storage solar water heaters: Arbitrary demand pattern

Ener#y Con. & Mgmt Vol. 21, pp. 229 to 238, 1981 Printed in Great Britain. All rights reserved

0196-8904/81/040229-10502.00/0 Copyright © 1981 Pergamon Press Ltd

PERFORMANCE OF COLLECTOR/STORAGE SOLAR WATER HEATERS: ARBITRARY DEMAND PATTERN M. S. SODHA, P. K. BANSAL and N. D. KAUSHIK Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi 110 016, India (Received 31 January 1981)

Abstract--This paper presents a simple transient model for predicting the thermal performance of two novel water heaters which combine both collection and storage of solar energy in a single configuration. The proposed model is applicable for demand patterns, characteristic of various domestic and industrial applications. The model takes into account the appropriate heat transfer processes during sunshine and off-sunshine hours and predicts, for a particular case, the time dependence of water temperature which is in close agreement with experimental observations. The model also predicts the variation of water temperature for the withdrawal of hot water at constant flow rate. Furthermore, it determines the time dependence of flow rate corresponding to extraction of hot water at constant temperature; this is a desirable situation in process heating and other applications. Solar water heating

Collector/storage system

Process heating

NOMENCLATURE Cw = Specific heat of water, J/kg °C Mw = Mass of water filled in the tank per unit collector area, kg/m 2 rhw = Mass flow rate of water per unit collector area, kg/s.m 2 S(t) = Solar intensity incident on glass cover at time t, W/m 2 So = Average value of solar intensity S(t), W/m 2 So., = Amplitude of the ruth harmonic of solar intensity, W/m 2 T~(t) = Ambient air temperature, °C Tao = Average value of TA(t), °C TAm = Amplitude of ruth harmonic of Ta(t), °C Tin(t) = Inlet water temperature, °C Ti. 0 = Average value of Ti.(t), °C T~,m = Amplitude of the mth harmonic of T~,(t), °C T~,(t) = Temperature of water at time t, °C t = Time co-ordinate, s UL = Total heat transfer coefficient through the glass cover and bottom insulation, W/m 2 °C (~r)~rt. = Effective absorptance transmittance product of the system

of hot water at constant flow rates continuously or at intermittent intervals. In this communication, we present a simple transient thermal model for predicting the performance of these water heaters for arbitrary d e m a n d patterns of hot water. As an example of the application of this analysis to demands where the whole mass of hot water is withdrawn from the system in the morning hours of the day, we have considered two specific days corresponding to two water heaters for which limited experimental data is available. The observed variations of solar intensity and ambient air temperature are represented by a Fourier series of six harmonics; this enables an analytical solution of the heat balance equation. Subsequently, the transient thermal behaviour of water heaters for d e m a n d patterns involving the withdrawal of hot water at constant flow rate t h r o u g h o u t the day continuously or at intermittent intervals, have also been explicitly investigated.

INTRODUCTION

MATHEMATICAL FORMULATION

A m o n g the various collection/storage solar water heaters proposed so far, the shallow solar p o n d (Fig. l(a)) a n d built in storage solar water heaters (Fig. l(b)) have n u m e r o u s domestic a n d industrial applications. Various analyses [1-5] for the transient thermal performance of these water heaters have been proposed. Sodha et al. have pointed out the inadequacies of earlier analyses and proposed rigorous [6-7] as well as simple [8] transient models. All these analyses are, however, applicable in situations where large m o r n i n g hot water demands exist and the whole mass of water is withdrawn from the system in the m o r n i n g hours of the day. In actual practice, this is not valid; a variety of applications involve withdrawal

The solar radiation, after transmission t h r o u g h the top glass is absorbed by the blackened t o p / b o t t o m surface of the tank and the hot water is withdrawn from the tank by displacement (continuously or intermittently). The heat balance for such collector/storage water heaters is, thus, given by

E.C.M,21/4

A

dT,~(t) MwC~ - dt = ( ~ r ) a f S ( t ) - U L [ T w ( t ) - TA(t)] -- thw(t)Cw[Tw(t) - T~.(t)].

