Effects of dye on the performance of a solar still

AppliedEnerg)7 (1980) 147 162

EFFECTS OF DYE ON THE P E R F O R M A N C E OF A SOLAR STILL* M. S. SODHA,A. KUMAR,G. N. TIWARI ('entre of Energy Studies, Indian Institute ~[" Technology, Hau: Khas, New Delhi, 110029, India

and G. C. PANDEY School o! Environmental Science, Jawaharlal Nehru University, New Delhi, 110016, India

SUMMARY

Experiments on the effects of dye on the output of a solar still were carried out during June, 1979. Black and violet dyes wereJound to be more effective than other dyes, and the perJormanee with a dye present was Jound to improve, the deeper the still.

NOMENCLATURE

C Cp hca hra

Specific heat of the insulation (J/kg °C). Specific heat of the water (J/kg °C). Convective heat transfer coefficient from the glass to the ambient environment ( W / m 2 °C). Radiative heat transfer coefficient from the glass to the ambient environment ( W / m 2 °C).

he.,

Convective heat transfer coefficient from the water surface to the glass ( W / m 2 °C).

hr.,

Radiative heat transfer coefficient from the water surface to the glass ( W / m 2 °C).

hell h3

h3

Evaporative heat transfer coefficient from the water surface to the glass (W/m 2 °C). Heat transfer coefficient from the absorbing surface to the w a t e r ( W / m 2 °C), Heat transfer coefficient from the water mixed with dye to the basin surface ( W / m 2 °C).

h4

Heat transfer coefficient from the insulation to the ambient environment (W/m 2 °C).

* Work partially supported by TERI (India).

147 Applied Energy 0306-2619/80/0007-0147/$02.25 ~ Applied Science Publishers Ltd, England, 1980 Printed in Great Britain

EFFECTS OF DYE ON THE PERFORMANCE OF A SOLAR STILL

149

Rajvanshi and Hsieh 18 recently reported a more systematic investigation of the effects of dyes on solar distillation. A complicated analytical mode] considering the energy exchanges between different layers of the water dye system and the bottom surface has been developed: a corresponding computer program was evolved and used to predict the performance of the still. The results of the computer program were in good agreement with experimental observations. In this paper we present a theory for the behaviour of the solar still with a dye dissolved in water; the analysis assumes a large water mass and linear]sat]on of Dunkers ~2 equations in the range of the operating parameters. In the present investigation--unlike the analysis of Rajvanshi and Hsieh~S--conduction through the insulation has been taken into account and the water has been assumed to be at a uniform temperature. The experimental results, corresponding to two identical stills fabricated at the liT, New Delhi, are in good agreement with theory. At this stage it is interesting to consider the physics underlying the enhancement in distillate output by the addition of a dye. In a conventional still, most of the solar radiation is absorbed by the base, which becomes the hottest region of the still; heat is transferred by the bottom surface to the water by convection and to the outside atmosphere by conduction through the insulating layer. When a dye is mixed with water almost all the solar radiation is directly absorbed by the water: the water transfers part of the heat to the bottom and it is then conducted to the outside atmosphere, through the insulation. Hence the water in a still when a dye is used is at a higher temperature than that in a still when a dye is not present; this accounts for the higher distillate output from the still when a dye is used.

ANALYSIS

Following the depiction of the different heat transfer modes given in Fig. 1, the energy balance conditions for the glass cover, basin water and absorbing surface can be written as: dT~ M ~ = T,H~ + (Qrw + Qcw + Oew) dt

Mdr -w dt

aa

(1)

= rzH~ + Qw - (Q,w + Q ~ + Q~w)

(2)

=Qw+Qi.~

(3)

-

-

where: rl

= (1 -

Rg)o~g P

r 2 = (l - Rg)(1 - : ( . ) ~ r 3 = (I - Ro)(1 - ~o)(1 - ~w)o%

(4)

150

M. S. SODHA, A. KUMAR, G. N. TIWARI, G. C. PANDEY

Transparent

Cover

Distillate Drain ~2Hs GL~ cw

~ Black Surface

7//// / / / / / ix¢/ / / / / / /

Insulation

To Fig. I.

