advanced solar pond basic theoretical aspects

7/28/2019 Advanced Solar Pond Basic Theoretical Aspects

http://slidepdf.com/reader/full/advanced-solar-pond-basic-theoretical-aspects 1/10

..'

Solar EMrgy Vol. 43. No. I, pp. 35-44, 1989Printed in the U.S.A.

0038-092X/89 S3.00 + .00

Copyright C>1989Maxwell Pergamon Macmillan pic

THE ADVANCED SOLAR POND (ASP): BASIC THEORETICALASPECTS *

HILLELRUBIN and GIORGIOA. BEMPORAD

Coastal and Marine Engineering Research Institute (CAMERI) Department of Civil Engineering,Technion-Israel Institute of Technology, Haifa 32000, Israel

Abstract-This manuscript concerns the possible improvement of the conventional solar pond (CSP)

performance by applying a multiselective injection and withdrawal procedure. We apply the term ad-vanced solar pond (ASP), for a solar pond (SP) in which such a procedure is applied. The multiselective

injection and withdrawal procedure creates in the SP a stratified thermal layer, namely a flowing layerwhich is subject to salinity and temperature stratification. This phenomenon is associated with reduction

of heat losses into the atmosphere and an increase of the temperature of the fluid layer adjacent to theSP bottom.

In the framework of this study transport phenomena in the ASP are analyzed and simulated by ap-plying a simplified mathematical model. The analysis and simulations indicate that the multiselective

and withdrawal procedure may significantly improve the performance of the SP.

1. INTRODUCTION

The solar pond (SP) is a shallow water body being

virtually a trap for solar radiation. The trapped solar

radiation is converted into thermal energy which is

accumulated in the deep water layers of the SP. The

thermal energy can be accumulated due to the sta-

bilizing salinity gradients existing in the SP, which

prevent thermal convection in the water body.

Proper operation of the SP depends on the ability

to withdraw hot water by a selective withdrawal while

preserving the density profile of the pond.

In the conventional solar pond (CSP) we identify

three major fluid layers as shown in Fig. lea): surfacelayer, barring layer, and thermal layer.

The surface layer is completely mixed due to at-

mospheric effects. The barring layer is comprised of

a stagnant fluid; it separates the thermal layer from

the surface layer. Heat is accumulated in the thermal

layer; this layer is subject to horizontal flow needed

in' order to utilize the thermal energy. The thermal

layer is almost completely mixed due to the selective

withdrawal, injection, and thermal convection.

In the advanced solar pond (ASP)[I] there is an

additional stratified thermal layer as shown in Fig.

l(b). This layer is comprised of several sublayers.

Each syblayer is equipped with injection and with-

drawal ports. Therefore a multiselective injection andwithdrawal characterizes the ASP. By making somebasic calculations it was claimed that the ASP overall

efficiency can be much higher than that of the CSP[l].

However the anticipated configuration shown in Fig.

1(b) requires adequate facilities that should be de-

veloped. The physical rationale for the ASP, as stated

by one of the reviewers of this paper, is to create

flowing layers in the upper portions of the thermal

*Parts of this study were presented at the Annual Meet-

ing of the American Solar Energy Society, June 20-24,1988.

layer and lower portions of the barring layer of a CSP,

so as to remove heat from the layers, lower their tem-

perature, reduce diffusive losses through the barring

layer, and leave more thermal energy for storage andextraction.

This study concerns the basic theoretical aspects

of the ASP performance. We develop a mathematical

approach leading to a numerical model by which the

engineering feasibility of the ASP can be evaluated.

2. ABSORPTION OF THE SOLAR RADIATION

The solar radiation is absorbed in the SP and con-

verted into thermal energy. The heating process canbe represented as an effect generated by a line source

whose strength is distributed exponentially along the

SP depth[2-5]. The strength of the thermal energy

source, qT, eventually represents the rate of absorp-

tion of the solar energy in the water body.

The solar radiation arriving at the bottom of the

SP is completely absorbed by the pond bottom, pro-

vided that it is completely black. If the pond bottom

is insulated the energy absorbed in the bottom leads

to heat flux which enters the thermal layer.

