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Performance characteristics of solar ponds operating at different latitudes
Applied Energy 17 (1984) 97-115
Performance Characteristics of Solar Ponds Operating at Different Latitudes M. N. A. H a w l a d e r Department of Mechanical & Production Engineering, National University of Singapore, Kent Ridge, 0511 (Singapore)
SUMMARY The results of a comparative study of the performance of solar ponds operating at different latitudes are presented in this paper. A mathematical tool was developed to study the behaviour of solar ponds operating under different physical and operational conditions. The hourly meteorological data of Kew and Singapore were used. The pond operating at higher latitudes, i.e. Kew, where seasonal variations are significant, must be deeper than a pond operating near the equator. The deeper pond can also act as an interseasonal storage device. A pond operating at a location in Singapore attains a fairly high temperature but the temperature requirements are also high for the desired applications such as space cooling in conjunction with a refrigeration cycle. Although a pond at Kew attains a lower temperature compared with a pond at Singapore, it can supply considerable amounts of thermal energy to support a space heating load. Both the ponds require auxiliary heating, the magnitude being dependent upon the nature of the load.
NOMENCLATURE Ds h hl H F Is
Thickness of a storage zone. Depth of the pond from the surface. Depth of surface mixed layer. Depth of surface mixed layer and insulating layer. Fraction of the solar radiation absorbed in a small layer 6 at the surface. Irradiance just below the surface.
97 Applied Energy 0306-2619/84/$03.00 © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
98
N T
To rs
:r w Cp k
U,,s QLs QLG QD Z
0 0r P
M. N. A. Hawlader
Irradiance at depth h. Irradiance at depth h, due to radiation reflected from the bottom of the pond. Day of the year. Temperature. Reference temperature. Temperature of the water in the surface mixed layer. Temperature of the water in the storage zone. Ground water temperature. Specific heat. Thermal conductivity. Conductance for the bottom heat loss. Conductance for the surface heat loss. Heat losses from the surface. Heat loss to the ground. Load demand. Dimensionless depth, h/H. Thermal diffusivity. Absorptance of the bottom liner of the pond. Thickness of a small layer at the surface where the fraction F of the solar radiation is absorbed. Dimensionless temperature, T/To. Angle of refracted radiation. Effective extinction coefficient. Density. Fourier number, ott/H2. INTRODUCTION
The salt-gradient solar pond represents one of the feasible solar systems for collection and storage of solar energy for various applications. Its simplicity of design and construction is likely to make it economically viable where adequate land is available for the construction of the pond. This system has a unique method of combining collection and storage of solar energy in the same system. While simple in principle, the successful design of such ponds requires information on various parameters under the local meteorological conditions. A solar pond is a large body of water consisting of three layers, a surface mixed layer at the top, caused by wind action and evaporation,
Performance characteristics of solar ponds operating at different latitudes
99
below which is an insulating layer where a density gradient prevents free convection effects, and underneath this insulating layer, another mixed layer known as the storage layer. A black liner at the bottom of the pond collects solar energy incident upon it and this energy is stored in the storage zone. Solar ponds have been investigated both theoretically and experimentally ever since the initiation of Tabor's work. In Israel, solar ponds have been considered for power production. 1,2 Nielsen,3 Bryant and Colbeck, 4 and Hawlader and Brinkworth5 have studied the possibility of using solar ponds for space heating applications. Hawlader 6 has investigated the possibility of using solar ponds for space cooling applications in the tropics, and Shah et al.7 have used solar ponds for greenhouse heating. The possible use of solar ponds for space heating as well as heating for food and paper processes has been considered by Styris et al. 8 The steady-state performance characteristics of solar ponds were studied by Kooi 9 and Hawlader. 10 Weinberger 11 developed a mathematical model to describe the performance of solar ponds. A similar technique was used by Akbarzadeh and Ahmadil 2 to predict the performance of a solar pond in Iran. This analytical approach is limited by the boundary conditions. The mathematical formulation of the behaviour of the solar pond presented by Rabl and Nielsen 13 and subsequently followed by Hull~4 considered the presence of a convective storage zone at the bottom of the pond. Hawlader and Brinkworth 5 developed a mathematical model where the governing equation of the solar pond was solved numerically giving a greater degree of freedom to include appropriate initial and boundary conditions. In the present paper, the mathematical formulation of Hawlader and Brinkworth 5 has been improved further by incorporating the ground storage effect and the reflectance of the bottom liner. This model is then used as a tool to study the performance of solar ponds at different latitudes. The sites selected for this study are located at Kew (51 o 29' N, 0°19'W), U K and Singapore (1°22'N, 103°55'E). The hourly meteorological data of Kew and Singapore have been used in this comparative study. THEORETICAL M O D E L When solar radiation falls on the surface of the pond, there will be a region of rapid attenuation where the infrared component of the
M . N . A. H a w l a d e r
100
SOLAR
RADIATION
////// SURFACE MIXED
H
Ds
/
INSULATING
ZONE
STORAGE
ZONE
LOAD I
I
LAYER
LOAD
SALINITY
Fig. I.
Schematic diagram of a solar pond.
radiation will be absorbed. After passing through this layer, the radiation will suffer an exponential decay 5'11'13 until it reaches the bottom of the pond. At the bottom of the pond, part of the radiation will be absorbed, depending on the nature of the pond's bottom or liner, and the rest will be reflected. In this analysis, the pond is considered to have three layers, as shown in Fig. 1. The pond is sufficiently large so that the wall losses may be neglected. An energy balance for a thin layer of fluid of thickness dh at a depth h in the insulating zone gives 5'6'1° 00
t~20 H2(Ih + It) #sec 0 r
~z - ~Z-~ 4
kT °
(1)
where I h = Is(1 - F ) e x p [-/~(h - 3) sec 0r] /r = (1 -- ~1 )Is(1 -- F) exp [ -- # ( H + D s - 3) sec Or] x exp [ - / K H + Ds - h) sec Or] In this analysis, the radiation reflected from the bottom is considered to be specular in nature. In reality, the reflected radiation from the bottom
Performance characteristics of solar ponds operating at different latitudes
101
will be neither specular nor diffuse, but somewhere between the two extreme conditions. For a deep pond, it m a y be reasonable to assume that the reflected radiation is specular in nature, as shown by Hull.15 In order to obtain a solution of the above equation, it is necessary to have an initial condition and two boundary conditions. The surface boundary condition is obtained by making an energy balance for the surface mixed layer and is given by the following equation .6 ~0~
(I~, + I'r)H 2
-
h,kro
H//00s'~
H 2 ULS
+hl
kh,
(0s- 0.)
