Membrane stratified solar ponds

Solar E~ergl Vol. 25. pp. 317. 325 Pergamon Press Ltd, 1980 Printed in Great Britain

MEMBRANE STRATIFIED SOLAR PONDSf JOHN R. HULL Department of Physics and Ames Laboratory-USDOE, Iowa State University, Ames, IA 50011, U.S.A. (Received 18 January 1980; revision accepted 31 March 1980) Abstract The membrane stratified solar pond is a body of liquid utilizing closely spaced transparent membranes to quench convective heat transfer in the top part of the pond. Membranes may be configured as horizontal sheets, vertical sheets or vertical tubes. Several suitable liquids and membrane materials are discussed. Conditions for suppression of convection are described, and transmission of solar radiation through the pond is discussed for each of the three membrane configurations. The steady state thermal efficiencyis calculated for the horizontal sheet configuration. Thermal behavior is similar to that of salt gradient solar ponds, but much deeper heat storage layers are feasible. In some cases aquaculture farming may be suitable in the storage layer.

I. INTRODUCTION Solar ponds have great potential as an inexpensive low-temperature heat source. Solar ponds with seasonal heat storage are especially attractive at higher latitudes, where heat load is often seasonally mismatched with available insolation. Most of the research on this subject has been confined to the salt gradient solar pond (SGSP)[1-3]. However, the SGSP has several inherent attributes which may limit its widespread use. Maintaining the salt gradient in a SGSP usually requires vigilance, and in many locations salt contamination is potentially a large environmental hazard due to the persistence of positive ions in soils. In addition to the toxicity to vegetation caused by overly salty soil, a major danger of salt pollution involves the detrimental effect of salt in drinking water upon people susceptible to hypertension and other salt related diseases [4, 5]. Rabl and Nielsen [1] have suggested several alternatives to salt water as methods for suppressing convection in solar ponds. This article develops one of their suggestions in the form of the membrane stratified solar pond (MSSP). The MSSP has the low cost and seasonal heat storage advantages of the SGSP, but can be configured to be environmentally sound and nearly maintenance free. The MSSP is illustrated schematically in Fig. 1. Like the SGSP, the MSSP is a body of liquid consisting of a nonconvective zone (NCZ), which serves as an insulating layer, on top of a lower convective zone (LCZ), which serves as a heat storage layer. Solar radiation is absorbed throughout the pond and on its darkened bottom and is converted to thermal energy by raising the temperature of the pond liquid. The temperature of the LCZ rises significantly above the ambient surface temperature, tPortions of this article were presented in a paper at the International Solar Energy Society Congress, 28 May-1 June 1979, Atlanta, Georgia, U.S.A. 317

resulting in a vertical temperature gradient across the NCZ. To minimize thermal losses through the sides and bottom, the pond must be large, or else the pond walls must be insulated. Then, since the pond liquid (e.g. fresh water) is opaque to infrared radiation, and since convection in the NCZ is suppressed, the thermal energy stored in the LCZ can only escape via conduction through the NCZ. Since most common liquids have moderately low thermal conductivities, if the thermal mass of the LCZ is large, then the time constant is long, and seasonal heat storage is possible. The major difference between the MSSP and the SGSP is in the mechanism for maintaining nonconvection in the NCZ. In the SGSP the salt concentration increases with depth in the NCZ, and the stabilizing effects of the resulting density gradient overcome the destabilizing effects of the temperature gradient. In the MSSP the NCZ is composed of liquid layers separated by transparent membranes. The distance between adjacent membranes is small enough that sufficient friction is available between the relatively fixed membrane surface and the fluid to com-

NCZ

-

MEMBRANE

REGION

Lczc C

%.\ \ \ \\\ ~\\-\\\~\~'~\\\\\\\\\\\\\\\\\\\\\\\\\t Fig. 1. Schematic diagram of membrane stratified solar pond (MSSP).

