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Performance of a salt-gradient solar pond for power production
Energy Convers. Mgmt Vo]. 25, No. 3, pp. 323-330, 1985 Printed in Great Britain. All rights reserved
0196-8904/85 $3.00+0.00 Copyright © 1985 Pergamon Press Ltd
PERFORMANCE OF A SALT-GRADIENT SOLAR POND FOR POWER PRODUCTION M . M . H A W A S and A. S. E L A S F O U R I * Faculty of Engineering, Garyounis University, Benghazi, Libya (Received 20 July 1984)
AMtract--An analytical model of the thermal behaviour of salt-gradient solar ponds is presented. The model is used to predict the performance of a solar pond intended for electric power production. The pond dimensions and heat extraction rate are optimized to achieve a high mean value and small amplitude of pond temperature. The upper convective zone is found to have a deleterious effect on pond performance. The optimum depth of the non-convective zone which maximizes the pond yield is presented. It is found that careful selection of the depth of the lower convective zone and of the amplitude and phase lag of the heat extraction rate can lead to a small value of the amplitude of the pond temperature fluctuations. Solar ponds
Salt-gradient ponds
Non-convecting ponds
NOMENCLATURE
~o = 2 x frequency = 2 x 10-7 rad/s Pw = heat capacity per unit volume of water, J/m 3 °C
a = L~/a w
H = /7 = /-7 = H0 kg, kw = Lc= Li = Li.m = Ls = P = /~ = /~ = P0 r= T= T = T= TO To = =
~
T,.0 t= v= x = ctg, ctw= 6= &o= 6p = ~/, = #n = /~, = = o=
Solar power production
daily-average solar radiation, W/m 2 mean (annual average) value of H, W/m 2 sinusoidally varying component of H, W/m: amplitude of/-7, m -2 thermal conductivity of ground water, W/m °C depth of lower convective zone, m depth of non-convective zone, m optimum depth of non-convective zone, m depth of upper convective zone, m daily-average heat extraction rate, W/m 2 mean (annual average) value of P, W/m 2 sinusoidally varying component of P, W/m 2 amplitude of P, W/m 2 effective angle of refraction, radians daily-average temperature of lower convective zone, °C mean (annual average) value of T, °C sinusoidally varying component of T, °C amplitude of ~, °C daily-average ambient temperature, °C mean (annual average) value of Ta, °C sinusoidally varying component of Ta, °C amplitude of T~, °C time measured from 21 June, s temperature of the non-convective zone, °C vertical co-ordinate (positive downwards) measured from top of non-convective zone, m thermal diffusivity of ground, water, m2/s phase lag of temperature of lower convective zone relative to solar radiation, radians phase lag of ambient temperature relative to solar radiation, radians phase lag of heat extraction rate relative to solar radiation, radians fraction of solar radiation having absorption coefficient absorption coefficient from nth portion of solar spectrum, m - ~ effective absorption coefficient, m -~ coefficient of transmission = 1 - reflective losses [2~/tO]1/2, m
*Presently on leave from Cairo University, Guiza, Egypt. 323
1. INTRODUCTION The merit o f solar p o n d s lies in their ability to collect solar energy o n a large scale a n d provide long-term h e a t storage. Solar ponds, as a source o f low-grade heat, have been f o u n d to be competitive with conventional heat sources in m a n y applications. Electric power p r o d u c t i o n f r o m large p o n d s is economically viable in s u n n y areas [1], while house heating is viable even in less favourable conditions [2]. This emphasizes the i m p o r t a n c e o f b o t h experimental a n d theoretical investigations on solar ponds, especially at low latitudes. T h e salt-gradient solar p o n d consists basically o f three zones: u p p e r convective zone, non-convective zone a n d lower convective zone. In the n o n convective zone, the salt c o n t e n t increases with depth, resulting in a n increase in density in the d o w n w a r d direction, which inhibits free convection. T h e lower convective zone serves as a heat storage from which heat can be extracted eventually. T h e u p p e r convective zone is caused mainly by surface wind mixing. Analytical models for salt-gradient solar p o n d s have been given by W e i n b e r g e r [3] a n d R a b l a n d Nielsen [4]. Weinberger considered a totally n o n convective p o n d , while R a b l a n d Nielsen presented a model for a m o r e general case in which a lower convective zone is considered. Hull [5], using computer simulation, investigated the validity o f the a s s u m p t i o n s employed in the R a b l a n d Nielsen model a n d f o u n d t h a t these a s s u m p t i o n s yield good m o d elling results. However, the R a b l a n d Nielsen model does n o t include a n u p p e r convective zone which is u n a v o i d a b l e in practice a n d has a negative effect on the p o n d performance. Also, the applicability o f their
324
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND
model, in its presented form, is limited to house heating. The purpose of this study is to modify the Rabl and Nielsen model to include an upper convective zone and to allow for a more general form of heat extraction. The modified model will then be used to predict the performance of a solar pond for electric power production, situated in Benghazi, Libya.
Solar radiation
/ L.,~
Upper convecl'ive zone
x~
Non-convechve zone L/
Lower
L~
convective zone
2. MATHEMATICAL MODEL The mathematical model presented here is based on the Rabl and Nielsen [4] formulation of the problem.
Fig. 1. The solar pond with its different zones.
at a rate given by (a) Input data
The mean daily insolation reaching the topmost
P = / ~ + P0 cos(cot - 8p),
(4)
surface of the pond, H, is taken to be purely sinusoidal so that H = / 7 +/-I, (1)
where /~ is the mean value, P0 is the amplitude of fluctuations and 8p is its phase lag with respect to the insolation.
where/7 is the mean value and/-7 is a time-dependent
The upper convective zone is assumed to be at ambient temperature. As the two subsequent zones are affected by the insolation and the ambient temperature, the quasi-steady-state temperature at any
term expressed as = H0 cos cot where/40 is the amplitude, to = 27t yr-~ and the time t = 0 at 21 June. This radiation is absorbed as it penetrates through the water layers. The solar pond treated here is as shown in Fig. 1, containing upper and lower convective zones. Following the Rabl and Nielsen model, the radiation reaching a depth x in the nonconvective zone is given by 4 Hx = z H ~, ~l,e -;"
point in the system will have a constant term and a sinusoidal term with the same frequency as that of the insolation. The temperature v in the non-convective zone will be v = f + iT,
(5)
where f is the mean value and ~ is the time-dependent term. The temperature of the lower convective zone will be
n=l
where z is the transmissivity = 1 - reflective losses, and /i, =/~,/cos r, where r is the effective angle of refraction (at 2 p.m. at equinox), and t/, is the fraction of solar radiation having absorption coefficient/~,, r/, and/~, have the following values rh = r/2 = r/3 = r/4 =
0.237 0.193 0.167 0.179
#j = #2 = #3 = #4 =
0.032 m-1 0.45 m ' 3 m -1 35 m t.
(6)
The goal of the following parts of this section is to determine the mean temperature T, the amplitude To and the phase lag 8. It will be assumed that all the relevant physical properties are constant, independent of both temperature and salinity and that there are no side losses. (b) The mean temperature T
The ambient air temperature is assumed to vary as Ta = Ta + T~,0 cos(cot - 8a),
T = T + T0 cos(cot --8).
