *C I I *C ~-

- 6 --3 --I 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 3 5 37 39 o~r t ~ m p e r o t _ u r e ( C )

Fig. ]0. Possibilities of condensation according to the altitude, tb is the temperature of a surface heated by natural convection, and cooled by radiation toward the sky.

154 X . B E R G E R et al.

solution is to have an edge (black body) at a tem- perature near that of the masked band. That can be obtained if the basin so constituted is not too deep: Figure 11 shows, for the case of a radio-sounding in Colomb B6char, that a band 15 ° high from the horizon is at a radiative sky temperature lff~C lower than the air, and 14~C lower if 30 ° high. The air maintained in the basin is at a temperature near that of the radiator at the bottom, and near that of the environmental air at the top. The hemispherical sky radiative tem- perature may be 30°C (or more) lower than the environmental air one. Thus, if the radiator is well back insulated, the mean air temperature inside the basin can be up to 15c'C lower than that of the environ- mental air. This air refreshes the border (which also faces the sky), and this last, at its turn, irradiates as the masked sky solid angle.

Such a condenser is what has been conceived in Lanzarote (Canary Islands) to water vineyards (a plant which does not require too much water) (Fig. 12). The altitude is about 1 km, which is an optimum to condensate. The soil is volcanic, i.e. black, dug and insulating. Each plant is at the bottom of a hole (diameter 4~5 m) which masks a 20 ° high horizontal band. No water supply exists, and a few liters can be condensed every night!

8. CONCLUSION

Sky radiation is the source which faces solar radi- ation. Its hemispherical and permanent (24 h/day) characteristics make it "diluted". Nevertheless, this source is responsible lbr the thermal equilibrium of the Earth, and for dew condensations. The altitude influence was the last study to undertake [1,2] to have a complete knowledge. A few results were surprising, as the coldest sky at midday in dry countries, at sunset in Mediterranean climates [8], and the maximum dew condensation possibilities for sites at about 1 km high in altitude.

The applications attached to sky radiation do not fail: they constitute what is called the "traditional systems". These systems are resulting from a knowl- edge patiently acquired over years, even centuries. The actual attitude should be to study them, and to improve them by using our scientific models and modern materials. Many cheap solutions could be set up with natural cooling by doing so. They would be aesthetic, non-polluting and easy to work. The system presented above is an example which should be repeated, volcanic soils existing in some desert sites, and dew having been observed as far as Tenere (Niger).

c~. (73

o t-,

oo

Mid lat i tude summ

Colomb Bechar ...-...mm~ i ~

~°

-14"C

W4 o . Q o.s 1.o

Fig. 11. Directional atmospherical emissivity for the mid-latitude summer model, and deduced from the measurements coming from a radio-sounding in Colomb B~char. Brim temperature of a basin equivalent

to the temperature of the part of sky occulted.

Clear sky radiation as a function of altitude 155

Fig. 12. In Lanzarote (Canary Islands), vineyards are watered only by dew. [Reprinted with kind permission from White Star, Hamburg, published in Geo. Rev.]

Clear sky radiation as a function of altitude 157

REFERENCES

1. X. Berger, A simple model for computing the spectral radiance of clear skies. Solar Energy 40, 321 (1988).

2. X. Berger and J. Bathiebo, From spectral clear sky emis- sivity to total clear sky emissivity. Int. J. Solar Wind Technol. 6, 551 (1989).

3. X. Berger, B. Cubizolles and I. Donet, Radio-sounding data for the determination of the infra-red sky radiation. Int. J. Solar Wind Technol. 5, 353 (1988).

4. W. L. Wolfe and G. J. Zissis, Infrared Handbook. IRIA

Environmental Research Center of Michigan (1978). 5. R. A. Craig, The upper atmosphere, in Meteorology and

Physics. Academic Press, New York (1965). 6. J. J. Morcrette, Sur la param6trisation du rayonnement

dans les modules de la circulation g~n6rale atmosph6rique. Th~se de doctorat d'~tat, Universit6 de Lille (1984).

7. E. G. Glueckauff, CO2 content of the atmosphere. Nature 153, 620 (1944).

8. X. Berger, C. N. Awanou and J. Bathiebo, Expressions for computing the radiative cooling flux from the sky in different climates. Int. J. Ambient Energy 9, 155 (1988).