Analytical model of solar pond with heat exchanger

Energy Convers. Mgmt Vol. 23. No. 1, pp. 23-31. 1983 Printed in Great Britain. All rights reserved

0196-8904/83/010023-09$03.00/0 Copyright ~ 1983 Pergamon Press Ltd

ANALYTICAL MODEL OF SOLAR POND WITH HEAT EXCHANGER S. K. RAO and N. D. KAUSHIKA Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India

(Received 10 February 1982) Abstract--This paper presents an analytical model of a three zone solar pond with heat exchange pipes laid in its bottom convective zone. Explicit expressions for the transient rate of heat extraction and the temperature at which heat can be extracted are derived as a function of geometrical and operational parameters of the system. The transfer of heat from the pond bottom convective zone to the heat exchange fluid is expressed in terms of a heat removal factor, F R. Analytical results, characteristic of the optimum performance of the pond, are presented and the criteria for the size and heat transfer characteristics of the heat exchanger are investigated. The annual average efficiency of heat extraction exhibits the asymptotic increase with the increase of length per unit pond area of heat exchange pipe. Solar pond

Heat extraction

Heat exchanger

S(x, t) = solar intensity at point x at time t, W/m 2

NOMENCLATURE

S~,,, = amplitude of mth harmonic of solar intensity S(0, t), W/m-' So = average value of solar intensity, S(x = O, t) T(x, t) = temperature distribution in pond system at point specified by x at time t. ' C T(x) = average value of T(x, t), C T,.(x) = amplitude of ruth harmonic of T(x, t), 'C TA(t) = ambient air temperature at time t, ' C 7~40= average ambient temperature, ~C T . . = amplitude of mth harmonic of ambient air temperature, C Ti(t)= inlet temperature of fluid (taken as ambient air temperature) at time t Tt(l, t) = temperature of fluid at length 1 of heat exchange tube and at time t T/o(t ) = outlet temperature of fluid at time t Tlo(t) = average value of T/~(t) T,,(t) = water temperature in bottom convective zone, C T,, = temperature of pond in constant temperature mode T~.~= average value of T.,(t), C T,,,,, = amplitude of mth harmonic of T,,(t), C = transmitivity of water surface = average efficiency of heat extraction

(7 : specific heat of water, J/kg °C C~ ~ specific heat of ground, J/kg c'C 3l = elemental length along the length of the heat exchanger tube, m F R : heat removal factor G ~ mass flow rate of water per unit collector area (kg/s. m-') h0 = sum of radiative and convective heat transfer coefficients (surface to ambient), W/m 2 °C h~ = heat transfer coefficient between nonconvective zone water and upper convective zone water, W/m2C h~ = heat transfer coefficient between nonconvective water and lower convective water, W/m2 °C h 3 = heat transfer coefficient between lower convective water and blackened ground, W/m s ' C h,.= evaporative heat transfer coefficient at pond surface h, ~ heat transfer coefficient from bottom convective zone of pond to fluid flowing inside heat exchanger, W/m 2 ' C K ~ thermal conductivity of water, W/m 'C Kz ~ thermal conductivity of ground, W/m ~C l r = total depth o f pond, m /~ = depth of upper convective zone. m 12= depth of nonconvective zone, m /~ = depth of lower convective zone, m L = total length of heat exchange tube, m = absorptance of blackened surface n i = absorption coefficient for j t h portion of solar spectrum, m t #j= fiaction of solar radiation having absorption coefficient nj p = perimeter of heat exchange tube, m ~) ( t ) - retrieved heat flux from bottom convective zone per unit time, W/m 2 .~(t) = average value of Q(t), W/m 2 ~ ( t ) - retrieved heat flux at constant temperature Q,(t) = average value of Q,.(t) p = density of nonconvective water, kg/m s p~ - density of ground, kg/m ~ p , . - heat capacity per unit volume of convective water, J/m ~ C

