Energy Vol. 15, No. 12, pp. 11611169, 1990 Printed in Great Britain. All rights reserved
0360~5442/90 $3.00 + 0.00 Copyright @ 1990 Pergamon Press plc
ENERGY BALANCES FOR UPWARDTYPE, DOUBLEEFFECT SOLAR STILLS HOMING YEH~
and NIENTUNG MA$
t Department of Chemical Engineering, Tamkang University, Tamsui, Taiwan and $ Department of Chemical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China (Received
10 November 1989; received for publication 18 May 1990)
AhstraetA multipleeffect solar still is more effective than a singleeffect unit because it uses the available energy more than once. Moreover, since an upwardtype solar still may
create a freeconvection effect, it is more efficient than a downwardtype unit. The energy balances for upwardtype, doubleeffect solar stills are given in this paper. The use of a computer may facilitate the determination of the theoretical performance under various operating and design conditions.
INTRODUCTION
In conventional, singleeffect solar stills, we use the collected solar heat only once since the heat of condensation of the vapor is transmitted through the cover and discharged into the atmosphere by convection and radiation. Application of a doubleeffect concept to the utilization of solar energy in the atmospheric evaporation of saline water has been investigated previously. l4 In the doubleeffect solar still developed by Dunkle,3 solar energy is used indirectly. The disadvantages of such an indirect system are the additional cost of the collector waterstorage system and heat loss occurring due to the transport of heat energy from the solar collector through the waterstorage tank to the first plate. A modified doubleeffect solar still using solar energy directly may be developed if the upper side of the enclosure is made of transparent material which transmits solar radiation.’ Thus, the upper side of the first plate serves as a solar collector, thereby making operation of the still directly responsive to the solar energy input and taking the place of a separate solar collector and storage tank. Both theoretical and experimental studies on downwardtype, doubleeffect solar stills have been conducted recently. Considerable improvements in productivity are obtained because of the reuse of latent heat, especially for high levels of insolation.6.7
UPWARDTYPE,
DOUBLEEFFECT
SOLAR
STILL
Figure 1 shows a schematic diagram of an upwardtype, doubleeffect solar still. The still consists of one transparent glass cover at the top and one absorbing plate at the bottom, as well as one transparent glass plate in the middle; all of these plates are parallel. In order to collect more insolation, the equipment is tilted from the horizontal, with the cover surface inclined toward the Sun. Accordingly, several weirs are installed in parallel on the upper surface of the glass plate to hold the saline water, while a blackened wet jute cloth is placed on the upper surface of the absorbing plate to form the liquid surface. Except for the glass cover, all parts of the enclosure are carefully insulated with a thick layer of asbestos to make the heat loss as low as possible. During operation, the transparent glass cover and glass plate transmit solar radiation and the absorbing plate is then heated directly by solar radiation. The saline water feeds are introduced on the upper surfaces of both the absorbing and glass plates, where some water evaporates, while the remainder is collected at the bottom and discarded as concentrated brine. The vapor produced from the saline water rises from the absorbing plate and is condensed on the lower side of the glass plate. The heat of condensation given up by the condensing vapor on the lower 1161
HOMINGYEH and NIENTUNGMA
1162
Fig. 1. Schematic diagram of an upwardtype,
doubleeffect distiller.
side of the glass plate is conducted through this plate and furnishes heat to evaporate an equivalent amount of water from the saline brine sliding down the upper side of the glass plate. Finally, the heat of condensation given up by the condensing vapor on the lower side of the glass cover is transmitted through this cover to the ambient. The condensates produced from the first and second effects are collected as fresh water in a trough from the enclosure. The main disadvantage of a downwardtype, doubleeffect solar still is that water vapor must be transferred from the lower side of the upper plate downward to the upper side of the lower plate for condensation and give up latent heat for reuse. Between the plates, there is a downward concentration gradient, which is necessary for the transfer water vapor. In the same direction, however, there also exists a temperature gradient, which suppresses free convection and thus also the mass transfer of water vapor. Therefore, an upwardtype, double effect solar still may be more effective than a downwardtype unit. Application of the doubleeffect concept to the direct utilization of so!ar energy in an upwardtype still has also been investigated experimentally by us. It was found that an upwardtype, doubleeffect solar still is more effective than a downwardtype unit because it may create a freeconvection effect while preserving the effect of reused energy, thereby yielding improving performance.8*15 It is our purpose to derive the energy balances for an upwardtype, doubleeffect solar still.