We apply a steady state simulation of the SP per-

formance by referring to the average annual values

of the physical parameters governing the SP opera-tion. It was shown by various studies[6] that steady

state simulations based on average values of param-

eters are quite accurate for calculations referring to

long times of operation of the SP.Our calculations of solar radiation refer to the Dead

Sea area in Israel, where some CSPs are operational.

We consider that the solar radiation energy penetrat-ing the SP surface is 200W m -2. This value is ob-

tained by assuming that half of the daily energy ar-

rives at the SP surface during the middle third of the

day, and that the radiation at 2 p.m. of October 21

can represent the strength of the solar radiation of the

35



36 H. RUBIN and Go A. BEMPORAD

Plastic net

Velocity

Salinitytemperature

°t

Salinitytemperature

it

---L--r

Surface layer

Barring layer

Thermal layer

( a)

Plastic net

~ -..2..-

-rSurface layer

Barring layer

Stratified thermal layer

Homogeneous thermal layer

(b)

Fig. 1. Distribution of velocity, salinity, temperatUre and density in solar ponds. (a) the conventional

solar pond. (b) the advanced solar pond.

middle third of the day. The refraction angle is as-

sumed to be 32.5 degrees.

3. THE FLOW FIELD

The surface layer. This layer is subject to atmo-

spheric effects and the wash flow. Some water quan-

tities evaporate from this layer into the atmosphere.

Therefore the flow' rate of the surface layer decreases

along the pond as follows

Q(T) =Q~)

-qx

where Q(T)is the surface flow-rate per unit width;

Q~) is the entrance value of Q(T);q is the rate of

evaporation from the ASP surface; x is the horizontal

coordinate. It should be noted that with regard to some

aspects the nonuniform velocity profile of the surfacelayer should be taken into account. However, in the

present study, we mainly concern the differences be-tween the CSP and ASP. For such considerations the

assumption of uniform surface flow is acceptable.

Some studies[7,8] refer to mixing effects gener-

ated by the atmosphere. Such effects cause the sur-

face flow to be assumed for practical purposes as being

(1)

almost uniformly distributed. Long experience in the

laboratory and field operations showed that the sur-

face of the SP should be protected against wind ef-fects. Such effects include waves and various kinds

of currents. Plastic nets, as shown in Figs. l(a) and

l(b) were found to be excellent means for the SP sur-

face protection. Eventually in every operational SPsuch means are utilized.

The barring layer. This layer is stagnant. Large

salinity gradients existing in this layer insulate the

thermal layers from the mixing effects existing in the

surface layer.

The stratified thermal layer. This layer is subjectto horizontal flow, and its salinity gradient avoids the

formation of circulating currents of thermohaline

convection. Therefore in each sublayer the following

condition should be satisfied[9,1O];

-ac~ (v -t K

)apaT

jap

ay v + D aTay ac(2)

where C is salinity; T is temperature; p is density; v

is kinematic viscosity; Kis heat diffusivity; D is salt

diffusivity; y is the vertical coordinate. ...

The flow in the stratified thermal layer is carried



I

ttJ-'if;;.;:~

ri'A;1'~~'"

.~,~

Theoreticalaspectsof the ASP

out by injection and withdrawal ports creating severalsublayers. The total thickness of the stratified ther-

mal layer is given as follows:

M

deS) = 2: dji-I

where djis the thickness of each flowing sublayer; Mis the number of sublayers. At the injections ports

each sublayer has its particular temperature and sa-

linity. However, due to the small thickness of the

sublayers, it is expected that, after a short distance,

continuous salinity and temperature profiles are es-

tablished in the stratified thermal layer.

The shear stress distribution in the i-th sublayer is

given as follows:

T = TiB - Y(TiB - T/T)/dj

where TiBand T,Tare shear stresses at the bottom and

top of the sublayer, respectively; Y is a local vertical

coordinate, namely Y = 0 at the bottom of the sub-layer, and Y = dj at the top of the sublayer.