(3)
Where Ta is the mean value, To,0is the amplitude of fluctuations, and 8, is the phase lag of the temperature with respect to the insolation. In the present treatment, only the quasi-steady-state temperature response of the pond is sought. By quasi-steady-state, it is meant that the initial starting transient has been damped out and the cyclic body temperature is established. The pond is intended to be used for power generation and the energy is assumed to be extracted
Consider an infinite pond (to neglect edge losses), under steady-state conditions, exposed to constant insolation and air temperature and exhanging no heat with the ground. The radiation reaching the lower convective zone (after traversing a distance of L, + L~) is completely absorbed there and will be equal to the heat outflow to the non-convective zone plus the heat extracted, all time-independent quantities are per unit area per unit time. Thus we have /TLi = k
df
'° d X x = L ,
+ P,
(7)
where kw is the thermal conductivity of the water.
HAWAS and ELASFOURI:
A SALT-GRADIENT SOLAR POND
The time-independent heat conduction equation for the non-convective zone, d2ff
sinh fi.(Li - x) + e ;.L, sinh//..~] 2 sinh/~.Li
J
1 dR x
dx2-kw
dx '
/' x ~cos ogt
(8)
with the boundary conditions t s ( 0 ) = T ~ f(L~) = T, can be easily solved, Substitution in equation (7) yields
and
T = Ta+ z R L t/" e-;"L'( 1 --e-~"L') - L i p k . . . . i//.
._-T=-----ff(p.a)o~)+ (B + E e -;'L' )
( 2coscot'~] x \ s i n ogt + ~ ] j ,
" (9)
Referring to equation (7), the heat balance equation of the lower convective zone for the timedependent part must include heat exchange with the
(14)
cos a sinh a cos(a - b) sinh(a - b) + sin a cosh a sin(a - b) cosh(a - b)
A =
B =
/7(L~, t) = kw~xx 05 x - L + P0 cos(cot - 6p)
cosh 2a - cos 2a sin a cosh a eos(a - b) sinh(a - b) - cos a sinh a sin(a - b) cosh(a - b) cosh 2a - cos 2a
ground as well as the change of the internal energy of the zone itself. Thus
cos a sinh a cos b sinh b + sin a cosh a sin b cosh b cosh 2a - cos 2a
D =
t
g OXix=L,+Lc
2 sin cot'~
where
k.,
(c) Time-dependent characteristics (amplitude To and phase lag 8)
--
325
sin a cosh a cos b sinh b
dT
Pw ~--dt'
(10)
-cosasinhasinbcoshb
E=
cosh 2a - cos 2a where kg is the thermal conductivity of the ground, Pw is the heat capacity of water per unit volume, and fig is the time-dependent part of the ground temperature. The quasi-steady-state temperature in the ground below the pond is given by [6]. (
~)
Vg(y) = T + To e--~'/°` cos cot - 6 - y
,
(11)
with y = x - (L~ + L~), and ag = x/(2~g/co), where ~g is the thermal diffusivity of the ground, The term 5 o f the non-convective zone temperature is obtained from the solution of the time-dependent heat conduction differential equation for this zone, viz. ~2~ c3xz
1 &5 aw~t
1 8/~(x, t) k., c3x '
(12)
with a = - L~ -, r%
b = - -x. rr.,
Knowing ~(x, t), we can substitute now in equation (10) and solve it for the two required unknowns T O and 5. In fact, equation (10) yields the required two equations because it must hold for all values of t and also because sin cot and cos cot are linearly independent. Actually, the two equations are obtained by putting t = 0 and 2n/co. Finally, we obtain tan(3 - 6.) - ct + fiR
(15)
and
subject to the boundary conditions: f ( x = 0) = T. and 5(x = L~) = T. aw Is the thermal diffusivity of the water, The solution [6] can be written as
and
To _ ~ cos(5 -- 6.) + fl sin(6 -- 6o)
(kg/rrg) + (kw/rr~)G+
'
where
g =5 + ~
R -
(kg/ag) + (k~/a.,)G+ (kg/rrg) + (k~/a.,)G
where 5 is the part due to the surface temperatures,
+ p.,L~o9
= C cos 6~ + S sin 6. - P0 cos(6p - 6~) + 2 Ta.0(k~/aw)r+
~(x, t) = 2T~,o[A cos(cot -- 6o) + B sin(cot -- 6.)] + 2T0[D cos(cot -- 6)
fl = S cos 5a -- C sin 5. -- P0 sin(6p - 6°) + E sin(cot -- ~)],
+ 2T~,o(k~/a.,)F ,
(13)
and v~, is due to the absorption of solar radiation
2zH o Z
e ;.L~
_
4
with
C = z H o ~ [l+¼(//.a~)4 ] e ~ ' t ' + ( f i . 6 ~ ) .=,
x {[ A+De
;.L,
x / [(fi.6~): - - ~2F + + F L
+ - 2e- ; . t ,
(16)
326
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND k/
)]}
x S
G
"cO 0
(/i"crw)2G+ 2
rl, e-~"Ls ((#,aw) 2 E4 [1 --~-l(finOw)4 ]~ e
-~,L,
n=l
....
d-
F(#,cr,.)_ -
e ~,L,
2
and
cosr=0.86.
-F+ + T
The properties of water and ground (average soil)
((/j,2w)2 G
)]}
are[4]
_ +G+
and sin a cosh a -I- sinh a cos a cosh 2a - cos 2a sinh 2a + sin 2a G±
z=0.974
t#,aW)L2r,-x - -
F± -
where t is the time in seconds measured from 21 June and co = 2 x 10 -7 rad/s. At 2 p.m. at equinox, the angle of incidence is 42.8 °, the angle of refraction is 30.7 ° and the reflective losses are 2.58~, hence
cosh 2a
~
COS
2a
kw=O.602W/m°C, kg = 0.96 W/m °C, aw = 1.2 m,
ag=2.15m, p~ = 4.18 kJ/m 3 °C.
(b) Time-independent parameters Equation (9) can be rewritten in the form zH 4 q,
°
3. APPLICATION The model presented in the preceding section can be used to predict the pond's performance for any application in which the rate of heat extraction varies sinusoidally, such as industrial process heating, space heating and electric power production. A solar pond for power production situated in Benghazi, Libya (latitude=32.1°N) is considered in this study, Tabor[l] has found that power production from large ponds is economically viable under weather conditions similar to that of Benghazi. Since solar ponds supply low-grade heat (at less than 100°C), the conversion to mechanical energy requires a lowtemperature heat engine. An organic vapour turbine has been developed for this purpose [7]. The conversion efficiency, being a function of Carnot cycle efficiency, increases with the temperature of the water supplied from the pond. Therefore, the pond should operate at its maximum possible temperature, and this temperature should be as steady as possible, i.e. fluctuating within a few degrees. In this section the performance of the pond is studied with the intention of determining the pond dimensions and the heat extraction pattern which yield as high a mean pond temperature and as small an amplitude of temperature fluctuations as possible.