(2)
where I[, = Is[1 - (1 - F ) e x p {-/~(hl - 6) see 0r}] I', = Is(l - ~ 1 ) ( 1 - F ) e x p [ - / ~ ( H + D s - 6)see 0,] × [exp { - / ~ ( H + Ds - h 1) sec 0 r } - exp { - # ( H + Ds) see 0, }] The second boundary condition is obtained by making an energy balance for the storage zone which gives: 6 aO b _ (1'~ - I ; ' ) H 2 toz
kDsT o
_-~[~/H/'a0b'~
(QD + QLG)H2
~- Ds \ C:Z:Iz= , --
(3)
Dsk T o
where I~ = ( 1 - F) I s( 1 - ~t1) exp [ - # ( H - 6) sec 0 r] /~' = (1 - F)(1 - ctI )I s exp [ - # ( H - D s - tO)sec Or] exp ( -/~D s sec Or) In Ref. 5, QLG was calculated by using ULG(Tb- TGW) in which Tb is seldom less than TGw and, therefore, the heat transfer from the storage zone to the ground becomes uni-directional. It is likely that sometimes the temperature of the storage zone will be lower than the soil in contact with it, depending on the load. This will lead to a heat transfer from the ground to the storage zone. This effect cannot be taken into account by U L c ( T b - TGW). In order to include these effects, the ground under the pond is considered as an infinite slab. This is obtained by the use of the following equation, assuming that the heat transfer is due to diffusion only, i.e.
80
~20
~- = ~ ~2
(4)
102
M. N. A. Hawlader
Like eqn (1), this also requires an initial condition and two boundary conditions to permit its solution. The soil under the pond may be assumed to have either a uniform temperature or a known temperature gradient. The interface between the pond and the ground is considered to be at a single temperature, namely 0(0, T) = 0b
(5)
The temperature in the soil at a depth L is assumed to be same as that of ground water temperature, giving
O(L, r) = 0Gw
(6)
The heat transfer QLG, between the storage zone and the pond is predicted using eqns (4), (5) and (6). The performance characteristics of the pond are evaluated by using eqns (1), (2) and (3) with the hourly meteorological data of Kew and Singapore. A brief description of the meteorological data of Kew and Singapore is given in the next section.
METEOROLOGICAL DATA Before a pond can be designed, it is necessary to understand the meteorological conditions of the locality and the type of applications required. The solar radiation available at the two locations considered in this study, namely Kew and Singapore, is shown in Figs 2 and 3. Figure 2 shows the net radiation available on the surface of the pond. The reflectance of the water surface has been taken into account in order to evaluate the net radiation. For Singapore, it is seen from the figure that radiation is evenly distributed over the entire period of the year. There is hardly any seasonal variation of the available solar radiation whereas for Kew, most of the radiation is available in the six months beginning in April. Figure 3 shows the monthly mean of daily totals of global radiation on a horizontal surface. It is seen from this figure that the Kew data show a considerable seasonal variation whereas for Singapore there is hardly any seasonal variation. In the UK, space heating is required in winter when the solar energy is least available. Hence, if a solar pond is designed for space heating application in the UK, it should possess interseasonal storage facilities. In Singapore, space cooling is required throughout the year, and because there is no marked variation in the available solar
Performance characteristics of solar ponds operating at different latitudes
103
7 1200 J::
SINGAPORE 2 Zo 800
< r,-
IX O
400
J
3 '.3
I ~00
i 200 JANUARY
7
Jc:
TO
I 300
I 400
DECEMBER
)200
KEW
800 .<
n.<(
~, 400 ul
¢n
9 0
I00 APRIL
Fig. 2.
TO
200 DECEMBER
. ,
300
400
. JANUARY
Daily average net global radiation on the surface of the ponds at Singapore and Kew.
104
M. N. A. Hawlader
2L,
SINGAPORE 20
z o D n~ [z:
12
8
u')
I F
i M
' A
I M
I J
I J
I A
I S
I 0
! N
I D
I 0
I N
I 0
MONTH
28
24
z o i..o
0:2
KEW
20
15
)2
8
L.
0
l J
I F
~ M
I A
i M
I J
I J
I A
i S
MONTH
Fig. 3.
Monthly mean of daily totals of global radiation on a horizontal surface at (a) Singapore (( -) 1978 values; ( . . . . ) 1979 values) and (b) Kew.
Performance characteristics of solar ponds operating at different latitudes
105
radiation over the year, the pond does not require any long-term storage facilities.
NUMERICAL EXPERIMENTS For a solar pond, the temperature of the storage zone is affected by: a. b. c. d.
the physical conditions of the pond; location of the pond; quality and clarity of the salt solution; the load characteristics.
For an efficient design and operation of a solar pond, it is essential to know how these different conditions affect the performance of the solar pond at a particular location. Numerical experiments were conducted to show the effects of these different conditions on the performance of the solar pond. The physical parameters affecting the performance of the pond are the dimensions of the pond, the depth of surface mixed layer, the thickness of insulating layer (3), and also the depth of the storage zone. The temperature of the pond is affected by the location of the pond. The pond should be located in areas where ground water movement is a minimum. The extinction coefficient, which is a measure of the quality and clarity of the salt solution, and the absorptance of the bottom liner are also factors affecting the temperature of the storage zone. The nature and magnitude of the load affects the temperature at which the load is available.
RESULTS AND DISCUSSIONS The results obtained from the numerical experiments using the hourly meteorological data of Singapore and Kew are analysed and presented in this section. It can be seen from Figs 4 and 5 that the temperature of the storage zone is affected by the extinction coefficient which is a measure of the quality and clarity of the water. The values of/z range from 0.05 to 1.0m-1. 5 After one year of operation, a pond in Singapore attains a maximum temperature of 110°C for/~ = 0.5 m - t but the maximum temperature is reduced to about 80°C when the extinction coefficient increases to
106
M. N. A. Hawlader
,,ol
OC!
hi = 0 . 1 m
0.95
H =I.5m
0. S0
(a) NO LOAD
Ds = 1 . 0 m H -0.Sin
< u'l
H = I.Om
-1
-1
u. O
'7O
uJ
o< n.