318

JOHN R. HULL

pletely suppress convection. Thus, no salt is needed anywhere in the MSSP, and convection suppression in the NCZ can be automatically achieved without maintenance procedures. The substitution of an assembly of membranes for the salt gradient in the NCZ produces other important differences between the MSSP and the SGSP. The SGSP is characterized by an upper convective zone (UCZ) which lies above the NCZ and in practice has a depth of at least 20 cm. Kooi [6] has shown that the UCZ decreases the overall efficiency of the SGSP and increases the depth of the NCZ necessary to achieve maximum efficiency under steady state conditions. The MSSP membranes can be placed very close to the surface, almost completely eliminating the UCZ. Another feature of the SGSP is that the LCZ usually contains a relatively concentrated salt solution. Salt is usually the major cost of the SGSP, and the depth of the LCZ is limited by economic considerations determined by the cost of the salt. The major cost of the MSSP is that of the membranes in the NCZ, and the depth of the LCZ can be much deeper than in the SGSP without significantly adding to the total cost. To deliver a reasonable amount of heat to the load, a minimum temperature in the LCZ must be maintained during periods of heat demand. A deeper heat storage region has the advantage of a smaller seasonal temperature fluctuation, so that a minimum LCZ temperature can be maintained with a lower average LCZ temperature, This results in a smaller temperature difference across the NCZ, so that the required depth of the NCZ is less, resulting both in higher collector efficiency and a lower cost for the membranes. A further advantage of a deep heat storage region is that the small temperature fluctuations in the LCZ may make aquaculture farming or biomass conversion at the bottom of the MSSP possible year round, even at higher latitudes.

2. M E M B R A N E C O N F I G U R A T I O N S A N D M A T E R I A L S

There are basically three types of membrane configurations for the MSSP: (1) horizontal sheets (HS), illustrated in Fig. 2; (2) vertical tubes (VT), illustrated in Fig. 3; and (3) vertical sheets (VS), illustrated in Fig. 4. In order to minimize cost and maximize solar transmission through the NCZ, it is desirable to make

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. . . . . . . . . . . . . . . . . . . . . . . .

~\\\\\

\\\\\\\\\\\\\

\\\\\\

~ \

\ "-.\\\\

\ \ \ \ \ \ \ \ ~

Fig. 2. Schematic diagram of MSSP in horizontal sheet configuration (HS).

Fig. 3. Schematic diagram of MSSP in vertical tube configuration (VT).

~.,~.,ii,,,

flLh i: iL]lt,.

,!t:

~

Fig. 4. Schematic diagram of MSSP in vertical sheet configuration (VS). most of the membranes as thin as possible. To implement this design feature a relatively thick horizontal membrane is placed across the upper pond surface in all configurations. This surface membrane prevents dust and debris from entering the pond and optically fouling the lower membranes. It also helps to minimize the mechanical stress induced on the lower membranes by surface wind effects. For all configurations minimal absorption and scattering are necessary membrane characteristics. In HS it is further necessary to match the index of refraction of the liquid to that of the membranes in order to minimize Fresnel reflection losses at the many membrane/liquid interfaces. If the NCZ liquid is water or ethanol, TeflonTM FEP film must be used in HS, since it is the only plastic whose index of refraction closely matches that of these liquids. Teflon is also possibly the best membrane material from an environmental criteria. It is inert to virtually every chemical, does not change properties after long periods of outdoor exposure, and may be recycled back into Teflon. 12.7-/~m (0.0005-in.) thick solar grade Teflon is commercially available. For vertical configurations, index of refraction is not a consideration, since specular reflection from the vertical membranes still propagates downward. This allows very inexpensive plastics to be used, some currently available in film thickness of 2.5 ttm (0.0001 in.).

3. S U I T A B L E L I Q U I D S

The MSSP liquid in the NCZ should have maximum transparency over the solar spectrum. Transparency of the LCZ is not as critical, but this liquid should be at least as dense as the NCZ liquid to avoid