FRACTION
OF INCIDENT ENERGY REMAINING AFTER THROUGH THE WATER LAYER
PASSING
For water thicknesses in the range from 1 cm to 10 m, a relation for the fraction of incident energy f(t) remaining after passing through a thickness of z cm was obtained by Bryant and Colbeck,’ viz.
f(z) =a’b’lnz,
(I) where a’ =0.73 and b’=0.08 (a’= 0.36 if z is in meters). This representation is simpler for use than that of Rabl and Neilsen.” However, for a water thickness 4 cm, it is not accurate, especially for the region very close to the water surface. For this reason, a modified expression for f (z) in this particular region is introduced as follows:
f t9 =a  b ln(z + c),
(2) where the constants a, b and c may be determined from the experimental data shown in Table 1.” The results are presented in Table 2. The comparisons of Eq. (2) with Eq. (l), as well as with Rabl and Neilsen’s expression, are presented in Fig. 2. It is seen from this figure that Eq. (2) is most accurate for the range of water thickness from 0 to 1 cm.
Upwardtype,
doubleeffect
1163
solar stills
Table 1. The fraction of incident energy remaining after passing through thickness of z, in various range of wave length f(z) X 102.
a
Table 2. Values of u, b and c for various water thicknesses. z, cm
a
b
c, cm 6.0x10'
0 
0.1
0.749
0.0492
o
1.0x102
0.716
0.0754
0.0256
0.716
0.0768
0.0262
0~10.0x10*
I
100
so 60 70 60
l
Erp.
data
1. Rob\
30
et al’s
CxDrCssion
2.f Lz)*o'b'In(z) 3.f(z).ob
20
In
(z+c)
10 t
01
0
, 0.00 1
I
I
0.01
0.1
I
1
Z (cm) Fig. 2. Fraction of incident energy remaining after passing through a thickness z.
HOMING YEH and NIENTUNGMA
1164 ABSORPTIVITY
AND
TRANSMITTANCE
OF WATER
LAYER
If the reflection of solar insolation at the surface of the water layer on the upper surface of the glass plate is neglected, then rw+(v,=l.
(3)
As shown in Figs. 1 and 3, we consider the optimal design case in which the heights of weirs, as well as the distances between weirs, are so installed that the depth of the water layer at one end is exactly zero. Thus, the depth of the water layer is related to the y axis by 2 =p(l
y/L).
(4)
Since the solar energy flux transmitted through the glass cover is r&, that transmitted the water layer on the glass plate may be calculated from
(rr,hv =
l/WL[ ~4, j=f(z)W
through
dy ].
0
After substitution
of Eqs. (2) and (4) into Eq. (5), we obtain (rlo)rw = rZo(a  b{[(p + c)/p] ln(p + c)  (c/p) In(c)  1)).
(6)
The solar energy flux absorbed by the water layer is obtained by using Eq. (3) and the fact that f(0) = 1 = a  b In c. The result is ( r&)a; = rZo(1  tw) = rlob{[p + c)/p][ ln(p + c)/c]  I}.
ENERGY
(7)
BALANCES
The energy flow diagram for upwardtype, doubleeffect solar stills is presented in Fig. 1. Neglecting the temperature drops through the cover and plates and assuming that the cover area is equal to the basin area, as well as equating the effective sky temperature for radiation to the ambient atmospheric temperature, the following steadystate energy balances are reached. Energy balance for the cover qJ0 + 4.2 + 4q + qq = 4m+ 4m
(8)
where qez = W&As, qcz = (&tar + LA,)
(9) (G  T,)I&
(IO)
41, = 0.90( T;  TZ),
(11)
4 ca = h,(T,  G),
(12)
4 ra = s,o(C
 7%
Nu, = h, W/k, = 0.028 ( VW/pJ”.*.
Fig. 3. Water layers on the glass plate.
(13) (14)
1165
Upwardtype, doubleeffect solar stills
Hollands obtained an empirical expression for the heattransfer coefficients in free convection of an inclined air layer of high aspect ratio, heated from below;13 this result is Nu2t
=
h21H2/ k, = 1 + 1.44[1
(1708/Ra2, cos @)I’  [l  1.708 sin(1.8$)1.6/Ra,,
+ [Razt cos $/5.830)‘” Razt = @@IM(G
cos $1
 l]+,
(15) (16)
 T,) + ((pS.2  F,P,,,)l(2.65p=P,,,))T,1,
where x+ = ( 1x1 +X)/2.
(17)
Since the water vapor at the water surface may be considered to be saturated, while that of air at the condensing surface may be unsaturated or supersaturated. Therefore, the modified factors F, and F2 are introduced into Eqs. (16) and (26), respectively.‘* For free convection from a vertical plate, the correlation equations are obtained by Churchill and Chu.14 Nuzs = h,p/k,
= 0.68 + 0.67Ra2s1’4/[1 + (0.492/Pr)9’16]4/9,
(18)
Pr = C,,Cr,lk,,
(19)
RazS = g cos
[email protected]( &  TJIpcL,a.
(20)
Energy balance for the glass plate +
(~w&v + (~lo)G&
~ZCpw(4
=
T,)
t&2
+ 
4cl +
4cl+ 4)C,w(T,
4rl