The following conditions should be satisfied at the

interfaces existing between the various sublayers:

T,T = T(i+I)B UiT= U(i+I)B

According to various studies [11-13] we may assume

that if the thermal layers are subject to a continuous

laminar velocity profile then the following conditionis satisfied:

TMT= -ttTw tt ==0.62

where Twis the shear stress at the bottom of the SP;

- TMTs the shear stress at the top of the stratified ther-

mal layer; tt is a coefficient.

The thickness of each flowing sublayer depends

on its flow-rate Q(i)and its density gradient as fol-

lows[I4]:

~=K (~ )O.27

Lj gO.sLl's ; Lj =

p(i)

(iJp/iJY)j

where pmis the average density of the sublayer fluid;

Lj is the buoyancy characteristic length; K is a con-

stant; g is the gravitational acceleration. Some otherexpressions for dj were suggested for laminar as well

as inertial buoyant layers[I5,I6]. However, all these

expressions provide similar results in the range of

Reynolds numbers about 1000 which is of our inter-

est with regard to the SP.

According to various experimental studies[ 17]

performed in Reynolds numbers of approximately 1000it is obtainedthat K == 0.2.

Reynolds number of the thermal layers flow can

be defined by various expressions. Here we adopt the

following definition:

Re=Rv""

(3)

,where Q is the discharge per unit width; v"" is theaverage kinematic viscosity. Considering laminar flow

of the thermal layers an integration of eqn (4) yields

the following velocity proflle:

I

[

y2

]U =- TiBY - - (TiB - TiT) + UiB

J.Lj 2dj

where IJ.jis the viscosity of the fluid comprising the

i-th sublayer; UiBs the velocity existing at the bottom

of the sublayer.

By integration of (9) we obtain the following

expression for the sublayer flow-rate:

(4)

[tf

Q(i) = udY =~ (2TiB+ T,T)+ u;sdj° 6IJ.j

(5)

The homogeneous thermal layer. This layer is sub-

ject to horizontal flow and thermal convection stem-

ming from heat absorption by the black bottom of the

pond. In our calculations we consider that the flow

is basically laminar. However the effect of thermalconvection which enhances momentum transfer in the

fluid layer is represented by an increased effective

viscosity .

The shear stress and velocity distribution in the

homogeneous thermal layer are given respectively asfollows:

T = Tw- Y(Tw - TIB)/do

(6)1

[

y2

]= J.Ltff TwY- 2do (Tw- TIB)

where do is the homogeneous layer thickness; IJ.tffs

the effective viscosity. By integrating (12) over the

thickness of the homogeneous thermal layer we ob- .

tain the following expression for its flow-rate:

(7)Q(O)=..! (

d~ d~

)w + TIB-

fl-tff 3 6

The thickness of the homogeneous therma1layer should

be controlled artifically by various means, like tem-

porary increase of the flow-rate of the stratified ther-

mal layer. In our calculations we assume that do has

a given value. By applying eqns (4)-(13) we deter-mine the distribution of shear stresses and velocities

in the thermal layers as shown in the Appendix.

4. THE TEMPERATURE FIELD

We assume that, in each elementary fluid volume of

the water body, heat convection is the dominant

transport mechanism in the horizontal direction, and

molecular heat diffusion is the dominant transportmechanism in the vertical direction. In order to con-

37

(8)

(9)

(10)

(11)

(12)

(13)



38 H. RUBINand G. A. BEMPORAD

sider effects of thennal convection on heat transfer

in the homogeneous thennal layer, it is possible to

assume that heat diffusivity is increased by the ther-mal convection[ 18]. Calculations indicate that heat

transfer from the pond bottom into the homogeneous

layer increases heat diffusivity in several orders of

magnitude. Therefore we may assume that with re-gard to heat transfer this layer is eventually fully

mixed. The barring layer is stagnant. Thus in this

layer only molecular diffusion of heat takes place.

Due to comparatively large temperature gradients in

the vertical direction and small temperature gradients

in the horizontal direction we ignore heat diffusion

in the latter direction. Mixing effects in the surface

layer and intimate contact with the atmosphere cause

its temperature to be unifonnly distributed in the ver-tical and horizontal directions.