L,)
=To_L
which shows that, for a given location, the mean value of the heat extraction rate P depends on the mean pond temperature T, the depth of the upper convective zone L~ and the depth of the nonconvective zone Li. The effect of Li o n / ~ is shown in Fig. 2. For fixed values of T and L,, an increase of Li reduces the upward heat loss. However, it also decreases the amount of radiation reaching the lower convective zone. There exists an optimum depth Li.m which maximizes the mean heat extraction rate. The concept of optimum depth of the non-convective zone was first introduced by Weinberger [3] and recently studied in more detail by Kooi [8] and Wang and Akbarazadeh [9]. By setting OP/OL~ = 0, the optimum depth L~,m is given implicitly by 4 ~, e -4"(L' +L,)[fi, L~ + 1 - e - 4 . L i ] "= ~#" kw -- ~ ( T , - T). (17)
~
The optimum depth L~,~ and the corresponding maximum value of the mean heat extraction rate P,, are ~oo 90
L,° 0 ~5
80
(a) Input data
_
The data required for the model are the radiation and ambient temperature for the location and also the water and ground properties. The pond is proposed to be located in Benghazi, Libya (latitude 32.1°N). Measured values of daily average total horizontal radiation and ambient temperature for the year 1981 (Meteorological Department, Tripoli, personal communication) were fitted to sinusoidal curves having the following expressions:
~ 60 -, ~ 50 to. 40 3o, zo
H = 211 + 101 cos(ogt) W/m 2. T~ = 20 + 8.25 cos(cot -- 0.775)°C,
kw
- Z ( r - to),
7"
/ / / 9 o .7c0 ° C
7o
// ~
~ j ] I t
"~'i ~o °o
1
~
JI j I 2 3I
4I
5I
L~tm) Fig. 2. Effect of the depth of the non-convective zone on the mean value of heat extraction rate.
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND
60 \
50
~
40
327
E E 2
~ ~
5 0 --
3 c
i~~
2O
7~90oc 10 o 30
i
]
i
I
[
I
40
50
60
70
80
90
4o 100
~'(°C)
shown as a function of the mean pond temperature /~ in Fig. 3. As the pond temperature increases, the optimum depth increases and the maximum value of the mean heat extraction rate drops, The effect of L~ on/~ is shown in Fig. 4. The upper convective zone has a deleterious effect upon the pond performance. An upper convective zone of 0.15 m reduces the heat extraction rate from 45.7 W/m 2 (corresponding to Ls = 0) to 38 W/m 2. If L s increases to 0.5 m, P would drop to 28 W/m 2. Similar results were found by other investigators [8, 9]. The negative effect of the upper convective zone is obvious since this zone absorbs a portion of the radiation while not contributing to the thermal insulation of the pond. Figure 5 shows the effect of L s on the optimum depth L~,m and the maximum value of the mean heat extraction rate/~m. The results shown in Figs 4 and 5 emphasize the importance of reducing the thickness of the upper convective zone as much as is practicable. Some ways have been proposed to reduce L , such as using floating plastic pipes and plastic grids [1]. In the following calculations, L~ is assumed to have the value of 0.15 m which seems to be a reasonable average value of the thickness of this zone [l, 8,9].
5o
e4
~ ° 90 ° c L,=O95m
~ 40 ~
I~ 3o
J 02
2 03
Fig. 5. Effects of the depth of the upper convective zone (Ls) on the optimum depth of the non-convective zone (L,.m) and the maximum heat extraction rate (/~,,). As mentioned above, the pond temperature should be as high as possible to achieve a relatively high conversion efficiency. On the other hand, the pond temperature should not be allowed to exceed the water boiling temperature. Therefore, 90°C seems to be a reasonable value for the pond mean temperature /'. Referring to Fig. 3, at this temperature the optimum depth Li.,, equals 2.75 m which corresponds to the maximum value of the mean heat extraction rate/~m of 52.3 W/m E. This depth may be too large from an economic point of view. Figure 2 shows that a reduction of the depth Li to one-half of its optimum value results in decreasing the heat extraction rate by only 10%. This shows that the depth of the non-convective zone cannot be determined without considering the economic aspects of the problem. Economic analysis is beyond the scope of the present study. However, in this study, the depth L~ is selected as that which maximizes the heat extraction rate per unit pond depth. Figure 6 displays the variation of P/(L~ + L,) with L~ for :T = 90°C and Ls = 0.15 m (note that Lc does not affect /v). The figure shows that the maximum value of P/(L~+ L J occurs at L~ = 0.95 m. Under these conditions the mean value of the heat extraction rate /~ is 38 W/m:. It may be interesting to note that a value of L, of about I m was used in most of the solar ponds already constructed and operated. The preceding analysis indicates that a pond of Ls of 0.15 m and L~ of 0.95 m operating at :£ of 90°C could provide a heat extraction rate /~ of 38 W/m E.
(c) Time-dependent parameters [
200 6-
I 0.1
Ls(m)
Fig. 3. Effect of the mean pond temperature (T) on the optimum depth of the non-convective zone (L~,,) and the maximum heat extraction rate (P,,).'
_
-. O0
ol
L_
I o3
I o4
I o5
o2 Ls Fig. 4. Effect of the depth of the upper convective zone (L,) on the mean value of the heat extraction rate (:P).
Part (c) of the mathematical model shows that the amplitude of temperature fluctuations To depends on the depth of the lower convective zone L, and the amplitude and phase lag of the heat extraction rate P0 and 5p besides the dependence on L~ and L,. The values of L~ and Ls have been selected in the preceding
HAWAS and ELASFOURI:
328 60
A SALT-GRADIENT SOLAR POND
60
-/
50
~
40
~
50
PO = l O W / m e
L¢ =10
173 40
T-90*C
A
Ls=Of5m
I1~
~
172
%
171
~ I
I
I
I
I
I
~
~0
i
~
I
I
PO = 3 0 W / m 2
..
o~
'
67
10
E
~ -4-
O
I
10
66
O
39
'
,..o ~:
PO = 3 0 W / r n 2
E
Lc=30m
-4"
~
I
~ I
I
1
2
3
4
5
L,.(m) 3.7
Fig. 6. Variation of/~ and
[P/(Li+
L,)] with L,.
36
I 0
002
I
I
I
006
I
f
010
014
I
I
I
018
~p (rod)
section. In this section, the effects of Lc, P0 and 6e on To are studied with the intention of minimizing the temperature amplitude To. The effect of 6p on To is shown in Fig. 7. The amplitude To first decreases with an increase of fp and then rapidly increases. There is an optimum value of fp which minimizes To. Increasing 5p from its optim u m value of 0.12 to 1 rad results in an increase of TO from 6.6 to 17.4°C. This shows that fp should be selected to achieve a small value of To. Figure 8 shows the effect of fie on TO for various values of Po and Lc. The optimum value of fp was found to be 0.12 tad in all cases. It is also evident from Fig. 8 that the minimum value of To depends on both Lc and PoFigures 9 and 10 display the variation of To and 5 with P0 and Lc. It is to be noted that the amplitude of the heat extraction rate Po cannot exceed 38 W / m 2, which is the mean value of the heat extraction rate P
Fig. 8. Effect of the heat extraction
phase
lag (5p) on the
temperature amplitude (To) for various values of (Lc) and (P0) (7` = 90°C, Ls = 0.15 m and L, = 0.95 m). (as determined by the values of Ls, Li and To), otherwise it would lead to a negative heat extraction rate in some days. Figure 9 shows that the temperature phase lag f does not depend on P0. The dependence of f on L c is displayed in Fig. 10. Actually, the primary concern is the temperature amplitude TO rather than its phase lag 6. Figures 9 and 10 show that an increase in either Po or Lc results in reducing To. There is no optimum value of either Po or L c in the sense that it results in a minimum value of To. To reduce the amplitude of temperature fluctuations, one may extract heat with higher ampli24 -
16
22 20
20
18
18
14
~P = O ~ 2
X " k
X ~
I
rod
15
2°
14 "O
,..,
12
~
12
t
1o
~o
~o
k°
~
8
6
6
~
4
~
I 01
I 02
I I 03 0 4
I 05
I 06
/m
,o
_
1.1
z
2 I 07
L~,2 rr
-1.2
4
Po - 3 0 W Lc=lOm
2 0
___-~
--1.3
I 08
I 0.9
I 10
8p(rod) Fig. 7. Effect of the heat extraction phase lag (hp) on the temperature amplitude (To) (T = 90°C, L~ = 0.95 m and Ls = 0.15 m).
