The absorption of radiation in solar stills

The absorption of radiation in solar stills

Solar Energy. Vol. 12. pp. 3 3 3 - 3 4 6 . THE Pe rgamon Press, 1969. ABSORPTION OF Printed in G r e a t Britain RADIATION IN SOLAR STILLS P...

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Solar Energy. Vol. 12. pp. 3 3 3 - 3 4 6 .


Pe rgamon Press, 1969.



Printed in G r e a t Britain





P. 1. COOPER* (Received 2 May 1968; in revised form 6 October ! 968) Abstract-A method of calculating the fraction of incident solar radiation which is productively used in solar stills is presented. The number of variables which can influence this make it necessary to standarize on set values of glass thickness and extinction coefficient and liner reflectance. The variables considered are the day of the year, latitude, cover slope, orientation, percentage diffuse radiation and insolation intermittency. The complexity of the resulting functions, together with the number of variables, makes it difficult to accurately predict what the still mean effective absorptance will be for a given set of conditions. It is found that insolation interminency has an insignificant effect and that the greater the daily proportion of diffuse radiation, the lower the absorptance. Using the equations presented, a mean effective absorptance can be calculated for any given combination of the variables. R6sum6- On pr6sente une m6thode de calcul de la fraction de rayonnement solaire incident qui s'utilise de mani6re productive dans les appareils de distillation solaire. Le grand nombre de param6tres variables dont il convient de tenir compte le rend n6ssaire d'6tablir des valeurs normalis6es pour 1'6paisseur du verre ainsi que le coefficient d'extinction et de r6flexion du rev6tement de I'appareil. Les param6tres variables que Yon consid6re sont le jour de I'an, la latitude, la pente de couverture, l'orientation, le pourcentage de rayonnement diffus et la nature intermittente de I'insolation. La nature complexe des fonctions r6sultantes, ainsi que le grand nombre de param6tres variables le rend diflicule de pr6voir de mani6re pr6cise I'absorption moyenne effective de I'ap~areil selon les conditions de fonctionnement. On 6tablit que la nature intermittente de I'insolation a un effet peu important et que plus la proportion journali/.'re de rayonnement diffus est grande, plus I'absorption est faible. En utilisant les 6quations pr6sent6es, on est en mesure de calculer une absorption moyenne effective pour toute combinaison donn6e de param6tres variables. R e s u m e n - S e presenta un m6todo para calcular la fracci6n de radiaci6n solar incidente que se utiliza en r6gimen productivo en los destiladores solares. Debido al nfimero de variables que pueden influir en ello, es necesario establecer valores fijos de espesor de vidrio, coeficiente de extinci6n y reflectancia dei revestimiento. Las variables consideradas son el dia del afio, latitud, pendiente de la cubierta, orientaci6n, porcentaje de radiaci6n difusa e intermitencia de insolaci6n. La complejidad de las funciones resultantes, unido al nfimero de variables, hace dificil pronosticar con certeza Ctlfiiserfi la absortancia efectiva media del destilador frente a una serie de condiciones dadas. Se ha encontrado que la intermitencia de insolaci6n ejerce un efecto insignificante y que, cuanto mayor sea la proporci6n diaria de radiaci6n difusa, menos serfi ia absortancia. Usando las ecuaciones presentadas, se puede calcular una absortancia efectiva media para cualquier combinaci6n concreta de las variables.

CONVENTIONAL s o l a r stills c o n s i s t o f a t r a n s p a r e n t c o v e r i n g o v e r a b a s i n o f w a t e r h e l d in a b l a c k liner. T h e c o v e r is s u b s t a n t i a l l y t r a n s p a r e n t to so l ar r a d i a t i o n an d o p a q u e to i n f r a r e d and s e r v e s as a c o n d e n s e r f o r the s a t u r a t e d v a p o r w i t h i n the still. D u e to the n a t u r e o f o p e r a t i o n , t h e m e t h o d o f c o l l e c t i n g s o l a r e n e r g y an d its s u b s e q u e n t d i st r i b u tion c a n b e e f f e c t i v e l y u n c o u p l e d a n d e a c h t r e a t e d i n d e p e n d e n t l y . T h e f r a c t i o n o f t h e i n c i d e n t r a d i a t i o n that a c t u a l l y t a k e s part in the s i m u l t a n e o u s h e a t an d m a s s t r a n s f e r p r o c e s s e s is d e p e n d e n t o n m a n y f a c t o r s . T h e m a n y d i f f e r e n t still d e s i g n s that h a v e e v o l v e d m a k e it n e c e s s a r y to define a still as u s e d in this paper. T h e g e n e r a l c a s e to be c o n s i d e r e d is o f a still with a g a b l e - t y p e glass c o v e r , n o t n e c e s sarily i n c l i n e d at e q u a l angles a n d c o v e r i n g a s u b s t a n t i a l l y h o r i z o n t a l b a s i n o f w a t e r h e l d in t h e liner. T h e v a r i o u s p a r a m e t e r s t h a t m a y i n f l u e n c e t h e e f f e c t i v e c o l l e c t i o n o f s o l a r r a d i a t i o n are *Department of Mechanical Engineering, University of Western Australia, Nedlands, Western Australia 6009. 333

