Effect of ground heat extraction on stability and thermal performance of solar ponds considering imperfect heat transfer

Solar Energy 198 (2020) 596–604

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Effect of ground heat extraction on stability and thermal performance of solar ponds considering imperfect heat transfer☆ Sunirmit Verma, Ranjan Das

T

⁎

Department of Mechanical Engineering, Indian Institute of Technology Ropar, Punjab 140001, India

ARTICLE INFO

ABSTRACT

Keywords: Solar pond Ground heat recovery Analytical solution Imperfect heat transfer Stability Entropy generation

This work revisits the idea of ground heat extraction in solar ponds by addressing various limitations associated with the conventional assumption of perfect heat transfer. For this, an overall heat transfer coefficient, U has been introduced to analyse the heat extraction taking place from the ground beneath the pond. The pond’s thermal performance is analysed for different values of U using closed form solutions and significant departure is seen with respect to the conventional assumption of an infinite U. Temperature distributions across various zones are obtained which exhibit deviation from the ideal distributions. It is also observed that the optimum size of non-convective zone yielding maximum efficiency depends on the effectiveness of ground heat extraction. Further, calculations are made for the minimum salt diffusion rate required to sustain stable pond operation. It is observed that the idealized exchanger performance assumption underestimates this critical value and a design based on it may initiate instability. Finally, calculations are made for the net entropy production rate which is also seen to be under-predicted with the existing theory. The present work can therefore prove useful for making an accurate assessment of the performance and stability of solar ponds involving ground heat extraction.

1. Introduction Solar ponds have long been studied as a cheap and useful means of harnessing solar thermal energy (Weinberger 1964). The initial studies on these ponds were concerned only with their individual performance (Srinivasan and Guha, 1987b; Wang and Akbarzadeh, 1982, 1983; Weinberger, 1964) but over the years, researchers have explored their utility in several other applications too. For example, the temperature gradient existing in solar ponds can be used to generate electricity by coupling them with thermoelectric generators (Ding et al., 2016; Kumar et al., 2018). Desalination is yet another area where these ponds can prove beneficial, which can be done by coupling them with water purification devices, such as solar still, multi stage flash distillation system and many more (Velmurugan and Srithar, 2007; El-Sebaii et al., 2008; Suárez et al., 2015; Abdallah et al., 2016; Mohamed and Bicer, 2019; Rostamzadeh et al., 2019). Some recent studies have also brought to light the potential of solar ponds in refrigeration applications (Rostamzadeh and Nourani, 2019). An appreciable portion of heat energy stored inside the lower convective zone (LCZ) of submerged solar ponds gets conducted to the ground below as waste. Recovering this

waste heat by installing heat exchangers inside the ground can enhance the output of solar ponds quite significantly (Ganguly et al., 2017). Many studies have been devoted to the purpose of analysing ground heat storage beneath solar ponds and its recovery. Some of the cardinal ones are discussed below. Akbarzadeh and Ahmadi (1979) used a 1-D transient heat transfer model to calculate the time-dependent temperature distributions in the ground below solar ponds. The distributions were used to evaluate pond-ground heat transfer process (during hot seasons) and groundpond heat transfer (during cold seasons). Hull et al. (1984) made numerical calculations for side and bottom heat losses from solar ponds to the ground surrounding them under steady-state operation. Two cross-sectional shapes were considered, namely circular and square, and the simulation results were verified with experiments conducted on a 400 m2 pond at Ohio state university. Hull (1985) calculated both steady and transient state loss of thermal energy from the pond to ground consisting of a moving water table. The results were based on analytical solutions and concluded the mass flow rate of water table as a significant parameter affecting the ground loss. Beniwal et al. (1987) estimated the ground heat loss in solar ponds

☆ The monetary assistance acquired through project EEQ/2016/000073 titled Design and Development of a Solar Pond and Biomass Driven Thermoelectric Unit for Domestic Power Generation using Inverse Method supported by Science & Engineering Research Board, India is thankfully recognized. Authors acknowledge IIT Ropar for giving access to other research facilities. ⁎ Corresponding author. E-mail address: [email protected] (R. Das).

https://doi.org/10.1016/j.solener.2020.01.085 Received 17 October 2019; Received in revised form 26 January 2020; Accepted 30 January 2020 0038-092X/ © 2020 International Solar Energy Society. Published by Elsevier Ltd. All rights reserved.