-
0 0
[
-
I 10
I 20
I 30
I 40
J1 0 50
Po(W/m 2 ) Fig. 9. Effect of the amplitude of the heat extraction rate (P0) on the temperature amplitude (To) and phase lag (6) (7" = 90°C, Ls = 0.15 m and L~ = 0.95 m).
HAWAS and ELASFOURI: 20
- 1.5
I e
,DO - 20 and 38 W/m22._ /
I6 ~ I 4
k
I"
~
12i
p
A SALT-GRADIENT SOLAR POND
/
/
10
8 /i/
/ / / ~"~Po
300
1.3
~E 250
1.2
TO
1.1
A
~: 150
:2 : ~I"2 2
'10
o
t00
0.9
~
~
018
4 , "~...~..~2
017
1I
~ 200
OW/m--~
6
0
35O
II4
Bp-012rod
~ / /
329
38 W/m 2
0.6
2I L~(m)
3I
30
~j *-.~b
lO
4 05
Fig. 10. Effect of the depth of the lower convectivezone (Lc) on the temperature amplitude (To) and the phase lag (&)
(T=90°C'Ls=O'15mandL~=O'95m)" tude and/or increase the depth of the lower convective zone. A higher amplitude of heat extraction rate, resulting in a higher amplitude of produced electric power, necessitates the use of an electric storage of higher capacity. On the other hand, a deeper pond might be more expensive. It can be concluded that the optimum values of P0 and L c can only be determined through a detailed economic analysis. However, thermal analysis can lead to important conclusions. It can be seen, from Figs 9 and 10, that the effect of P0 on To is more pronounced than that of Lc. A fixed value of To, say 3°C, can be achieved by various combinations of P0 and Lc. Figure 11 shows the values of Lc, required to maintain To at 3°C, as a function of P0An increase of P0 from 30 to 38 W/m 2 (nearly 1.3 times) would reduce the required Lc from 4 to 0.4 m (10 times). It is, therefore, quite reasonable to select a value of 38 W/m 2 for P0 which corresponds to L c of 0.4 m. Under these conditions, the phase lag of the
25! 20 ~5
A
95
,o 90 ~ k 85 ~ 10050
~
~
E o
1
2
3
4
5
6
7
8
9
10 11 1-2
Time in months ( starting from dune 21) Fig. 12. Variation of daily-average values of solar radiation, ambient temperature, pond temperature and heat extraction rate throughout the year (Ls = 0.15 m, L~= 0.95 m, L~.= 0.4 m,/~ = 38 W/m2, P0 = 38 W/m2 and tip = 0.12 rad).
pond temperature from equation (15) would be 0.873 tad. From the above discussion, it can be seen that the most reasonable dimensions of the pond are Ls = 0.15 m, L~ = 0.95 m, and L~ = 0.4 m. This pond would, under Benghazi weather conditions, operate at a mean temperature T of 90°C with an amplitude TOof 3°C, and would provide energy extraction at a mean rate /~ of 38 W/m 2 with an amplitude P0 of 38W/m 2 and phase lag 6p of 0.12rad. Figure 12 shows the variation of the daily average values of ~1total radiation, ambient temperature, pond tem%° 3 o *c perature and rate of heat extraction through the year for the proposed pond. 4 It may be noted that, during the present study, an article [10] has been published presenting an ana3 lytical model for salt-gradient solar ponds similar to -~ the one given in our study. We are in doubt about the results published in that article for two reasons. First, ..4 ~ 2 we have checked that model, after correcting some printing errors, by inserting the data of Rabl and I Nielsen [4] and obtained their results. However, the results published in Ref. [10] are inconsistent with its model. Second, the solar radiation data given in that I J I I I 030 32 34 36 38 40 article are not realistic. A mean annual radiation 2 ) (denoted b y / 7 in the article) of 538.8 W/m 2 is given Po( W / m for a location at 30°N latitude. This is a highly Fig. 11. Effect of the amplitude of the heat extraction rate (P0) on the depth of the lower convective zone (Lc) for a exaggerated value compared with 244.7 W/m 2 for fixed value of the temperature amplitude (To) (7" = 90°C, 32°N reported by Tabor and Weinberger [11] and L~= 0.95 m and Ls = 0.15 m). with 211 W/m 2 for 32. I°N in the present work.
[
330
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND 4. CONCLUSIONS
An analytical model for predicting the thermal performance of salt-gradient solar ponds has been developed. This model represents a modification of the Rabl and Nielsen model. It considers an upper convective zone, and its application is not limited to space heating only. The model has been used to predict the performance of a solar pond proposed for electric power production in Benghazi, Libya. The pond dimensions and the pattern of the heat extraction rate have been determined such that the pond should operate at as high a mean temperature and as small an amplitude of temperature fluctuations as possible. It has been found that the upper convective zone has a serious negative effect on the pond performance and should be kept as thin as possible. For given values of the mean pond temperature and depth of the upper convective zone, there is an optimum depth of the non-convective zone which maximizes the mean heat extraction rate. However, this optimum depth seems to be economically too large. The depth of the non-convective zone has, therefore, been selected as that which maximizes the heat extraction rate per unit pond depth. The dependence of the temperature amplitude on the depth of the lower convective zone and the
pattern of the heat extraction rate has been studied. It has been found that there is an optimum value of the phase lag of the heat extraction rate which minimizes the temperature amplitude. It has been also found that an increase of the amplitude of the heat extraction rate and/or the depth of the lower convective zone reduces the temperature amplitude, the effect of the amplitude of the heat extraction rate being more pronounced. REFERENCES I. H. Tabor, Solar Energy 27, 181 (1981). 2. H. C. Bryant and I. Colbeck, Solar Energy 19, 321 (1977). 3. H. Weinberger, Solar Energy 8, 45 (1964). 4. A. Rabl and C. E. Nielsen, Solar Energy 17, 1 (1975). 5. J. R. Hull, Solar Energy 25, 33 (1980). 6. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn, pp. 65, 105. Oxford University Press, O x f o r d(1959). 7. H. Tabor and L. Bronicki, U.N. Conf. New Sources of Energy,Rome, Paper 35/S 54 (1961). 8. C. F. Kooi, Solar Energy 23, 37 (1979). 9. Y. F. Wang and A. Akbarzadeh, Solar Energy 30, 555 (1983). 10. E. A. Asian Abdel Salam, S. T. Probert, P. W. O'Kallaghan, M. Hussein and B. Norton, Appl. Energy 16, 283 (1984). 11. H. Tabor and H. Z. Weinberger, Non-convecting solar ponds. In Solar Energy Handbook (edited by J. F. Kreider). McGraw-Hill, New York (1980).