SE VoL 12 No. 3 - l)


P. I. C O O P E R

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Day of the year, Latitude, Cover slopes, Orientation, Percentage diffuse radiation, Departure of the solar radiation curve from a clear-day profile, i.e. intermittency, Glass thickness, Extinction coefficient of the glass, Reflectance of the still liner.

After allowance has been made for reflection and unproductive absorption of solar radiation, an effective still absorptance a, can be defined as that which determines the fraction of the total incident radiation which actually takes place in the mass and heat transfer processes. When making an analysis of this effective absorptance it is convenient to subdivide the process into three sections; transmission and absorption, astronomical relationships and still geometry. TRANSMISSION AND ABSORPTION Incident radiation suffers reflection and absorption at the cover, reflection at the water surface and basin liner and absorption within the water and at the liner. This is indicated in Fig. 1. "

ption ...... I.~........... ~f!-. ~ "


Fig. I. Reflectionand absorption in a solar still. Condensation on the underside of the cover glass results in a thin condensate film which is assumed to be non-absorbing (see the Appendix). The overall effect of this water layer is to increase the transmittance of the system by reducing reflectance. Reflection takes place at the air-glass, glass-water film, water film-air, air-water and waterliner interfaces. The reflectance at each interface is governed by Fresnel's relations for the parallel and perpendicular components of reflection. For natural or unpolarized light, these components are similar, and thus R = ½{RI,+R±} /~

l [ t a n 2 (0~- 0,.) + sin 2 ( 0 ; - 0A / = 21tan 2 (O~+Or) sin 2 (O~-O, iJ"

(1) (2)

The relationship between 0~and 0r is given by Snell's Law, sin 0i = n___! sin0,, nz where nl and n._,are the absolute refractive indices of the two media.


The absorptionof radiationin solar stills


The absorption of radiation in glass is controlled by the extinction coefficient which relates the initial and final intensities, ! = to e -~.


The more familiar form is based on the absorptance a, where a = i - e -kx.


The use of an extinction coefficient implies that the glass does not exhibit any significant absorption bands in the solar spectrum. This is the case for most commercial window glasses of low iron content [ 1]. This distribution of an incident ray,is shown diagrammatically in Fig. 2. Addition of the transmitted components yields the transmittance to the desired degree of accuracy depending on the number of terms chosen and the angle of incidence, viz. r = (1 - - a ) ( l -- R,)(1 --R2)(I--R3) + (1 - a ) ( 1 --R,) (1 --R2) (1 - R a ) R 2 R 3 + ( 1 - or) ( I - R ~ ) ( 1 - - R 2 ) ( 1 - R~)R.zZR33

+ ( I - or)s{ 1 --R~) (1 - - R 2 ) ( 1 - R 3 ) R I R 3 +






In a similar manner, the absorptance and reflectance can be found. For all but the largest angles of incidence, the sum in each case is rapidly converging and by the inclusion of only a few additional terms sufficient accuracy is attained. AIR R~


7 /


/ /

7" X ~ /\/ // V / , / / /







¢ , / (I-~


I-e))(l-~) (I- ~) R, R 2

Fig. 2. Distribution of a ray incident on the cover glass.