Solar Energy 198 (2020) 596–604

S. Verma and R. Das

Nomenclature

Qg QLCZ Qnet r S1 S2 S3

pond’s cross sectional area = 50 m2 specific heat of water = 4186 J⋅kg−1⋅K−1 salinity at location x underneath the UCZ-NCZ junction, kg⋅m−3 C1 indefinite integral constant appearing in NCZ temperature profile function, K⋅m−1 C2, C3, C4 arbitrary constants of integration appearing in the ground and ground exchanger temperature profile functions, K⋅m−2, K, K d LCZ size = 6 × 10-1 m D diffusion coefficient at location x underneath the UCZ-NCZ junction, m2⋅s−1 D1, D2 constants appearing in the expression giving temperature dependency of diffusion coefficient = 5.838 × 10 10 m2·s - 1 and 4.03 × 10 11 m2⋅°C−1⋅s−1 (Srinivasan and Guha, 1987a) h NCZ height, m I radiation intensity reaching a location x underneath the UCZ-NCZ junction, W⋅m−2 I0 average solar intensity falling on the pond = 200 W⋅m−2 Jx rate of salt transfer at location x underneath the UCZ-NCZ junction, kg·s−1 J steady state salt transfer rate, kg⋅s−1 Jcrt minimum steady state salt transfer rate for sustaining stable pond operation, kg⋅s−1 k thermal conductivity of pond water = 6 W⋅m−1⋅K−1 (Husain et al., 2003) kg thermal conductivity of dry clay ground = 1.28 W⋅m−1⋅K−1 (Wang and Akbarzadeh, 1982) LCZ lower-convective zone mg flow rate of water within ground heat exchanger =5 kg⋅s−1 mLCZ flow rate of water within LCZ heat exchanger, =5 kg⋅s−1 NCZ non-convective zone

A cp C

p3 , p4

the terms mgrUcp +

−1

respectively, m

rU mg cp

2

+

2 rU kg A

rU g cp

and m

-

( ) rU mg cp

2

+

S4

Snet tg

T Ta Tg Tf

U

ULCZ

UCZ x y

power derived from the ground exchanger, W power derived from the LCZ exchanger, W net power derived from both ground and LCZ exchangers, W radius of cross section of either heat exchanger = 1.6x10-2 m NCZ conduction entropy production rate, W⋅K−1 ground conduction entropy production rate, W⋅K−1 entropy production rate through heat transmission across LCZ exchanger, W⋅K−1 entropy production rate through heat transmission across ground exchanger, W⋅K−1 rate of net entropy production, W⋅K−1 distance of ambient temperature ground isotherm underneath pond base = 5 m (Ganguly et al., 2017) temperature at location x underneath UCZ-NCZ junction, K average temperature of environment = 20 °C ground temperature at location y underneath LCZ-ground junction, K ground exchanger temperature at a location y underneath LCZ-ground junction, K overall heat transmission coefficient ascribed to groundground exchanger heat transfer, W⋅m−2⋅K−1 overall heat transfer coefficient describing heat transfer from LCZ to LCZ exchanger = 53.748 W⋅m−2⋅K−1 (Rao and Kaushika, 1983) upper convective zone distance measured beneath the UCZ-NCZ junction, m depth measured from the LCZ-ground junction, m

Greek symbols 1.

.. 4; . 1 .. 4

2 rU kg A

g f

considering the soil to be a dispersion of solid and gaseous particles in an effectively continuous medium. The ground beneath solar ponds was classified into three distinct regions: hot water region (near the pond), cold water region (near the water table) and vapour region (in between the above two). Using the theory of coupled heat, moisture and vapour transport, the effective soil thermal conductivity was calculated and was verified experimentally for moist soil. Zhang and Wang (1990) studied how the transient state ground heat loss in solar ponds is influenced by various parameters such as storage zone thickness, water table depth, pattern of heat extraction, and thermophysical properties of ground. Prasad and Rao (1993) proposed installation of deep trenches at the base of solar ponds. This approach reduced the thermal energy lost to the ground underneath the pond and also reduced the salt requirement of the storage zone. However, the viability of the technique was restricted to applications where depth of water table below the ponds is greater than 10 m. Singh et al. (1994) came up with a simulation model for a solar pond and assessed the temperature profile and stability for different heat extraction patterns and at different times. Sezai and Taşdemiroğlu (1995) used a 2-D transient heat transfer model to evaluate the effect of pond base reflection on the ground heat loss in solar ponds. The results revealed that the ground loss is almost independent of the bottom reflectivity. Kayali et al. (1998) developed empirical relations involving time-dependent air and soil temperatures for Cukurova region of Turkey to make accurate predictions regarding thermal performance of solar

constants ascribing decrement of solar intensity in Eq. (1), Rabl and Nielsen (1975) UCZ height = 2 × 10-1 m effectiveness of the LCZ exchanger efficiency excess temperature T -Ta , K excess temperature Tg -Ta , K excess temperature Tf -Ta , K

