An analysis of the non-convecting solar pond

Solar Energy Vol. 27, No. 3, pp. 195-204, 1981 Printed in Great Britain.

0038-092X/81/090195-10502.00/0 Pergamon Press Ltd.

AN ANALYSIS OF THE NON-CONVECTING SOLAR POND M. N. A. HAWLADERand B. J. BRINKWORTH Solar EnergyUnit, UniversityCollege, Cardiff, Wales

(Received 27 April 1981) Abstract--The thermal behaviour of the non-convectingsolar pond is examined by numerical solution of the dynamic equations, incorporating detailed representation of the losses from the surface and using hourly meteorologicaldata for a site in southern England. Temperature histories for the first few years of operation are given, showing the influence of the leading physical characteristics of the pond. It is shown that the pond temperatures are stronglydependent on the effective extinction coefficientfor solar radiation and the thermal losses from the pond bottom. The temperature history approaches a quasi-steady form within two to three years of operation, depending on the load demand. Using realistic assumptions for the main determinants of pond behaviour, it is shown that modest loads (around 10 per cent of the average insolation) can be served in this climate at temperatures appropriate for practical applications.

INTRODUCTION

The concept of the solar pond for simultaneous collection and storage of solar energy has been known for some time[I,2]. Most of the early investigations were confined to its use for power production[3]. In the early 1960's, a considerable amount of work was done, mostly in Israel, before it was concluded that solar ponds were not economically competitive with conventional power sources existing at that time [4]. The fuel crisis of 1973 and subsequent concern about the availability of fossil fuel created renewed interest in solar ponds applications. Among these, use of solar ponds has been proposed for space heating[5,6], process heating[6,7] and desalination [8]. The mathematical formulation of the behaviour of solar ponds was first given by Weinberger[9], whose analytical solution of the partial differential equation for the transient temperature distribution was obtained by

superimposing the effects of radiation absorption at the surface, in the body of water and at the bottom, the effect of each being considered separately. This technique was subsequently followed by Dake and Harleman[10] and Akbarzadeh and Ahmadi[ll]. The results are restricted by the assumption of boundary conditions considered by Weinberger. In a real pond, there is a mixed layer at the surface caused by wind action and heat transfer. There may also be a mixed layer at the bottom forming a zone for storage and,extraction of energy, as shown in Fig. 1. An analytical treatment of this more general case has been reported by Rabl and Nielsen[5]. In the present study, the basic energy equation has been solved numerically, which gives greater freedom to incorporate appropriate initial and boundary conditions, and more realistic representation of climatic conditions, load variations and so on. Performance estimates have been obtained, using solar radiation

I

6

so~ace

---I-........ .II o,o,,o0 .o lr D layer

from load

.... t.......................... Os

storage zone

Fig. 1. The solar pond. 195

----F .

.

.

.

.

Surface absorption layer

.

to

196

M. N. A. HAWLADERand B. J. BRINKWORTH

hourly meteorological data for Kew in West London flat. 51 28N, long. 00 19W). These are compared with earlier estimates for operation in the UK and provide a pointer to potential solar pond performance in other high-latitude maritime climates.

ing form provides an adequate representation of the essential features of absorption: In =/~(1 - F) exp (-/xz)

(1)

where the path length to a depth h is given by A~ORPTION OF RADIATION IN TIlE POND

Penetration of solar radiation into an outdoor pond is a matter of some complexity. Part of the incident radiation is reflected at the surface, determined largely by the solar elevation and the degree of surface agitation. The radiation which enters the surface is gradually attenuated by absorption in the water and by scattering by suspended particles. Absorption is dependent on the wavelengths; the short and long wave parts of the spectrum are absorbed within a few centimetres of the surface, the rest penetrating more deeply. Calculations have been made of the attenuation of solar radiation in water by considering it to be divided into spectral bands with known attenuation coefficients[5,9, 13]. For example Rabl and Nielsen[5] used 4 bands and more recently Hull obtained similar results using 40 bands[B]. Bryant and Colbeck[12] and Hull[B] have fitted simple analytic functions to the resulting irradiance data for use in calculations of pond performance. The general form of the variation of direct solar irradiance with distance in water is shown by the data of Fig. 2. This gives measurements made with fresh and salt water in the authors' laboratory, compared with others from various sources[9, 14,26]. It is indicated that reflection and absorption near the surface is followed by an approximately exponential decay. Variations in irradiance at a given depth are due to differences in the quality of the water. As far as the direct radiation is concerned, it is considered that an exponential approximation of the follow-

z = (h - 8) sec Or.