Schematic diagram of single basin type solar still.

and: %, = [1 - e x p ( - m f l m y ) ] The expressions for (2,, Q .... Ocw and Q .... following Dunkle, 12 are given in the Appendix. The following assumptions have been used in eqns (1) to (3). (i) A constant mass of water is maintained by the continuous addition of water to achieve a constant level in the basin; therefore, the rate of addition of water equals that of evaporation. It is further assumed that the energy required to heat the water from the ambient temperature (before the addition to the basin) to the temperature of the water in the basin is negligible compared with that required to evaporate the same mass, i.e.: cv(rw - L ) ~ hw

The above inequality is, in general, reasonably valid. (ii) There is no vapour leakage from the still. (iii) The temperature gradients along the thickness of the glass cover and the water depth have been assumed to be negligible. (iv) The area of the cover glass and the still and the surface area of the water are considered to be equal. The observed dependence of the vapour pressure of the water on the temperature can be expressed to a good approximation by the linear equation: 2° p = R 1T + R 2

(5)

where the constants R1 and R 2 are calculated from the saturation vapour pressure data by least-squares curve fitting in the temperature range of interest.

EFFECTS OF DYE ON THE PERFORMANCE OF A SOLAR STILL

15 1

The energy transferred from the absorbing surface to the water, and vice versa, can be written, respectively, as: Qw = h3(Ox~O - Tw)

(6(a))

Qw = h'3(Tw - 0x-o)

(6(b))

and:

while loss to the insulation from the absorbing surface is given by: Oi.~ =

(7)

\ ~ x /l =o

where O(x, t) is the temperature distribution in the insulation. Equations (1), (2) and (3) can now be rewritten as: M dTo O cl~- = r x H ~ + h ~ ( T w - To) - h 2 ( T ~ - 7".)

(8)

M,,. dT~, dt

(9)

-

r2Hs

÷ h3(Ox=°

-

Tw) - h l ( T " ' - To)

and" / ~ 0 "~

K |\ ~--] F~x/., = 0

"c3H s = h3(Ox= o - T . . ) -

( 1 O)

where h~, h: and h 3 are defined in the Appendix. The temperature distribution O(x, t) in the insulation is governed by the onedimensional heat conduction equation given by: ?20

~C?O

The energy balance for the surface of the insulation in contact with the air is given by: ?01

- K?.v ~=t"

= h4(Ox_ L -

Ta)

(12)

The solar intensity and ambient air temperature can be considered to be periodic and hence can be Fourier analysed in the form: x

H e = a o + Re>'a.

exp(ino~t)

(13(a))

b, e x p ( i m , ) t )

(13(b))

l.-..,d

and"

, =l T,, = b o + Re n='!

152

M . S . SODHA, A. KUMAR, G. N. TIWARI, G. C. PANDEY

where: o g = 2 r t / 2 4 x 60 × 6 0 s - 1 In view of eqns (13), we can assume the following periodic solutions:

O(x, t) = Ax

+ B + Re ~ a {C. exp (/~.x) + D. e x p ( - B . x ) } e x p n=l

To(t)

(inmt)

:t

= go + Re ~-~ g. exp

(in.t)

n=l

and:

Tw(t ) =

H o + Re

H. exp

(imot)

n=l

where A, B, go, Ho, C., D., g. and H. are the constants to be determined from eqns (8), (10) and (14) and are given by:

1

H o = - [(hi nl A1

~go

+ h2)g 0 -

rl + z2

=

nl =

Zlao]

h2b 0 -

K

a°

K

It h2,hl+3,} t,1+2,h1+',31th2,hl+3bol 1 +

~1~3

go--

hlh-3

[hKbfl]~ {(1_ +~fi-)M6exp(fl.L!- -

hlh3

(1 _ h ~ . ) M s e x p ( - ~ . L ) }

g. =

H. = Mlg . + M2 C. = Msg . + M~ D. = Mvg . + M s

EFFECTS OF DYE O N THE P E R F O R M A N C E OF A S O L A R S T I L L

153

M1 =[incoMg+( hi+ M 2

Vh2b. + ~ rla.] -L~j

=

M3=

1 + h3 + inc°Mw M1 - h3

M,=

l + ~ + ino~Mw M 2 - ~ a °

MS-2Kfl.