INTRODUCTION T h e r e are two m e t h o d s w h i c h have b e e n e m p l o y e d for h e a t e x t r a c t i o n f r o m solar salt p o n d s . T h e first is b a s e d o n the selective w i t h d r a w a l o f the b o t t o m layer o f h o t b r i n e f r o m the p o n d [1]. T h e s e c o n d m e t h o d m a k e s use o f heat e x c h a n g e pipes placed in the b o t t o m region o f the p o n d ; the h e a t e x c h a n g e fluid, n o r m a l l y water, while flowing t h r o u g h these pipes is h e a t e d a n d carries the h e a t to the s y s t e m w h e r e it is used. Jain [2] a n d Nielsen [3, 4] e x p e r i m e n t a l l y tested this p r o c e s s o f heat e x t r a c t i o n a n d f o u n d t h a t the h e a t e x c h a n g e by free c o n v e c t i o n o f p o n d w a t e r a r o u n d the heat e x c h a n g e r is quite g o o d . H i p s h e r a n d B o e h m [5] a n d Badger et al. [6] e x a m i n e d the heat 23

24

RAO and KAUSHIKA: SOLAR POND WITH HEAT EXCHANGER x=O ~--UCZ

-----_--_~ ( t ) -

m

NCZ

x=l~ +12

: Tro [t)

[ t t

l 1/

X " I I +l 2 + I 3

Fig. 1. Schematic diagram of three zone solar pond with heat exchanger.

transfer characteristics of thin walled tubing as a heat exchanger in a nonconvective solar pond and recommended their applicability. The overall thermal performance of such a system has yet to be evaluated. This paper presents an analytical model of the solar pond with tube heat exchangers in its bottom region. Results of the analytical model are given for two modes of operation: (i) heat extraction with a constant flow rate of heat exchange fluid, water; this will cause the temperature of the heat extraction zone to vary and (ii) heat extraction by keeping the temperature of heat extraction constant. This will, obviously, require the variation of flow rate of heat removal water. A comparison is also shown between the heat removal process using the in-pond heat exchanger and the method of circulating the pond brine to an external heat utilization system [1, 7, 8].

,,=~ S0,.e 'm'°']

S(x't)=IS°°+

x I j ~ i P/e-"~ ] where

S ~ = r S0 and So, , = z So, , . Also 6

T.(t)=T.~+

~ ttl

T.,md.......

(2a)

TAn, e i....

(2b)

I 6

T4(t) = TAo + ~ m=

TIIE MODEL AND APPROXIMATIONS A salt gradient solar pond normally consists of three zones: a convective zone at the surface, a second convective zone at the bottom and a nonconvective zone in between them [9]; the convective-nonconvective zone boundaries, in general, are variable with time. However, this motion is very slow [3, 10] and for appropriate temperatures and salinities in the upper and lower convective zones, the zone boundaries are rather stable. We, therefore, assume the zone boundaries to be stationary with time. A schematic diagram of a solar pond with stationary boundaries and having the heat exchanger in its bottom convective zone is illustrated in Fig. 1. The absorption of solar radiation in the pond water has been represented by Kaushika et al. [11] as a summation of five exponential terms. The solar intensity and ambient air temperature are assumed to be periodic and are represented by six harmonics of a Fourier series; consequently T..(t), TA(t) and T(x, t) will also be periodic. So we have

(1)

1 6

T(x, t) = T(x) + ~ m-

T.,(x) e '"~''

(2c)

I

where o9 = 2n/(365.25 x 24 x 60 x 6 0 ) s - k The temperatures and heat fluxes in the nonconvective and ground zones are governed by the Fourier heat conduction equation and the appropriate boundary conditions at the interfaces; the temperatures and heat flux in the convective zones are governed by energy balance considerations. The Fourier heat conduction equation for the nonconvective zone contains a heat source term that results from the absorption of solar radiation and is given by K

~2T(.v. t) ?;T(x, t) ~S(.r, t) ~ - pC + ?x ~ ~t (?x

(3)

The appropriate boundary conditions at the interfaces are: At x = l~

RAO and KAUSHIKA: SOLAR POND WITH HEAT EXCHANGER . OT(x,

t) I

- - I~. ~

= h, [r,(t) -

T ( x = I,,

t)] (4a)

Ix=If

At x = 1~+ 12 • ~ T ( x , t) -- i

l

~

r=/l+l 2

= hz[T(x = 1, + 12, t) -- Tw(t)].