ponds. It was observed that using average values can lead to significant errors in the simulation results. Saxena et al. (2009) studied the effect of water table depth on thermal behaviour of solar ponds to determine that the table depression beyond a certain extent does not increase the storage zone temperature any further. Also, the time of transient operation is independent of how deep the water table is. Ganguly et al. (2017) proposed recovery of a part of the ground heat loss in solar ponds by a ground heat exchanger. The exchangers in the lower convective zone and ground were first considered to be in series and then in parallel connection and the results of the transient model concluded that the former arrangement is more productive. Amigo and Suárez (2017) studied the effect of ground water table on a solar pond functioning under constant load. A 1-D transient model revealed that the ground with a deep water table acts as an energy storage reservoir that allows stable operation, while shallow water table leads to enhanced ground loss and consequently a lesser efficient pond. Amigo et al. (2017) carried out a comprehensive investigation of underground heat storage for various types of solar ponds like submerged and non-submerged, along with artificially and naturally heating provisions. They validated the results with a 28 day long experiment. Verma and Das (2019) developed an analytical solution for assessing the performance of submerged solar ponds. The model considered heat extraction from non-convective zone (NCZ), LCZ as well as the ground beneath. The role of various factors on the thermal behaviour under a fixed pond volume was studied. Verma and Das 597

Solar Energy 198 (2020) 596–604

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2.2. Assessment of NCZ temperatures Since solar radiative intensity is assumed to decay as per the expression of Rabl and Nielsen (1975), therefore the intensity at a distance x below the UCZ-NCZ junction, I is given by

I = I0

4

i (x + ) .Using

ie

i=1

this expression, and applying energy balance

for an infinitesimal NCZ element (Fig. 2a) under steady state gives (Rabl and Nielsen, 1975; Tundee et al., 2010) d (IA) dx dx

d dx

+

d dx

(kA ) dx = 0 dT dx

4

I0 A

i (x + )

ie

i=1

dx +

d dx

(kA ) dx = 0 dT dx

Introduction of excess temperature T Eq. (1) yields,

d2 I0 = dx 2 k

Fig. 1. Solar pond with heat extraction from both ground and LCZ.

and simplification of

4 i (x + )

i ie

(2)

i=1

Integrating Eq. (2) once with respect to x gives,

(2020) carried a detailed investigation of the effect of imperfect heat extraction from the gradient zone in solar ponds with respect to thermal efficiency, stability and exergy destruction. It is observed that heat extraction from ground beneath solar ponds using heat exchangers has been analysed earlier, but, none of those studies considered the temperature drop across the surface of the exchanger in this context. Instead, the earlier studies considered an infinite overall heat transmission coefficient, U at the surface of the ground exchanger, which is not convincing under a real situation. A model that considers actual heat transfer across this exchanger via a finite overall heat transfer coefficient has not been reported yet. The present work is therefore aimed at filling this research lacuna with respect to temperature distributions, efficiency, stability and entropy generation within the solar pond.

4

d I = 0 dx k

i (x + )

ie

+ C1

(3)

i=1

where C1 is indefinite integral constant. Integration of Eq. (3) with respect to x helps us obtain the NCZ temperature profile function as shown below

d = 0

x 0

=

I0 k

I0 k 4

4

i (x + ) dx

ie

+

i=1

x

C1 dx

0

ie

i (1

e

i x)

i

i=1

+ C1 x

(4)

2.3. Temperature profile in ground and the ground exchanger water stream

2. Mathematical model

The steady-state energy conservation for an infinitesimal element of thickness, dy in the ground, and the ground exchanger water stream respectively yield the following (Fig. 2b) (Ganguly et al., 2017),

Fig. 1 shows the schematic diagram of a solar pond assisted with ground heat extraction. Solar radiation undergoes absorption inside the NCZ to eventually reach the LCZ. A fraction of the solar energy reaching the LCZ is lost by conduction from the LCZ-NCZ junction, and the LCZground junction, while the remaining part is extracted as useful energy from LCZ. The energy lost by LCZ through conduction to the ground beneath is also not completely wasted as its some part is recovered by heat extraction from the ground. Therefore, the effective output from the pond is the sum of power outputs from both LCZ and ground exchangers, which are connected in parallel arrangement.

d dy

kg A

U (2 rdy )(Tg

dTg dy

dy

Tf ) =

U (2 rdy )(Tg

Tf ) = 0

(5a) (5b)

mg cp dTf

Substituting Tg from Eq. (5b) in (5a) and introducing excess temperature Tf Ta = f gives,

d3

f dy 3

2.1. Assumptions

2 rU d 2 f mg cp dy 2

2 rU d f =0 k g A dy

(6)