T,)
+
QC,w(T,

a
+
qe2
+
qc2 +
qr2r
(21)
where 44
qcl = 4r1=
=
hl(T,
=
C.&BIv4[(K
(22)

wc’:
NUI = h,Hllk, Ral
Al&,

73

= +
(23)
G),
(24)
m
Nu2tI
Ra2,+Ra,>
(25)
(pS.1  F2pS.2)Tl(2.6%

pS.dl.
(26)
Energy balance for the absorbing plate (d3)L~
+
~lqnv(T

T,)
=
4.1
+
qcl
+
qrl
+
qb
+
@I

~l&v(T,

T,),
(27)
in which qb denotes the heat loss from the outside surface of the absorbing plate to the ambient. Practically, except for the glass cover, all parts of the enclosure are carefully insulated. Hence, qb may be neglected in Eq. (27). In addition, the absorptivity of water film on the absorbing plate with blackened wet jute cloths, is assumed to be unity. Energy balance for the evaporation rates
We consider first the second effect, viz. 4cpav2

mL2
=
@,,A,
+
A,As)(T,

T,)IAb
=
qc2,
(28)
where X2 = ww,,24!4 Combination
= U{(~wI~a)[P,,2I(~T
 Ps.2)  ws,cIv% 
F,P,,c)I~.
(29)
of Eqs. (28) and (29) gives 4 = WwW2A = 2.59[@2,4
+ h,A,)/M,C,,A,l[P,,2/(P, + kb)/AdP,,/(&
 pS.2) Ws.,/(~,
 Ws.,)]
 p5.2)  F,P,,I(PT  F,P,,JI.
(30)
Similarly, D, = 2.59h,[P,,,l(F~
 P,,,)  &pS,2/U‘~ W’s,2)1.
(31)
Consequently, there exist five unknown variables T,, T2, T,, L&, and D2 which may be obtained simultaneously from Eqs. (8), (21), (27), (30), and (31).
HOMINGYm
1166
DETERMINATION
and NIENTUNGMA
OF MODIFIED
FACTORS
F, AND
F2
The modified factors introduced in Eqs. (16) and (26) may be affected by the distances from the evaporating surfaces to the condensing surfaces and by the temperature differences between these surfaces. Accordingly, we define F, = a,[(Tz  T,)lHP,
(32)
4 = Qz[(T,  G)141b2,
(33)
doubleeffect Table 3. Experimental results for upwyzd\T, solar stills on 2 May 1988 (1, = 330 kgm h , I,  29.6”c and V = 1.7 mset‘). Upwardtype, Time
tl,
t2,
OC
OC
t c' oc
7:00
26.0
26.0
26.0
8:00
32.5
29.7
27.8
9:oo
43.1
39.3
34.7
1O:OO
48.3
43.5
37.2
11:00
56.0
51.2
43.6
12:OO
60.9
55.9
47.5
13:OO
70.4
63.4
54.5
14:00
63.0
58.4
47.4
15:00
52.7
48.7
38.7
16:00
47.1
44.0
36.6
17:oo
41.8
39.8
33.3
18:00
36.3
35.1
30.3
19:oo
31.9
31.7
doubleeffect Dl' kgm'hr'
still %'
kgm*hr'
0
0
0.012
0.031
0.071
0.079
0.169
0.192
0.192
0.212
0.295
0.425
0.319
0.551
0.165
0.342
0.098
0.161
0.053
0.110
0.018
0.090
0.018
0.018
29.7
Table 4. Experimental results for upwardtyr doubleeffect solar stills on 4 July 1988 (I, = 493 kc&m*h , I, = 305°C and V = 2.3 msetI). Upwardtype,
I Time
still
t2,
to’
%'
oc
oc
oc
kgm'hr'
kgm'hr'
0
0.018
0.047
0.049
0.139
0.175
7:00
27.5
27.5
27.5
8:00
40.3
35.5
32.3
9:oo
50.1
43.9
37.1
10:00
59.3‘ 51.4
43.6
ll:oo
66.8
49.1
58.7
12:00
71.5
63.3
51.7
13:00
72.2
63.6
52.2
14:00
70.2
62.4
50.7
15:00
66.7
59.9
48.9
16:00
59.6
54.6
45.3
17:00
50.7
47.8
39.7
18:OO
42.3
40.2
33.9
19:00
doubleeffect
tls
30.1
29.5
26.5
Da*
0.285
0.360
0.368
0.458
0.405
0.582
0.329
0.564
0.278
0.486
0.168
0.360
0.090
0.277
0.043
0.174
0.031
0.102
Upwardtype, doubleeffect solar stills
1167
in which the constant a,, u2, b, and b2 can be determined by employing Eqs. (32) and (33), coupled with experimental data. The procedure will be described as follows. First, the values of F, and 4 are calculated from Eqs. (34) and (39, respectively, by employing the experimental data with the results (02H/k,){2.59[p,,/(p,~