0196-8904/85 $3.00+0.00 Copyright © 1985 Pergamon Press Ltd
PERFORMANCE OF A SALT-GRADIENT SOLAR POND FOR POWER PRODUCTION M . M . H A W A S and A. S. E L A S F O U R I * Faculty of Engineering, Garyounis University, Benghazi, Libya (Received 20 July 1984)
AMtract--An analytical model of the thermal behaviour of salt-gradient solar ponds is presented. The model is used to predict the performance of a solar pond intended for electric power production. The pond dimensions and heat extraction rate are optimized to achieve a high mean value and small amplitude of pond temperature. The upper convective zone is found to have a deleterious effect on pond performance. The optimum depth of the non-convective zone which maximizes the pond yield is presented. It is found that careful selection of the depth of the lower convective zone and of the amplitude and phase lag of the heat extraction rate can lead to a small value of the amplitude of the pond temperature fluctuations. Solar ponds
Salt-gradient ponds
Non-convecting ponds
NOMENCLATURE
~o = 2 x frequency = 2 x 10-7 rad/s Pw = heat capacity per unit volume of water, J/m 3 °C
a = L~/a w
H = /7 = /-7 = H0 kg, kw = Lc= Li = Li.m = Ls = P = /~ = /~ = P0 r= T= T = T= TO To = =
~
T,.0 t= v= x = ctg, ctw= 6= &o= 6p = ~/, = #n = /~, = = o=
Solar power production
daily-average solar radiation, W/m 2 mean (annual average) value of H, W/m 2 sinusoidally varying component of H, W/m: amplitude of/-7, m -2 thermal conductivity of ground water, W/m °C depth of lower convective zone, m depth of non-convective zone, m optimum depth of non-convective zone, m depth of upper convective zone, m daily-average heat extraction rate, W/m 2 mean (annual average) value of P, W/m 2 sinusoidally varying component of P, W/m 2 amplitude of P, W/m 2 effective angle of refraction, radians daily-average temperature of lower convective zone, °C mean (annual average) value of T, °C sinusoidally varying component of T, °C amplitude of ~, °C daily-average ambient temperature, °C mean (annual average) value of Ta, °C sinusoidally varying component of Ta, °C amplitude of T~, °C time measured from 21 June, s temperature of the non-convective zone, °C vertical co-ordinate (positive downwards) measured from top of non-convective zone, m thermal diffusivity of ground, water, m2/s phase lag of temperature of lower convective zone relative to solar radiation, radians phase lag of ambient temperature relative to solar radiation, radians phase lag of heat extraction rate relative to solar radiation, radians fraction of solar radiation having absorption coefficient absorption coefficient from nth portion of solar spectrum, m - ~ effective absorption coefficient, m -~ coefficient of transmission = 1 - reflective losses [2~/tO]1/2, m
*Presently on leave from Cairo University, Guiza, Egypt. 323
1. INTRODUCTION The merit o f solar p o n d s lies in their ability to collect solar energy o n a large scale a n d provide long-term h e a t storage. Solar ponds, as a source o f low-grade heat, have been f o u n d to be competitive with conventional heat sources in m a n y applications. Electric power p r o d u c t i o n f r o m large p o n d s is economically viable in s u n n y areas [1], while house heating is viable even in less favourable conditions [2]. This emphasizes the i m p o r t a n c e o f b o t h experimental a n d theoretical investigations on solar ponds, especially at low latitudes. T h e salt-gradient solar p o n d consists basically o f three zones: u p p e r convective zone, non-convective zone a n d lower convective zone. In the n o n convective zone, the salt c o n t e n t increases with depth, resulting in a n increase in density in the d o w n w a r d direction, which inhibits free convection. T h e lower convective zone serves as a heat storage from which heat can be extracted eventually. T h e u p p e r convective zone is caused mainly by surface wind mixing. Analytical models for salt-gradient solar p o n d s have been given by W e i n b e r g e r [3] a n d R a b l a n d Nielsen [4]. Weinberger considered a totally n o n convective p o n d , while R a b l a n d Nielsen presented a model for a m o r e general case in which a lower convective zone is considered. Hull [5], using computer simulation, investigated the validity o f the a s s u m p t i o n s employed in the R a b l a n d Nielsen model a n d f o u n d t h a t these a s s u m p t i o n s yield good m o d elling results. However, the R a b l a n d Nielsen model does n o t include a n u p p e r convective zone which is u n a v o i d a b l e in practice a n d has a negative effect on the p o n d performance. Also, the applicability o f their
324
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND
model, in its presented form, is limited to house heating. The purpose of this study is to modify the Rabl and Nielsen model to include an upper convective zone and to allow for a more general form of heat extraction. The modified model will then be used to predict the performance of a solar pond for electric power production, situated in Benghazi, Libya.
Solar radiation
/ L.,~
Upper convecl'ive zone
x~
Non-convechve zone L/
Lower
L~
convective zone
2. MATHEMATICAL MODEL The mathematical model presented here is based on the Rabl and Nielsen [4] formulation of the problem.
Fig. 1. The solar pond with its different zones.
at a rate given by (a) Input data
The mean daily insolation reaching the topmost
P = / ~ + P0 cos(cot - 8p),
(4)
surface of the pond, H, is taken to be purely sinusoidal so that H = / 7 +/-I, (1)
where /~ is the mean value, P0 is the amplitude of fluctuations and 8p is its phase lag with respect to the insolation.
where/7 is the mean value and/-7 is a time-dependent
The upper convective zone is assumed to be at ambient temperature. As the two subsequent zones are affected by the insolation and the ambient temperature, the quasi-steady-state temperature at any
term expressed as = H0 cos cot where/40 is the amplitude, to = 27t yr-~ and the time t = 0 at 21 June. This radiation is absorbed as it penetrates through the water layers. The solar pond treated here is as shown in Fig. 1, containing upper and lower convective zones. Following the Rabl and Nielsen model, the radiation reaching a depth x in the nonconvective zone is given by 4 Hx = z H ~, ~l,e -;"
point in the system will have a constant term and a sinusoidal term with the same frequency as that of the insolation. The temperature v in the non-convective zone will be v = f + iT,
(5)
where f is the mean value and ~ is the time-dependent term. The temperature of the lower convective zone will be
n=l
where z is the transmissivity = 1 - reflective losses, and /i, =/~,/cos r, where r is the effective angle of refraction (at 2 p.m. at equinox), and t/, is the fraction of solar radiation having absorption coefficient/~,, r/, and/~, have the following values rh = r/2 = r/3 = r/4 =
0.237 0.193 0.167 0.179
#j = #2 = #3 = #4 =
0.032 m-1 0.45 m ' 3 m -1 35 m t.