Integrating once with respect to y, we get,

d2

Based on the above description, following are the assumptions used in developing the present analytical model of the solar pond based heat extraction system,

f

dy 2

2 rU d f mg cp dy

2 rU kg A

= C2

f

(7)

where C2 is another integral constant like C1. Eq. (7) is a linear, nonhomogenous ordinary differential equation involving coefficients of constant nature. Thus, its general solution is the summation of particular integral and complementary function. The particular integral is found using the method of variation of parameters giving the general solution as shown below,

• UCZ remains at ambient temperature. • Pond’s cross sectional area is large as compared to its side wall • • • •

Ta as

(1)

surface area and therefore, side wall heat losses can be neglected (Ganguly et al., 2017) Solar radiative intensity is assumed to undergo attrition according to the exponential expression given by Rabl and Nielsen (1975). Harvesting of useful heat from the system is done via heat exchangers located in ground and LCZ. Inlet of water in each exchanger occurs at the ambient temperature The ground is assumed to be isothermal at 5 m below the pond at the value of ambient temperature (Ganguly et al., 2017)

f

= C3 e p3 y + C4 e p4 y + e p4 y

(p4

e p3 yC2 p3 ) e (p3 + p4 ) y

e p3 y

(p4

e p4 y C2 p3 ) e (p3 + p4 ) y

(8)

where, p3 =

rU mg cp

+

( )+ rU m g cp

2

2 rU kg A

and p4 =

rU m g cp

-

( ) rU mg cp

2

+

2 rU kg A

and C3 and C4 are arbitrary constants of integration. Simplification of Eq. (8) gives the following, 598

Solar Energy 198 (2020) 596–604

S. Verma and R. Das

have assumed 9 coils of this exchanger, making its total length = 10d, where d is the LCZ thickness. Also, ULCZ is the overall heat transmission coefficient prevailing at LCZ exchanger surface, whose value for a 16 mm radius pipe has been taken from Rao and Kaushika (1983). Using the above boundary conditions, i.e., Eq. (11), the constants C1, C2, C3 and C4 are found out to be p4 p3

e p4 tg

+

e p4 tg p42 e p3 tg p3 (p3 + p4 )

p3 p3 + p4

I0 A kA + mLCZ cp h

h

C1 = I0 k

1

e p4 tg

p4 p3

ie

i (1

4

1

+

e

i h)

ie

i (1

e

i h)

i e p4 tg p4 ) (p3 + p4 )(kA + mLCZ cp h) e p3 tg p3

e p4 tg p4 2 e p3 tg p3 (p3 + p4 )

p3 p3 + p4

4

i=1 kg Ap3 p4 h (e p3 tg p3

i

i=1

h

(12a) LCZ cp h

{e ( p4 tg

f

C2 p3 p4

= C3 e p3 y + C4 e p4 y +

p4 tg

(9)

{

g

=

C2 + p3 p4

p3 + p4

C4 =

=

C2 p3 p4

p4 p3

1

)} +

ie

i (1

e

i h)

i

i=1

4 i=1

=0

QLCZ + Qg mLCZ cp Qnet = = I0 A I0 A

(11a)

4

I0 k

mLCZ cp =

ie

i (1

= QLCZ

i (h + )

kg A

( ) d g dy

y=0

+ kA

C1 (kA + mLCZ cp h) =

mLCZ cp

where

I0 k

4 ie i=1

kg Ap3 p4 h (e p3 tg p3 e p4 tg p4 ) (p3 + p4 )(kA + mLCZ cp h) e p3 tg p3

C3 p4 + C4 p3 p3 + p4

+

C2 p3 p4

4 i=1

(11c)

ie

i

(

+ C1 h + mg cp C3 + C4 +

C2 p3 p4

)

i (1

i h)

e

=

i

{e ( p4 tg

(11d)

where, p3 =

is the effectiveness of the LCZ exchanger. It is given by

{

2 r (10) ULCZ d mLCZ cp .

=1 e The factor of 10 here arises because of the assumption of coiled LCZ exchanger (to increase its effectiveness). We

p4 = 599

rU mg cp

-

p4

rU mg cp

)

p4

1 + mg cp

p3

p3 e p3 tg

+ mLCZ cp e p4 tg

1

p3

{

(

mg cp p4 e p4 tg

kA + mLCZ cp h

p3 + p4 i h)

(13)

I0 A

(C3 + C4) p3 p4 e

i h)

mg cp e p4 tg

( )

i (1

+ mg cp ( f ) y = 0

I0 A

Substitution of Eq. (12) in Eq. (14) finally yields the below mentioned expression of pond efficiency,

d dx x = h

kg A

x=h

(14)