(1)

The only approximation in writing this equation is that there is negligible heat storage in the insulation. 229

230

SODHA, BANSAL AND KAUSHIK: COLLECTOR/STORAGE SOLAR WATER HEATERS

Solar Radiot ion

r

~

Air gap

~

Black surface of tank

~

Water

ig

Inlet

: ~

Bottom surface of tank -- Wooden box Outlet Insulation

Fig. l(a).

Solar Radiation

I I I I

._•_

__------

Inlet

Glass cover

----~

•

------1/]~ -

Wooden box Water

-

, o . o ,o , ,o .

Fig. l(b). Fig. 1. Sketch of (a) built-in storage solar water heater, and (b) shallow solar pond water heater. This equation is solved to obtain the time dependence of water temperature [T~(t)] for the following two cases.

the solution of equation (2) is, thus, given by

Tw(t)=[~(b.So+a.

Case I: Non withdrawal of hot water [rhw(t) = 0]

TAo).(1-e-"')

m=l

The heat balance equation (1) reduces to dTw(t)

- dt

x(b.

+ a. Tw(t) = b. S(t) + a" TA(t),

' )

+ Tw0"e-"' + S o m + a T A m ) ( e i. . . .

(2)

e-at)}],

where a = (UL/MwCw) and b = [(~z)af/MwCw-I.

where o9=2~/24.h-1 and initial water temperature.

Since S(t) and TA(t) are assumed to be represented by a Fourier series of six harmonics as below

For off-sunshine hours. From equation (2), the water temperature during off-sunshine hours may be given as

the

P

T~(t) = / T~(z) e- ~('- ') + { 1 - e- ~('- o TAo

6

S(t) = So +

Two= Tw(t=O),

(4)

~ So,.e i"'~t

(3a)

m=l

L

+

~ (

a

~{ei . . . .

e-.(t-o

and 6

rA(t) = TAo +

Y~ TAm e ~"~', m=,

(3b)

x e ira°'} Tam-], J

(5)

SODHA, BANSAL AND KAUSHIK: COLLECTOR/STORAGE SOLAR WATER HEATERS where z is the time of sunset, a is in accordance with night time values of U L, and Tw(z) is given by equation (4), for time t -- z. Case II: Withdrawal of hot water at constant flow rate

231

where z is the time of sunset; a' and c' are in accordance with night time values of UL; and Tw(z) is given by equation (8) for time t = z. NUMERICAL COMPUTATION AND RESULTS

The heat balance equation (1) may, now, be written as

dT~(t) M~C~ - dt ~- ( ~ ' [ ' ) e f f S ( t ) -

UL[Tw(t) - Ta(t)] (6)

- rhwCwETw(t) - T~.(t)], or

dTw(t) d ~ + a'. Tw(t) = b', S(t)

The variation of solar intensity and ambient air temperature, utilized in this paper are those of the experiments of Sodha et al. [6, 7] and are reproduced in Figs 2(a) and 2(b) for built-in storage solar water heater and shallow solar pond water heater, respectively. We have made computations, assuming the inlet water at ambient air temperature for the following set of parameters, corresponding to those in the experiment and calculations of Sodha et al. [6-8]. Built-in storage solar water heater

+ c'. TA(t) + d'. Tin(t).

(7a)

Assuming the periodic variation of inlet water temperature Tin(t) as below 6

Tin(t) = Ti°o +

~ Ti,m e imp°',

(7b)

(~'~)cff

0.8

M w = 100 kg/m 2 C w = 4190 J/kg °C Two = 34°C fillw = 0.0025 to 0.05 kg/s.m 2

6.2 W/m 2 °C (for day) 1.54 W/m 2 °C (for night) [corresponding to glass wool insulation of 5 cm thickness at the bottom (during sunshine hours) as well as on the top (during off-sunshine hours)]

m=l

the solution of equation [7a] may be given as Tw(t) = [Tw(O)e-"" + l ( b ' . S o + c'.TAo + d"Tino)'(1 -- e-"")

