Analysis of natural circulation solar water heating systems

Energy Con. & Mgmt Vol. 21,283 to 288, 1981

0196-8904/81/040283-06502.00/0 Copyright© 1981 Pergamon Press Ltd

Printed in Great Britain. All rights reserved

ANALYSIS OF N A T U R A L C I R C U L A T I O N SOLAR WATER H E A T I N G SYSTEMS* M. S. SODHA and G. N. TIWARI Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India

(Received 16 April 1981) Abstract--This paper presents an analysis of the performance of a solar water heating system with natural thermosyphon circulation between the collector and the storage tank. The analysis is based on the formulation by Ong except that provision for withdrawal of hot water from the tank (for domestic/ industrial use) has been made in the energy balance equation; further in contrast to the use of the finite difference method by Ong, explicit expressions have been obtained. The results of the present analysis (in the absence of withdrawal of hot water from the tank) are seen to be in better agreement with experiments than the corresponding results of Ong, obtained by use of the finite difference method. Numerical results, corresponding to hot water retrieved from the storage tank, have been presented for two modes of hot water withdrawal viz. the constant flow rate and constant mean storage tank water temperature. Water heating

Flat plate collector

Solar energy

NOMENCLATURE A= Ac = Ai = B= C = d~ = f~ =

constant in equation (6) collector area, m 2 insulated storage tank surface area, m 2 constant in equation (6) specific heat of water, J/kg °C internal diameter of riser, m equivalent friction factor for tubes, headers and connecting pipes [6] Fc = plate efficiency factor Y0 = gravitational constant, m/sec 2 H = solar intensity, W/m z hi, h2, ha ~ heights corresponding to points 1, 2, 3, h4, hs, h6 f = 4, 5 and 6 respectively, m Ke = equivalent head loss for tubes, headers and connecting pipes respectively L = tube length, m ph = water mass flow rate due to natural circulation, kg/sec M(t) = water flow rate at which water is collected from storage tank. kg/sec N = number of tubes Ta = ambient temperature, °C T,. = mean system temperature during sunshine hours, °C T~, = mean system temperature during off sunshine hours, °C t = time, s U~, U. Ut = overall heat transfer coefficient between plate, box insulation, storage tank and outside air respectively, W/m 2 °C W~, W~, W~ = thermal capacities of plate, connecting pipes and storage tank respectively.

j/re

Wwc, Wwp, Ww, = thermal capacities of water stored in tubes, connecting pipes and storage tank respectively, J/°C 7c = plate absorptivity p = water density, kg/m 3 z = glass transmittivity * Work supported by Department of Science and Technology, Government of India.

INTRODUCTION Close [2] has presented a simple mathematical model for a solar hot water system, with natural circulation between the collector and the storage tank, assuming solar irradiation as a sine function of time; no allowance was made for the withdrawal of hot water from the tank. Based on Close's [2] analysis, G u p t a & Garg [3] have analysed the system incorporating a plate efficiency factor and expressing solar intensity and ambient temperature as a Fourier series of time. A finite difference method has been used by Tzafestas et al. [4] and O n g [5] to predict the performance of solar water heating systems. A parallel plate absorber type solar water heating system has been analysed by G r o s s m a n et al. [6] for quasi-steady state conditions with forced or natural circulation of water between the absorber plate and storage tank. The effect of heat transfer and frictional loss in connecting pipes and storage tank and relative height of the storage tank above the absorber plate were not considered [6]. Recently, H a u n g [7] has developed a more general theory for parallel plate absorber water heating systems with natural circulation of water between the plate and the tank (for no draw off of hot water from the tank) by representing solar radiation as a sine function of time. The dynamic performance of solar water heating systems has also been investigated by various workers [8-11] without much improvement over earlier work. In this communication, we have considered a solar water heating system with natural t h e r m o s y p h o n circulation of water. The effect of withdrawal of hot water from the tank has also been considered. The analysis is based on the formulation of O n g [1] except that provision for withdrawal of hot water from the tank (for domestic and industrial use) has 283

284

SODHA AND TIWARI: NATURAL CIRCULATION HEATING SYSTEMS

It

t

6

and

-l-j !i "

Connecting pipes

I

I

Ww,

w ; = w, +

3[

w = w~+ wp+ w,+ ww~+ ww,+ ~;~,.

Storage tank

II

Equations (1) and (3) can be used to determine the mean system temperature during sunshine and off sunshine hours while equation (2) can be used to evaluate the mass flow rate on account of natural circulation. Solving equations (1) and (3) with the initial conditions T . , = T,.o

IIIII

Fig. l. Schematic diagram of solar water heater.

at

t=0

and

T~,= T~,o at

t = t',

one obtains,

been made in the energy balance equations. In contrast to the use of a finite difference method by Ong [1], explicit expressions have been obtained in the present investigation. Numerical calculations have been made for the same parameters as used by Ong [1]. The mean water temperature calculated by the present model (in the absence of withdrawal of hot water from the tank) are seen to be in better agreement with experiments than the corresponding results of Ong [-1] obtained by the use of a finite difference method. Further, numerical results have been presented for the cases when hot water is withdrawn from the storage tank at constant flow rate and constant mean system temperature respectively.