Membrane stratified solar ponds stressing the horizontal membranes. Other desirable properties of MSSP liquids include high viscosity, large heat capacity, low thermal conductivity, and stability under both sunlight and moderately high temperatures. NCZ liquids discussed here include fresh water, concentrated sugar solution, and ethanol. Fresh water is an excellent MSSP liquid, since it is very stable, has moderately good values of viscosity, thermal conductivity and heat capacity, and in many locations is quite abundant and cheap. It would be necessary to initially prepare tap or ground water before it enters the NCZ to eliminate dust and microorganisms which might scatter or absorb the incoming solar radiation. This preparation can be accomplished with standard swimming pool technology and would include filtration and probably chlorination or bromination [7]. Once initially treated, little further treatment should be required, as the NCZ is sealed off from the environment. Except for the above considerations, the water need not be especially "pure". The only salts that are significantly absorbing in the visible are those of the transition metals, which exhibit weak absorption due to ligand field splitting [8]. Iron, sometimes found in groundwater in concentrations as high as 7 ppm [9], has a molar absorptivity of about 0.1 1. mole -x cm -~ in the bluegreen part of the spectrum [10], where pure water is most transparent, Copper, sometimes used at slightly lower concentrations to control algae, has approximately the same molar absorptivity in this spectral region [10]. Comparable concentrations of either of these ions result in an increase of radiation attenuation in this spectral region of approx. 0.1 per cent through 1 m of water. Complex organic compounds are usually strongly absorbing at visible wavelengths and should be excluded from the NCZ. One class of MSSP liquid is characterized by the addition of a thickening agent to water. An example of this is concentrated sugar solution. When mixed with water, sugar sufficiently raises the index of refraction of the solution so that cheaper plastics can be used in HS. In addition, the viscosity is significantly increased, so that compared to fresh water, approx, one-tenth the number of membranes are required. Further, sugar at high concentrations is an antimicrobial agent. One disadvantage of sugar solution is that since sugar solution is heavier than water, and since the membranes will not support much stress, the entire pond must be composed of sugar solution, or an even denser liquid must be found for the LCZ. Another possible liquid is ethanol, a by-product of many agricultural industrial processes. Ethanol has about the same index of refraction as water but has a much lower thermal conductivity. Compared to water, the NCZ need be only 29 per cent as deep for the same insulating value, and approximately half as many membranes are required. This, together with ethanol's comparable transparency, results in more solar radiation reaching the LCZ. Since ethanol is less

319

dense than water, it will float on top of a fresh water LCZ without stressing the membranes. Thus, only a relatively small part of the MSSP need be ethanol. The UV absorption spectrum of ethanol is similar to that of w a t e r [ i l l , so ethanol should exhibit long term stability in solar radiation. Like sugar solution, high concentrations of ethanol are toxic to biologi.:al organisms. Problems with ethanol include flammability at high concentrations, surface instability to rainfall, and evaporation through the top membrane. Diffusion of ethanol through a 76-,um (0.003-in.I Teflon membrane, however, is estimated at less than 1 mm/yr. A combination of water and ethanol could serve to overcome several potential engineering problems in the MSSP. With a homogeneous liquid the temperature gradient across the NCZ results in a density gradient, due to volume expansion with increasing temperature. Colder dense fluid is on top, the warmer lighter fluid is below. This situation is unstable and will mechanically stress the membranes in HS, e~en though convection between the membranes is efl;ectively suppressed. A simple way to eliminate this macroinstability is to use progressively higher concentrations of ethanol at successively higher levels in Ihe NCZ. The concentration increase is large enough that each horizontal layer in the NCZ will be denser than the layer above it for all NCZ temperature profiles encountered during the year. This results in absolute macrostability, since a less dense fluid will float or~ a more dense fluid without stressing the membrane separating the fluids. A further advantage in having a relatively high concentration of ethanol towards lhe pond surface is to prevent freezing during the winter. This eliminates deterioration of the membrane structure associated with ice expansion.

4. CONVECTION BETWEEN MEMBRANES

The theory of convection between membranes of different geometries has been previously examined :Eor the case of honeycomb panels that suppress convection in flat plate collectors [12 14]. Convection o~ a fluid contained between a pair of horizontal membranes heated from below is governed by the equation R = "q~ fld ~,

(1)

VK

where R is the Rayleigh number, g is the acceleration of gravity, ~ is the coefficient of thermal expansion, v is the kinematic viscosity, K is the thermal diffusivity, fl is the vertical temperature gradient across the fluid, and d is the vertical distance of the fluid between the membranes. When R is less than the critical Rayleigh number Rc (for the case of rigid horizontal membranes, Rc = 1708), then convection in an initially stagnant fluid is suppressed. As R increases above R,., convective heat transport increases, with the heat flux