=0

+ C1 h =

e

i

i=1

+

ie

e

i h)

i

e p4 tg p42 e p3 tg p3 (p3 + p4 )

p3 p3 + p4

Energy balance in LCZ : I0 A

i (1

4 ie i=1

I0 A kA + mLCZ cp h

( )x = h = ( g ) y = 0 I0 k

kg Ap3 p4 h (e p3 tg p3 e p4 tg p4 ) (p3 + p4 )(kA + mLCZ cp h) e p3 tg p3

e p4 tg p42 e p3 tg p3 (p3 + p4 )

p3 p3 + p4

(11b) 4

e

i

Substituting Eqs. (4) and (9) in Eq. (13) results in the following,

Ground exchanger water stream enters at ambient temerature, i.e., ( f ) y = tg = 0 C3 e p3 tg + C4 e p4 tg +

i (1

4 ie i=1

(12b) i h) p4 e p4 tg p3 e p3 tg

(12d)

( g ) y = tg = 0 p3 + p4

)} +

}

The pond’s thermal efficiency is defined by the fraction of solar power falling on the pond that is converted to exchangers’ power output, i.e.,

At y = tg ,ground temperature equals ambient temperature, i.e., C2 p3 p4

e p4 tg p4 )

(10)

The boundary conditions needed for the evaluation of the unknown constants mentioned above are,

+

{e ( p4 tg

2.4. Expression for efficiency

C3 p4 e p3 tg + C4 p3 e p4 tg

k g Ap3 p4 h (e p3 tg p3

(p3 + p4 )(kA + mLCZ cp h ) e p3 tg p3

1

i h)

e p3 tg p3 (p3 + p4 )

p3 + p4

I0 A kA + mLCZ cp h

p4 p3

e

i

e p4 tg p42

p3

1

p3

i (1

(12c)

Substituting Eq. (9) in Eq. (5b) and introducing excess local ground temperature Tg Ta as g yields the temperature distribution in the ground, i.e.,

C3 p4 e p3 y + C4 p3 e p4 y

{e (

)} +

p4

Fig. 2. Magnified view of a (a) differential layer in NCZ, and (b) differential layer in ground and the ground exchanger water stream. C3 =

4 ie i=1

I A

p3 ) p4 e p4 tg kA + m0

(p4

C2 =

e p4 tg p42

p3

e p3 tg p3 (p3 + p4 )

p3 + p4

e p4 tg p4 )

(p3 + p4 )(kA + mLCZ cp h ) e p3 tg p3

( ) rU mg cp

+

( ) rU mg cp

2

+

2 rU kg A

2

+

)}

1

e p3 tg p3 (p3 + p4 )

p3 + p4

k g Ap3 p4 h (e p3 tg p3

p4 p3

mLCZ cp e p4 tg p42

mLCZ cp p3

)} +

(

2 rU kg A

;

}

(15)

Solar Energy 198 (2020) 596–604

S. Verma and R. Das

2.5. Effect on stability

Jcrt

The steady-state salt transfer expression in the NCZ is,

(J

x

+

dJx dx dx

)

1

2 x=h

dT dx x = h

1

2 x=0

dT dx x = 0

(22)

Jx = 0 2.6. Influence on entropy production

(16)

Jx = constant, say J

So, under steady-state, the rate of salt transfer across the NCZ is a constant. We must continuously remove brine from the UCZ and supply the same to the LCZ for maintaining this steady diffusion rate. Therefore, we can also term J as the steady-state brine requirement. According to Fick’s law, rate of mass transfer is related to the concentration gradient as,

Jx = AD

{1. 18A (D + D T ) ( ) }, = max. {0. 44A (D + D T ) ( ) }

dC dx

Entropy produced in the present model can be categorized into four broad categories: S1 which is the entropy production rate by conduction in the NCZ, S2 representing the entropy generation rate due to conduction in the ground, S3 denoting the rate of entropy production by heat transfer across the LCZ exchanger, and S4 signifying the rate at which entropy gets generated by heat transport across the ground exchanger. The expressions for each of these are discussed below (Sahin, 2011)

(17)

where D is the diffusion coefficient of the medium. The diffusion coefficient depends significantly on the local temperature. In the present work, the linear expression used by Srinivasan and Guha (1987a) has been used to describe this dependence, i.e.,

S1 =

0

J = A (D1 + D2 T )

dC dx

=

(19)

dC dx

dT : at the UCZ , and dx

J > 1.18A (D1 + D2 T )

dT : at the LCZ dx

(

(

tg

S4 =

(20b)

=

0 tg

0

1 Tg

I0 A

h

+

i=1

i ie

I0 A

i=1

i ie

dx

i (x + )

dx

+ Ta

tg 0

i (x + )

T

4

1 Tg + dTg

+

4

0

0

= kg A

Tg 2

(23a)

)

dTg dy

Tg

2

dy

2

dy

g + Ta

(

)(

dTg

d g dy

{

) h

dx +

+ Ta

)