Fonnulating the assumptions represented in the

preceding paragraph we obtain the following expres-

sions for the surface, barring, stratified thennal and

homogeneous thennal layers, respectively:

T(T) = canst

tiT qT - 0+--ayZ KpCp

" aT a2T qT--=-+-K ax ayZ KpCp

do

(0)aTo =1

qTdy

pCpQ ax 0 aT

)J<f)- (KpCp ay Y-do

where T(T) is the surface layer temperature; Cp is the

specific heat; To is the homogeneous layer tempera-

ture; K is heat diffusivity; J}B)is heat flux from the

SP bottom. Expressions (14)-(17) are employed

hereafter in order to develop a numerical model ofheat transfer in the ASP.

5. TIlE SALINITY FIELD

With regard to convection and diffusion dominance

we utilize the same assumption applied in the pre-ceding section with regard to heat transfer. We also

assume that the homogeneous thennal layer is fully

mixed with regard to salinity distribution. Vertical

molecular diffusion is the only salinity transfer mech-

anism considered in the stagnant barring layer. Due

to the vertical salinity diffusion process, the salinity

of the homogeneous thennal layer decreases along

the SP, and the salinity of the surface layer increasesalong the SP.

Fonnulating the assumptions represented in the

preceding paragraph we obtain the following expres-sions for the surface, barring, stratified thennal and

homogeneous thennal layers respectively:

(C(T»

) = -Q(T)acm + q eT)ay y-h D ax . D

a2c-=0ayZ

(18)

(19)

"ac a2c

vax=ayZ(20)

(aco

) = - Q(O) aco

ay y-do D ax

where eT) is the surface layer salinity; Co is the ho-

mogeneous thennal layer salinity; D is salt diffusiv-ity; h is the distance between the SP bottom and the

interface existing between the surface and barring

layers.

The expressions represented by eqns (18)-(21) are

employed hereafter in the development of the nu-

merical model which is able to simulate salinity transferin the ASP.

(21)

(14)6. THE NUMERICAL MODEL

Expressions (14)-(21) are subject to initial con-

ditions, which are physically the conditions existing

at the SP entrance. Expressions (15)-(16) and (19)-

(20) are parabolic differential equations. In order to

solve these equations we apply an implicit finite dif-ference numerical model which utilizes variable mesh

size in the vertical direction. Applying such an ap-

proach for (15) and (16) we obtain the following setof linear equations: .

(15)

(16)

(17)

[2

] [ "..,.(m+1) + T (m+1) -lJ-1 J(ilYj + ilYj-l)ilYj_, Kjilx

2 2

]+ +

(ilYj + ilYj-l)ilYj (ilYj + ilYj-l)ilYj-1

- ..,.(m+)

[

2

]'+1

J (ilYj + ilYj-l)ilYj

= T;m)-!i.. + qTj~ KjPCp

(22)

where m is a superscript referring to the longitudinal

position of the nodal point; j is a subscript referring

to the vertical position of the nodal point.Expressions (14) and (17) represent boundary

conditions of the numerical grid being expressed re-spectively as follows

~m) = T(T) atj = N (23)

[Q

<O). 1

]

1T~m+l)- - - + Tjm+I)-

Kilx ilYI ilYI

= Tdm)Q(O) + (<py=doKilx PKCpQ(O)

. where <P is the intensity of the solar radiation.

(24)



1!~.f.".

~i~:~'

~~

Theoretical aspects of the ASP

Expressions (22)-(24) in conjunction with given

initial conditions, representing the temperature pro-

file at the SP entrance, yield the development of the

temperature profile along the SP.

Applying the variable mesh size for the finite dif-

ference approximation of (19) and (20) we obtain the

following set of linear equations:

[

2

] [

U.-c~m+I) + C(m+ I) -L)-1 j

(~Yj + ~Yj-I)~Yj-1 DJu

2 2

]+ +

(~Yj+ I1Yj-I)~Yj (I1Yj+ I1Yj-I)~Yj-1

[

2

]

u.- C(m+1) - C (mJ-Lj+1 - j

(~Yj + I1Yj-I)~Yj Dlu

Expressions (18) and (21) represent boundary con-

ditions of the numerical grid being expressed respec-

tively as follows:

1

[

1 Q(7)(mJ

]C(m+l)-

+ C(m+IJ

+

q

N-I N - - --~YN-I I1YN-1 DNtu DN

Q(7)(mJ='C(mJ-

N DNtu

[

Q(OJ 1

]

1c~m+1)- - - + c\m+IJ-

D,ju ~YI I1YI

Q(OJ-c(mJ-- 0 D,ju

where N is the number of nodal points in the vertical

direction. As a subscript N refers to the interface be-

tween the surface and barring layers.