320

JOhN R. HULL

across the fluid given by

1.0

Nu =

h/hc,

where Nu is the Nusselt number, h is the total heat flux, and hc is the heat flux due to conduction only. From a set of parameters at Re, Nu increases from 1 to 1.7, if fl is doubled; and increases from 1 to 2.9, if d is doubled [15]. The calculation of the parameters that just suppress convection are similar for the other configurations. Convection for long circular tubes is governed by (1), except that d is the radius of the tube. Rc for the first convective mode is 67.9 [16], so in VT the tube diameter is about 0.9 that of the HS spacing for the same temperature and temperature gradient. For VS, d is the perpendicular distance between the parallel vertical sheets. In this case Rc = 500.6 [16], requiring 0.74 the membrane spacing of HS. The fluid physical parameters, ~t, v and K, are functions of the temperature. Hence, for a given temperature gradient, the distance between membranes that just suppresses convection, d~, is also a function of temperature. This is illustrated graphically in Fig. 5 for the case of water in HS. The corresponding case of ethanol in HS is illustrated in Fig. 6. To suppress convection at an equivalent temperature gradient, concentrated sucrose solution with the same index of refraction as Tedlar TM requires a membrane spacing approx. 10 times that of water. The dashed lines in Fig. 5 denote arbitrarily chosen extremes for the temperature profiles that might be found in typical solar ponds. They illustrate an important design distinction between HS and vertical configurations. In vertical configurations the spacing between membranes is constant throughout the depth of the NCZ. The spacing must be chosen to suppress convection at the most unstable depth of the NCZ. Throughout most of the NCZ depth, the vertical membranes are then spaced closer together than what is necessary to just prevent convection. In HS, however, the membrane spacing can be varied according to the temperature conditions at a given depth. For example, according

i

1.0

i

,

i

,

i

\ . . . . .

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i

-

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c

:

0.3 CM

0. I t)

0.01

20

0

40

60

t (°C)

Fig. 6. Distance dc between horizontal membranes in ethanol that just suppresses convection--as a function of temperature t and temperature gradient ft. to Fig. 5, a MSSP operating at low temperature (40°C) would space its horizontal membranes closer together at the top of the NCZ than at the bottom. 5. RADIATION TRANSMISSION

In general the transmission of radiation through the NCZ depends on the angle of incidence, radiation wavelength and polarization, membrane geometry, and the index of refraction, absorption, and scattering properties of both the membranes and the liquid. The present discussion is confined to water. At 2 = 632.8 nm, ethanol has an absorption constant approximately half that of water [17], and visual comparisons indicate that the transparency of ethanol or sugar solution approximates that of water. We assume that all reflections are specular and that there is no scattering in the liquid or the membranes.

Horizontal sheets We consider a set of N layers, each layer consisting of one membrane and one liquid layer. For now we ignore any absorption of radiation in the liquid layer. The transmission of light at wavelength ). through a set of horizontal layers is given by [18]

l

dc=O.4CM

PsPM)-1, (3) Ps+~(p, 3) = Ps + Pu T2 (1 - PsPM)-1, (4) TN+M(p,3) = Ts TM (1 --

where the transmission, TN (TM), is the fraction of incident radiation transmitted through N (M) layers, and PN (PM) is the total fraction reflected from N (M) layers. These variables are generated from

u o

----_.4

(2)

0. I

T t = 3(1 - p)2(1 - p2T2)-1, 0.01 0

20

40

60

80

l'Oi J I00

t(oc)

Fig. 5. Distance d, between horizontal membranes in water that just suppresses convection--as a function of temperature t and temperature gradient ft. The dashed lines denote extreme values of the temperature profile that might be found in some typical solar ponds.

P1 = p[1 + z2(1 - p)2(1 -

(5)

p2z2)-l],

(6)

where p is the reflectance from a single liquid-membrane interface, given by p = sin2(0i - 0,)/sin2(0i + 0,)

(7)

for radiation polarized perpendicular to the plane of

Membrane stratified solar ponds

321

0,5

incidence, and

i

i

_-- . . . .

p = t a n 2 ( 0 i - 0j/tan2(0~ + 0r) for radiation dence, where in water, and brane. 01 and

n~ sin0i = n, sin0,.