The factor F may be interpreted as a measure of that part of the radiation which is absorbed within a short distance, 8, of the surface. Consideration of the data of Fig. 2 suggests that the thickness 8 is about 0.06m; F appears to have an average value of about 0.4 and to be independent of time[16]. A time-dependent element enters through the quantity I,, equivalent to the irradiance just beneath the surface. This is represented by /~ = (1 - a)G,

(3)

where G is the solar irradiance at the surface and "a" is the albedo of the surface. The albedo is taken to depend only on the solar zenith angle 0,, in the manner shown in Fig. 3, after the data of Raphael[15]. The time-variation of solar zenith angle throughout the year is readily determinable as a function of latitude by well-reported methods [17]. A ray incident on a horizontal surface at an angle 0z to the normal is refracted to the angle 0, given by Shell's law: sin Or = (sin O,)/n,

(4)

where n is the refractive index of the water. Equations (1)-(4), together with meteorological data, make it possible to estimate the rate of absorption of energy at depth h as a function of time. The adequacy of

I.o 0.8

0.6

QN"•

0

0

0

O.

0

g

~k = 0.32 m-I

'Z: x ref. 26

e

.e-,

-..< ""O"e

0.2

author

I

0.5

(2)

I

1.0 path length, m

Fig. 2. Penetration of solar radiation in waters.

ret g range for 6 sites. ~

= I

1.5

1.0rn -I

An analysis of the non-convectingsolar pond

197

0.3

0.2 O

r~ O

m

0.1

!

1

I

60 solor zenith ongle Oz , degrees 30

9O

Fig. 3. Variation of surface albedo with zenith angle (after Raphael, Ref. [15]).

the approximations cannot be determined without further measurement under a wide range of climatic conditions. For instance, the effects of climatic variables on the albedo " a " is not well established. A particular difficulty attaches to the treatment of the diffuse component of the solar radiation. This has been addressed recently by Hull[13], who showed for a particular location that the penetration of the diffuse component could be represented by assuming it combined with the direct component at a particular incidence angle. Curves of irradiance vs depth very similar to those of Fig. 2 are found for the combined radiation. Pending further study of this question, the irradiance G in eqn (3) is taken to be the hourly mean global irradiance. It appears that the effects of variation in the diffuse fraction may reasonably be consolidated in the empirical constants of the penetration function eqn (1). In particular, variation of the effective attenuation coefficient # provides a means of investigating the sensitivity of pond behaviour to several factors whose influence is uncertain. It has been shown elsewhere[18] that/~ has a powerful effect on pond temperature distribution and collection efficiency. From Fig. 2, the range of values of encompassing most of the data run approximately from 0.32 to 1.0 m-', and these values have been used in the work now reported. MATHEMATICAL FOMULATION

fer in the vertical direction. The lateral dimensions of the pond are assumed to be large compared with its depth, and, consequently, the heat transfer through the surrounding walls is negligible compared with the vertical heat flux. In this analysis, it is assumed that radiation reaching the bottom is absorbed there. This is equivalent to the assumption that the absorptance of the bottom liner is unity. In a real pond it may be difficult to ensure that the bottom remains unreflecting. However, the assumption may be robust, since any reflected radiation, which will be of a diffuse form, would be likely to be absorbed mainly within the storage zone, which would typically be of the order of one metre in depth. The effects of imperfect absorption at the bottom liner would then depend mainly on the relative values of the conductance for heat transfer upwards and downwards from the liner. Equation (5) may be written in a dimensionless form in the following way: 00 020 + D2Ih¢~ sec O, Or - aZ~ kTo _

(6)

where O = T] To,

Z = h/D

and

A heat balance for a small layer in the non-convecting insulating zone gives the following equation[18, 19]:

at

r = ~----~,the Fourier number. /3"

aT [a2T\ --at :'~[-~)

+ i'

h~(pc,)-i sec 0r

(5)

This equation is based upon one-dimensional heat trans-

The solution of eqn (6) requires two boundary conditions and an initial condition. The initial condition can be set up in any way and is taken to be known. The pond is

M. N. A. HAWLADERand B. J. BRINKWORTH

198

assumed to have a surface mixed layer of thickness h~, in which heat balance results in the following equation: 0Ts Ot

Is

ph~Cp

[1 - (1 - F) exp {- ~(h - 6) sec 0r}]

k

QLs = ULs(Os - 0,).