&O
i 100
I0 0
NUMBER
g
I ~.00
I 300 N
H I
W < 0
i 200 OF DAYS.
hi = 0 . 1 m H
1oo
:
0.95
0.5m
....
(b)
0.50
NO LOAD
Ds = 0 . 5 m
= 0,5 m "1
g. C) n'
"70 w
=- 1,0 m -
<
40
nuJ <
10
I
I
I
I
I00
200
300
~00
NUMBER
Fig. 4.
OF
DAYS,
N
Variation of storage temperature due to changes of absorptance of the bottom liner of the pond at Singapore.
Performance characteristics of solar ponds operating at different latitudes
120
t/d z
o N
iii
Os = H =
t.Om
hI =
0.1 m
(a)
d-, 0.95
1.5 m .....
107
NO
LOAD
0.50
~ 90 o if)
N
a0
112 ,u = 0.5 m -~
UJ
-, o
30
P - = I . 0 m -~
< 0
0
100
I
1
I
200
300
~00
NUMBER OF DAYS N 120 I.IJ
Ds
= 0.6m
N
H
= 0.5m
hi
= 0.1m
g
~ 90
....
~'-i 095 0.50
(b) NO L O A D
,, 60 o f12 I[ 30 o u.i I
I
1
I
100
200
300
t.00
NUMBER OF Fig.
5.
DAYS, N
Variation o f storage temperature due to changes o f absorptance o f the b o t t o m liner o f the pond at Kew.
108
M. N. A. Hawlader
(a) SINGAPORE
,,, 130 Z O N
H - 1.0m
LU
DS = 1.0m
'~
NO LOAD
~J = 1.0m -1
100
o(~= 0 9 5
O
F~
hl:O.Im ~
70
~
0.3m
w
10
0
I
I
I
I
I00
2O0
300
400
NUMBER
uJ g
OF DAYS,
N
120
(b)
I'M UJ
~
KEW NO LOAD
90
QC ffl
~ 60 112 IE uJ ,,,
30
~
h = ,3 O n rI. 0 ~ . ' . 0 S .n r
0
>
I
1
I
100
200
300
NUMBER
Fig. 6.
OF
DAYS,
I
1-00
N
Temperature variation due to surface mixed layer at (a) Singapore and (b) Kew ponds.
1 "0 m - 1. For the same depth of surface mixed layer, when Fig. 4(a) is compared with Fig. 4(b), it is seen that the influence of the extinction coefficient on the temperature of the storage zone is significantly reduced as the depth o f the pond is reduced. Similar effects can be seen in a pond in the UK, as shown in Fig. 5. The effects of absorptance of the bottom liner of the pond on the
Performance characteristics of solar ponds operating at different latitudes
109
temperature of the storage zone can also be seen in Figs 4 and 5. For a deeper pond,/~ has a greater effect on the temperature than 0tI as shown in Figs 4(a) and 5(a). However for a shallow pond the absorptance of the bottom liner has a greater effect on the storage temperature than the extinction coefficient. A higher value of the extinction coefficient # implies a less transparent pond. Hence less radiation reaches the bottom of the pond to be absorbed and stored. For a higher value of#, the change in the value of the absorptance of the bottom of the pond does not affect the temperature of the deep pond, as shown in Figs 4(a) and 5(a). For a lower value of #, more radiation reaches the bottom of the pond to be absorbed and stored and thus the effects of different values of ~1 are more significant. From Fig. 6(a) and (b), it can be seen that when the depth of the surface mixed layer was increased, keeping the other parameters constant, the temperature of the storage zone decreased. For a fixed value of H, the effective thickness of insulation decreased when the depth of the surface mixed layer was increased. Although, the same amount of energy reaches the storage zone, due to the decreased insulation effect, more energy is lost through the insulating zone, thereby reducing the temperature of the storage zone. When the extinction coefficient of the salt water and thicknesses of the surface mixed layer and storage zone are maintained constant, an increase in depth from the surface of the pond to the storage zone, H, increases the insulation effect and reduces the amount of radiation reaching the storage zone of the pond. Up to a certain value of depth H, the insulation effects have stronger influences on the storage temperature than the reduction in the radiation input to the store, xo Beyond this point, an increase in H causes a reduction in the storage temperature. This effect is shown in Fig. 7(a). A deeper pond is likely to show less fluctuation in storage temperature due to changes in the environmental condition, although it takes a longer time to attain the steady-state condition, as shown in Fig. 7(a) and (b). For a higher latitude, a slightly deeper pond is better, since it reduces the seasonal fluctuation of temperatures, as is shown in Fig. 7(b). The thickness of the storage zone indicates the thermal capacity of the pond. A shallow storage zone attains a quasi-steady state quicker with a large fluctuation of temperature due to loads and environmental conditions because of the small thermal capacity. 5"6 Figure 8(a) and (b) shows the interaction of the pond with the soil below
M. N. A. Hawlader
110
(a) SINGAPORE
c J,
o 130
~-~=~---,~-~--~-
NO LOAD
LU
2.5 m
--
g N t.U
o ~00 <
g
'
~
=
0
.
Sm
o: ;io
w
~0 0 <
~
w
l
250
0
l
095 = 0.5 rn-'
:
I
I
500 750 NUMBER OF DAYS, N
1000
I
II00
(b) uJ Z 0 N
KEW
120
NO ~ A D
UJ
o
9O
OC
1 , / " ~X.,
~7,<.. \ ~,'~
~ 60 el.
,,,
30
0 <
d
y"
Off
u.,I
0
Fig. 7.
I
I
I
250
500
750
I
I
1000 liO0
NUMBER OF DAYS N Effect of insulation thickness on the temperature of the storage zone at (a) Singapore and (b) Kew ponds.
Per/ormance characteristics o/ solar ponds operating at d(fferent latitudes TEMP
DISTRIBUTION
BELOW
POND
111
(a)
80 SINGAPORE \
k'
)u
- 1.0m-
H
= I.S m
\ ~
\\
60
o,-,om h, . o , m
Q II 0
40
I LU p-
'-.. .... ~.~ "--_ ~.20 20 0
~
~
------_.~._~ . . . . .
---__...._._=
I
I
I
0.8
1.6
2.4
DEPIH
BELOW
._:_..-~ , "
POND
~ -
3.2
~
i
4.0
m
(b)
60
or.,
KEW
Z
,~ ,-0
\-\ t, 0
el ~"
20
p-
N = 20
"--__
1 0.8 DEPTH
Fig. 8.
--...-.~._=....