The radiation transmitted by the glass-water system undergoes reflection at the water surface, absorption in the water and absorption and reflection at the liner. The amount absorbed in the water is dependent upon the path length. With normal incidence, a water depth of 1 in. will absorb approximately 34 per cent of the radiation, assuming that the water film on the underside of the cover has not significantly altered the spectral distribution [2]. This was, in fact, the assumption for no absorptance of the



film. Because of the band absorption of the basin water and the difficulty of accurately predicting the reflectance of the base of the still, an approximate liner reflectance is defined. This represents the radiation reflected back from the base after repeated interreflections and absorptions, and is applicable to the radiation passed at the air-water interface. It will be a function of the turbidity of the water, the colour of the liner and the water depth. Having taken into account all the losses both by unproductive absorption and reflection, the net result is the effective absorptance of the still ~., which is applicable to the total incident radiation. ASTRONOMICAL RELATIONSHIPS F r o m a knowledge of the quantities transmitted, it is possible to calculate the transmittance for a range of angles of incidence. T o calculate the angles of incidence of the radiation, the following astronomical relationships are stated without proof but are readily found in any relevant almanac. The solar altitude or angle of incidence on a horizontal surface is given by, sin ~' = cos L cos H cos 8 + sin L sin 8 = cos L,*.


A convenient approximate relationship for solar declination in terms of the day of the year n (i.e. I st or 200th) is •l"(284+ n) } 8 = 23.45 sin [ ~ x 2~r .


For a general surface of any orientation and slope/3, the angle of incidence 1,, is given by cos !,, = cos o~' cos Z sin fl + sin a ' cos ,8. (9) T h e sign convention used in this paper is that ,8 is positive if the surface of interest is sloped toward the south, in either hemisphere. Z is the absolute difference between the solar azimuth and the horizontal projection of the normal to the surface projected from the surface of interest. It is often referred to as l !',..~v :dl-solar a fimuth angle, hence





where y., is the azimuth angle of the Sun referred to lhe north direction and is increasingly positive east of north and negative west of north. ~, is the angle between the horizontal projection of the normal to the surface and the north meridian. It is positive clockwise from north and negative anticlockwise. The hour angle of the Sun, measured from solar noon. is (i I) where T~ is the time from sunrise to sunset and 0 is the time after sum'ise. H is zero at noon and positive in the morning. *Here lh should not be confused with its more general induction of the intensity of radiation on a horizontal surface (likewise I.).

The absorption of radiation in solar stills

3 37

From Eq. (7), at sunrise a equals zero, therefore T, -- ~ cos-' (--tan 8 tan L).


With the use of the preceding equations it is now possible to calculate, at any instant, the angle of incidence of solar radiation on a surface of any slope and orientation. The procedure is as follows: (l) Consider the surface lying in an east-west plane and sloped toward the north or south at the appropriate angle/3 to the horizontal, observing the correct signs; (2) Rotate the surface about a vertical axis through an angle ~ so that the surface of interest is correctly oriented, once again paying attention to the sign of ~0. With the substitution of the appropriate parameters into Eqs. (7-12), the angle of incidence can be found. Note that these equations are not applicable in the arctic and antarctic circles. STILL GEOMETRY

In general, at any particular time~ the transmittance of the covers will differ depending on the inclination of each side. Consider the configuration in Fig. 3. Of the total incident radiation on the still, the fraction on cover i is Z P / Y Z and that on cover 2 is Y P / Y Z .


1 _X




Aa Fig. 3. Still geometry.

From the sine rule for the solution of triangles, YP/YZ =


sin/32 sin ($+/3,) sin/3, sin (iV - ~2)


and ZP/YZ =


1 + sin.: B~ sin (gl- fl~) s m ~ sin (g, + ~,)"



P.I. COOPER p = sinfl, sin (0--/3.,) sin/3., sin (0+/3,)

then the fraction through 1,f~, is I/( I + P) and that through 2,fz, is P/( 1 + P). N o t e that these relationship are only valid for/3., < 0 < (180-/3~) when 0 ~ (180-/31), fl = 0,f2 = 1. Angle 0 is the projection of the solar altitude angle a' on the vertical plane of the normals to the two sloping surfaces• Geometrical considerations give the following relationship: tan 0 = tan c~' sec Z'


Z ' = lY.~-}¢[I.



N o t e that fl, is applicable to that surface whose product B × ¢ is negative or zero. On the average, different amounts of energy are transmitted by either cover, and to avoid the complications of allowing for different energy absorption rates in different regions of the basin water, a mean transmittance is defined• With reference to Fig. 3. i f Q is the insolation rate per unit horizontal area, then Total incident Fraction incident on Fraction incident on Amount transmitted Amount transmitted

= Q(A,, + A , ) 1 =.f~Q(A.,+A,) 2 = f_,Q (A._,+ A, ) by I = rtj;Q(A2 + A , ) by 2 = r..,f_,Q (A._,+ At ).