(6)
The goal of the following parts of this section is to determine the mean temperature T, the amplitude To and the phase lag 8. It will be assumed that all the relevant physical properties are constant, independent of both temperature and salinity and that there are no side losses. (b) The mean temperature T
The ambient air temperature is assumed to vary as Ta = Ta + T~,0 cos(cot - 8a),
T = T + T0 cos(cot --8).
(3)
Where Ta is the mean value, To,0is the amplitude of fluctuations, and 8, is the phase lag of the temperature with respect to the insolation. In the present treatment, only the quasi-steady-state temperature response of the pond is sought. By quasi-steady-state, it is meant that the initial starting transient has been damped out and the cyclic body temperature is established. The pond is intended to be used for power generation and the energy is assumed to be extracted
Consider an infinite pond (to neglect edge losses), under steady-state conditions, exposed to constant insolation and air temperature and exhanging no heat with the ground. The radiation reaching the lower convective zone (after traversing a distance of L, + L~) is completely absorbed there and will be equal to the heat outflow to the non-convective zone plus the heat extracted, all time-independent quantities are per unit area per unit time. Thus we have /TLi = k
df
'° d X x = L ,
+ P,
(7)
where kw is the thermal conductivity of the water.
HAWAS and ELASFOURI:
A SALT-GRADIENT SOLAR POND
The time-independent heat conduction equation for the non-convective zone, d2ff
sinh fi.(Li - x) + e ;.L, sinh//..~] 2 sinh/~.Li
J
1 dR x
dx2-kw
dx '
/' x ~cos ogt
(8)
with the boundary conditions t s ( 0 ) = T ~ f(L~) = T, can be easily solved, Substitution in equation (7) yields
and
T = Ta+ z R L t/" e-;"L'( 1 --e-~"L') - L i p k . . . . i//.
._-T=-----ff(p.a)o~)+ (B + E e -;'L' )
( 2coscot'~] x \ s i n ogt + ~ ] j ,
" (9)
Referring to equation (7), the heat balance equation of the lower convective zone for the timedependent part must include heat exchange with the
(14)
cos a sinh a cos(a - b) sinh(a - b) + sin a cosh a sin(a - b) cosh(a - b)
A =
B =
/7(L~, t) = kw~xx 05 x - L + P0 cos(cot - 6p)
cosh 2a - cos 2a sin a cosh a eos(a - b) sinh(a - b) - cos a sinh a sin(a - b) cosh(a - b) cosh 2a - cos 2a
ground as well as the change of the internal energy of the zone itself. Thus
cos a sinh a cos b sinh b + sin a cosh a sin b cosh b cosh 2a - cos 2a
D =
t
g OXix=L,+Lc
2 sin cot'~
where
k.,
(c) Time-dependent characteristics (amplitude To and phase lag 8)
--
325
sin a cosh a cos b sinh b
dT
Pw ~--dt'
(10)
-cosasinhasinbcoshb
E=
cosh 2a - cos 2a where kg is the thermal conductivity of the ground, Pw is the heat capacity of water per unit volume, and fig is the time-dependent part of the ground temperature. The quasi-steady-state temperature in the ground below the pond is given by [6]. (
~)
Vg(y) = T + To e--~'/°` cos cot - 6 - y
,
(11)
with y = x - (L~ + L~), and ag = x/(2~g/co), where ~g is the thermal diffusivity of the ground, The term 5 o f the non-convective zone temperature is obtained from the solution of the time-dependent heat conduction differential equation for this zone, viz. ~2~ c3xz
1 &5 aw~t
1 8/~(x, t) k., c3x '
(12)
with a = - L~ -, r%
b = - -x. rr.,
Knowing ~(x, t), we can substitute now in equation (10) and solve it for the two required unknowns T O and 5. In fact, equation (10) yields the required two equations because it must hold for all values of t and also because sin cot and cos cot are linearly independent. Actually, the two equations are obtained by putting t = 0 and 2n/co. Finally, we obtain tan(3 - 6.) - ct + fiR
(15)
and
subject to the boundary conditions: f ( x = 0) = T. and 5(x = L~) = T. aw Is the thermal diffusivity of the water, The solution [6] can be written as
and
To _ ~ cos(5 -- 6.) + fl sin(6 -- 6o)
(kg/rrg) + (kw/rr~)G+
'
where
g =5 + ~
R -
(kg/ag) + (k~/a.,)G+ (kg/rrg) + (k~/a.,)G
where 5 is the part due to the surface temperatures,
+ p.,L~o9
= C cos 6~ + S sin 6. - P0 cos(6p - 6~) + 2 Ta.0(k~/aw)r+
~(x, t) = 2T~,o[A cos(cot -- 6o) + B sin(cot -- 6.)] + 2T0[D cos(cot -- 6)
fl = S cos 5a -- C sin 5. -- P0 sin(6p - 6°) + E sin(cot -- ~)],
+ 2T~,o(k~/a.,)F ,
(13)
and v~, is due to the absorption of solar radiation
2zH o Z
e ;.L~
_
4
with
C = z H o ~ [l+¼(//.a~)4 ] e ~ ' t ' + ( f i . 6 ~ ) .=,
x {[ A+De
;.L,
x / [(fi.6~): - - ~2F + + F L
+ - 2e- ; . t ,
(16)
326
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND k/
)]}
x S
G
"cO 0
(/i"crw)2G+ 2
rl, e-~"Ls ((#,aw) 2 E4 [1 --~-l(finOw)4 ]~ e
-~,L,
n=l
....
d-
F(#,cr,.)_ -
e ~,L,
2
and
cosr=0.86.
-F+ + T
The properties of water and ground (average soil)
((/j,2w)2 G
)]}
are[4]
_ +G+
and sin a cosh a -I- sinh a cos a cosh 2a - cos 2a sinh 2a + sin 2a G±
z=0.974
t#,aW)L2r,-x - -
F± -
where t is the time in seconds measured from 21 June and co = 2 x 10 -7 rad/s. At 2 p.m. at equinox, the angle of incidence is 42.8 °, the angle of refraction is 30.7 ° and the reflective losses are 2.58~, hence
cosh 2a
~
COS
2a
kw=O.602W/m°C, kg = 0.96 W/m °C, aw = 1.2 m,
ag=2.15m, p~ = 4.18 kJ/m 3 °C.
(b) Time-independent parameters Equation (9) can be rewritten in the form zH 4 q,
°
3. APPLICATION The model presented in the preceding section can be used to predict the pond's performance for any application in which the rate of heat extraction varies sinusoidally, such as industrial process heating, space heating and electric power production. A solar pond for power production situated in Benghazi, Libya (latitude=32.1°N) is considered in this study, Tabor[l] has found that power production from large ponds is economically viable under weather conditions similar to that of Benghazi. Since solar ponds supply low-grade heat (at less than 100°C), the conversion to mechanical energy requires a lowtemperature heat engine. An organic vapour turbine has been developed for this purpose [7]. The conversion efficiency, being a function of Carnot cycle efficiency, increases with the temperature of the water supplied from the pond. Therefore, the pond should operate at its maximum possible temperature, and this temperature should be as steady as possible, i.e. fluctuating within a few degrees. In this section the performance of the pond is studied with the intention of determining the pond dimensions and the heat extraction pattern which yield as high a mean pond temperature and as small an amplitude of temperature fluctuations as possible.