+ h3 J-

M6 - 2Kfl. M4 1 +

MY-2Kfl.[M3(

]/3 J ÷ ~ 3 -

M2

1 _ -h--~-] M, 1

and:

M~-

2K~

M4 1 - h 3 j +

h3 - M 2

Thus the heat flux per unit area corresponding to evaporation is:

Qew = hwmw = heff(Tw - To) = heffIHo - go + 2 ( H n - gn) exp(in~ot)]

(14)

n- 1 where heyf is the evaporative heat transfer coefficient from the water surface to the glass and is defined in the Appendix. The mass of water, M, evaporating per day in kg/m2/day can be obtained by integrating the right-hand side of eqn. (14) over 24 h. Thus:

M

=

f2 4×60×b0 hef f (T~h-Tg) d , [

__ h~H(Ho - go) x 24 x 60 x 60kg/m2/day

hw

In the case of water with dye present,

h3 =

- h ' 3.

(15)

154

M. S. S O D H A , A. K U M A R , G. N. T I W A R I , G. C. P A N D E Y

EXPERIMENTAL DETAILS

The experimental set-up consists o f two identical single sloped stills (0-8 m × 0.86 m × 0.2 m) m a d e u p o f galvanised iron sheet (24 gauge) with an angle o f 10 ° and encased in a w o o d e n box. T h e base and walls o f the basins were initially p a i n t e d b l a c k a n d white, respectively; a 3 m m thick glass cover was fixed on the t o p o f the a s s e m b l y with the help o f a frame m a d e u p o f ' a l u m i n i u m T' a n d r u b b e r gaskets to m a k e the system airtight. T h e space between the basin a n d the w o o d e n box was filled with a 5-cm layer o f glass wool. A V-drain o f galvanised iron sheet was used for the d r a i n a g e o f distillate w a t e r with a slight angle so as to enable the distillate to flow out. The still was south-facing for 3 d a y s in o r d e r to simulate the periodic c o n d i t i o n s for every set o f o b s e r v a t i o n s . The a m o u n t s o f distillate water and solar intensity as well as the a m b i e n t air t e m p e r a t u r e - - w e r e r e c o r d e d at h o u r l y intervals. T h e t e m p e r a t u r e s o f the glass cover a n d the basin water were m e a s u r e d h o u r l y by c o p p e r C o n s t a n t a n t h e r m o c o u p l e s with the aid o f a dc m i c r o v o l t m e t e r : o t h e r j u n c t i o n s o f the t h e r m o c o u p l e s were kept in ice. T h e water t e m p e r a t u r e was n o t e d as the a r i t h m e t i c average o f the readings o f v a r i o u s t h e r m o c o u p l e s fixed at different depths.

DISCUSSION

N u m e r i c a l p r e d i c t i o n s were m a d e for the p e r f o r m a n c e o f an e x p e r i m e n t a l solar still. T h e relevant p a r a m e t e r s were: Without D y e o~= 7-2722 × 10 -5 s -1 p = 64"08 k g / m 3 C = 670 J / k g °C K = 0.04 W / m °C L =0.05 m M o = 5226 J / m z °C M w = 672634 J/m 2 °C r I =0"1 r2 = 0 " 0 273 = 0"6 h I = 22-52 W / m 2 °C h 2 = 5 0 W / m 2 °C h i = 135.05 W / m 2 °C22 h 4 = 22.08 W / m 2 °C23 = 6.2472 per m hey f = 14.01 W / m 2 °C

With Dye ~ = 7'2722 x 10 -5 s -1 p = 64-04 k g / m 3 C = 670 J / k g °C K = 0.04 W / m °C L = 0.05 m M o = 5226 J/m 2 °C Mw = 672634 J/m e °C z 1 =0"1 r 2 = 0'8 T 3 = 0"06 h I = 24-42 W / m 2 °C h 2 = 5 0 w / m 2 °C h i = 67-53 W / m 2 °C 22 h 4 = 22.08 W / m 2 °C23 ~ --- 6-2472 per m hey s = 15-51 W / m 2 °C

EFFECTS OF DYE ON THE P E R F O R M A N C E OF A SOLAR S T I L L

155

In the temperature range of the experimental observations (15 to 65°C), the following linear relationship, obtained from a least-squares fitting of the data, is used for calculating the saturation vapour pressure: p = 420.69T-

1.22239 x

l0 s

where p is expressed in N/m 2 and Tin K. This relationship and data taken from steam tables 24 are shown in Fig. 2, in the temperature range 288-337 K.

the

0032 Q. ~E

D 03 o3

0"024

o

o

Iz

o

g n°

0'016

o

o

o

o

o

0008 o

293

o

o

o

o

o

o

I

I

I

I

1

I

I

I

297

301

306

309

313

317

321

325

1

329

I

I

333

337

TEMPERATURE (K) Fig. 2.