Expressions (25)-(27) in conjunction with a given

salinity profile at the SP entrance yield the devel-

opment of the salinity profile along the ASP.

7. SIMULATION OF THE ADVANCED SOLAR POND

PERFORMANCE

Theoretically the numerical model developed in this

study can be applied for any velocity distribution

stemming from different injection procedures. How-

ever only a procedure forming the smooth and con-

tinuous laminar profile avoids the Kelvin-Helmholtz

type of instability[18] generating circulating currents

and mixing phenomena between the moving sublay-ers. Therefore we refer here to smooth laminar ve-

locity profiles.

Practically the value of the Reynolds number de-

pends on the availability of thermal energy and the

temperature of its exploitation stemming from the

strength of solar radiation and the size of the SP. For

long SP the Reynolds number should be high enough

to allow appropriate use of the accumulated thermal

energy. We apply here several examples referring toRe = 500. However the reference to higher Reynolds

number does not change the basic implications of this

study.

39

~~

Figure 2 shows the velocity profile for thermal layer

discharge of 0.5 litlseclm in which Reynolds num-

ber is 500. This profIle of velocities is associated with

the development of temperature and salinity profiles

as shown respectively in Figs. 3 and 4. In these fig-

ures the profiles referring to entrance represent theinitial conditions of the numerical model.

Figures 3 provide some information about the pos-sible advantages of the ASP with regard to the CSP.

It is assumed that the surface layer temperature is al-

most identical to the atmospheric temperature. There-

fore, heat losses into the atmosphere depend on the

temperature gradients existing in the barring layer.We have also to consider that some solar radiation is

obsorbed in the barring layer. Most of it is lost into

the atmosphere. As a result of the phenomena dis-

cussed in the preceding sentences, by increasing the

average temperature of the thermal layer existing ina CSP we cause an increased heat loss into the at-

mosphere. We apply Fig. 3, and compare the per-

formance of a CSP and an ASP, whose Reynolds

number and total thickness of the thermal layers areidentical. If the fluid adjacent to the SP bottom is

subject to the same temperature in both ponds, then

the temperature existing in the interface between the

thermal layers and the barring layer is higher in the

CSP than in the ASP. Therefore identical tempera-

ture existing at the bottom of both ponds leads to largerheat losses in the CSP than in the ASP.

There are various manners to calculate the effi-

ciency of the SP performance[ 19]. Here we represent

this parameter as the ratio between the thermal en-

ergy gained in the thermal layers and the energy of

the solar radiation which penetrates the SP surface.The thermal energy gained in the thermal layers

is equal to the difference between the heat flux con-vected at the pond exit and this flux convected at the

pond entrance. In the particular examples represented

by Fig. 3(a) and Fig. 3(b) we referred to bottom en-

trance temperature of 80°C in a CSP and an ASP,

and surface temperature of 35°C. The figures indicate

that the exit, bottom temperatures for the CSP and

ASP are 85°C~and 94°C, respectively. The net energy

output of the CSP and ASP are 15.7kW1m and

21.7kW1m, respectively. These energy outputs are

obtained with efficiencies of 8 and 11 percent, re-

spectively. Following the-suggestion of one of the

reviewers of this manuscript, we performed simula-

tion with CSP subject to the same initial and bound-

ary conditions asthose of Fig. 3, whose thermal layer

thickness is 25 cm. The output temperature was 88°C

and the efficiency was 14 percent. This phenomenon

is typical to the CSP operation, where significant in-

crease in efficiency can be obtained provided that low

output temperature and small heat storage are ac-

ceptable. However the general outcome of all sim-

ulations was that with the ASP it is possible to obtain

significant increase in the combination of the main

basic parameters of the SP utilization: output tem-

perature, efficiency and heat storage.