A mean transmittance r,,, is defined as follows:

r,,(Q(A2 + A,) = r,fjQ(A.2 + A,) +r.,~Q(A., + A,) •


(17) (18)

When r,,, equals either r, or r2, only one surface is receiving radiation and the underside of the opposing surface will reflect a portion of this transmitted radiation that normally would not reach the water, so adding to the total received. This form of internal reflection may be disregarded because it can only become apparent at low solar altitudes when the solar radiation is greatly attenuated by atmospheric turbidity. A mean effective absorbtance a,,, applicable to the total horizontal radiation for the day may be defined as 1. O/m

= f a.~Q/Qr



where Qr is the total radiation for the day and is made up of the total direct and diffuse components. The problem is ideally suited to solution by digital computer because of its repetitive nature and considerable arithmetic complexity.

The absorption of radiation in solar stills



The variables determining the mean effective absorptance may be separated into two groups; those which are a function of constructional materials and those which are not. Glass thickness, extinction coefficient and liner reflectance fall into the first category. These variables also have a uni-directional effect in that an increase in the arithmetical value of any has a detrimental effect on absorptance. The actual choice is influenced by other factors; the thickness by structural strength, the extinction coefficient by cost and the liner reflectance by the amount of deposits which have built up on the liner. Thus, the six variables to be investigated for their effect on energy collection are day of the year, latitude, cover slopes, orientation, percentage diffuse radiation and intermittency for set average values of glass thickness, extinction coefficient and liner reflectance. To further simplify the problem the restriction of equal cover slopes is imposed. SOLAR RADIATION

It is normally assumed that an analog for the solar radiation on a horizontal surface is given by the following function:

o.T l Q = - ~ sin \ T,}\ + cos \ T, }}


where V simulates the effect of intermittent cloud cover. A closer approximation is achieved by using a value of Ts somewhat shorter than that calculated from Eq. (12), because of the air mass attenuation in the morning an d evening flattening the ends of the curve. Based on the fact that pyranometers require an unobstructed view above about 5 deg elevation and that in turbidity measurements values are generally obtained from the zenith to about 60 deg from the zenith, the actual time must lie between 20 min and 2 hr shorter than Ts. As a first approximation, a value 1 hr shorter has been used in this paper. The total radiation Qr is the sum of the total direct and diffuse components Qo and Qa, where (.)i, = fl, Qr and Qd ft Qr" Diffuse radiation is that energy scattered out of the direct solar beam by air molecules and aerosol (dust, smoke particles etc.), and is typically of short wavelengths. Thus, the energy distribution in the diffuse spectrum will differ from that of the direct, but the transmittance will remain the same for glasses of approximately constant spectral transmission, as assumed in this paper. The assumption has been made that the diffuse radiation is isotropic, leading to an apparent angle of incidence of 60 deg for a horizontal surface. The apparent angle of incidence will change with cover slope between about 55 and 60 deg, but as the transmittance does not usually vary significantly over this range, 60 deg is used. =



Because of the number of variables, it is only possible to observe cross-sections of the response, in this case the mean effective absorptance, at particular points in a multidimensional space. Initially, three values of each variable were chosen and curves drawn showing absorptance as a function of one variable for different values of another, based on the



central values of the remaining four. This entailed 73 sets of calculations to produce 15 graphs of nine points each. F o r only an additional 12 calculations, it was possible to include four extra axial points on each. The first results indicated that the functions were more complex than originally thought and it was necessary to consider one quarter and three quarter values of each variable, as well as the upper and lower limits and the central value. Thus, using five values of each variable and producing 15 graphs of 17 points each. entailed 145 calculations. The "standard" values chosen for gla~s thickness. extinction coefficient and liner reflectance were those generally applicable to common stills, namely 0-125 in., 0.4 in. -~ and 5 per cent, respectively. The five values chosen for each variable are given in Table I. An upper value of 60 deg was chosen for the c o v e r slope as at this angle, for a given still width, the glass area Table I Variable value Lower Quarter Central Three quarter U pper

Day of year

Latitude (deg)

Cover slope tdeg)

Orientation ldeg)

Diffuse radiation lper cent)