Variation o f the saturation vapour pressure with temperature. O O O O Data from steam tables.

Found by curve fitting.

The calculated hourly variations of distillate per unit basin area, with and without dye present, are shown in Fig. 3. Figure 4 (a) and (b) present the calculated hourly variation of glass and water temperatures, respectively; the experimental points are shown by circles. Figure 5 shows the variation in the ratio of distillate outputs, with and without dye (black, LHS scale), and the distilled water productivity in the two cases (RHS scale), with water depth. The calculated effects of insulation thickness and the absorptivity of the dye are shown in Figs 6 and 7, respectively. The hourly variations of the solar intensity on the glass surface are plotted in Fig. 8. It is evident that the six harmonics used in our calculations are sufficient for the convergence of the Fourier series (see Tables 1 and 2).

156

M. S. S O D H A , A. K U M A R , G. N. T I W A R I , G . C. P A N D E Y 11

o

270

I

Without

TT With

Dye

Dye

240

%

210

x

o

"E a~

o

180

150

-

--

-

o

o

o

o

o

c

>

o

o

120 o o :2: 0_

9O

60 3O

I

1

I

14

16

18

I

0

2

4

G

8

10

12

I

I

I

20

22

24

TIME (h)

Fig. 3.

H o u r l y v a r i a t i o n o f t o t a l distillate per s q u a r e metre. -...... T h e o r e t i c a l . © © © O E x p e r i m e n t a l . ! W i t h o u t dye. II W i t h dye.

0

65

55

Without

o

o

Dye

o

t.) /.5

n I.fl I-

35

25 0

I

I

I

I

I

I

I

I

I

I

I

2

4

6

8

10

12

14

16

18

20

22

24

TIM£ (h)

Fig. 4(a).

H o u r l y v a r i a t i o n o f the w a t e r a n d glass t e m p e r a t u r e s o f the still w i t h o u t a dye. -. . . . . T h e o r e t i c a l . © O © C) E x p e r i m e n t a l .

65

o

o

With Dye

0

0

55

45 o

35

Z5

1 2

I 4

I 6

1 8

1 10

I 12

I 14

1 16

I 18

I 20

1 22

I 24

TIME fh )

Fig. 4(b).

Hourly variation of the water and glass temperatures of the still with a dye present. Theoretical. O O O O Experimental. . . . . .

With

dye ( B l a c k )

,With

no

dLJe -5.5

\

o \

5.0 R.H.S. S c a [ e

\

1.5

\

4.5 W

\

"-

1.4

4.0

1.3

3.5

-J

1.2

3.0

~n r-,,

t.t~-

2.5

1.0

Fig. 5.

L H S. Scale

I 2

I 4

1 6

I 8

I 10

1 12

2.0 14

D E P T H OF W A T E R ( c m ) Variation of the ratio of outputs with and without black dye (R) and distillate water with and without dye, with water depth. - - - - With dye. - - - Without dye.

158

M . S . SODHA, A. KUMAR, G. N. TIWARI, G. C. PANDEY

1

0.6 0.4 0.2

%

0.0

,9

I 2

l 4

I 6

l B

I 10

I 12

D E P T H ( m ) x t02 -

Fig. 6.

l 14

I 16

=

Variation of the daily distillate output of the still for different water absorptivities with thickness of insulation.

L: i

50

0.10m 0.05m

C3 Q-

30

I

I

I

I

I

L

[

I

I

1

1

2

3

4

5

6

7

~

9

10

ABSORPTIVITY --

Fig. 7.

Variation of the daily productivity for different insulation thicknesses with ~lbsorptivit~.

159

EFFECTS OF DYE ON THE PERFORMANCE OF A SOLAR STILL

900 ~

800

Solar

Intensity e

700

43

600 4o SO0

~

37 ~

z

400

w z

34 ~

300

z w m

28

100

I

2

I

/-

I

6

I

8

l

I

10

I

12

I/-,

l

16

I

18

l

20

I

22

I

24

25

TIME (h) Fig. 8.

H o u r l y ~ariations of the solar intensity and a m b i e n t air t e m p e r a t u r e for 19th of June, 1979.