f’s.2)

F,P,,c/&
(OlHik,){2.59[pS,1/(pT

&f’s,c)l)l
=
 P,,i)  F,P,,,I(&
Pu2tA2,
+
 E&,2)1)’
Nu2sWIWL]I&>
(34)
= Nu,.
(35)
Equation (34) is derived by combining Eqs. (15), (18) and (30) to eliminate h2t and h2s, while Eq. (35) is derived by combining Eqs. (25) and (31) to eliminate h,. The final values of F, and F2 are then plotted in the figures as functions of ln[( T2  T,)/H,] and ln[( T1  T,)H,], respectively, and the correlated values of (1, and b, as well as u2 and b, may be obtained, using the method of least squares. As an illustration, we employed the following experimental data for upwardtype, doubleeffect solar distillers, as shown in Tables 3 and 4;8 Applying the specified method, Figs. 4 and 5 were constructed and the following empirical correlations obtained: F, = 0.606[(T2 
T,)/H,y9,
(36)
4 = 0.512[(T,  T2)/H,]o.273.
(37)
CONCLUSIONS
It has been found from the previous experimental results that an upwardtype, doubleeffect solar still is more effective than a downwardtype unit because it may create a freeconvection effect while preserving the effect of reused energy.15 This can be also shown by Figs. 4 and 5 since both F, and F2 are greater than unity. It means that the water vapors are supersaturated on the condensing surfaces, while, in the downwardtype unit, the water vapors are unsaturated on the condensing surface, as shown by Fig. 3 in another previous paper.’ Furthermore, it is believed that the cost for constructing an upwardtype solar still will be cheaper than a downwardtype solar still because the former only needs two working chambers, while the latter needs three.
1.0
In
F, = 0.5 c’O.606
0.5
+ 0.119
In
I(T2Tc
I/H21
C(T,Tc)/HpIO”g
= 4
5
In C(T,T, I/H,1 Fig. 4. Relation between F, and (T,T,)/H,.
6
HISMINGYEHand NIENTUNGW
1168
In F2*0.669 F~ ao.512
+ 0.273 [(T,T~
ln [(T, TZ)/H,l
I/H,
lo273
4
3
5 In
C(T, T2 I/H,
6
1
Fig. 5. Relation between F, and (T’T’)/H,.
Under various conditions, the water temperatures and the rate of evaporation of products in the first and second effects, as well as the cover temperature, may be predicted by solving Eqs. (Ig), (211, (27), (3Oh and (31) simultaneously. AcknowtedgemenfThe authors wish to express their thanks to the Chinese National Science Council for financial support through Grant No. NSC 760413EOO201. REFERENCES
1. D. C. Ginings, U.S. Put. No. 24453.50(1948). 2. M. Telkes, Office of Saline Water, Res. and Dev. Rep. No. 13, U.S. Dept. of the Interior,
Washington, DC (1956). 3. R. V. Dunkle, Znt. Dew. Heat Transfer, Conference at Denver, CO, Part 5, p. 895 (l%l). 4. M. K. Seluck, Sol. Energy 8,23 (1964). 5. E. D. Howe and B. W. Tleimate,
Fundamentals of Water Desalination, pp. 431468,
A. A. M.
Sayigh ed., Academic Press, New York, NY (1977). 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