I
l
t
t
1.6
2.t.
3.2
4.0
BELOW
POND,
m
Temperature of soil beneath the pond at (a) Singapore and (b) Kcw.
it, i.e. the temperature variation of the soil at different depths with respect to time. For a pond in Singapore, the ground water temperature was assumed to be 20 °C and located at a depth of 4 m from the bottom of the pond, whereas the ground water temperature at Kew was assumed to be 7 °C. The soil under the pond acts as a thermal stabilizer receiving heat at times when the storage temperature is higher than the ground and
112
M. N. A. Hawlader
(o) SINGAPORE TEMP
VARIATIONS
~
130
WITH
LOAD
DALLY LOAD NO LOAD
100
U o
~
LU
z o N
70
2 M J m -2
t.U
L9 -o t-.-
Ds = l . O m
&O
H
=
hi =
9
I0
0
~I = 0. S m
Z0m
ml-- 0 . 9 5
0.2 m
I
I
I
I
250
500
750
1000
UJ
NUMBER
OF
DAYS,
N
¢y
(b)
,~o
TEMP
~E
VARIATIONS
W I T H LOAD
KEW
UJ
LU
o (t n... u.i
90 D A ILY L O A D NO LOAD
60
30
0
0
LOAD
Fig. 9.
i
I 25O
I
150
_ .o.
°_
LOAD
_ NO LOAD
I
I000 LOAD
NUMBER OF DAYS, N Effects of load on the temperature of the storage zone of the ponds at (a) Singapore and (b) Kew.
PerJbrmance characteristics of solar ponds operating at different latitudes
113
delivering it when the temperature of the storage zone is lower than the soil. Considering the effects of all the factors described earlier, the depth of the pond at Singapore was selected to be 3 m while that of the Kew pond was considered to be 3.5 m. Figure 9 shows the dimensions of such ponds and the effects of load on the temperature distribution. For the Singapore pond the load was assumed to be continuous. This pond has been considered for space cooling applications. A pond at Kew has been considered mainly for space heating applications. For space cooling, the minimum temperature required for an efficient operation of an absorption chiller is 75 °C. From Fig. 9(a), it is seen that this pond can supply 2 MJ m - 2 per day without reducing the temperature of the storage zone below 75 °C. This pond can also supply 3 MJ m - 2 per day, but the temperature will be lower than the desired value. In this case, an auxiliary heater may be needed. A higher load can be supported if the extinction coefficient of the salt water is reduced further, making the pond more transparent. Under this condition, this pond should be able to support a higher load and the pond area required for any particular application will be further reduced. The pond at Kew is likely to be able to support a space heating load in winter and thus operate seasonally, collecting energy during the summer and supply heating load during the winter, thereby acting also as an interseasonal storage device. This pond can support a load of about 1.5MJm -2 day, which is slightly smaller than that delivered by the Singapore pond. But for a space cooling application, an absorption refrigeration system is used and the coefficient of performance of such systems is quite low, e.g. COP = 0.7. No such system is required for a space heating application. Although, the ponds are considered for space heating and cooling applications, they can also supply domestic hot water and industrial process heat. In the case of industrial process heat, an auxiliary heater may be necessary depending on the temperature requirements. It is interesting to note that a pond in Singapore located near the equator attains a much higher temperature compared with a pond at Kew but the temperature requirements for the load are also high for Singapore as far as the supply thermal energy is concerned. CONCLUSION This paper compares the performances of solar ponds at different latitudes. The mathematical model 5'6 has been further improved and
114
M. N. A. Hawlader
used as a tool to facilitate this comparative study. The effects of different parameters affecting the operating conditions of the pond at the two locations considered in this study have been investigated with the hourly meteorological data of the two locations. This study indicates that a pond at Kew should be deeper than one at Singapore. The pond at Kew also acts as an interseasonal storage device. These ponds can support loads at the temperatures demanded by local conditions. Auxiliary heaters are sometimes required for both the ponds depending on the type and magnitude of the loads. ACKNOWLEDGEMENT The author would like to thank Miss Lee Pui Chan and Mr Yeoh Seng Chye for their help in the computational work.
REFERENCES 1. H. Tabor, Solar ponds--large area solar collector for power production, Solar Energy, 7(1) (1963), p. 190. 2. L. Bronicki, J. Lev-Er and Y. Porat, Large solar electric power plant based on solar ponds, Worm Power Conf., Munich, 1980. 3. C. E. Nielsen, Experience with a prototype solar pond for space heating, Proc. ConJl Sharing the Sun, Vol. 5, ISES (USA) and Solar Energy Soc. of Canada, Winnipeg, 1976, pp. 169-82. 4. H. C. Bryant and I. Colbeck, A solar pond for London? Solar Energy, 19 (1977), p. 321. 5. M. N. A. Hawlader and B. J. Brinkworth, An analysis of the non-convecting solar pond, Solar Energy, 27(3) (1981), pp. 195-204. 6. M. N. A. Hawlader, The use of solar pond for air conditioning, Proc. Conf. on Utilization of Solar Energy for Refrigeration and Air Conditioning, March 14-19, International Institute of Refrigeration, 1982, pp. 61-74. 7. S.A. Shah, T. H. Short and R. P. Fynn, Modeling and ~testinga salt-gradient solar pond in northeast Ohio, Solar Energy, 27(5) (1982), pp. 393-401. 8. D. L. Styris et al., The non-convecting solar pond applied to building and process heating, Solar Energy, 18 (1976), p. 245. 9. C. F. Kooi, The steady state salt gradient solar pond, Solar Energy, 23 (1979), pp. 37-45. 10. M. N. A. Hawlader, The influence of extinction coefficient on the effectiveness of solar ponds, Solar Energy, 25 (1980), p. 461. 11. H: Weinberger, The physics of the solar pond, Solar Energy, 8 (1964), pp. 45-62.
Performance characteristics of solar ponds operating at different latitudes
115
12. A. Akbarzadeh and G. Ahmadi, Computer simulation of the performance of a solar pond in the southern part of Iran, Solar Energy, 24 (1980), p. 143. 13. A. Rabl and C. E. Nielsen, Solar ponds for space heating, Solar Energy, 17 (1975), p. 1. 14. J. R. Hull, Computer simulation of solar pond thermal behaviour, Solar Energy, 25 (1980), p. 33. 15. J. R. Hull, Calculation of solar pond thermal efficiency with a diffusely reflecting bottom, Solar Energy, 29(5) (1982), pp. 385-9.