With regard to salinity transfer there are some dif-

(26)

~n



1.2

H. RUBINand G. A. BEMPORAD

0.0o 2

40

:3 4

Velocity, (mm/sec)

5

Fig. 2. Velocitydistributionin the solar pond, Re = 500.

ferences between the development of the salinity pro-

file along the CSP and ASP as indicated by Fig. 4.However these differences have a minor effect on the

SP performance. It should be noted that the signifi-

cant difference between the increase of salinity of the

surface layer and decrease of salinity of the thermal

layer stems from water evaporation.

8. DISCUSSION

The simulations represented in the preceding sec-

tion demonstrate some possible advantages of the ASP.

Such advantages can be summarized as an increase

of the pond bottom temperature, an increase of the

SP efficiency and an increase in the heat storage.

However in the preceding section we only considereda single design procedure of the ASP, in which the

stratified thermal layer is created on account of the

upper portion of the homogeneous thermal layer of

the CSP. However various other design and utiliza-

tion procedures are also attractive. It is possible to

expand the stratified thermal layer on account of the

barring layer. In such a manner we increase the amountof solar radiation which can be utilized. We also in-

crease the heat storage of the SP in such a manner.

A very attractive procedure suggests a very thick

stratified thermal layer being comprised of several

sublayers. The flow-rates of all sublayers are iden-

tical. The flow-rate withdrawn from the first sublayer

is injected into the second sublayer and so on as shown

schematically in Fig. 5. The flow of the lowest sub-layer can be either transferred into the heat exchanger

of the heat utilizing system, or injected into the ho-

mogeneous thermal layer (Fig. 5). However in such

a case the interface existing between the stratified and

homogeneous thermal layers may represent a location

of discontinuity in the velocity profile as shown in

Fig. 5. Therefore this interface may be subject toKelvin-Helmholtz instability. Then it is eventually

represented by a thin mixing layer. The schematics

of the figure suggest that the heat exchangers system

withdraws hot water from the homogeneous thermal

layer, and diverts water of comparatively low tem-

perature into the top of the stratified thermal layer.

However Fig. 5 also shows some of the practical dif-ficulties associated with the ASP. The ASP requires

a lot of piping and pumping, inlets and outlets ar-

ranged laterally to induce laterally uniform flow and

minimize mixing. All these topics have been beyond

the scope of this study. However some other positiveissues of the ASP should also be considered.

The high temperature'of the homogeneous thermal

layer enables its salinity to be very high, provided'

that salts like magnesium chloride are utilized.

Therefore the surface layer salinity can also be higher

than the salinity of that layer in the CSP. The in-

creased salinity of the surface layer decreases the rate

of evaporation from the SP surface. This phenome-

6

-1.01

Om<Entrance)-..E . ---- 200m,g

Ol-.-.- 400m

-..-.- 600m...

0.6. -...- 800m'0

ItI ---- JOOOm(Exit)

Q)0.4

ucc

-CI)

0 0.2



1.2

1.0

0.8

0.6

0.41

0.2

0.020

1.2

1.0

0.8

0.6

0.4

0.2

0.0

20


---- Om (Entrance)

200m

-.-.- 400 m

600 m..-..-

-...- 800 m

1000 m(Exit)---

ave. ~radient 80CYm

100

o m(Entrance)

--- - 200m

400m.-.-

-..-..- 600m

800m

80 90 100

41




1.2Om (Entrance)

- 1.0200m

400m.-.-

E-..Eo 0.8-omQ).c. 0.6l-

EelL. 0.4

-..--- 600m

800m...-

---- 1000 m(Exit)

Fig. 4. Salinity profile development along the solar pond, Re = 500. (a) in the conventional solar pond.(b) in the advanced solar pond.