25

the interface at x = It +/2 and may be evaluated by solving the Fourier heat conduction equation (3) with boundary conditions (4) and (5). Similarly, the fourth term is obtained by solving the equations (6) and (7). The second and third terms are obtained from equations (1) and (2). So we have

0 (t)= {(~)o+s0o- U , . ( T + - T~,0]}

(4b)

+I(

In equation 4(a), T~(t) is the temperature of the upper convective zone and it may be expressed in terms of atmospheric air temperature by consideration of the energy balance in this zone as in Kaushika and Rao [8]. So we have

+±

+

+-,) SomlJjn)

: K(#~-.~)(kh+

ASCx, t) = ho[T, (t) -- TA(t)]

K/

x e-"J (h + t2)Kfl,.Ds..

+ h+[(T, (t) - 7 T, (t)) C, - C21 + h~[T,(t) - r ( x = Ii, t)] + pwl~

dTl(t) dt

(5)

where A S ( x , t) = S ( x = O, t) - S ( x = 11, t) and C~ = 2.933 and C2 = 39.11505. The temperature distribution in the ground zone is governed by the following Fourier heat conduction equation K+

32T(x, t) C OT(__x, t) ~x 2 : Pg g ~t

.

Kgfl,,,g]

•

tim - - n i

tJ

e

.

(9)

.....

It L, (a~ )+~, T'AO, D3,,, Dam, T'Am and h~m are expressed in Appendix I [8].

+:/,+t2+l+. = h3[T'(t) = I t q- 12 +

\h3/

~/../n2 e._ np(ll+l -2-----f -

(6)

OT(x, t)]

-- T ( x

, )

- - n c - n J Ih+/2) r-]

Corresponding boundary conditions are --/~*~

+So+ ,+,w(1)+

13,

t)l

+ ++S(x = l~ + 12 + ls, t)

(7a)

Heat extraction at constant flow If the heat is extracted by the heat exchange fluid, water, flowing at a constant rate, then

and

rh,, C, Q (t) = ~ [TinCt) - TM)]. &T(X,ox t) . . . . = 0.

The energy balance in the bottom convective zone of the solar pond may be written as . OT(x, t) x=/,+12

-P'13

The temperature of the fluid inside the heat exchanger tube Tf(l, t) can be expressed in terms of the pond bottom convective zone temperature. T~,.(t) by considering the heat balance of an element between l and 1 + ~/along the length of the heat exchanger as below

dTw(t) dt

rh,,,C (d/L, + t , dl

+ S ( x = l~ +/2,

t)

. OT(x, t)

+ t~g--

OX

(10)

(7b)

6l = h,[T,,,(t) - TrCl, t)lp61

or

(8) x=11+12+13

Q (t) is the beat absorbed by the heat exchange pipe system; this heat is carried away by the heat exchange fluid. The first term in equation (8) represents the heat flux coming out of the nonconvective zone through

dTt (/, t) Tl(d, t ) - Tw(t)

-h,p rh+C.

which, on integration, yields C~[Tr(I, t ) -

T,,,(t)]=exp(-

h,pl

)

26

RAO and KAUSHIKA: SOLAR POND WITH HEAT EXCHANGER

The constant of integration C~ is obtainable from the initial condition that, TI(0 , t ) = Tit). So we have

TI(/, t)

=

T,,(t)

-

[T,,,(t)

-

T,.(t)]

exp(-

x

h,pl

\ ,~.cd

TI(I = L, t) = Tto(t) = T..(t)F R + Tz(t)(1 -- F~) or

TMt) = T,(t)+ [ T . ( t ) - r,(t)]F.