0"3 0.2

~ t

0.1 0.0

r = e x p ( - K~ L/cosO,),

(10)

where K)~ is the membrane attenuation coefficient at wavelength 4, and L is the thickness of the membrane. The transmission for each 2 and polarization in (3) is built up by iteration from (5), (6) and (4). The effect of a difference in index of refraction between the liquid and the membranes can be determined by setting Ka = 0 in (10). The results of this analysis for both Teflon and Tedlar TM membranes in water are illustrated graphically in Fig. 7, where transmission is plotted versus the number of layers for several angles of incidence. The index of refraction is a function of 2 for both water and the plastics, and so the results shown in Fig. 7 will change slightly as a function of 4. The values shown in Fig. 7 are typical for the middle part of the visible spectrum. According to Snell's law, light entering the top of the pond from air will refract and have a maximum angle of incidence in water of 48.6". Then, as indicated by the results shown in Fig. 7, Fresnel losses for Teflon in water are negligible. When the appropriate absorption constant for the membranes is used in (10), the results of (3) then give the transmission through the NCZ as a fraction of the transmission through an equally deep NCZ of liquid alone. The total transmission for a given 2 at a depth z from the surface, T(2, z), is given by

T(;t, z) = Tt(2, z) TN(2),

. . . . . . . . . . . . .

(11)

oq

6o*1

o.81-\ ".. ~7o 1

I-

ONLY

(9)

For our case, n~ is the index of refraction of water, and n, is the index of refraction of the membrane. Further,

I1 " ~ - - ~

WATER

(8)

polarized parallel to the plane of inci0~ is the angle of incidence, in this case 0, is the angle of refraction in the mem0, are related by Snell's law

,.4.

i f i

~

21- " -"----°°

,

0 o

I

20 °

i

i

L

40 o

I

60 °

I

_1

~1

80 °

0a Fig. 8. Total transmission T of solar radiation through a 1.0 m deep NCZ containing 200 horizontal Teflon membranes in water as a function of angle of incidence in air 0,. L is the thickness of each membrane. where Tt is the transmission through the liquid only, and TN is the transmission through the set of N membranes. The total transmission through the NCZ is found by integrating T(;,, z) over the solar spectrum and summing on both polarizations. As an example, the transmission through a 1.0 m deep NCZ consisting of 200 Teflon membranes in water is calculated as a function of the angle of incidence in air at the pond surface. We assume Kz = 0.59cm-1, independent of wavelength. The total transmission T as a function of depth z and angle of incidence in air 0, is given by

T(z, 0,) = z,(O,) TN f

TI(A,Z/COSOi) dZ,

112)

where %(0,) is the coefficient of transmission at the pond surface due to Fresnel reflection. The integral in (12) can be evaluated by the method of Hull [19], which divides the solar spectrum for 2 < 1200nm into 40 wavelength regions. The results of this calculation are shown in Fig. 8 for the case of 2.54 #m thick membranes (upper curve) and 12.7/~m thick membranes (lower curve). The dashed line is the transmission through pure water with no membranes. For all cases, the two polarizations have been averaged. The above analysis probably overestimates the transmission slightly, because scattering in the membranes and non specular reflection is neglected. This scattering would tend to produce more light at higher angles of incidence, resulting in more Fresnel losses and longer path lengths through the membranes.

Vertical sheets •

0

50

I00

150

200

N

Fig. 7. Transmission T as a function of membrane number N and angle of incidence 0~ in water for Teflon and Tedlar membranes. In all cases the transmission is the average of both polarizations, and the attenuation coefficient of both water and the membranes is assumed to be zero. nw.t, = 1.333, nT¢no,= 1.343 and nr,dl~ = 1.455.

The transmission of light through a set of vertical Sheet membranes can be calculated using the geometry of Fig. 9. Membranes of thickness L are located in the yz plane and separated from adjacent membranes by horizontal length x. Light in water is incident from above at polar angles 0 and 05. 0~ is the angle of incidence with respect to the membranes and

322

JOHN R. HULL

/ -

x

-

i

Fig. 9. Geometry for vertical sheet membrane configuration. is given by cos0i = sin0 sin49.

(13)

Part of the light that is incident upon a single membrane will be reflected, and part of the light will be transmitted through the membrane according to (5)-(10). The light that is transmitted will be incident on the next membrane at the same angles 0 and tk. The light that is reflected will be incident on the next membrane at angles 0 and - 49. In either case 0i is the same, and the problem may be treated as if the transmitted and reflected light propagate together. The total transmission TN through a set of N membranes is given by TN = (Pa + 7"1)N,

(14)

with Pt and Tx given by (6) and (5) respectively. If the depth of the NCZ is D, then the total number of membranes which light will strike while traversing the NCZ is

N(O, 49) = D tan0 sin49/x. 0.5

i

I

I

i

i

i

i

i

(15)

i

30% 2.54

0"

' o.,°:~2L=,ZT~M--'--..~~ 0.3

0.0

i

0"

i

20*

i

i

40*

i

I

800

i

i

u

80*

80 Fig. 10. Transmission T of solar radiation through a 1.0 m deep NCZ containing vertical sheet Teflon membranes spaced 0.25 cm apart in water. 0o is the angle of incidence in air at the pond surface, q~ is the angle between the plane of incidence and the plane of the membranes. L is the thickness of an individual membrane.