I nterminency

356 36 81

0 -I 5 -30

0 15 30

0 22.5 45

0 25 50

0 15 30

127 173

-45 -60

45 60

67.5 90

75 100

45 60

is double that of zero inclination. The cover glass usually represents a significant proportion of the total capital investment, l f a large improvement in mean effective absorptance is not evident for a 60 deg c o v e r slope, then the increased slope will not be justified. T o represent midsummer, midwinter, the equinoxes and the one quarter and three quarter 'days of the year', values for these seasons in the southern hemisphere have been used and, consequently, the accompanying latitudes are negative. T o make it generally applicable in both hemispheres, the days are not referred to by their actual number but only by their seasonal names. DISCUSSION

The mean effective absorptance plotted as a function of the variables is shown in Figs. 4-13. It was found that intermittency had an insignificant effect on the absorptance and, as a consequence, these five graphs have not been included. It is open to conjecture whether the function chosen to represent departure from clear-day profiles is in fact a reasonable analogue. One way of comparison is by visual inspection of the profile with reference to an actual intermittent day. T h e curve for an intermittency factor of 30 and a 12-hr day is shown in Fig. 14. It is evident that the insolation should not fall to zero; this implies zero diffuse radiation. This will not alter the effect of intermittency since variability and not absolute value is the factor of interest. From the point of view of having a more direct analog, it would be better represented by superimposing a higher frequency variation on a carrier. The variable that appears to have most effect is the fraction of diffuse radiation. This is seen from Figs. 7, 10, 12 and 13. F o r any given values of the other variables, the

The absorption of radiation in solar stills

3 41

• lS

















Fig. 4. Mean effective absorptance for different latitudes and seasons.

Fig. 5. Mean effective absorptance for different cover slopes and seasons.

variation of mean effective absorptance with diffuse fraction is linear. The necessary assumption that the diffuse radiation is a constant percentage of the total at any instant of the day is true at the upper and lower diffuse fractions (representing clear and completely overcast days) and, on the average, will be true for intermediate values. For a given total radiation, the greater the diffusion fraction the lower the absorptance. The effect of cover slope is depicted in Figs. 5, 8, 11 and 12. A factor that has not been taken into account when evaluating cover slope is the increase in the effective area

iI m

(.J X


tO t.) Z



O.. o.. O


~n "6

tn en




~E "5









Fig. 6. Mean effective absorptance for different orientations and seasons.








Fig. 7. Mean effective absorptance for different diffuse fractions and seasons.


P. !. C O O P E R







n,. o


















Fig. 8. Mean effective absorptance for different cover slopes and latitudes.

Fig. 9. Mean effective absorplance for different

orientations and latitude~.

of condensation. As a consequence, glass temperatures are lower and glass-water temperature differences increase• This leads to a greater distillation rate. T h i s effect has been investigated by the author and found to be relatively minor[3]. From Figs. 6, 9, l 0 and 13, it would appear that the best year-round performance is found with the still axis oriented in a north-south direction, but that for the wtlues of the variables used in the graphs, in some circumstances, an orientation of zero deg is favourable.



i~'E rio















Fig. I 0, Mean effective absorptance for different

Fig. I I. Mean effective absorptance lk~r different

diffuse fractions and latitudes,

orientations and cover slopes.

The absorption of radiation in solar stills


f • .

• '25







U Z <[ I.-



0 m


5! I













Fig. 13. Mean effective absorptance for different diffuse fractions and orientations.

Fig. 12. Mean effective absorptance for different diffuse fractions and cover slopes.

0~=1000 l 30


hi fit:










1 12

Fig. 14. Insolation profile for the intermittency function.


The rather complex forms of the functions and the number of variables used make it impossible to predict an actual mean effective absorptance for any particular combination, Not only will it be a function of the five variables found to effect performance, but it will also be a function of those variables dependent on constructional materials. It has been shown that intermittency (as defined) has only a small effect, and can be disregarded, while the percentage of the total radiation which is diffuse is the largest determining factor.


P . I . COOPER Although

from the graphs presented

in t h e p a p e r , t r e n d s m a y b e n o t e d , u s i n g t h e

e q u a t i o n s p r e s e n t e d a n d w i t h t h e a i d o f a s m a l l d i g i t a l c o m p u t e r , it is a s i m p l e j o b t o a s s e s s w h a t t h e e x p e c t e d m e a n e f f e c t i v e a b s o r p t a n c e o f a still, u n d e r a n y s p e c i f i e d c o n d i t i o n s , m a y be.

,4ckmnrledgements-This paper represents a portion of the work being carried out on solar stills with a CSI RO Research Grant and supporled by the CSI RO Division of Mechanical Engineering. The author also thanks J. A. Appleyard for project supervision.