(1 l)

Thus, the heat flux obtainable at a constant flow rate is expressed by equations (12) and 03).

Heat extraction at constant temperature The heat that can be extracted at constant temperature of the heat extraction zone may be evaluated from equation (9) by substituting T,,m=0 and 7".o = T... So we have Q~(t) = (~r)~rrSoo- U,~(T.. - T'Ao)

where

--~[T'AmD4,,

+ ~

is called the heat removal factor. From equations (10) and (11) we get

X

S°"nn;

/=1 K flm(fl~ - n2)

rh. C.,/

Kflm Dom e-",u~ +121

--

O. (t) = GC.[T..(t) -- T,(t)]F~ 1

or

t

-- e-,,tKflmD 4

O(t)

= G'C,,[Tw(t) - T~(t)]

(12)

where

"J-Som~ (o(~,e i=,

\hff

G ' = G F R.

h3

When the flow rate of the heat exchange fluid is maintained constant, the temperature of the heat extraction zone, T,,(t), will vary, and this may be evaluated by substituting the value of Q(t) from equation (12) in (9). Evaluating the time independent and dependent parts separately, we have

rw,.

Pm -- Hd

i

_ ( G + ~)KflmD4me-n, i} 5

-So.,,.,£ikhd L

(14)

Thus, the annual average efficiency of heat extraction at constant temperature is given by -

Soo

(15)

NUMERICAL COMPUTATIONS AND RESULTS

x Kfl,.Dsm e -#~u~ +12) 1

/

Q,.(t) (13a)

= [TAmG'C.,- D4mT'Am

n~

Ke fl,,e

~.[7~2 It"a2 me- nj(It+12)a __ irmot .2 / e

TwO = [(~'C)effS00 -~- ULT'~o + TAoG'C,,]

/(G'C., + U~)

n'(11+12+/3

I +-- 1

:..,.,2+,3.

G'C., + D3m + ime)pwl3 +

(13b)

Numerical computations of the heat removal factor (FR), constant flow heat flux, Q(t) and outlet temperature of the fluid, Tlo(t ) have been carried out for different flow rates and L/A (length of the heat exchanger tube per unit area) values, using the ICL 2960 computer at IIT Delhi. The efficiency (~/) of heat retrieval from the solar pond at constant temperature, T/o, has also been calculated for different F R values using equations (11), (14) and (15). Both the sets of calculations were performed in the same program. In these calculations, the solar intensity and ambient air temperature values of the year 1974, at New Delhi, have been used. The complex amplitudes of harmonics (TAo, TAm, So, S~,,) of observed annual variations of S ( t ) and TA(t ) were obtained from Sodha et al. (1981). The heat transfer coefficients h~, h2, h3, hey and h0 were also obtained from Sodha et al. [12]. The coefficient of heat transfer from pond water to heat exchange fluid was calculated using considerations of free and forced convective heat transfer as shown in Appendix II. The heat transfer

RAO and KAUSHIKA:

SOLAR POND WITH HEAT EXCHANGER

coefficients and other thermophysical parameters of the pond used in this model are given below C .... 4190 J/kg Cx = 1840 J/kg K = 0.569 W/m °C Ke = 0.519 W/m °C p = 1000 kg/m 3 Px = 2050 kg/m 3 p,,. = 4190 J/m 3 °C lr = 4.0 m l~ =0.1 m 12= variable from 0.0 m to ( I t - l ~ ) in steps of 0.25 m h0 = 10.84 W/m 2 °C hi = 56.58 W/m 2 °C h2 = 48.28 W/m 2 °C h3 = 78.92 W/m 2 °C h~v = 32.34 W/m 2 °C h, = 94.534, 53.748, 29.316, 20.237 W/m 2 °C =0.9 7 =0.6 r = 0.94 G = 2 × l0 -4, 5 × 10 -4 , 1 × 19 -3kg/m2/s. The heat removal factor, F~ is a parameter characteristic of heat transfer from the bottom convective zone to the heat exchange fluid, FR = 1 corresponds