Performance Characteristics of Solar Ponds Operating at Different Latitudes M. N. A. H a w l a d e r Department of Mechanical & Production Engineering, National University of Singapore, Kent Ridge, 0511 (Singapore)
SUMMARY The results of a comparative study of the performance of solar ponds operating at different latitudes are presented in this paper. A mathematical tool was developed to study the behaviour of solar ponds operating under different physical and operational conditions. The hourly meteorological data of Kew and Singapore were used. The pond operating at higher latitudes, i.e. Kew, where seasonal variations are significant, must be deeper than a pond operating near the equator. The deeper pond can also act as an interseasonal storage device. A pond operating at a location in Singapore attains a fairly high temperature but the temperature requirements are also high for the desired applications such as space cooling in conjunction with a refrigeration cycle. Although a pond at Kew attains a lower temperature compared with a pond at Singapore, it can supply considerable amounts of thermal energy to support a space heating load. Both the ponds require auxiliary heating, the magnitude being dependent upon the nature of the load.
NOMENCLATURE Ds h hl H F Is
Thickness of a storage zone. Depth of the pond from the surface. Depth of surface mixed layer. Depth of surface mixed layer and insulating layer. Fraction of the solar radiation absorbed in a small layer 6 at the surface. Irradiance just below the surface.
97 Applied Energy 0306-2619/84/$03.00 © Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain
98
N T
To rs
:r w Cp k
U,,s QLs QLG QD Z
0 0r P
M. N. A. Hawlader
Irradiance at depth h. Irradiance at depth h, due to radiation reflected from the bottom of the pond. Day of the year. Temperature. Reference temperature. Temperature of the water in the surface mixed layer. Temperature of the water in the storage zone. Ground water temperature. Specific heat. Thermal conductivity. Conductance for the bottom heat loss. Conductance for the surface heat loss. Heat losses from the surface. Heat loss to the ground. Load demand. Dimensionless depth, h/H. Thermal diffusivity. Absorptance of the bottom liner of the pond. Thickness of a small layer at the surface where the fraction F of the solar radiation is absorbed. Dimensionless temperature, T/To. Angle of refracted radiation. Effective extinction coefficient. Density. Fourier number, ott/H2. INTRODUCTION
The salt-gradient solar pond represents one of the feasible solar systems for collection and storage of solar energy for various applications. Its simplicity of design and construction is likely to make it economically viable where adequate land is available for the construction of the pond. This system has a unique method of combining collection and storage of solar energy in the same system. While simple in principle, the successful design of such ponds requires information on various parameters under the local meteorological conditions. A solar pond is a large body of water consisting of three layers, a surface mixed layer at the top, caused by wind action and evaporation,
Performance characteristics of solar ponds operating at different latitudes
99
below which is an insulating layer where a density gradient prevents free convection effects, and underneath this insulating layer, another mixed layer known as the storage layer. A black liner at the bottom of the pond collects solar energy incident upon it and this energy is stored in the storage zone. Solar ponds have been investigated both theoretically and experimentally ever since the initiation of Tabor's work. In Israel, solar ponds have been considered for power production. 1,2 Nielsen,3 Bryant and Colbeck, 4 and Hawlader and Brinkworth5 have studied the possibility of using solar ponds for space heating applications. Hawlader 6 has investigated the possibility of using solar ponds for space cooling applications in the tropics, and Shah et al.7 have used solar ponds for greenhouse heating. The possible use of solar ponds for space heating as well as heating for food and paper processes has been considered by Styris et al. 8 The steady-state performance characteristics of solar ponds were studied by Kooi 9 and Hawlader. 10 Weinberger 11 developed a mathematical model to describe the performance of solar ponds. A similar technique was used by Akbarzadeh and Ahmadil 2 to predict the performance of a solar pond in Iran. This analytical approach is limited by the boundary conditions. The mathematical formulation of the behaviour of the solar pond presented by Rabl and Nielsen 13 and subsequently followed by Hull~4 considered the presence of a convective storage zone at the bottom of the pond. Hawlader and Brinkworth 5 developed a mathematical model where the governing equation of the solar pond was solved numerically giving a greater degree of freedom to include appropriate initial and boundary conditions. In the present paper, the mathematical formulation of Hawlader and Brinkworth 5 has been improved further by incorporating the ground storage effect and the reflectance of the bottom liner. This model is then used as a tool to study the performance of solar ponds at different latitudes. The sites selected for this study are located at Kew (51 o 29' N, 0°19'W), U K and Singapore (1°22'N, 103°55'E). The hourly meteorological data of Kew and Singapore have been used in this comparative study. THEORETICAL M O D E L When solar radiation falls on the surface of the pond, there will be a region of rapid attenuation where the infrared component of the
M . N . A. H a w l a d e r
100
SOLAR
RADIATION
////// SURFACE MIXED
H
Ds
/
INSULATING
ZONE
STORAGE
ZONE
LOAD I
I
LAYER
LOAD
SALINITY
Fig. I.
Schematic diagram of a solar pond.
radiation will be absorbed. After passing through this layer, the radiation will suffer an exponential decay 5'11'13 until it reaches the bottom of the pond. At the bottom of the pond, part of the radiation will be absorbed, depending on the nature of the pond's bottom or liner, and the rest will be reflected. In this analysis, the pond is considered to have three layers, as shown in Fig. 1. The pond is sufficiently large so that the wall losses may be neglected. An energy balance for a thin layer of fluid of thickness dh at a depth h in the insulating zone gives 5'6'1° 00
t~20 H2(Ih + It) #sec 0 r
~z - ~Z-~ 4
kT °
(1)
where I h = Is(1 - F ) e x p [-/~(h - 3) sec 0r] /r = (1 -- ~1 )Is(1 -- F) exp [ -- # ( H + D s - 3) sec Or] x exp [ - / K H + Ds - h) sec Or] In this analysis, the radiation reflected from the bottom is considered to be specular in nature. In reality, the reflected radiation from the bottom
Performance characteristics of solar ponds operating at different latitudes
101
will be neither specular nor diffuse, but somewhere between the two extreme conditions. For a deep pond, it m a y be reasonable to assume that the reflected radiation is specular in nature, as shown by Hull.15 In order to obtain a solution of the above equation, it is necessary to have an initial condition and two boundary conditions. The surface boundary condition is obtained by making an energy balance for the surface mixed layer and is given by the following equation .6 ~0~
(I~, + I'r)H 2
-
h,kro
H//00s'~
H 2 ULS
+hl
kh,
(0s- 0.)