Once TN is computed, then the total transmission through the NCZ, with respect to the solar radiation incident upon the pond surface, can be calculated from (11) and (12). As an example, we calculate the transmission of solar radiation through a 1.0 m deep NCZ containing vertical sheet Teflon membranes spaced 0.25 cm apart in water. The transmission is relative to the insolation in air at the pond surface. The results are shown in Fig. 10. 0a is the angle of incidence in air. 49 is the same as in Fig. 9. The results do not take into account losses from any additional horizontal membranes, nor any losses at the upper edges of the vertical membranes. For the upper curve, 49 = 0 °, the plane of incidence is parallel to the plane of the membranes. For this case, transmission of light passing through the NCZ is not affected by the membranes at all, but is determined only by the Fresnel reflections at the surface and the absorption of the water. As q~ increases, absorption in the membranes becomes significant, and the transmission decreases. As 49 increases above 30 °, however, the transmission remains nearly constant. At these angles the increase in the number of membranes traversed and the lower reflectance at each membrane is balanced by the decrease in path length through each membrane. This effect is illustrated by comparing the bottom two curves of Fig. 10, q5 = 30 ° and ~b = 90 °, for the case L = 12.7/~m. For~ the case L = 2.54/~m, the transmission for any given 0o does not differ by more than 0.01 for 30 ° < 49 < 90 °.

Vertical tubes The transmission of light through a set of vertical tube membranes with square cross section can be calculated by considering two sets of independent vertical sheet membranes. In the geometry of Fig. 9, additional membranes are placed in the xz plane. F o r every TN(O, 49) for the membranes in the yz plane; there is a corresponding TM(O, 49) for the membranes in the xz plane. The two functions are related by

Tu(O, 49) = TN(O,90 ° -- 49).

(16)

The total transmission through a NCZ of depth D is then given by

T(O, Oa, (9) = ~,(0,) Tu(O, 49)Tu(O, 90 ° - q~) × f Tl(2, D/cosO)d~

(17)

where 0 and 0a are related by Snell's law. The result of this calculation for the example of a 1.0 m deep NCZ consisting of square vertical tube membranes in water is shown in Fig. 11. The horizontal width of each tube is 0.5 cm. 0, is again the angle of incidence in air, and 49 is the angle between the plane of incidence and the plane of either set of parallel membranes. The top two solid curves are for Teflon membranes of thickness L = 2.54#m. The bottom two solid curves are for Teflon membranes of

Membrane stratified solar ponds thickness L = 12.7 #m. The two dashed curves are for Tedlar membranes of thickness L = 12.7 ~m. At a given 0,, the transmission varies continuously from a maximum at th = 0 °, to a minimum at ~b = 45 °. As in the vertical sheet example, losses from additional horizontal membranes or from the top edges of the vertical tubes are neglected.

6. T H E R M A L

PERFORMANCE

The results of the previous section have shown that, for any of the three MSSP configurations, transmission of solar radiation through the NCZ can be almost equivalent to that through an equal amount of pure water. It follows that the thermal performance of the MSSP, in general, is approximately equivalent to that of the SGSP. The time dependent thermal performance of the SGSP has been detailed by Weinberger 1,-20] and Rabl and Nielsen [1]. More recently, Kooi I-6] has analyzed the steady state SGSP as an analog to a flat plate collector, developing an efficiency equation of the Hottel-Whillier-Bliss type. We adopt this latter method to calculate the thermal efficiency of a steady state MSSP for the case of HS in water. We assume that the solar radiation is incident at 50° in air. This angle of incidence would be appropriate as a yearly average for a SGSP at about 40 ° latitude [1, 19]. We further assume that the NCZ is composed of Teflon membranes with a 25 #m Teflon membrane at the bottom of the NCZ, and a 76 #m Teflon membrane at the top of the NCZ. There is no UCZ, so the top of the NCZ corresponds to the top of the pond. The MSSP is always assumed to be configured to give the maximum efficiency for a given design operating temperature. According to Kooi [6], in a steady state solar pond with no UCZ, for a given insolation H and the temperature difference At across the NCZ, the depth D of the NCZ which gives the maximum efficiency is given