Tm

=

exp

(f;);; f(t) dt

-

g(t)exp

+ T.oexp(-flf(t)dt

(;;) f(t)dt

)

dt

(4)

and

T',.=exp(- f,:F(t)dt) f[G(t)exp( f[F(t)dt)dt + T ' , . o e x p ( - f [ F(t)dt)

(5)

where, ANALYSIS

f(t) =

Following Ong [1], referring to Fig. 1, and making provision for withdrawal of hot water from the tank (by displacement) the energy balance equations can be written as (i) Sunshine

9(t) -

u + M(t)C W rctcFcA~ W

W dTm - - + U(T,. - to) + ~4(t)C(T,. - Ta) = z~cFcAcH dt Heat stored

Heat lost

Heat lost due to withdrawal of hot water from tank

(1).

Solar radiation absorbed by the absorber

and for storage tank

(2) &c(T3 - Ts) = W; dT~ Heat collected

Heat stored

+ U,(Tm- ta) + M(t)C(Tm - L) Heat lost

and (ii) Off sunshine (H = 0)

F(t) -

W; dT~,

d r - + U,(T',. - t~) + M(t)(T'm - Ta) = 0 (3)

where,

Heat lost due to withdrawal of hot water from the tank

u, + M(t)C W't

and

U = FcAc(Uc + Ut) + AiUi (neglecting connecting pipe losses)

G(t)= (U~ to+

M(t)C ) ~T°

.

SODHA AND TIWARI: NATURAL CIRCULATION HEATING SYSTEMS

285

When ~/" or T,. is constant, an explicit expression for q can be obtained by using the corresponding expression for M(T,. - T,) (which is periodic). Following Ong [1], the mass flow rate of water due to natural circulation can be expressed as

Since H, T, and t, are in general periodic we may express g(t) and G(t) as g(t) = go + ~ g . cos(nwt - a.) n

G(t) = Go + ~ G. cos (nwt - er'.). i1

f(s)(2AT,. + B) DC

[~/(t)] 3 =

For a constant value of Mw(t), f and F are also constant; thus substituting for 9(t) and G(t) as above in equations (4) and (5), one obtains

1

k

dt + U,(T,. - t.) + M(t)C(T., - To)

T , . = e - - C ' [ ~ (eS' - 1) + 7".,o]+ e - I ' ( e I ' - 1)9" f(1 +

(10) where

f2 ]

(h 3 -

x

cos(nwt - a.) + ~

sin(nwt - a.)

f(s) = 2(h3 - hi) - (h2 -- hi)

(6)

h5) 2

(h 6 -- h5)

and

and T~, = e_V<,_,,, [ Go F (eF, - er'') + T~,o] +

e-F('-V)(e v" F(1

I

dT~

- 1)G.

Substituting for T., and d t

n2w2"] +

x

"~

D = 16(f~L/di + K,,)/(zt'gop di4N2).

can be computed from equation (10).

t72 }

]

cos (nwt - a~,) + ~ - sin (nwt - a~,) .

N U M E R I C A L RESULTS AND DISCUSSIONS

(7)

The following parameters, corresponding to Ong's [13 experiment, have been used in the calculations:

The rate of withdrawal of hot water from the tank corresponding to a constant temperature Tm during

dT,.

sunshine can be obtained by putting ~ f - = 0 in

~ c = 0 . 9 , r =0.9, F c = 0 , 7 7 , A~= 1.5m 2, W = 6 . 4 7 × 105J/°C, U = 9 . 3 W / C , c = 4.19 × 103 J/kg C , T,,o = 29.44cC.

equation (1); thus 1

(M) ...... -

(T= -

L)c

[rct¢F~A~H - U ( T , . - G)].

(8)

ta ~

fI

' M C ( T . , - T~)dt + W,,(T,.(t') - T,.o)

(9) f i" A¢H dt

I

A

I

i

I

i

Ta .

The hourly variation of solar intensity and ambient air temperature has been shown in Fig. 2. Equation (4) has been used to compute the mean system temperature and its hourly variation a shown in Fig. 3. Experimental and theoretical results of Ong have been shown by Q and continuous curve while results of the present model are shown by dotted line.

The efficiency of utilization of solar energy is given by

q=

from equation (1), til(t)

I

I

I

I

I

I

945

"e 63O

3z

g

26.6

3P5

21

o n-

0

t

I 9

l Time

I I[

I of

day,

i 13

a

I 15

J

I 17

l

15.5 19

h

Fig. 2. Hourly variation of solar intensity and ambient air temperature for a typical day in Kuala Lumpa.