OLs

+ ph--~ \ - ~ - ] , =h~ oh,Cp"

(7)

The first term on the r.h.s, represents the effect of the fraction of the radiation absorbed in the surface mixed layer. The second term represents heat transfer by conduction between the surface mixed layer and a layer below it. The third term represents heat losses due to convection, radiation and evaporation from the surface to the atmosphere in contact with it. Equation (7) may be cast in the dimensionless form 00,

0r

H2ls ..

k-Tohoh~U - ( 1 - F ) e x p { - ~ ( h , - 8 ) s e c 0,}]

+ foo,

hi ~,'~'/Z=ZI

VLsO "~1 (Os -- Oa)"

(8)

The bottom boundary condition is obtained from a heat balance for the storage zone, giving 0Tb _ (1 - F)L k O-"-t-- pDsCp exp [-/z(D - 8) sec Or] pDsCp X \ ah/h=O

pDsCp

pDsCp"

The sum of the surface heat loss rates QLs determines the surface heat loss coefficient ULs in eqn (8) through the identity

(9)

The terms on the right represent respectively the effects of solar radiation reaching the storage zone, conduction to the insulating layer, losses to the ground beneath tile pond and heat delivered to the load. In dimensionless form eqn (9) becomes

(11)

However, the empirical equations used to determine QLs have been developed from measurements of losses from a body of convecting fluid at nearly uniform temperature, above that of the ambient air. In the case of the solar pond, the losses occur from a thin mixed layer near the surface which is resting on an insulating layer in which convection is suppressed. It is not yet clear to what extent the bulk loss equations might apply to this case, particularly when the liquid surface temperature is equal to, or less than the ambient air temperature. Pending further work it has been assumed that in these conditions the loss could not exceed the rate at which heat could be conducted across the surface layer, giving ULS = k/(D - hD.

(12)

Uncertainties remain, also, about the extent to which the ground beneath the pond can act as a reservoir for heat, returning to the pond at times of low temperature a proportion of that lost at times of high temperature. In practice the bottom loss will vary between installations, and may be dominated by transfer to moving groundwater which would be likely to convect heat away from the pond at all times. In the studies reported, the bottom loss is represented by a conductance ULo and a constant sink temperature of 70C is assumed; the significance of the bottom loss is examined by varying the value of ULo assumed. Further work on the bottom loss is in hand.

RESULTS AND DISCUSSION

Few previous assessments of solar pond behaviour d0b _ D2(1 - F)Is exp [-/z(O - 8) sec Or] D (d0b have been made for British conditions. An early study by a---;k ToO, - ~ C ~ / z =, Brinkworth, contributed to a UK-ISES collective ULGD2 QDD2 survey[25] concluded that a pond could serve a load (Oh - Oow) kToDs . (10) equivalent to 10 per cent of the insolation (i.e. about 350 MJ/m 2 per annum) at temperatures suitable for space Equation (6), with the surface boundary condition eqn (8) heating. However, Bryant and Colbeck[12], using the and lower boundary condition eqn (10), is solved methods of Rabl and Nielson[5], found that loads of 20 numerically by a finite difference procedure, using the per cent of the insolation could be served at mean annual Crank-Nicholson approximation. To minimise round-off temperatures rising to over 70°C, depending on the errors, step sizes are determined in accordance with the thicknesses of the insulating layer and the storage zone. criteria recommended by Richtmyer and Morton[20]. These predictions are for the quasi-steady periodic behaviour reached after initial transients have died away. HEAT LOSSES FROM THE POND The initial behaviour found in the present study is shown Heat losses from the surface of the pond take place in Fig. 4 for the first year of operation, beginning in April through radiation, convection and evaporation. Empirical (data for 1963).This is for the no-load condition and shows equations for these processes have been developed and the strong influence of the extinction coefficient. It is shown are summarized elsewhere[21]. Radiation losses are that a maximum temperature of 63°C is reached with a referred to ambient temperature by incorporating the reasonable value of/z = 1.0 m-~, but that if it were possible relation between this and the effective sink temperature to maintain a value of 0.32 m-~, the maximum temperature for the atmosphere given by Idso and Jackson[22]. Con- in the storage zone would approach 100*C, with a storage vective and evaporative losses are evaluated by well- zone thickness of 1.0 m. Figure 5 shows how the initial known methods relating these to the ambient tem- temperature peak is affected by the thickness of the storage perature and wind speed [23, 24]. zone.