27

to heat extraction by the circulating brine layer [1, 8]. Figure 2 illustrates the dependence of FR on the size and diameter of the heat exchanger pipe and the flow rate of heat exchange fluid. FR increases with an increase in length and diameter of the heat exchange pipe and with a decrease in flow rate. The variations in annual average values of the pond bottom convective zone (zone of heat extraction) temperature, Two, and outlet temperature TIo(t ) have been investigated in Fig. 3(a) and (b). T,o is always greater than Tj0(t), and the magnitude of the differences depends on the geometrical and operational parameters of the pond as well as the heat exchanger system; the difference between the temperature of the heat exchange fluid of the outlet and the pond bottom convective zone temperature ranges between 5 and 3ffC. Nielsen (1980) has reported that an average temperature difference in 12'~C between the heat exchange fluid and pond water was observed in the experimental pond (200 m 2) at Ohio University using an exchanger consisting of 36 m of length and 3.2 cm of diameter. This observed value falls well within the range predicted by the present model. The variations in average heat flux, Q(t) with the depth of nonconvective zone for different values of FR are depicted in Fig. 4(a) and (b). Similar variations in the annual curves of Q (t), T,(t) and Tio(t ) are given in Fig. 5(a) and (b). The retrieved heat flux increases

o9f 08

////

////

o,

////

q- I ~ ~,q I ~

a4

Q3 Q2

OV .I o.o ! 0.0

I --D

= 0.096

m

2--D

= 0.064

m

3--D

= 0.032

rn

4--D

= 0.016

m

1

[

I

I

[

1

I

[

[

0.~

0.2

0.3

0.4

0.5

0.6

0.7

o.e

0.9

L /A , m-~ Fig. 2. Variation of heat removal factor (FR) with size of heat exchanger and flow rate of heat exchange fluid.

28

RAO and K A U S H I K A :

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--

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07246

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0.5 0 4 ~ --~

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o2 o3

////

// I// II/

LIA

09242 ~ I - ~ - - - - - - - - ~ "// i ~ 08555-

/

5c

........

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/ //

- - -

/1~

/

/

D .0032m

o -oo32 m

/

/I / I

1(30

G - 2 x l O 4kg / m2 / S

G=2xlO-4kg/m21s

04752

/

/

6£

O05m-i, FR-02756

iLiA.

120

S O L A R P O N D WITH H E A T E X C H A N G E R

~ 0.9242 0.9602 0.9602 0.9242 0 , 3 / - - - ' - - 0.8555 0 2 - - 07246

....

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../

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005

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--

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7f/ 20

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1.0

00

Depth of

the

I

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2.0

3.0

nonconvective

zone,

I 4.0

m

o

1.0 Depth of

(a)

130

L/A

FR

005,

01210

/

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~i 1 /J': ,111// ~ //../ /f

-----

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....

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o~,

7

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(a)

/ 7

z

I

I 0.0

I

I

I

I

I0

20

30

40

Depth

of

the

nonoonvective

zone,

m

(b)

o I[

00

I

Depth

Fig. 3. Variation of average outlet temperature ( ) and pond temperature (. . . . . ) with depth of nonconvective zone for different values of F R at flow rates; (a) G = 2 × 10-4 kg/m2 s; (b) G = 5 x 10-4kg/m2s.

I

I0

20

of

the n o n c o n v e c t i v e

I

30 zone,

m

(b) Fig. 4. Variation o f average heat flux with depth o f nonconvective zone for different values o f F R at flow rates; (a) G = 2 × 10 4kg/m2s; ( b ) G = 5 x 10 4 k g / m ; s .

RAO and KAUSHIKA:

POND

WITH

HEAT

EXCHANGER

29

temp. temp,

G:2xIO -4 kg/mZ/s

flux

D =0.032 m

Pond Outlet

120 - -x---x--x-- H e a t

SOLAR

[ z : 2 . 5 m. 80

------~

LIA ~

0.2

IlO

/

/ I00

90

P

8(]

E

l-

7C ''~'-- ~ ' - ' ~"x~ \

x~.