L,)
=To_L
which shows that, for a given location, the mean value of the heat extraction rate P depends on the mean pond temperature T, the depth of the upper convective zone L~ and the depth of the nonconvective zone Li. The effect of Li o n / ~ is shown in Fig. 2. For fixed values of T and L,, an increase of Li reduces the upward heat loss. However, it also decreases the amount of radiation reaching the lower convective zone. There exists an optimum depth Li.m which maximizes the mean heat extraction rate. The concept of optimum depth of the non-convective zone was first introduced by Weinberger [3] and recently studied in more detail by Kooi [8] and Wang and Akbarazadeh [9]. By setting OP/OL~ = 0, the optimum depth L~,m is given implicitly by 4 ~, e -4"(L' +L,)[fi, L~ + 1 - e - 4 . L i ] "= ~#" kw -- ~ ( T , - T). (17)
~
The optimum depth L~,~ and the corresponding maximum value of the mean heat extraction rate P,, are ~oo 90
L,° 0 ~5
80
(a) Input data
_
The data required for the model are the radiation and ambient temperature for the location and also the water and ground properties. The pond is proposed to be located in Benghazi, Libya (latitude 32.1°N). Measured values of daily average total horizontal radiation and ambient temperature for the year 1981 (Meteorological Department, Tripoli, personal communication) were fitted to sinusoidal curves having the following expressions:
~ 60 -, ~ 50 to. 40 3o, zo
H = 211 + 101 cos(ogt) W/m 2. T~ = 20 + 8.25 cos(cot -- 0.775)°C,
kw
- Z ( r - to),
7"
/ / / 9 o .7c0 ° C
7o
// ~
~ j ] I t
"~'i ~o °o
1
~
JI j I 2 3I
4I
5I
L~tm) Fig. 2. Effect of the depth of the non-convective zone on the mean value of heat extraction rate.
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND
60 \
50
~
40
327
E E 2
~ ~
5 0 --
3 c
i~~
2O
7~90oc 10 o 30
i
]
i
I
[
I
40
50
60
70
80
90
4o 100
~'(°C)
shown as a function of the mean pond temperature /~ in Fig. 3. As the pond temperature increases, the optimum depth increases and the maximum value of the mean heat extraction rate drops, The effect of L~ on/~ is shown in Fig. 4. The upper convective zone has a deleterious effect upon the pond performance. An upper convective zone of 0.15 m reduces the heat extraction rate from 45.7 W/m 2 (corresponding to Ls = 0) to 38 W/m 2. If L s increases to 0.5 m, P would drop to 28 W/m 2. Similar results were found by other investigators [8, 9]. The negative effect of the upper convective zone is obvious since this zone absorbs a portion of the radiation while not contributing to the thermal insulation of the pond. Figure 5 shows the effect of L s on the optimum depth L~,m and the maximum value of the mean heat extraction rate/~m. The results shown in Figs 4 and 5 emphasize the importance of reducing the thickness of the upper convective zone as much as is practicable. Some ways have been proposed to reduce L , such as using floating plastic pipes and plastic grids [1]. In the following calculations, L~ is assumed to have the value of 0.15 m which seems to be a reasonable average value of the thickness of this zone [l, 8,9].
5o
e4
~ ° 90 ° c L,=O95m
~ 40 ~
I~ 3o
J 02
2 03
Fig. 5. Effects of the depth of the upper convective zone (Ls) on the optimum depth of the non-convective zone (L,.m) and the maximum heat extraction rate (/~,,). As mentioned above, the pond temperature should be as high as possible to achieve a relatively high conversion efficiency. On the other hand, the pond temperature should not be allowed to exceed the water boiling temperature. Therefore, 90°C seems to be a reasonable value for the pond mean temperature /'. Referring to Fig. 3, at this temperature the optimum depth Li.,, equals 2.75 m which corresponds to the maximum value of the mean heat extraction rate/~m of 52.3 W/m E. This depth may be too large from an economic point of view. Figure 2 shows that a reduction of the depth Li to one-half of its optimum value results in decreasing the heat extraction rate by only 10%. This shows that the depth of the non-convective zone cannot be determined without considering the economic aspects of the problem. Economic analysis is beyond the scope of the present study. However, in this study, the depth L~ is selected as that which maximizes the heat extraction rate per unit pond depth. Figure 6 displays the variation of P/(L~ + L,) with L~ for :T = 90°C and Ls = 0.15 m (note that Lc does not affect /v). The figure shows that the maximum value of P/(L~+ L J occurs at L~ = 0.95 m. Under these conditions the mean value of the heat extraction rate /~ is 38 W/m:. It may be interesting to note that a value of L, of about I m was used in most of the solar ponds already constructed and operated. The preceding analysis indicates that a pond of Ls of 0.15 m and L~ of 0.95 m operating at :£ of 90°C could provide a heat extraction rate /~ of 38 W/m E.
(c) Time-dependent parameters [
200 6-
I 0.1
Ls(m)
Fig. 3. Effect of the mean pond temperature (T) on the optimum depth of the non-convective zone (L~,,) and the maximum heat extraction rate (P,,).'
_
-. O0
ol
L_
I o3
I o4
I o5
o2 Ls Fig. 4. Effect of the depth of the upper convective zone (L,) on the mean value of the heat extraction rate (:P).
Part (c) of the mathematical model shows that the amplitude of temperature fluctuations To depends on the depth of the lower convective zone L, and the amplitude and phase lag of the heat extraction rate P0 and 5p besides the dependence on L~ and L,. The values of L~ and Ls have been selected in the preceding
HAWAS and ELASFOURI:
328 60
A SALT-GRADIENT SOLAR POND
60
-/
50
~
40
~
50
PO = l O W / m e
L¢ =10
173 40
T-90*C
A
Ls=Of5m
I1~
~
172
%
171
~ I
I
I
I
I
I
~
~0
i
~
I
I
PO = 3 0 W / m 2
..
o~
'
67
10
E
~ -4-
O
I
10
66
O
39
'
,..o ~:
PO = 3 0 W / r n 2
E
Lc=30m
-4"
~
I
~ I
I
1
2
3
4
5
L,.(m) 3.7
Fig. 6. Variation of/~ and
[P/(Li+
L,)] with L,.