Q)

0c0 0.2n

a

0.00 10 20 30 40 50

(a) Concentration,(0/0)

1.2 .- Om(Entrance)

---- 200m

- 1.0.E I -.-.- 400m-..E I -..-..- 600 m

0.81 -..-- 800m0m r- ---- 1000 m (Exit)Q)

-{:. 0.6

EeLL 0.4Q)0c0-. 0.2a

0.030 40 5010 20

(b) Concentration., (0/0)



- ,


'V--:r

From heatexchange rs

--c-

Additionalsalt

Fig.5. A schematic of an advanced solar pond in which the withdrawn flow-rate of a particular sublayeris injected into the adjacent lower sublayer.

non is associated with an increase in the surface layer

temperature; namely it reduces the temperature gra-

dient existing in the barring layer, and thereby it re-

duces heat losses into the atmosphere.

The present study covered only some basic as-

pects of the ASP performance. Some more careful

studies accompanied with experimental investiga-

tions should be performed before any pilot plant of

such a SP is designed. However very important prac-tical issues should be taken into account like piping,mixing system, settlement and solution of salts, etc.

All such subjects should be dealt before a cost ef-fective ASP can be envisioned.

9. SUMMARY AND CONCLUSIONS

There is a possibility to improve the performance

of the CSP by applying a multi-injection-withdrawal

procedure. Such a procedure creates in the SP an ad-

ditional stratified thermal layer. The SP with such a

layer is termed an ASP. The basic aspects of the ASP

operation are analyzed in this paper by applying a

simplified mathematical model considering major

transport phenomena in the solar pond. This modelleads to analytical calculations of the momentumtransfer and numerical simulations of heat and salin-

ity transfer in the SP. Such simulations indicate that

the ASP advantages are implied by higher bottom

temperatures and higher efficiency. Furthermore the

ASP suggests a variety of procedures for its utiliza-tion, some of them are discussed in this paper. The

positive theoretical results of this study with regard

to the ASP operation suggest the performance of some

laboratory studies relevant to this subject.

NOMENCLATURE

C salinity, dimensionlessCo salinity of the homogeneous thennal layer, dimen-

sionless

c:n salinity of the surface layer, dimensionlessCp specific heat, Jkg"' °e"'d, thickness of the i-th thennal sublayer, mdo thickness of the homogeneous thennal layer, m

D mass diffusivity, m2s"'

g gravitational acceleration, ms-2h distance between the pond bottom and the surface layer,m

JT diffusive heat flux, Wm-2J'f) diffusive heat flux at the solar pond bottom, Wm"2

K coefficient, dimen$ionless

L, characteristic buoyancy length of the i-th sublayer, m

q rate of evaporation, ms"1qT strength of the heat source, Wm-3Q volumetric flow-rate per unit width, m2s-'

(to flow-rate of the i-th sublayer, m2s"' .

Q(T) flow-rate of the surface layer, m2s"1Q?;> entrance flow-rate of the surface layer, m2s"1

Q'OI flow-rate of the homogeneous thermal layer, m2s"'Re Reynolds number of the thennal layers fow, dimen-

sionless

T temperature, °e

T(T) temperature of the surface layer, °eTo temperature of the homogeneous thermal layer, °eU flow velocity, ms"1

U/B flow velocity at the bottom of the i-th sublayer, ms"'U,T flow velocity at the top of the i-th sublayer, ms-Ix horizontal coordinate, m

y vertical coordinate, mY local vertical coordinate, ma ratio between shear stresses, dimensionlessK heat diffusivity, m2s.11.1.viscosity, Pas

1.1..//effective viscosity, Pasv kinematic viscosity, m2s-'

v... average kinematic viscosity, m2s-1

p density, kg/m-3

43




p(Q average density of the i-th sublayer, kg/m-JT shear stress, Pa

Tw shear stress at the solar pond bottom, PaTiS shear stress at the bottom of the i-th sublayer, Pa

TiT shear stress at the top of the i-th sublayer, Pac/I energy of the solar radiation, Wm-2

Acknowledgment-This research was supported by theMinistry of Energy and Infrastructure, Israel.

REFERENCES

1. A. Ozdor,Methodof trappingandutilizingsolarheat,U.S. patent No. 4,462,389 (1984).

2. H. Weinberger, The physics of the solar pond, SolarEnergy 8, 45 (1964).

3. R. A. Tybout, A recursive alternate to Weinberger'smodel of the solar pond, Solar Energy 11, 109 (1966).

4. A. Rabl and C. F. Nielsen, Solar pond for space heat-ing, Solar Energy 17, 1 (1975).

5. H. Rubin, B. A. Benedict and S. Bachu, Modeling the

performance of a solar pond as a source of thermalenergy, Solar Energy 32, 771 (1984).