+

1 T

2

d dx

dTg

dTg

tg 0

1 T + dT

k g A dy

k g A dy

= mLCZ cp ln

from Eq. (19) in Eq. (20), we get,

J > 0.44A (D1 + D2 T )

0

)(

kA

S3 = mLCZ cp ln

(20a)

dC dT > 1.18 : at the LCZ dx dx

tg

= kg A

Further, Wang and Akbarzadeh (1982) have suggested the following conditions for stability of a solar pond,

dC dT > 0.44 : at the UCZ , and dx dx

tg 0

S2 =

where D1 and D2 are constants characteristic of the medium. Their values for saline water are 0.5838 × 10 9 m2 ·s - 1 and 0.0403 × 10 9 m2·o C - 1·s 1, respectively (Srinivasan and Guha, 1987a). Combining Eqs. (16) (17) and (18) we get,

(

dT kA dx

h 0

S1 =

(18)

D = D1 + D2 T

Substituting

h

(23b)

Ta + (Tx = h Ta

Ta + x = h Ta

)

U (2 rdy )(Tg

U (2 rdy )

(Tg

Tf )2

Tg Tf

Ta )

}

mLCZ cp (Tx = h Tx = h

Ta )

mLCZ cp x = h x = h + Ta

Tf ) =

( tg 0

1 Tg

+

(23c) 1 Tf

)

2 2 rU ( g f) dy ( g + Ta )( f + Ta )

(23d)

The net entropy generation rate, Snet is given by the summation of the above-mentioned four components.

(21a)

3. Validation (21b)

The authentication of the present model has been done with the work of Giestas et al. (1996). The reference model of Fig. 3 gives the steady-state NCZ temperature profile when the pond is operating without any heat extraction and is devoid of any ground heat loss. The

If we define Jcrt as the critical minimum value of steady-state salt transfer rate required to sustain stable pond operation, then we can write,

Fig. 3. Validation analysis of proposed method. 600

Solar Energy 198 (2020) 596–604

S. Verma and R. Das

UCZ and the NCZ thicknesses were taken to be 0 and 1 m, respectively, and the absorption of solar radiative intensity within the pond was assumed to be described by an extinction coefficient µ as Ix = I0 e µ (x + ) . Two different values of I0 and µ were taken: (50 W·m−2, 0.2 m−1) and (66 W·m−2, 0.8 m−1). The surface of the pond was assumed to lose energy governed by an overall loss coefficient, Usurface of 100 W·m−2·K−1 Therefore, the reference model can be generated from the present model by incorporating the following boundary condition,

I0 A (1

e

µ

) + kA

d dx

= Usurface A x=0

x= 0

temperature at the exchanger. Since there is no temperature difference, so no heat gets transferred, despite a very high U. For the remaining distance of the exchanger length, we have a ground temperature profile in the neighbourhood from which heat can be extracted. So for this much distance, the exchanger stream does show a temperature increase, and because of extremely high U, it shows an exceptional increase in even such a small distance. Two conclusions can therefore be drawn from this discussion, (i) for most part of tg, the behaviour of f with respect to U depends on the range of U considered and (ii) just very close to the LCZ, the trend is strict, which is that more U implies more f .Since the outlet of exchanger is at y = 0, so, it is safe to say that the outlet temperature of exchanger stream and thus, the energy gained by it will increase as we use larger and larger values of U. Fig. 6 reveals that larger values of U cause decreased local NCZ temperatures. This can be explained as follows. Under steady-state, the rate at which ground loses heat to the ground exchanger water stream must equal the rate at which it receives energy from the LCZ. Now, a larger U means more energy extraction from ground. As explained above, this would cause more energy being lost by LCZ to the ground. Increased rate of conductive loss to the ground would mean lesser storage zone temperature and thus, lesser NCZ local temperatures. Fig. 7 represents the pond efficiency, as a function of its NCZ thickness, h. As revealed by several other studies, an optimum value of NCZ thickness exists in solar ponds that yields maximum efficiency. The reason for the existence of this optimum value is also well-known to be the trade-off between useful solar energy available at the LCZ and the net conductive heat loss from the LCZ. However, the present study reveals another important aspect of this optimum h value, which is its dependence on U. It is evident from Fig. 7 that the assumption of infinite U clearly underestimates the optimum NCZ thickness, and overestimates the corresponding maximal efficiency. The magnitude of the error in this prediction is not ignorable. For example, in Fig. 7, the assumption of U predicts that the pond will perform best for h 0.9 m , while a practically existent value, say U = 200 W·m 2·K 1 predicts the same at h 2.1 m .Hence, if this thickness is designed based on the idealized heat transfer assumption, then efficiency of pond will be below its actual maximum value. The reason for this underestimation can be elucidated as follows. The optimum value of h is the one at which a trade-off is achieved between the following two effects, (i) useful solar energy reaching the LCZ and (ii) net conductive heat loss from the LCZ (which includes conductive heat loss from the NCZ-LCZ interface and from LCZ-ground interface). Under steady state, the total energy extracted from the ground equals the energy received by ground from the LCZ. A large value of U means more heat energy extracted from the ground, and therefore, a large amount of conductive heat loss at the LCZ-ground interface. This in turn implies that lesser energy is dT stored within the LCZ, therefore, lesser temperature gradient dx is