323

by

f : T(z) dz - DT(z) = kAt/H,

118)

where k is the thermal conductivity of the NCZ liquid. For this particular depth the radiation absorbed in the NCZ produces a temperature profile such that dt/dz = 0 at the NCZ-LCZ boundary. No heat is conducted upwards out of the LCZ, and all radiation penetrating into the LCZ is available for use by the load. The thermal efficiency ~/is then ~/= FR T(D),

119)

where F a is the heat removal efficiency factor. Once the membrane spacing is determined, T(z) in (18) is calculated via the methods of the previous section. As a simplifying assumption, we assume the membrane spacing is always 0.5 cm. For most cases this is a conservative value. We then obtain the graph of ~/FR vS At/H, as shown in Fig. 12. The top efficiency curve is for L = 2.54#m. The bottom curve is for L = 12.7 #m.

7. E N G I N E E R I N G

CONSIDERATIONS

There are several engineering details which must be considered before a working MSSP can be constructed. While it would be inappropriate to suggest a working design without initial test experience, many of the potential engineering problems can be identified. Some of these have been mentioned in previous sections. Others are discussed in this section.

0.7

r i ,11 "1

i i Ill, Oo = 5 0 *

0.6 ~ 4 0.5

0"51 ' ~b=~3°,L; 2'5 4 ~ M' 40',

2.54

0.4

0.4

a:o.3

0.3

0.2 0.2

p.M

]-

la. o. 2 t~

40 °, 12.]

12.7/~

J

0.1F -TEFLON _---TEDLAR 0.01 I t , I 0* 20 ° 40 *

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0.1

I

60*

80 *

8o Fig. 11. Transmission T of solar radiation through a 1.0 m deep NCZ consisting of 0.5 cm wide square vertical tube membranes in water. 0o is the angle of incidence in air. ~bis the angle between the plane of incidence and the plane of one set of membranes. L is the thickness of an individual membrane. s.c. 25/4-

D

i

0.0

o.oi

i

i

i iiii]

i

i

i

0.I AI/HIeC MZ W-I}

iiiit 1.0

Fig. 12. Thermal efficiencyof a MSSP in HS as a function of At/H. The NCZ consists of a bottom membrane of thickness 25/~m (1 ml), a top membrane of thickness 76 #m (3 ml), and other membranes of thickness L spaced 0.5 cm apart. The angle of incidence in air is 0° = 50°.

324

JOHN R. HULL

The intrinsic difficulty with the MSSP is the fragility of the membranes in the NCZ. Most of these membranes must be thin in order to maximize optical transparency. This thinness produces difficulties in the fabrication and deployment of a membrane assembly that can withstand practical operating conditions. These difficulties increase as the pond size becomes larger. Clearly, the membrane material must be strong enough to avoid tear and rupture under any of the stress conditions likely to be encountered. Further, the membrane material must not adhere to itself in the MSSP liquid, as this will disrupt the membrane spacing and increase the stress on the membranes. Membrane sag will cause irregular spacing and angles between the membranes. In severe cases this results in onset of convection in some parts of the pond and a reflection of some of the light into a more horizontal path. The membrane support structure must then prevent the membranes from appreciably sagging, while interfering minimally with the transmission of solar radiation through the NCZ. Accordingly, the support structure may have to occasionally adjust the tension on the membranes as they change size with time and temperature. Further structural considerations include sealing the membranes at the sides of the pond to prevent contamination of the NCZ liquid, and configuring the membranes in sections to make rapid repairs possible. Furthermore, if the LCZ is used for aquaculture farming, then service ports must be provided that interfere minimally with solar collection. Another problem is the optical degradation and membrane stress which results from the formation of air bubbles on the membranes. The top of the MSSP poses additional difficulties. One consideration is the stability of the surface membrane to intense rain and wind. Cold water above the surface membrane will cause macroinstability of the type discussed in Section 3. While surface tension and edge supports will help to secure the surface membrane, it may be necessary to provide restraining members across the surface at spaced intervals for security during strong winds. Another surface problem is the accumulation of dust and debris on the surface membrane. This membrane may require periodic cleaning with either water or compressed air spray. Choice of MSSP liquid and membrane materials is an economic as well as an engineering consideration. Water and Teflon FEP film is a combination that provides long term stability to both temperature and sunlight, is currently available, and provides reasonable thermal efficiencies. It may prove more economical, however, to use a less expensive membrane material and replace it occasionally, or to use a more expensive liquid in order to use fewer membranes. To approach the theoretical limit of maximum thermal efficiency, it is desirable to use membranes that are extremely thin. For some materials this may necessitate the manufacturing of films that are significantly thinner than what is currently commercially available.