36
I 0
002
I
I
I
006
I
f
010
014
I
I
I
018
~p (rod)
section. In this section, the effects of Lc, P0 and 6e on To are studied with the intention of minimizing the temperature amplitude To. The effect of 6p on To is shown in Fig. 7. The amplitude To first decreases with an increase of fp and then rapidly increases. There is an optimum value of fp which minimizes To. Increasing 5p from its optim u m value of 0.12 to 1 rad results in an increase of TO from 6.6 to 17.4°C. This shows that fp should be selected to achieve a small value of To. Figure 8 shows the effect of fie on TO for various values of Po and Lc. The optimum value of fp was found to be 0.12 tad in all cases. It is also evident from Fig. 8 that the minimum value of To depends on both Lc and PoFigures 9 and 10 display the variation of To and 5 with P0 and Lc. It is to be noted that the amplitude of the heat extraction rate Po cannot exceed 38 W / m 2, which is the mean value of the heat extraction rate P
Fig. 8. Effect of the heat extraction
phase
lag (5p) on the
temperature amplitude (To) for various values of (Lc) and (P0) (7` = 90°C, Ls = 0.15 m and L, = 0.95 m). (as determined by the values of Ls, Li and To), otherwise it would lead to a negative heat extraction rate in some days. Figure 9 shows that the temperature phase lag f does not depend on P0. The dependence of f on L c is displayed in Fig. 10. Actually, the primary concern is the temperature amplitude TO rather than its phase lag 6. Figures 9 and 10 show that an increase in either Po or Lc results in reducing To. There is no optimum value of either Po or L c in the sense that it results in a minimum value of To. To reduce the amplitude of temperature fluctuations, one may extract heat with higher ampli24 -
16
22 20
20
18
18
14
~P = O ~ 2
X " k
X ~
I
rod
15
2°
14 "O
,..,
12
~
12
t
1o
~o
~o
k°
~
8
6
6
~
4
~
I 01
I 02
I I 03 0 4
I 05
I 06
/m
,o
_
1.1
z
2 I 07
L~,2 rr
-1.2
4
Po - 3 0 W Lc=lOm
2 0
___-~
--1.3
I 08
I 0.9
I 10
8p(rod) Fig. 7. Effect of the heat extraction phase lag (hp) on the temperature amplitude (To) (T = 90°C, L~ = 0.95 m and Ls = 0.15 m).
-
0 0
[
-
I 10
I 20
I 30
I 40
J1 0 50
Po(W/m 2 ) Fig. 9. Effect of the amplitude of the heat extraction rate (P0) on the temperature amplitude (To) and phase lag (6) (7" = 90°C, Ls = 0.15 m and L~ = 0.95 m).
HAWAS and ELASFOURI: 20
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Fig. 10. Effect of the depth of the lower convectivezone (Lc) on the temperature amplitude (To) and the phase lag (&)
(T=90°C'Ls=O'15mandL~=O'95m)" tude and/or increase the depth of the lower convective zone. A higher amplitude of heat extraction rate, resulting in a higher amplitude of produced electric power, necessitates the use of an electric storage of higher capacity. On the other hand, a deeper pond might be more expensive. It can be concluded that the optimum values of P0 and L c can only be determined through a detailed economic analysis. However, thermal analysis can lead to important conclusions. It can be seen, from Figs 9 and 10, that the effect of P0 on To is more pronounced than that of Lc. A fixed value of To, say 3°C, can be achieved by various combinations of P0 and Lc. Figure 11 shows the values of Lc, required to maintain To at 3°C, as a function of P0An increase of P0 from 30 to 38 W/m 2 (nearly 1.3 times) would reduce the required Lc from 4 to 0.4 m (10 times). It is, therefore, quite reasonable to select a value of 38 W/m 2 for P0 which corresponds to L c of 0.4 m. Under these conditions, the phase lag of the
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1
2
3
4
5
6
7
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Time in months ( starting from dune 21) Fig. 12. Variation of daily-average values of solar radiation, ambient temperature, pond temperature and heat extraction rate throughout the year (Ls = 0.15 m, L~= 0.95 m, L~.= 0.4 m,/~ = 38 W/m2, P0 = 38 W/m2 and tip = 0.12 rad).
pond temperature from equation (15) would be 0.873 tad. From the above discussion, it can be seen that the most reasonable dimensions of the pond are Ls = 0.15 m, L~ = 0.95 m, and L~ = 0.4 m. This pond would, under Benghazi weather conditions, operate at a mean temperature T of 90°C with an amplitude TOof 3°C, and would provide energy extraction at a mean rate /~ of 38 W/m 2 with an amplitude P0 of 38W/m 2 and phase lag 6p of 0.12rad. Figure 12 shows the variation of the daily average values of ~1total radiation, ambient temperature, pond tem%° 3 o *c perature and rate of heat extraction through the year for the proposed pond. 4 It may be noted that, during the present study, an article [10] has been published presenting an ana3 lytical model for salt-gradient solar ponds similar to -~ the one given in our study. We are in doubt about the results published in that article for two reasons. First, ..4 ~ 2 we have checked that model, after correcting some printing errors, by inserting the data of Rabl and I Nielsen [4] and obtained their results. However, the results published in Ref. [10] are inconsistent with its model. Second, the solar radiation data given in that I J I I I 030 32 34 36 38 40 article are not realistic. A mean annual radiation 2 ) (denoted b y / 7 in the article) of 538.8 W/m 2 is given Po( W / m for a location at 30°N latitude. This is a highly Fig. 11. Effect of the amplitude of the heat extraction rate (P0) on the depth of the lower convective zone (Lc) for a exaggerated value compared with 244.7 W/m 2 for fixed value of the temperature amplitude (To) (7" = 90°C, 32°N reported by Tabor and Weinberger [11] and L~= 0.95 m and Ls = 0.15 m). with 211 W/m 2 for 32. I°N in the present work.
[
330
HAWAS and ELASFOURI: A SALT-GRADIENT SOLAR POND 4. CONCLUSIONS
An analytical model for predicting the thermal performance of salt-gradient solar ponds has been developed. This model represents a modification of the Rabl and Nielsen model. It considers an upper convective zone, and its application is not limited to space heating only. The model has been used to predict the performance of a solar pond proposed for electric power production in Benghazi, Libya. The pond dimensions and the pattern of the heat extraction rate have been determined such that the pond should operate at as high a mean temperature and as small an amplitude of temperature fluctuations as possible. It has been found that the upper convective zone has a serious negative effect on the pond performance and should be kept as thin as possible. For given values of the mean pond temperature and depth of the upper convective zone, there is an optimum depth of the non-convective zone which maximizes the mean heat extraction rate. However, this optimum depth seems to be economically too large. The depth of the non-convective zone has, therefore, been selected as that which maximizes the heat extraction rate per unit pond depth. The dependence of the temperature amplitude on the depth of the lower convective zone and the
pattern of the heat extraction rate has been studied. It has been found that there is an optimum value of the phase lag of the heat extraction rate which minimizes the temperature amplitude. It has been also found that an increase of the amplitude of the heat extraction rate and/or the depth of the lower convective zone reduces the temperature amplitude, the effect of the amplitude of the heat extraction rate being more pronounced. REFERENCES I. H. Tabor, Solar Energy 27, 181 (1981). 2. H. C. Bryant and I. Colbeck, Solar Energy 19, 321 (1977). 3. H. Weinberger, Solar Energy 8, 45 (1964). 4. A. Rabl and C. E. Nielsen, Solar Energy 17, 1 (1975). 5. J. R. Hull, Solar Energy 25, 33 (1980). 6. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edn, pp. 65, 105. Oxford University Press, O x f o r d(1959). 7. H. Tabor and L. Bronicki, U.N. Conf. New Sources of Energy,Rome, Paper 35/S 54 (1961). 8. C. F. Kooi, Solar Energy 23, 37 (1979). 9. Y. F. Wang and A. Akbarzadeh, Solar Energy 30, 555 (1983). 10. E. A. Asian Abdel Salam, S. T. Probert, P. W. O'Kallaghan, M. Hussein and B. Norton, Appl. Energy 16, 283 (1984). 11. H. Tabor and H. Z. Weinberger, Non-convecting solar ponds. In Solar Energy Handbook (edited by J. F. Kreider). McGraw-Hill, New York (1980).