^ "~'x

~

~x

",x

--x~ xf~X

x\

~

~

^

..4/x~f-x~x~x-~x~X f

x...~- . ~

60 _

~ - - - ×-- x ~ x _.--^S . ~ "'f'-x-x~x'-''~ ~

x

xlx

xi

~

x ~'I

x

x

- ~ U ", =

/'-

f~

50

40 D

I

I

I

I

I

I

I

I

1

I

1

d

F

M

A

M

d

d

A

S

0

N

4O u

Month

(a)

I/

G=Sxl(~4kglm2/s

D'O.O32m i001_ / 2 . 2 , 5 m

L/,4

--x--x~--

Heat

. . . . . ~

Pond temp Outlet temp.

:\

.._. - -

flux

~

~__

0 5 m -{ ,,,~x~.4.!~

___- ~ - - - -

70

~ x . 2 ~ x ~ .

I

_ --"

9o: ~ . x . X .

-%\,,,

./

X

8o

=..

/

--

./.x"-

~-~-~--

i

~.x/ x ~

\.~.- ~

--

/

3

i

x'i ^

~ / ~ ~

x

- ~

--

~__

/~

\

~'"

~ -

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0.2

x

...~x~

__ ~

_~/x'>-'-~o.-~

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6o

"

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! ~05 50

50 ~

40 O

~

I

I

I

I

I

I

I

I

1

,I

I

d

g

M

A

M

d

d

A

S

0

N

40 D

Month

(b)

Fig. 5. Annual variation of outlet temperature ( f o r different

values

o f F R at flow

), pond temperature (. . . . . ) and heat flux ( × - -

rates: (a) G = 2 × 10-4 k g / m 2 s ; (b) G = 5 x 1 0 - 4 k g / m 2 s .

×)

30

RAO and KAUSHIKA:

SOLAR POND WITH HEAT EXCHANGER G = 2 X fO 4 kg/mZ/s

30

25

2O

o ud B5

0

1.0 Depth of

the

20

30

nonconvective

zone~

3.75 m

Fig. 6. Variation of annual average efficiency (r/) of heat extraction of constant temperature with depth of nonconvective zone for different values of F R.

with increase in Fn; the magnitude as well as range of variation of the temperature of the heat exchange fluid at outlet, Tlo(t ), is always less than the corresponding values of Tw(t). Figure 6 illustrates the annual average efficiency of heat extraction at constant temperature for various values of heat removal factor, FR. The efficiency of heat extraction is more for larger FR values which correspond to the larger length and diameter of the heat exchanger. Such heat exchangers would require more material and maintenance and would be cost-intensive. Furthermore, a longer heat exchange pipe would correspond to the larger pressure difference between the inlet and outlet and hence would require higher cost for circulating the heat exchange fluid. Acknowledgements--The authors are thankful to Professor M. S. Sodha, Dr P. K. Bansal, Mr Subhash Chandra and Mr S. N. Sukla for useful discussions. REFERENCES