A~ A~ ]; f. fr,

fa g Ih I,, lo ! K L Mr n n, n2 Q Q,.

Qr Q,

Qd R.r R~ /~ T, V L',,, X x Y Z Z' a as a,,, /3 /3~ /3z t,' y., 8 0 0i 0r /z p r~ r~ T,,,

NOMENCLATURE projected area ofcover I,ft ~ projected area of cover 2, ft = fraction of the total radiation passing through cover I fraction of the total radiation passing through cover 2 fraction of the daily radiation total which is direct fraction of the daily radiation total which is diffuse gravitational constant, ft hr -~angle of incidence of radiation on a horizontal surface, deg angle of incidence of radiation on a general surface, deg initial intensity of incident radiation, B.t.u. ft --~hr-' finalintensity of transmitted radiation, B.t.u. ft -2 hr -* extinction coefficient forglass, in. -* latitude, deg mass flow of water film per unit width, Ib ft -* hrday of the year absolute refractive index of medium l absolute refractive index of medium 2 solar radiation intensity per unit area, B.t.u. ft -~ hr -1 rate of condensation of vapour, Ib ft-2 hr-, total radiation for the day, B.t.u. ft-z direct radiation for the day, B.t.u. fl-z diffuse radiation for the day, B.t.u. ft -~ parallel component ofreflectivity at each interface perpendicular comPonent of reflectivity at each interface combined comPonent ofreflectivity at each interface time from sunrise to sunset, hr radiation intermittency factor mean velocity of condensate film, ft hr -* distance from apex of still, ft path length of radiation through glass, in. thickness of condensate film at distance X, fl wall-solar azimuth angle, deg absolute wall-solar azimuth angle, deg absorptance of cover glass for solar radiation effective absorptance of a still for solar radiation mean effective absorptance for the day inclination to the horizontal, deg slope of cover 1,deg slope of cover 2, deg altitude angle of the Sun, deg Azimuth angle of the Sun, deg solar declination, deg time after sunrise, hr angle of incidence, deg angle of refraction, deg dynamic viscosity, lb ft -~ hr -~ density, lb ft '~ transmittance of cover I for solar radiation transmittance of cover 2 for solar radiation mean transmittanceofthe cover orienlation of still axis. deg

The absorption of radiation insolar stills


REFERENCES [1] A. G. H. Dietz, Diathermous materials and properties of surfaces. Space Heating with Solar Energy, p. 33. Cambridge (1954). [21 N. E. Dorsey, Properties o f Ordinary Water Substance. Reinhold, New York (1940). I31 P. !. Cooper, Digital simulation of transient solar still processes. Solar Energy 12, 313 (1969). [41 M. Jakob, Heat Transfer, Vol. I. Wiley New York (1949). APPENDIX The mean velocity of a condensate film on a surface inclined at an angle/3 is[4] gpsinfl 3/~





for a vapor velocity equal to zero. Considering unit width, the mass flow at point x is gpZsin/3 3 p. ' Y:~"

M~ = pYr,,, =


At a point (x + dx) the flow is larger by an amount dM~ which must come from condensation in distance d~ dM ~ = g pZ sin/3 -





dY and dM ~ = Q,d~ .'.






integrating, x=

g p~ sin/3 _ . 3/~Q,.




Thus, at a distance x from the start of the film, the film thickness is given by Y = ~ l \ g p 2 sinfl/ft.


For a peak distillation rate of 0. i Ib ft -2 hr -1, a mean film temperature of 100°F, a cover slope of ! 5 deg, and considering a point 2 ft from the still apex, substitution of the appropriate parameters into Eq. (7) gives a film thickness of approximately 0-02 ram. From the graph of percentage absorptance versus log of film thickness, Fig. A-l,[2], the absorptance is about I per cent. At a point 4ft from the apex, this has risen to only 1.3 per cent. 100


,,,8O tO Z ¢1-

60 ,¢C W

~ 4o Z U,I U ¢Y b~ a.



Fig. A - 1. Percentage absorptance vs. log of film thickness.



These considerations have been for normal incidence; for other angles, the path length will be greater than the film thickness. For low cover slopes and Or large. Q,. will be small learly morning, late afternoonl, for high cover slopes sin/3 will be large and hence Y small. Thus, considering other unknowns, the absorptance of the water film may be neglected.