TABLE 1 FOURIER COEFFICIENI"S OF SOLAR INTENSITY AVAILABLE ON THE GLASS SURFACE

n

0

1

2

3

4

5

6

An

306"0554

457"165

157"995

18"917

22"051

18"089

7"6910

188"961

20"850

4"459

235"161

183"168

( W / m 2)

on (degrees)

4~'247

TABLE 2 FOURIER ('OI!Ft.I('ItiNTS OI- AMBItiN! A1R IIMPt!RAJt RI! n

0

Bn ('C) t~n (degrees)

36.6958

1

2

3

4

5

6

6"666

1-065

0.045

0.300

0.298

0.157

230"105

351"466

77-879

155"915

299.941

25"516

160

M. S. SODHA, A. KUMAR, G. N. TIWARI, G. C. PANDEY TABLE 3

Dye used

Red Violet Black

Daily productivity Jor 0.1 m depth/per square metre With dye Without dye (litres) (litres) 3.3037 3.799 3.862

Output with dye Output without dye

Total insolation (kJ/m2/day )

1-08 1-23 1-26

2.477 x 104 2.8713 × 104 2.477 x 104

3-066 3.09 3.066

Table 3 shows the daily distillate water output for different dyes. Figures 3 and 4 show that the calculated results are in close agreement with the experimental observations. CONCLUSION

(1) The theory is valid only for a large basin water mass because, in such cases, the evaporative losses are small compared with the actual water mass. This corresponds to the asymptotic part of Fig. 5 (see also Fig. 3 of Bloemer et al. 1 and Fig. 8 of Cooper3). If, however, smaller water depths are maintained, as is necessary for maximum productivity, the amount of distillate cannot be estimated by the present theory. However, from a maintenance point of view, large water depths are preferred. (2) The black dye injected in the water increased the productivity by 48 ~o for a 14cm depth (Fig. 5). (3) The black and violet dyes are recommended for increasing the productivity of the still (Table 3). This is easily understandable in terms of the spectral distribution of solar energy. (4) The productivity increases rapidly with increasing insulation thickness up to 4 cm and then more slowly in both cases (with or without the dye being present). (Fig. 6). (5) The productivity increases rapidly for low absorptivity systems and then tends to saturate with increasing absorptivity (Fig. 7). (6) Not more than one harmonic has been used in earlier analyses of the solar still. 16 17 It is impossible to reproduce solar intensity and ambient air temperature with one harmonic only. Hence one needs to consider more harmonics. As has been mentioned earlier, six harmonics are found to be a good representation of the observed variation. ACKNOWLEDGEMENTS

The authors are grateful to Professor S. P. Sabberwal, Mr J. K. Nayak and Mr Alok Srivastava for fruitful discussions and help during the work described in this paper.

EFFECTS OF DYE ON THE PERFORMANCE OF A SOLAR STILL Thanks

are also due to the Indian

supplying

Meteorological

Department,

New

161 Delhi, for

the relevant data.

REFERENCES

1. J. W. BLOEMER,J. R. IRWIN, J. A. EIBLING and G. O. G. LOF, A practical basin type solar still, Solar Energy, 9 (1965), p. 197. 2. R. N. MORSE and W. R. W. READ, A rational basis for the engineering development of a solar still, Solar Energy, 12 (1968), p. 5. 3. P. I. COOPER, Digital simulation of transient solar still processes, Solar Energy, 12 (1969), p. 313. 4. P. I. COOPER, The absorption of radiation in solar stills, Solar Energy, 12 (1969), p. 333. 5. R. N. MORSE, W. R. W. READ and R. S. TRAYFORD, Operating experiences with solar stills for water supply in Australia, Solar Energy, 13 (1970), p. 99. 6. J. A. EIBLING, S. G. TALaERT and G. O. G, LDF, Solar stills for c o m m u n i t y use Digest of technology, Solar Energy, 13 (1971), p. 263. 7. P. 1. COOPER, M a x i m u m efficiency of single effect solar stills, Solar Energy, 15 (1973), p. 205. 8. P. 1. COOPER, Digital simulation of experimental solar still data, Solar Energy, 14 (1973), p. 451. 9. P. I. COOPER and W. R. W. READ, Design philosophy and operating experience for Australian solar stills, Solar Energy, 16 (1974), p. 1. 10. E. D. HOWE and B. W. TLEIMAT, Twenty years of work on solar distillation at the University of California, Solar Energy, 18 (1974), p. 97. 1 I. V.A. BAUMand R. BAIRAMOV,Heat and mass transfer processes in solar stills of hot-box type, Solar Energy, 8 (1964), p. 78. 12. R. V. DUNKEL, Solar water distillation The roof type still and multiple effect diffusion still. P r o c