1. C. Elata and O. Levin, llth Cong. Int. Assoc. Hydraulics Res., Leningrad (1965).

2. G. C. Jain, U.N.E.S.C.O. Cong. Sun Service Mankind, Paris (1973). 3. C. E. Nielsen, Proc. Joint Conf. Am. Can. Solar Energy Soc., Winnipeg, Vol. 5, pp. 169-182 (1976). 4. C. E. Nielsen, Hand Book of Solar Energy, Part A (edited by C. W. Dickinson and P. N. Cheremisinoff). Marcel Dekker, New York (1980). 5. M. S. Hipsher and R. F. Boehm, Proc. Winter Ann Meet A.S.M.E., Solar Energy Division (1976). 6. P. C. Badger, W. L. Roller and T. H. Short, Joint Meet. Am. Soc. Agricult. Engnrs Ca. Soc. Agricult. Engng, University of Manitoba, Winnipeg, paper No. 79-4027 (1979). 7. H. Weinberger, Solar Energy 7, 189 (1963). 8. N. D. Kaushika and S. K. Rao, Int. J. Energy Res. (1981). To be published. 9. C. E. Nielsen, LS.E.S. Cong., New Delhi, Vol. II, p. 1176 (1978). 10. C. E. Nielsen and A. Rabl Cong. LS.E.S., Los Angeles, extended abstract 35/5, pp. 271 (1975). 11. N. D. Kaushika, P. K. Bansal and M. S. Sodha, Applied Energy 7, 169 (1980). 12. M. S. Sodha, N. D. Kaushika and S. K. Rao, Int. J. Energy Res. 5, 321 (1981). 13. W. H. McAdams Heat Transmission. McGraw Hill, New York (1954). 14. H. Y. Wong, Heat Transfer for Engineers. Longman, London (1977).

R A O and K A U S H I K A : APPENDIX

SOLAR POND WITH HEAT EXCHANGER

1

31

and

AS,. = time dependent part of AS(x, t). 1

1

1

L

u, =j,,,o4-/, 4-K ' /= l

,)

L\h~) 4-

APPENDIX

+ \h~ n,KJ

.}/2,

and T~0 =

ASo + h,, C2 + 7Q4o(ho+ h,,TCt) ho + h,,Ci + ht

1

h,

where

and

1

1 ~

:

tl u,+ll

l - + --I e " D,~\h',m Kfl,.] _ b,::;

2

hforc e D

K,

1

1

L

/2t

\

/~d

where

ft., = (ima)pC./ K)l 2. fl,,,e = (irnegpe C~/ Kx)l z, \ i,,, Kfl,,d\h2

jjd

where L = characteristic dimension (diameter of the tube), p = density of fluid, fl = volumetric expansion coefficient of fluid, /2 = viscosity of fluid, K = thermal conductivity of fluid, C r = specific heat of fluid, D = diameter of tube, At = temperature difference, v = mean velocity of flowing fluid, m, e and K' are constants (for the flow rates considered in this paper, these constants are 0, 0, 3.66 and 1.0 respectively); f stands for fluid, water in the present case. The calculated values of h, for different diameters of the heat exchange pipe are tabulated below.

Ih(ho + h,,Ci + im~op,,,l 1) ho+h,.Cl+hl+irnogp,,.l ]

l

- CRe,~p~K'

[\ # J~ \ K

and

V

.

I1, h free hf....

/

K o,/

1

1

ht,....

and forced convective heat transfer coefficient, hf,,,~eis given by [14]

Kfl,,J " AS + TA,,(h o + h,,TCt) TI4 = .... ho + h,,Ci + hi 4- imoJp, ll

+

4-

hK,L = O.4 7 [ £ pfi g flt ( CK# ~ ~0,:,

D2,,,\h2

h'~, '

1

+

The heat transfer coefficient due to free convection, hf,e~ is given by [13]

"

Kfi,J -

1

hl~ ~ tl,

1

AS 0 = Time independent part of AS(x, t). 1 /

=

We assume that the thickness of the tube is very small. Hence the heat transfer coefficient due to conduction, h,. is very large and 1/h, may be neglected. So, we have

th(ho + h,,CO h i ° - h o + h,,C~ + h t

D~,,, =

II

Transfer of heat from pond water to heat exchange fluid is due to (a) free convection at the outer surface of the tube, (b) conduction through the thickness of the tube and (c) forced convection inside the tube. Therefore, the total heat transfer coefficient is given by

1

Kfl,.]

1

Kfl,,d\h2 D:,=(hl,,+Klfl,,)(:+~)e

KflmJ

" ~,,,,, + L,,

- ( hl~,,- l{iJ::,) ( ~ - Klfl~) e-/"''I'll2'

D (m)

h, (W/m 2 C )

0.016 0.032 0.064 0.096

94.534 53.748 29.316 20.237