(24)

In addition to the above, the present model is simplified based on the parameters as indicated in Table 1. The two profiles obtained by the scaled-down version of the present model are found to be in satisfactory agreement with the reference model, thereby validating the current work. It is worthwhile to note that we have assumed the UCZ to be at ambient temperature, while the reference model considers heat transfer between UCZ and the ambience to be governed by an overall heat transfer coefficient Usurface = 100W·m - 2·K - 1 as per Eq. (24). This different boundary condition at the UCZ surface does not affect the validation, because consideration of heat loss from the surface reveals the UCZ to be just a little hotter than the ambience, the difference being near about 1 °C. Hence, the assumption of UCZ being at ambient temperature is a reasonably accurate one. 4. Results and discussion Figs. 4, 5 and 6 respectively show the temperature distribution in the ground, in the ground exchanger water stream, and in the NCZ. Results are presented for different values of the overall heat transmission coefficient at the surface of the ground heat exchanger, U. It can be observed from Fig. 4a that the local ground temperatures are reduced if the calculations are made using a larger U. An increase in U means more effective heat transfer at the ground exchanger surface. So, a larger quantity of energy is extracted from the ground, thereby lowering local ground temperatures. Fig. 4b reveals that the behaviour of local temperature gradient in ground with respect to U depends on the location under consideration. If the location lies close to the pond base (i.e., small y), then increasing U will increase the magnitude of local because of increased energy donation to the ground exchanger. On the other hand, if the location is close to the ambient temperature ground isotherm (i.e., large y), then the trend is opposite, because at larger U, temperature has already decayed almost completely to the ambient temperature and now there is not much change to be accomplished. However, at a smaller U, there is a slower rate of temperature decrease, meaning that it has yet to reach the ambient value. So, it still has a finite temperature gradient, which is greater than that of the former case. Fig. 5 shows that the local ground exchanger water stream temperature neither exhibits a strictly increasing nor a strictly decreasing trend with respect to U. From Eq. (5b), it is seen that each such local temperature depends on two factors: the corresponding Tg (this decides how much energy is available) and the value of U (this factor decides how much effectively the available energy will be transferred). An increase in U results in oppositely directed effects on these two factors. On one hand, it enhances the effectiveness of heat transfer from ground to the exchanger, but on the other hand, it also reduces local ground temperatures (Fig. 4a), meaning that there is now lesser energy to transfer. The cumulative consequence of increasing U in a practical range is that the former factor dominates and therefore, at any location, a higher f value is registered for a higher U. But, for very large values of U, the net outcome of these two effects is different. We see from Fig. 5 that when, U , then up to about 4 m from its inlet (i.e., y 1), the ground exchanger water stream does not show any increase in temperature. This is because, the ground is at the ambient temperature throughout this much distance, (Fig. 4a) which is the same as the inlet

( )

x= h

predicted. This in turn means that the effect of conduction heat loss from LCZ-NCZ interface gets reduced. So, while on one hand, conductive heat loss at the LCZ-ground interface increases, but, that at the LCZ-NCZ interface decreases. The collective effect is that the sum of both these losses increases. So, for the aforementioned two effects to Table 1 Parameter settings for comparison with literature results. Parameter

Value

mg

0 0 0

mLCZ kg 1

2,

3,

1

Usurface

601

4

1 0

0.2 m

1

(for I0 = 50 W·m 2)

0.8 m 1 (for I0 = 66 W·m 2) 100 W⋅m−2⋅K−1

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S. Verma and R. Das

Fig. 4. (a) Temperature distribution in ground and (b) its gradient for different values of U, h = 1.0 m.