An analysis of the non-convecting solar pond Figures 6 and 7 show the effect of variation in the thickness of the insulating layer. It is seen that the seasonal fluctuation is damped as the thickness is increased and that the maximum temperature at first increases and then begins to fall as less of the incident radiation reaches the pond bottom. For the range of extinction coefficient covered, maximum temperatures are reached for pond depths around 1-1.5 m. The effects of applying a load to a pond during the initial year of operation are illustrated in Figs. 8 and 9. In Fig. 9 it is seen that if heat extraction starts around the beginning of the space heating season, useful temperatures are maintained throughout the first season for

199

loads up to 1 MJ/m 2 per day. Figure 10 extends the operating time illustrated to a period of two years, showing that a quasi-steady periodic state is not reached in this time for the no-load condition, though a continuous extraction of heat at a rate of 1 MJ/m 2 per day from the autumn of the first year stabilises the pond more quickly. For space heating applications the load requirement varies seasonally, and the behaviour is illustrated in a general way in Fig. 11, which shows the effect of a discontinuous load applied only in the winter months. For this case, the temperature cycle becomes stabilised by the third year. Finally, the sensitivity of the behaviour to heat loss

120,

1011

3 80 S 0

E ~ 6O 0 C

~ 4o g g 2o 0o

I

S0

I

100

I

150 deys from

I

i

200 250 start {14 April)

I

300

i

I

350

400

Fig. 4. Storage zone temperature with no load. (h, = 0.1 m, D = 1.5 m, D, = 1.0 m).

100

3,o 80

60 ~ 40 g E 2

20

0 0

,60

I

,so days

2bo

I

I

250

300

from start (14 April}

Fig. 5. Storage zone temperature with no load. (h~ = 0.1 m, D = 1.5 m,/~ = 0.32 m-~).

I

350

4bo

200

HAWLADER and B. J. BRINKWORTH

M.N.A.

u 120[ 1001-

i,° ~ 6o

g 40 2

y

~

Insulation layer I J thickness, D ]

1

/////

\

///// //////

\

\ \\

2.4m

~~;.~=

"~

~

/o,.

~,,,/0.6m

m 20

I

I

50

100

I

i

I

150 200 250 days from start (14 April )

I

300

I

I

350

400

Fig. 6. Storage zone temperature with no load. (hi = 0.1 m, D, = 1.0 m, # = 0.32 m-l).

120

C.3 e

100

= 80

7 e~

~ 60

i

I Insulating layer //1.8m

40

E

o 20 I

so

~6o

,~o

2~o

I

2~

I

300

I

3so

days from start (14 April ) Fig. 7. Storage zone temperature with no load. (h, = 0.1 m, D, = 1.0 m,/~ = 1.0 m-I).

4'oo

An analysis of the non-convecting solar pond

,-, loo 2

8O

201

f Load in

E

-* 60 tO N

40

O

m

2C

Oo

I

I

50

100

I

150 days from

i

200 start

I

I

I

250 300 (1/, Aprit )

O0

350

4

Fig. 8. Storage zone temperature with load after 2 months. (h~ = 0.1 m, D = 1.5 m, D, = 1.0 m).

=; 80

~

e ~L 60 E

[ ,oad in 0.0 0.5 1.0 0.0 0.5 1.0

O

,- z,0

00

I

50

I

100

I

I

i

I

150 200 250 300 days from start (14 April)

I

350

Fig. 9. Storage zone temperature with load after 5 months. (h~ = 0.1 m, D = 1.5 m, Ds = 1.0 m).

400

1

202

M . N . A . HAWLADERand B. J. BRINKWORTH

o

I~ 10o 0

.No load

E

6O ¢O N

4O

O

2(1

1 MJim 2 per day I - April - December ----~--,---January - December

I -I--L

January ..........

Fig. 10. Storage zone temperature over 2yr. (hi =0.1 m, D = 1.5 m, D, = 1.0m, p. = 1.0m ~).

100 ~3

I

80 No Load

-

--

Load -~ No Load ~ I MJ/m2per day

Load 1MJ/m 2 per day

60

E 40 0 N

g

20

0

Apn'l -Dec

_

January - December

I -7-

January

Fig. 11. Storage zone temperature with intermittent load. (h~ = 0.1 m, D = 1.5 m, D, = 1.0 m, ~ = 1.0 m-l).