(2)
where I[, = Is[1 - (1 - F ) e x p {-/~(hl - 6) see 0r}] I', = Is(l - ~ 1 ) ( 1 - F ) e x p [ - / ~ ( H + D s - 6)see 0,] × [exp { - / ~ ( H + Ds - h 1) sec 0 r } - exp { - # ( H + Ds) see 0, }] The second boundary condition is obtained by making an energy balance for the storage zone which gives: 6 aO b _ (1'~ - I ; ' ) H 2 toz
kDsT o
_-~[~/H/'a0b'~
(QD + QLG)H2
~- Ds \ C:Z:Iz= , --
(3)
Dsk T o
where I~ = ( 1 - F) I s( 1 - ~t1) exp [ - # ( H - 6) sec 0 r] /~' = (1 - F)(1 - ctI )I s exp [ - # ( H - D s - tO)sec Or] exp ( -/~D s sec Or) In Ref. 5, QLG was calculated by using ULG(Tb- TGW) in which Tb is seldom less than TGw and, therefore, the heat transfer from the storage zone to the ground becomes uni-directional. It is likely that sometimes the temperature of the storage zone will be lower than the soil in contact with it, depending on the load. This will lead to a heat transfer from the ground to the storage zone. This effect cannot be taken into account by U L c ( T b - TGW). In order to include these effects, the ground under the pond is considered as an infinite slab. This is obtained by the use of the following equation, assuming that the heat transfer is due to diffusion only, i.e.
80
~20
~- = ~ ~2
(4)
102
M. N. A. Hawlader
Like eqn (1), this also requires an initial condition and two boundary conditions to permit its solution. The soil under the pond may be assumed to have either a uniform temperature or a known temperature gradient. The interface between the pond and the ground is considered to be at a single temperature, namely 0(0, T) = 0b
(5)
The temperature in the soil at a depth L is assumed to be same as that of ground water temperature, giving
O(L, r) = 0Gw
(6)
The heat transfer QLG, between the storage zone and the pond is predicted using eqns (4), (5) and (6). The performance characteristics of the pond are evaluated by using eqns (1), (2) and (3) with the hourly meteorological data of Kew and Singapore. A brief description of the meteorological data of Kew and Singapore is given in the next section.
METEOROLOGICAL DATA Before a pond can be designed, it is necessary to understand the meteorological conditions of the locality and the type of applications required. The solar radiation available at the two locations considered in this study, namely Kew and Singapore, is shown in Figs 2 and 3. Figure 2 shows the net radiation available on the surface of the pond. The reflectance of the water surface has been taken into account in order to evaluate the net radiation. For Singapore, it is seen from the figure that radiation is evenly distributed over the entire period of the year. There is hardly any seasonal variation of the available solar radiation whereas for Kew, most of the radiation is available in the six months beginning in April. Figure 3 shows the monthly mean of daily totals of global radiation on a horizontal surface. It is seen from this figure that the Kew data show a considerable seasonal variation whereas for Singapore there is hardly any seasonal variation. In the UK, space heating is required in winter when the solar energy is least available. Hence, if a solar pond is designed for space heating application in the UK, it should possess interseasonal storage facilities. In Singapore, space cooling is required throughout the year, and because there is no marked variation in the available solar
Performance characteristics of solar ponds operating at different latitudes
103
7 1200 J::
SINGAPORE 2 Zo 800
< r,-
IX O
400
J
3 '.3
I ~00
i 200 JANUARY
7
Jc:
TO
I 300
I 400
DECEMBER
)200
KEW
800 .<
n.<(
~, 400 ul
¢n
9 0
I00 APRIL
Fig. 2.
TO
200 DECEMBER
. ,
300
400
. JANUARY
Daily average net global radiation on the surface of the ponds at Singapore and Kew.
104
M. N. A. Hawlader
2L,
SINGAPORE 20
z o D n~ [z:
12
8
u')
I F
i M
' A
I M
I J
I J
I A
I S
I 0
! N
I D
I 0
I N
I 0
MONTH
28
24
z o i..o
0:2
KEW
20
15
)2
8
L.
0
l J
I F
~ M
I A
i M
I J
I J
I A
i S
MONTH
Fig. 3.
Monthly mean of daily totals of global radiation on a horizontal surface at (a) Singapore (( -) 1978 values; ( . . . . ) 1979 values) and (b) Kew.
Performance characteristics of solar ponds operating at different latitudes
105
radiation over the year, the pond does not require any long-term storage facilities.
NUMERICAL EXPERIMENTS For a solar pond, the temperature of the storage zone is affected by: a. b. c. d.
the physical conditions of the pond; location of the pond; quality and clarity of the salt solution; the load characteristics.
For an efficient design and operation of a solar pond, it is essential to know how these different conditions affect the performance of the solar pond at a particular location. Numerical experiments were conducted to show the effects of these different conditions on the performance of the solar pond. The physical parameters affecting the performance of the pond are the dimensions of the pond, the depth of surface mixed layer, the thickness of insulating layer (3), and also the depth of the storage zone. The temperature of the pond is affected by the location of the pond. The pond should be located in areas where ground water movement is a minimum. The extinction coefficient, which is a measure of the quality and clarity of the salt solution, and the absorptance of the bottom liner are also factors affecting the temperature of the storage zone. The nature and magnitude of the load affects the temperature at which the load is available.
RESULTS AND DISCUSSIONS The results obtained from the numerical experiments using the hourly meteorological data of Singapore and Kew are analysed and presented in this section. It can be seen from Figs 4 and 5 that the temperature of the storage zone is affected by the extinction coefficient which is a measure of the quality and clarity of the water. The values of/z range from 0.05 to 1.0m-1. 5 After one year of operation, a pond in Singapore attains a maximum temperature of 110°C for/~ = 0.5 m - t but the maximum temperature is reduced to about 80°C when the extinction coefficient increases to
106
M. N. A. Hawlader
,,ol
OC!
hi = 0 . 1 m
0.95
H =I.5m
0. S0
(a) NO LOAD
Ds = 1 . 0 m H -0.Sin
< u'l
H = I.Om
-1
-1
u. O
'7O
uJ
o< n.
&O
i 100
I0 0
NUMBER
g
I ~.00
I 300 N
H I
W < 0
i 200 OF DAYS.
hi = 0 . 1 m H
1oo
:
0.95
0.5m
....