6. V. Joshi and V. V. N. Kishore, Applicability of steady

state equations for solar pond thermal performance pre- .dictions, Solar Energy 11, 821 (1986).

7. J. R. Hull, Computer simulation of solar pond thermalbehavior, Solar Energy 25, 33 (1980).

8. J. F. Atkinson and D. R. F. Harleman, A wind mixedlayer model for solar ponds, Solar Energy 31, 243(1983).

9. G. Veronis, On finite amplitude instability in ther-mohaline convection, J. Marine Res. 23, 1 (1965).

10. D. A. Nield, The thermohaline Rayleigh-Jeffreys.problem, J. Fluid Mech. 29, 545 (1967).

11. A. T. Ippen and D. R. F. Harleman, Steady state char-acteristics of subsurface flow, U.S. Nat. Bur. of Stan-

dards, Cire. 521, Symp. on Gravity Waves, 79 (1951).

12. J. P. Raymond, Etude des courants d'eau boueuse dans

les retenues, 4th Congress on Large Dams, New Delhi,Trans. vol. 4, R48 (1951).

13. K. Lofquist, Flow and stress near interface between

stratified liquids, The Physics of Fluids 3, 158 (1960).14. P. Gariel, Recherches experimentales sue l'ecoulement

de couches superposees de fluides de densites differentes,La Houille Blanche 4, 56 (1949).

15. G. A. Lawrence, Selective withdrawals through a pointsink, 2nd International Symposium on Stratified Flows,

Trondheirn, Norway, 411, (1980).16. J. Imberger, Selective withdrawals: a review, 2nd In-

ternational Symposium on Stratified Flows, Tron-

dheirn, Norway, 411 (1980).

17. O. Levin and C. Elata, Selective flow of density strat-ified fluid, Tech. Report No. 5/133/62, Dept. of CivilEng., Technion, Haifa, Israel (1962).

18. S. Chandrasekhar, Hydrodynamic and hydromagneticstability, Oxford at the Clarendon Press, London (1961).

19. C. F. Kooi, The steady state salt gradient solar pond,Solar Energy 23, 37 (1979).

APPENDIX:

CALCULATION OF THE THERMAL LAYERS PHYSICAL PARAMETERS

According to eqns (5)-(13) we obtain for each sublayer

of the stratified thermal layer and the homogeneous thermal

layer the following expressions:

d, d, Q(o)- T;a + - T(I+I}8 + U(l-UT =-31L1 61L, d,

d, d,UiT=- T;a + - T(I+I}8 + U(I-UT

21-4 21L,

(A. 1)

(A.2)

1'08 = 'Tw; TMB= -aT... (A.3)

where the subscript i = 0 refers to the homogeneous ther-

mal layer.

The expressions represented in (A.I)-(A.3) are em-

ployed in order to obtain a set of M + 1 linear equations.From them one equation refers to the homogeneous thermal

layer (i =0) and M equations refer to the various sublayers

of the stratified thermal layer. The M + 1 equations arerepresentedas follows:

~(~ dJ-,

) (d, d,_,

).J- + - TJS + - + - T;a +J-O 2ILJ 21LJ-' 3ILl 21L,_,

d, Q(Q .

+-T{/+'}8 =- O:s I:SM - 161L1 d,At-I

( ) ( )dJ dJ-, d", d"'-I

L.J -+- TJB+-+- T",s+J-O 2ILJ 21LJ-' 3ILAt 21L"'-1

d", Q(/o()-a--rOB =-

61L", d",

The set of equations represented by (A.4) and (A.5) is as-

sociated with eqn (7) to provide relationships between theshear stress distribution, the partial flow-rates, thicknesses

of the various sublayers of the stratified thermal layer andthe density gradients existing in these sublayers.

If we also refer to a continuous laminar velocity profilethen values of the sublayers thicknesses and flow-rates aredirectly connected.

(A.4)

(A.5)

advanced solar pond basic theoretical aspects

Documents