H. M. Yeh and L. C. Chen, Energy 12, 1251 (1987). H. M. Yeh, S.W. Tsai, and N.T. Ma, Energy W, 115 (1988). L. C. Chen, Ph.D. Thesis, National Cheng Kung University, Tainan, Taiwan, R.O.C. H. C. Bryant and I. Colbeck, Sol. Energy 19, 321 (1977). A. Rabl and C. E. Nielsen, Sol. Energy 17, 1 (1975). A. Defant, Physical Oceanography, Vol. 1, p. 53 Pergamon Press, Oxford (1951). H. M. Yeh, L. W. Ten and L. C. Chen, Energy 10,683 (1985). H. M. Yeh, and L. C. Chen, Energy 10,1237 (1985). S. W. Churchill and H. S. Chu, Znt. 1. Heat Muss Transfer 18, 1323 (1975). H. M. Yeh and L. C. Chen, Energy 15, 123 (1990).
(1985).
NOMENCLATURE
A,, = Surface area of glass plate between two weirs (cm*) A, = Surface area of weirs (cm’) A,, = Surface area of the water layer between two weirs (cm2) a,b = Constant defined in Eq. (2) and shown in Table 2 c = Constant (in cm) defined in Eq. (2) and
c,;
C,,
D,; D2 F,; F2 f(z)
shown in Table 2 = Heat capacity of dry air; water (calg‘K‘) = Net distilled water production rate (gcn2sec‘) = Modified factor for glass cover; plate 2 = Fraction of incident energy remain
Upwardtype, doubleeffect solar stills
= = =
=
=
=
= =
ing after passing through a thickness z Gravitional acceleration (cmset‘) Chamber height of the first; second effect (cm) Convective heattransfer coefficient between the absorber and glass plates (Cal*_sec‘Kl) FZnvective heattransfer coefficient between the glass cover and weirs; between the glass cover and the surface of water layer (calcm2sec‘K‘) Convective heattransfer coefficient between the glass cover and air (calcm2sec‘K‘) Solar radiation incident per unit surface area of cover (Calcm‘set‘) Thermal conductivity of air (Calcm‘set‘Kr) Distance between two weirs (cm)
= Molkcular weight of dry air; water = Mass of dry air circulating between the evaporating and condensing surfaces in the first; second effect (gsee‘) Nu,, Nu,, Nuzt = h,H,IK,; h,plk,; h&lk, Pr = C,&k, = Vapor pressure of water at Z ; P,,,; e,2; ps,, G; TE(mmHg) = Total pressure (mm Hg) PT = Weir height (cm) P = Convective; radiative heat qEl; qr1 flux from the absorbing plate to the glass plate (calcm‘set‘) = Convective; radiative heat 4c2; qrz
1169
flux from the glass plate to the glass cover (Calcm‘set‘) = Convective; radiative heat flux from qen; qm the lass cover to the ambient (calcm‘set‘) qc1; qez = 1,D,; A2Dz (Calcm‘set‘) = Rayleigh number Ra T = Absolute temperature (K) T,; T2; T, = Temperature of the absorbing plate; glass plate; glass cover (K) =Temperature of the ambient; the T,; ‘I inlet water (K) = Wind speed in the atmosphere (cmV see‘) W = Width of solar still (cm) = Absolute humidity of air at the w, condensing surface of cover (g waterg air‘) = Absolute humidity of air saturated W,, with water vapor at T2 (g waterg air‘) = Axis parallel to the plate surface y (cm) = Depth of water layer (cm) z Greek letters
a (ue; % !? c*; sv# A P. cl r; rv v, p,
= Thermal diffusivity of air (cm’set‘) = Absorptivity of the glass cover; the water layer = Thermal expansion coefficient of air (Kl) = Emissivity of the glass cover; the water surface = Latent heat of water (Calg‘) = Air density (gcm’) = StefanBoltzmann constant (calcm2sec‘K4) = Transmittance of the glass cover; water surface = pa/p. (cm*seeI) = Absolute viscosity of air (gcm‘set‘)