(b)
0.50
NO LOAD
Ds = 0 . 5 m
= 0,5 m "1
g. C) n'
"70 w
=- 1,0 m -
<
40
nuJ <
10
I
I
I
I
I00
200
300
~00
NUMBER
Fig. 4.
OF
DAYS,
N
Variation of storage temperature due to changes of absorptance of the bottom liner of the pond at Singapore.
Performance characteristics of solar ponds operating at different latitudes
120
t/d z
o N
iii
Os = H =
t.Om
hI =
0.1 m
(a)
d-, 0.95
1.5 m .....
107
NO
LOAD
0.50
~ 90 o if)
N
a0
112 ,u = 0.5 m -~
UJ
-, o
30
P - = I . 0 m -~
< 0
0
100
I
1
I
200
300
~00
NUMBER OF DAYS N 120 I.IJ
Ds
= 0.6m
N
H
= 0.5m
hi
= 0.1m
g
~ 90
....
~'-i 095 0.50
(b) NO L O A D
,, 60 o f12 I[ 30 o u.i I
I
1
I
100
200
300
t.00
NUMBER OF Fig.
5.
DAYS, N
Variation o f storage temperature due to changes o f absorptance o f the b o t t o m liner o f the pond at Kew.
108
M. N. A. Hawlader
(a) SINGAPORE
,,, 130 Z O N
H - 1.0m
LU
DS = 1.0m
'~
NO LOAD
~J = 1.0m -1
100
o(~= 0 9 5
O
F~
hl:O.Im ~
70
~
0.3m
w
10
0
I
I
I
I
I00
2O0
300
400
NUMBER
uJ g
OF DAYS,
N
120
(b)
I'M UJ
~
KEW NO LOAD
90
QC ffl
~ 60 112 IE uJ ,,,
30
~
h = ,3 O n rI. 0 ~ . ' . 0 S .n r
0
>
I
1
I
100
200
300
NUMBER
Fig. 6.
OF
DAYS,
I
1-00
N
Temperature variation due to surface mixed layer at (a) Singapore and (b) Kew ponds.
1 "0 m - 1. For the same depth of surface mixed layer, when Fig. 4(a) is compared with Fig. 4(b), it is seen that the influence of the extinction coefficient on the temperature of the storage zone is significantly reduced as the depth o f the pond is reduced. Similar effects can be seen in a pond in the UK, as shown in Fig. 5. The effects of absorptance of the bottom liner of the pond on the
Performance characteristics of solar ponds operating at different latitudes
109
temperature of the storage zone can also be seen in Figs 4 and 5. For a deeper pond,/~ has a greater effect on the temperature than 0tI as shown in Figs 4(a) and 5(a). However for a shallow pond the absorptance of the bottom liner has a greater effect on the storage temperature than the extinction coefficient. A higher value of the extinction coefficient # implies a less transparent pond. Hence less radiation reaches the bottom of the pond to be absorbed and stored. For a higher value of#, the change in the value of the absorptance of the bottom of the pond does not affect the temperature of the deep pond, as shown in Figs 4(a) and 5(a). For a lower value of #, more radiation reaches the bottom of the pond to be absorbed and stored and thus the effects of different values of ~1 are more significant. From Fig. 6(a) and (b), it can be seen that when the depth of the surface mixed layer was increased, keeping the other parameters constant, the temperature of the storage zone decreased. For a fixed value of H, the effective thickness of insulation decreased when the depth of the surface mixed layer was increased. Although, the same amount of energy reaches the storage zone, due to the decreased insulation effect, more energy is lost through the insulating zone, thereby reducing the temperature of the storage zone. When the extinction coefficient of the salt water and thicknesses of the surface mixed layer and storage zone are maintained constant, an increase in depth from the surface of the pond to the storage zone, H, increases the insulation effect and reduces the amount of radiation reaching the storage zone of the pond. Up to a certain value of depth H, the insulation effects have stronger influences on the storage temperature than the reduction in the radiation input to the store, xo Beyond this point, an increase in H causes a reduction in the storage temperature. This effect is shown in Fig. 7(a). A deeper pond is likely to show less fluctuation in storage temperature due to changes in the environmental condition, although it takes a longer time to attain the steady-state condition, as shown in Fig. 7(a) and (b). For a higher latitude, a slightly deeper pond is better, since it reduces the seasonal fluctuation of temperatures, as is shown in Fig. 7(b). The thickness of the storage zone indicates the thermal capacity of the pond. A shallow storage zone attains a quasi-steady state quicker with a large fluctuation of temperature due to loads and environmental conditions because of the small thermal capacity. 5"6 Figure 8(a) and (b) shows the interaction of the pond with the soil below
M. N. A. Hawlader
110
(a) SINGAPORE
c J,
o 130
~-~=~---,~-~--~-
NO LOAD
LU
2.5 m
--
g N t.U
o ~00 <
g
'
~
=
0
.
Sm
o: ;io
w
~0 0 <
~
w
l
250
0
l
095 = 0.5 rn-'
:
I
I
500 750 NUMBER OF DAYS, N
1000
I
II00
(b) uJ Z 0 N
KEW
120
NO ~ A D
UJ
o
9O
OC
1 , / " ~X.,
~7,<.. \ ~,'~
~ 60 el.
,,,
30
0 <
d
y"
Off
u.,I
0
Fig. 7.
I
I
I
250
500
750
I
I
1000 liO0
NUMBER OF DAYS N Effect of insulation thickness on the temperature of the storage zone at (a) Singapore and (b) Kew ponds.
Per/ormance characteristics o/ solar ponds operating at d(fferent latitudes TEMP
DISTRIBUTION
BELOW
POND
111
(a)
80 SINGAPORE \
k'
)u
- 1.0m-
H
= I.S m
\ ~
\\
60
o,-,om h, . o , m
Q II 0
40
I LU p-
'-.. .... ~.~ "--_ ~.20 20 0
~
~
------_.~._~ . . . . .
---__...._._=
I
I
I
0.8
1.6
2.4
DEPIH
BELOW
._:_..-~ , "
POND
~ -
3.2
~
i
4.0
m
(b)
60
or.,
KEW
Z
,~ ,-0
\-\ t, 0
el ~"
20
p-
N = 20
"--__
1 0.8 DEPTH
Fig. 8.
--...-.~._=....
I
l
t
t
1.6
2.t.