8. CONCLUSION We have shown that the membrane stratified solar pond (MSSP) may be a viable alternative to the salt gradient solar pond as an inexpensive source of lowtemperature heat. The lower convective zone (LCZ) of the MSSP can be made deep with little economic penalty. The heat storage function of the LCZ can be supplemented by aquaculture farming on a year round basis at higher latitudes. The membranes are placed in the nonconvecting zone (NCZ) at the top of the pond and can be configured as either horizontal sheets (HS), vertical sheets (VS), or vertical tubes (VT). A variety of membrane materials can be used in the vertical configurations, but the index of refraction of the membranes must closely match that of the liquid in HS. A combination of water and TeflonTM FEP membranes has been suggested for use in the MSSP. This combination should perform at relatively high thermal efficiency with an effective lifetime of many years. Several MSSP engineering problems have been identified, and several simple construction and maintenance techniques have been suggested. The spacing between membranes that insures suppression of convection has been calculated as a function of temperature t and temperature gradient fl for both water and ethanol. For water in HS or VT, the spacing for practical values of t and fl is 0.4-1.0 cm. For water in VS the membrane spacing for similar thermal regimes is 0.34).7 cm. A method has been presented for calculating the optical transparency of the NCZ, assuming specular reflections and no scattering. We have used this method to calculate the transparency of the NCZ with respect to an air mass one solar spectrum for all three MSSP configurations. The use of commercially available 12.7/~m (1/2 ml) Teflon membranes and water in a 1.0 m deep NCZ produces a transparency 80-90 per cent that of pure water alone. The thermal efficiency of the steady state MSSP in HS has been calculated using the method of Kooi I-6]. The thermal efficiency is comparable to that of a well maintained SGSP.

Acknowledgements--The author has benefited on this project from discussions with L. Hodges, P. Sidles, J. McClelland, M. lies, H. Sparks, R. Struss and R. Szydlowski. This work has been partially supported by Iowa State University and U.S. Dept of Energy Contract W-7405-ENG-82.

HS LCZ MSSP NCZ SGSP UCZ VS VT D FR H Kz

NOMENCLATURE horizontal sheet configuration lower convective zone membrane stratified solar pond nonconvective zone salt gradient solar pond upper convective zone vertical sheet configuration vertical tube configuration depth of NCZ, m heat removal efficiencyfactor surface insolation, W m-2 membrane attenuation coefficient at wavelength ~,, m -1

Membrane stratified solar ponds

L Nu PN R R¢

T

T, T~ d

dc g h h~ k t

At X Z

r/ 0 0i G K V

P Ta

0

thickness of membrane, m Nusselt number reflection from set of N membranes Rayleigh number critical Rayleigh number total transmission transmission through liquid only transmission through set of N membranes distance between adjacent horizontal or vertical sheet membranes, effective radius of tube membranes, m distance between membranes that just suppresses convection, m acceleration of gravity, m sec- 2 total vertical heat flux, W m-2 vertical heat flux due to conduction, W m -2 thermal conductivity, W m - 1 °C- 1 index of refraction temperature, C vertical temperature difference, ~C horizontal length between adjacent vertical sheets, m depth from surface, m coefficient of thermal expansion, °Cvertical temperature gradient, °C m thermal efficiency polar angle, (') angle of incidence, (°) angle of refraction, f ) thermal diffusivity, m 2 secradiation wavelength, m kinematic viscosity, m 2 sec- 1 reflection from a single liquid/membrane interface transmission through a single membrane Fresnel transmission coefficient at pond surface polar angle. (r)

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