286

SODHA AND TIWARI:

NATURAL CIRCULATION HEATING SYSTEMS

0

0

56

0/

/

i

/

0

0

~

//

48

l

,f

? 40

/

32 ___0._~

24

-

-

-0~

i

i

I 9

7

I II

i

Time

I 13

of

day,

I h

I 15

I

[ 17

I

I 19

"

Fig. 3. Hourly variation of mean water temperature in tank for no withdrawal of water from the tank. , experimental results; theoretical results of Ong[1] by finite difference method; . . . . . theoretical results of author's analysis.

set. It is seen that higher temperature can be achieved when withdrawal of water starts sometime after sunrise. To get hot water at constant temperature, one has to vary the rate of withdrawal of water from the tank. This effect has been shown in Fig. 6 for different constant water temperatures. For higher temperature, the rate of withdrawal of water from the tank should be lower. Figure 7 shows the dependence of overall system efficiency (r/) on M, the rate of withdrawal of water. It is seen that r/increases with increasing .~ which cor-

F r o m this figure, it is seen that the present model gives better agreement with experimental results than the finite difference method. Figure 4 shows the hourly variation of mean water temperature for different constant rates of withdrawal of water from the insulated tank during sunny hours. As expected, it is seen that the available mean water temperature decreases with increasing rate of withdrawal of hot water from the storage tank. Figure 5 illustrates the hourly variation of water temperature, when water is withdrawn from the hot tank at constant rate, from 2/4/7/9/hours after sunrise to sun-

56

I M-o.o

I

48

32 ¸

24

I 7

I 9

I

I II

Time

I of

I 13 doy,

h

I

I 15

I

I f7

I

I 19

"

Fig. 4. Time variation of mean water temperatures for different rates of withdrawal of hot water.

S O D H A AND TIWARI:

NATURAL C I R C U L A T I O N H E A T I N G SYSTEMS

287

,~ • 13.5 ko/hr 52

44

20

9

II

13

Time of day Fig.

i

I 15

i

I 17

i

I 19

,-

5. Effect of time of starts of withdrawal of hot water on the time variation of water temperature.

80

7O

60 A

50

40

30

I0

-Ii

i

I

9

il

13

Time of day

15

17

i

I

19

,~

Fig. 6. Time vairation of rate of withdrawal of hot water in the constant mean water temperature mode.

288

SODHA AND TIWARI:

NATURAL CIRCULATION HEATING SYSTEMS

60

t ¢v 55

E te

g 50

"6 45

.u ILl I I0

40

[ 20

Flow rate of

water

3~1

through

tank

/0

(kg/h)

•

Fig. 7. Dependence of overall efficiency (r/) on the rate of withdrawal of water from the tank.

t

,~5o

56

A w

E @

g

T

55

40

•~ ¢:

30

W 2O

30

I

I

I

40

50

60

54

I

53

0

II

1.2

13

114

1.5

Height of storage tank (m]

Collection temperature (*C1----

16

17

.,

Fig. 8. Dependence of r/ on the mean collection temperature (for constant temperature mode).

Fig. 9. Effect of capacity of storage tank on q for constant rate of withdrawal of water from tank.

r e s p o n d s to lower T,, and hence lower losses. Fig. 8 shows the d e p e n d e n c e of r/ on the m e a n collection t e m p e r a t u r e (for c o n s t a n t t e m p e r a t u r e mode); as expected r/ decreases with increasing T,. c o r r e s p o n d ing to larger losses. The system efficiency increases by a small a m o u n t (say a b o u t 2°,;,) for considerable c h a n g e (60%) in the capacity of storage tank, the increase can be unders t o o d in terms of lower m e a n t e m p e r a t u r e and lower losses (Fig. 9).

movtsos, Finite difference modelling, identification and simulation of a solar water heater, Solar Eneryy 16, 25 (1974). K. S. Ong, An improved computer programe for the thermal performance of a solar water heater, Solar Energy 18, 183 (1976). G. Grossman, A. Shitzer and Y. Zvirin, Heat transfer analysis of a fiat plate solar energy collector, Solar Energy 19, 493 (19771. B. J. Huang, Similarity theory of solar water heater with natural circulation, Solar Eneryy 25, 105 (1980). Domestic water heating by solar energy, Building Research Establishment Digest, Digest 205, 1 (1977). Y. Zvirin, A. Shitzer and G. Grossman, The natural circulation solar heater models with linear and nonlinear temperature distribution, Int. J. Heat Mass Transfer 20, 999 (1977). G. L. Morrison and D. B. J. Ranatunga, Thermasyphon circulation in solar collectors, Solar Energy 24, 191 (1980). A. Shitzer, D. Kalamanoviz, Y. Zvirin and G. Grossman, Experiments with a fiat plate solar water heating system in thermosyphon flow, Solar Energy 22, 17 (1979).

REFERENCES

[1] K. S. Ong, A finite difference method to evaluate the thermal performance of a solar water heater, Solar Energy 16, 137 (1974). [2] D. J. Close, The performance of solar water heaters with natural circulation, Solar Energy 6, 33 (1962). [3] C. L. Gupta and Garg H. P., System design in solar water heaters with natural circulation, Solar Energy 12, 163 (1968). 1-4] S. G. Tzafestas, A. V. Spyridonos and N. G. Kou-

[5] [6] [7] [8] I-9]

[10] 1-11]