found to be much lesser than the actual one (considering a finite U). Fig. 8 represents the influence of U on the value of Jcrt . It is observed that increasing U leads to a lower value of Jcrt . A large value of U means more energy extraction from the ground, thus, less energy will go as internal energy of the pond water. This reduces both the temperature and its gradient at the LCZ. It follows from Eq. (22) that as a result, the critical rate of salt transfer Jcrt decreases. This means that the currentlypracticed assumption of infinite U will always underestimate the value of Jcrt . Furthermore, since Jcrt is the minimum value of J required for stable operation of the pond, therefore, even a minor error in its prediction can lead to destruction of the salinity gradient and pond performance. As an illustration, in Fig. 8, consideration of idealized heat transfer assumption (U ) at the ground exchanger surface predicts that the brine requirement is at least 1.493 × 10−6 kg/s. Suppose, we remove salt from the UCZ and supply to the LCZ at a rate of say, 3 × 10−6 kg/s, which is above the predicted critical limit. However, a practically existent value of U (say, 200 Wm−2 K−1) predicts a critical brine flow rate of 4.354 × 10−6kg/s. Therefore, design of the pond for stable operation assuming perfectly effective ground exchanger (U ) may wrongly lead to an unstable pond. Fig. 9 represents the influence of overall heat transfer coefficient, U on the net entropy production rate, Snet and its various constituents d (S1, S2, S3, S4 ). Increasing U decreases , g , dx (Figs. 4 and 6), while on

Fig. 5. Temperature distribution of ground exchanger water stream for different values of U, h = 1.0 m.

d g (Fig. dy

d

4b), its effect is location specific. For some y, g decreases with dy increasing U while at others, it increases. This produces different effects on various terms involved in each of the components (Eq. (23)). When U is increased, we discuss the effects below,

S1(Fig. 9a): The portion decreased ).

Fig. 6. Temperature distribution in NCZ for different values of U, h = 1.0 m.

The portion

h 0

kA

d dx

+ Ta

h 0

I0 A

4

i=1

i ie

+ Ta

i (x + )

dx increases (because of

2

dx decreases (because of decreased

Fig. 7. Efficiency as a function of h for different values of U.

neutralize each other, the available solar energy at the LCZ must increase. This will be achieved at a smaller h value, since less thickness means less absorption. Therefore, by adopting the conventional practice (U ), the optimum value of h is wrongly-predicted, and the same is

Fig. 8. Dependence of critical salt transfer rate, Jcrt on U, h = 1.0 m. 602

d ) dx

Solar Energy 198 (2020) 596–604

S. Verma and R. Das

increasing U (Fig. 9e). This is because the major contribution to Snet comes from S1, which always decreases with increasing U, thus, a similar trend is observed with Snet . Therefore, it is important to use a practical value of U to estimate the exegetic potential of a solar pond.

and increases (because of decreased ). S2 (Fig. 9b): Numerator increases at some locations and decreases at d others (due to similar trend of g ), and denominator always decreases (because of decreased

dy

g ).

S3 (Fig. 9c): Both the portions mLCZ cp ln

decrease. S4 (Fig. 9d): The portion

2 rU g + Ta

increased U) and the portion

( g

(

Ta + x = h Ta

) and

mLCZ cp x = h x = h + Ta

increases (because of decreased f)

f + Ta

2

g

5. Conclusion

and

decreases (because of decreased

This article presents analytical solution to assess the thermal performance of solar ponds assisted by heat recovery from the ground beneath considering an actually realizable condition. This is based on the assumption of a finite heat transfer coefficient at the ground exchanger surface rather than an infinite value conventionally used. The effectiveness of heat extraction from ground is seen to have considerable effect on various aspects of the performance of solar ponds, such as stability, entropy production and the optimum NCZ thickness yielding maximum efficiency. These effects are discussed briefly below

g

and increased f ). It is therefore clear that an entropy component can either increase or decrease with respect to increasing U, and this shall depend on which of the effects (on its various terms) predominates. This will in turn depend on the range of U under consideration for a given set of other parameters held constant (h, mg , mLCZ , kg etc.). However, we can still draw one conclusion, which is that Snet always decreases with

Fig. 9. Entropy production rates for distinct values of U, h = 1.0 m. 603

Solar Energy 198 (2020) 596–604

S. Verma and R. Das

• The local NCZ and ground temperatures exhibit strict decrease with • •

•

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increasing U, but, the behaviour of local temperatures inside the ground exchanger depends on the location and the range of U under consideration. The prevalent assumption of U underestimates the optimum NCZ thickness and may lead to attainment of an efficiency lesser than the actual maximum. This underestimation is attributed to the fact that this assumption over predicts the net conductive heat loss from the LCZ. The conventionally used assumption of U underestimates the minimum brine requirement needed to ensure stable functioning of the solar pond. A design methodology based on this assumption may therefore initiate instability. The reason for this is that value of U used in the calculations affects the local NCZ temperatures which in turn affects the local mass diffusion coefficient. The assumption of U is seen to predict much lesser exergy destruction than the realistic one. The entropy generation resulting from heat conduction in NCZ, and extraction from LCZ decrease with increasing U. However, entropy generated by heat conduction and extraction in the ground do not show any strict trend. Depending on the range of U, they can show an increase or decrease. However, the entropy produced by conduction in NCZ is the most dominant of all the constituents; thereby the net entropy production rate also follows a similar decreasing variation with U.

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