Shallow solar pond water heater 1

+

(~T)cff = 0 . 8

~=l(a,+imog) ( b ' s o m + c ' ' T A , , ,

+ d'. Ti,m)'(e im°'t - e-a't)],

d

(8)

where

M~ = Cw = Two = n~w = UL

a,=(UL+rhwC~,~

\

b,

M--~ /

(~Z)eff

MwCw'

and Tw(0) = Tw(t = 0), the initial water temperature. For off-sunshine hours. From equation (6), the water temperature during off-sunshine hours may be given as

Tw(t) = [ Tw(r) e-"'(t-~ + al ( c " r A °

+ d" Ti,o)

x (1 - e - " ' " . ° )

,.=l

a' -t- m ~

( c " TAm + d " Ti"")

x { e /. . . . . e - , V - , , , ei,.,o,}"~] ' )_1

(9)

100 kg/m 2 4190 J/kg °C 33°C 0.0025 to 0.05 kg/s.m 2 f 26.1 W/m 2 °C (for day) 0.8 W / m 2 °C (for night) [-corresponding to glass wool insulation of 10 cm thickness at the bottom (during sunshine hours) as well as on the top (during off-sunshine hours)].

To have a numerical appreciation of the validity of the proposed simple model for predicting the transient performance of collector/storage solar water heaters, we made calculations, corresponding to the situations where all the hot water is withdrawn from the system in the morning hours of the day, using equations (4) and (5) for sunshine and off-sunshine hours separately (since UL is different for these two periods). The results of this model are compared with earlier models as well as experimental observations in Figs 3(a) and 3(b) for built-in storage water heater and shallow solar pond water heater respectively. The agreement of the results of this model witl~ the experimental results corresponding to UL (day) = 10.2W/m z °C and UL (night) = 2.54W/m 2 °C for built-in storage water heater, and UL (day) -= 30.1 W/m e °C and UL (night) = 2.8 W/m 2 °C for shallow solar pond water heater is fairly good.

232

SODHA, BANSAL AND K A U S H I K :

COLLECTOR/STORAGE SOLAR WATER HEATERS

800

- 80

E 9.. o

600

--60

"6

o.

E ..go

== 400 c

.F.

I

==


20

200

0

~ 5

8

12

16

20

2/*

4

in hours ( t )

Time

Fig. 2(a). 800

6/.0

6o

/*80

f~

o

E

.=

>.

¢:

320 L0

o {/3

C

.~_ 20 <

160

0

A 6

,

I 10

,

I 1/.

,

I

\

,

18 Time in hours ( t )

I 22

,

I

,

2

Fig. 2(b). Fig. 2. Hourly variation of solar intensity and ambient air temperature for (a) built-in storage water heater corresponding to 26th May 1978 at New Delhi, and (b) shallow solar pond water heater corresponding to 28th April 1979 at New Delhi.

I0 6

SODHA, BANSAL AND K A U S H I K :

C O L L E C T O R / S T O R A G E SOLAR W A T E R HEATERS

233

BUILT IN STORAGE 70 O

O O 0

0

0 ~ 0

..

o

60 o (¢ D.

E S0

40

301

I

8

I

12

16

I

20

I

I

24

4

Time in hours (t) Fig. 3(a).

SHALLOW SOLAR POND 70

60

¢.) o

.= =o E

~o ' ~

°

o -"~o

....~.~6.~.b~

5C

O

/

4(;

30

8

I

12

I

16

I

20

I

24

1

4

Time in h o u r s ( t ) Fig. 3(b). Fig. 3. Hourly variation of water temperature for a (a) built-in storage solar water heater, and (b) shallow solar pond water heater. - simple transient model; . . . . . . . . rigorous theoretical results; O O O O O © experimental points; present model; ........ present model considering UL (day) = 10.2 and UL (night) = 2.54 for case (a), and UL (day) = 30.1 and UL (night) = 2.8 for case (b).

8

234

SODHA, BANSAL AND KAUSHIK:

COLLECTOR/STORAGE SOLAR WATER HEATERS

BUILT IN STORAGE

kg/s.m 2

60

o

=

50

E

~0

30[ 8

L

J

I

12

16

20

~ 2/.