International Heat TransJer ConJerenee, Part V, International Del:elopment.~' in Heat Tran~'/~,r, Unit,ersity of Colorado, 1961, p. 895. 13. M. A. S. MALIK and V. V. TRAN, A simplified mathematical model for predicting the nocturnal output of a solar still, Solar Energy, 14 (1973). p. 371. 14. H. P. GARG and H. S. MANN, Effect of climatic, operational and design parameters on the yearround performance of single sloped and double sloped solar stills under Indian and arid zone conditions, Solar Energy, 18 (1976), p. 159. 15. B. FRICK, Some new considerations about solar stills, Proceedings ~/ International Solar Energy Congress, Melbourne, 1970, p. 395. 16. J. R. HIRS('HMANN and S. K. ROEFLER, Thermal inertia of solar stills and its influence on performance, Proceedings oJ International Solar Energy Congre.s,s, Melbourne, 1970, p. 402. 17. V. A. BAUM, R. B. BAYARAMOVand V. M. MALEVSKY,The solar still in the desert, Proceeding,s ~l International Solar Energy ('ongre~s, Melbourne, 1970, p. 426. 18. ANIL K, RAJVANSHIand C. K. HSIEH, Effect of dye on solar distillation: Analysis and experimental evaluation. Paper presented at the 1979 International Congress of ISES, Georgia, USA. 1979, p, 327. 19. M. S. SODHA, G. N. TIWARI, A. KUMAR and R. C. TYA(;I, A low co,~lportable .~'olar~till, Technical Report TR-79-S-10, Centre of Energy Studies, liT, New Delhi, India. 2(I. M. S. SODHA, A. K. KHATRY and M. A. S. MAI.IK, Reduction of heat flux through a roof by waler film, Solar Energy, 20 (1978), p. 189. 21. W. C. MCADAMS, Heat transmi~sion (3rd ed.), McGraw-Hill, NY, 1954. 22. J. L. THRELKELD, Thermal ent,ironmental engineering. Prentice-Hall Inc., New Jersey, 1970. 23. J. A Dt/FFIE and W. A. BECKMAN,Sohlr ener,~y thermalproee,~.~es, John Wiley & Sons, NY, 1974. 24. L. SCHMII)T. Properties q! water and ~team in SI unit.s, Springer-Vcrlag, Berlin, 1969.

APPENDIX: HEAT TRANSFERS TO THE ATMOSPHERE

The combined radiative and convective loss to the atmosphere can be written as: Qa = h,.a(Tu -

Ta) + ~:,a[(T, + 273.15)'* - (T, + 261.15) 4]

(A.1)

162

M. S. SODHA, A. KUMAR, G. N. TIWARI, G. C. PANDEY

and %a[(Tg + 273.15) 4 - (T, + 261.15) 4] h 2 = hca +

L)

(A.2)

where: h¢,=5.7+3-8V,

Vin m/s

In evaluating h 2 from eqn. (A. 1), the mean values for the temperatures T~ and T,, can be used.

INTERNAL HEAT AND MASS TRANSFER

The radiation, convection and evaporation between the water surface and the cover glass can be approximated to those between two infinite parallel planes. The following equations, proposed by Dunkle, ~z are used to describe these modes. The radiative loss from the water to the glass is given by: Qr,. = 0"9~r[(T w + 273"15) 4 - (T o + 273"15) 4]

(A.3)

hrw = 09a[(Tw + 27315)4 - (T9 + 27315)4]

(A.4)

i.e."

(Tw- T0) which neglects successive specular reflections between the transparent cover and the water surface. Convective loss from the water to the glass is given by: [ RI(I~-x T°)(Tw + 273-15) -],..3 Qcw =0-884 (Tw-T0)+(268.9 - _ ~ - ~ _ R~(-Tw,+~73.15)J (T w - To) (A.5) i.e.: Ra(T ~, - Tu)(T w + 273.15) 11/3 R-Z-- i(~(-~w + 273'15)_]

hcw= 0-884 ( T w - To) q (268'9 x ~ - -

(A.6)

while the evaporative loss is given by: Q,w = 16.276 × I O - 3 Q ~ , R ~ ( T , , - To)

(A.7)

heH = 16"276 x 10-3h~wR~

(A.8)

and: h w = h~ss + hew + h,w

(A.9)

where h,w, hew and hess are calculated from equations (A.4), (A.6) and (A.8) by taking the mean values for temperatttres Tw and To.