An analysis of the non-convecting solar pond from the pond bottom is illustrated in a general way. In previous figures a bottom-loss coefficient ULO of 0.12W/m2K was used. Figure 12 shows the effect of a three-fold increase in this coefficient, which might result from a nearby groundwater flow. It is seen clearly that the bottom loss is a significant determinant of storage zone temperature. At the higher loss rate, the pond performance is close to that predicted by Brinkworth[25]. Results resembling those of Bryant and Colbeck[12] are obtained if low values are assumed for the bottom loss and extinction coefficient. CLOSURE

The study continues, with an examination of pond behaviour when serving loads with more detailed timehistories, as required by building space-heating and industrial processes, and of the technical and economic optima for leading pond design variables. The selection of results presented here show the strong dependence of the pond temperature on the extinction coefficient for solar radiation and the thermal losses from the pond bottom. Further measurements of both quantities are required to support design calculations. With realistic assumptions for the main determinants of pond behaviour, it is found that modest loads can be served at useful temperatures, even in an unpromising climate. Hour-by-hour simulation has made it possible to examine the transient behaviour at start-up, and it is indicated that under typical loads, a quasi-steady periodic state is likely to be reached after 2 yr of operation.

203

Acknowledgement--This study was carried out with support from the UK Science Research Council.

NOMENCLATURE

a h hl k n t z Cp D Ds F G Ih /~ Qo QLc

QLs Tb To Ts ULG

ULs Z a 0~

O, # p ~"

pond surface albedo depth from pond surface thickness of surface mixed layer thermal conductivity refractive index time path length of radiation within pond specific heat capacity depth of bottom of insulating layer thickness of storage zone fraction of solar energy absorbed near surface global solar irradiance irradiance at depth h irradiance just below surface rate of energy transfer to load, per unit area rate of energy loss from bottom, per unit area rate of energy loss from surface, per unit area temperature of storage zone initial (uniform) temperature temperature of surface mixed layer conductance for bottom heat loss conductance for surface heat loss dimensionless depth, hiD thermal diffusivity solar zenith angle angle of refracted radiation effective extinction coefficient density dimensionless time or Fourier number, at~D2

100 O

,=

80

'-I

O

~, 60

"

Load 1M Jim 2 per day

E .

Bottom loss conductance

ULG

40

O N

g

O

Z0

O

0

- - April- Dec.

=i~

January - December

-j=

January

Fig. 12. Storage zone temperature with load; effect of bottom loss. (h~ = 0.1 m, D = 1.5 m, Ds = 1.0 m,/~ = 1.0 m-I).

204

M.N.A. HAWLADERand B. J. BRINKWORTH

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15. J. M. Raphael, Prediction of temperature in rivers and reservoirs. Proc. ASCE, J. Power Div. PO2, 157 (1962). 16. J. P. Holman, Heat Transfer, 4th Edn, p. 487. McGraw-Hill, New York (1976). 17. J. A. Duffle and W. A. Beckman, Solar Energy Thermal Processes. Wiley, New York (1974). 18. M. N. A. Hawlader, The influenceof extinction coefficienton the effectiveness of solar ponds. Solar Energy 25,461 (1980). 19. M. N. A. Hawlader, Analyticaland experimental study of the temperature distribution in a solar energy storage system. Proc. Conf. Storage in Solar Energy Systems, pp. 44--54. ISES (UK) London (1978). 20. R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, 2rid Edn. Wiley Interscience, New York (1%7). 21. M. N. A. Hawlader and B. J. Brinkworth, An analysis of solar heated outdoor swimmingpools. Proc. Conf. Practical Experiences with Solar Heated Swimming Pools, pp. 71-83. ISES (UK), London (1978). 22. S. B. Idso and R. D. Jackson, Thermal radiation from the atmosphere. J. Geophys. Res. 74, 5397 (1%9). 23. IHVE Guide, Book C, Reference Data, 4th Edn. Institution of Heating and Ventilating Engineers, London (1970). 24. W. H. Carrier, The temperature of evaporation. Trans. Am. Soc. Heating & Ventilating Engrs 24, 24 (1968). 25. Solar Energy: A U.K. Assessment. ISES (UK), London (1976). 26. C. E. Nielson, Experience with a prototype solar pond for space heating. Proc. Conf. "Sharing the Sun", Vot. 5, pp. 169-182. ISES (U.S.A.) and Solar Energy Society of Canada, Winnipeg(1976).