3.2
4.0
BELOW
POND,
m
Temperature of soil beneath the pond at (a) Singapore and (b) Kcw.
it, i.e. the temperature variation of the soil at different depths with respect to time. For a pond in Singapore, the ground water temperature was assumed to be 20 °C and located at a depth of 4 m from the bottom of the pond, whereas the ground water temperature at Kew was assumed to be 7 °C. The soil under the pond acts as a thermal stabilizer receiving heat at times when the storage temperature is higher than the ground and
112
M. N. A. Hawlader
(o) SINGAPORE TEMP
VARIATIONS
~
130
WITH
LOAD
DALLY LOAD NO LOAD
100
U o
~
LU
z o N
70
2 M J m -2
t.U
L9 -o t-.-
Ds = l . O m
&O
H
=
hi =
9
I0
0
~I = 0. S m
Z0m
ml-- 0 . 9 5
0.2 m
I
I
I
I
250
500
750
1000
UJ
NUMBER
OF
DAYS,
N
¢y
(b)
,~o
TEMP
~E
VARIATIONS
W I T H LOAD
KEW
UJ
LU
o (t n... u.i
90 D A ILY L O A D NO LOAD
60
30
0
0
LOAD
Fig. 9.
i
I 25O
I
150
_ .o.
°_
LOAD
_ NO LOAD
I
I000 LOAD
NUMBER OF DAYS, N Effects of load on the temperature of the storage zone of the ponds at (a) Singapore and (b) Kew.
PerJbrmance characteristics of solar ponds operating at different latitudes
113
delivering it when the temperature of the storage zone is lower than the soil. Considering the effects of all the factors described earlier, the depth of the pond at Singapore was selected to be 3 m while that of the Kew pond was considered to be 3.5 m. Figure 9 shows the dimensions of such ponds and the effects of load on the temperature distribution. For the Singapore pond the load was assumed to be continuous. This pond has been considered for space cooling applications. A pond at Kew has been considered mainly for space heating applications. For space cooling, the minimum temperature required for an efficient operation of an absorption chiller is 75 °C. From Fig. 9(a), it is seen that this pond can supply 2 MJ m - 2 per day without reducing the temperature of the storage zone below 75 °C. This pond can also supply 3 MJ m - 2 per day, but the temperature will be lower than the desired value. In this case, an auxiliary heater may be needed. A higher load can be supported if the extinction coefficient of the salt water is reduced further, making the pond more transparent. Under this condition, this pond should be able to support a higher load and the pond area required for any particular application will be further reduced. The pond at Kew is likely to be able to support a space heating load in winter and thus operate seasonally, collecting energy during the summer and supply heating load during the winter, thereby acting also as an interseasonal storage device. This pond can support a load of about 1.5MJm -2 day, which is slightly smaller than that delivered by the Singapore pond. But for a space cooling application, an absorption refrigeration system is used and the coefficient of performance of such systems is quite low, e.g. COP = 0.7. No such system is required for a space heating application. Although, the ponds are considered for space heating and cooling applications, they can also supply domestic hot water and industrial process heat. In the case of industrial process heat, an auxiliary heater may be necessary depending on the temperature requirements. It is interesting to note that a pond in Singapore located near the equator attains a much higher temperature compared with a pond at Kew but the temperature requirements for the load are also high for Singapore as far as the supply thermal energy is concerned. CONCLUSION This paper compares the performances of solar ponds at different latitudes. The mathematical model 5'6 has been further improved and
114
M. N. A. Hawlader
used as a tool to facilitate this comparative study. The effects of different parameters affecting the operating conditions of the pond at the two locations considered in this study have been investigated with the hourly meteorological data of the two locations. This study indicates that a pond at Kew should be deeper than one at Singapore. The pond at Kew also acts as an interseasonal storage device. These ponds can support loads at the temperatures demanded by local conditions. Auxiliary heaters are sometimes required for both the ponds depending on the type and magnitude of the loads. ACKNOWLEDGEMENT The author would like to thank Miss Lee Pui Chan and Mr Yeoh Seng Chye for their help in the computational work.
REFERENCES 1. H. Tabor, Solar ponds--large area solar collector for power production, Solar Energy, 7(1) (1963), p. 190. 2. L. Bronicki, J. Lev-Er and Y. Porat, Large solar electric power plant based on solar ponds, Worm Power Conf., Munich, 1980. 3. C. E. Nielsen, Experience with a prototype solar pond for space heating, Proc. ConJl Sharing the Sun, Vol. 5, ISES (USA) and Solar Energy Soc. of Canada, Winnipeg, 1976, pp. 169-82. 4. H. C. Bryant and I. Colbeck, A solar pond for London? Solar Energy, 19 (1977), p. 321. 5. M. N. A. Hawlader and B. J. Brinkworth, An analysis of the non-convecting solar pond, Solar Energy, 27(3) (1981), pp. 195-204. 6. M. N. A. Hawlader, The use of solar pond for air conditioning, Proc. Conf. on Utilization of Solar Energy for Refrigeration and Air Conditioning, March 14-19, International Institute of Refrigeration, 1982, pp. 61-74. 7. S.A. Shah, T. H. Short and R. P. Fynn, Modeling and ~testinga salt-gradient solar pond in northeast Ohio, Solar Energy, 27(5) (1982), pp. 393-401. 8. D. L. Styris et al., The non-convecting solar pond applied to building and process heating, Solar Energy, 18 (1976), p. 245. 9. C. F. Kooi, The steady state salt gradient solar pond, Solar Energy, 23 (1979), pp. 37-45. 10. M. N. A. Hawlader, The influence of extinction coefficient on the effectiveness of solar ponds, Solar Energy, 25 (1980), p. 461. 11. H: Weinberger, The physics of the solar pond, Solar Energy, 8 (1964), pp. 45-62.
Performance characteristics of solar ponds operating at different latitudes
115
12. A. Akbarzadeh and G. Ahmadi, Computer simulation of the performance of a solar pond in the southern part of Iran, Solar Energy, 24 (1980), p. 143. 13. A. Rabl and C. E. Nielsen, Solar ponds for space heating, Solar Energy, 17 (1975), p. 1. 14. J. R. Hull, Computer simulation of solar pond thermal behaviour, Solar Energy, 25 (1980), p. 33. 15. J. R. Hull, Calculation of solar pond thermal efficiency with a diffusely reflecting bottom, Solar Energy, 29(5) (1982), pp. 385-9.