4

8

Time in hours (tl Fig. 4(a). SHALLOW SOLAR POND 60

.m 2

50E

g. E

1.0

30

8

/

,~

I

12

I

16

20

2/.,

/,

8

Time in hours [t]

Fig. 4(b). Fig. 4. Hourly variation of water temperature at different constant flow rates of water (0.0025-0.09 kg/s.m 2) for (a) built-in storage solar water heater, and (b) shallow solar pond water heater. These values of heat transfer coefficient (UL) are in better conformity with the wind velocity data for the days of experimental observations. The earlier authors have taken the wind velocity into account, rather arbitrarily.

Subsequent calculations were made for the case of withdrawal of hot water at a constant flow rate using equations (8) and (9) separately for sunshine and offsunshine hours. The results of calculations are exhibited in Figs 4(a) and 4(b) respectively. It is noted that

SODHA, BANSAL AND KAUSHIK: COLLECTOR/STORAGE SOLAR WATER HEATERS 0.051-

235

C O.OS Kg/s.m 2

0.0,~ -

t,a E ~4

_o_~_K~_~.

v

"-" 0.03--

c~

-

0

0.02-

2 o 0.01

0.01 Kg/s.m 2 - -

o oo2s 0.0 A 8

I 12

B

I

o

I 20 Time in hours (t)

16

I

I

F

2/,

t.

8

Fig. 5(a).

BUILT IN STORAGE

60

e.. 01 Kg/s.m 2

50 a.

E

~

0.03 Kg/s.m 2 0.05 Kg/s.m 2

40

3C

I

12

I

16

I

20 Time in hours (t) Fig. 5(b).

I

24

1

4

236

SODHA, BANSAL AND KAUSHIK:

COLLECTOR/STORAGE SOLAR WATER HEATERS

60

SHALLOW SOLAR POND

0.0025 Kg/s.m ~

o

P

50

I

0.01 Kg[s.m2

E

0.03 Kg/s.m 2 40

30

0.05 Kg/s.m 2

8

I

12

I

I

16

20

I

2&

I

4

Time in hours (t) Fig. 5(c). Fig. 5. (a) Demand pattern for withdrawal of hot water, characteristic of application in process industry. Hourly variation of water temperature corresponding to constant flow rates of water (0.0025-0.05 kg/s.m 2) from 14 hr to 18 hr for (b) built-in storage solar water heater, and (c) shallow solar pond water heater.

0.05

E

0.05 Kg/s.m 2

0.04

v

.E _~ 0.03 "6

~ 0.02

0.01

0.01 Kg/s.m 2

0.0025 Kg/s.m 2 0.0Le 8

12

16

20 Time in hours (~) Fig. 6(a).

24

SODHA, BANSAL AND KAUSHIK:

COLLECTOR/STORAGE SOLAR WATER HEATERS

237

70r-

~.n~2 6(]

vo

P E ¢¢

E

50

t.0

301

8

I

12

I

I

16

I

20

2/+

I

4

8

Time in hours (tl Fig, 6(b). 60

P =1 o

50-

0.05

Kg/s.m

/-t

E

4O

30

8

1

12

I

16

I

I

I

20 24 4 8 Time in hours (tl Fig. 6(c). Fig 6 (a). Demand pattern of hot water in Indian hospitals. Hourly variation of water temperature corresponding to constant flow rates of water (0.0025-0.05 kg/s.m 2) from 12 hr to 14 hr for (b) built-in storage solar water heater and (c) shallow solar pond water heater.

the temperature of the available hot water decreases with increase in flow rate. For certain applications, it is desirable to extract hot water at constant temperature. Obviously, for this case, one has to vary the flow rate hour to hour. An estimation of this required

variability of the flow rate may be made by drawing a horizontal line in Figs 4(a) and 4(b). It may be seen that the required variability is less for highter temperatures than for lower ones. In Fig. 5(a), we exhibit the demand pattern charac-

238

SODHA, BANSAL AND KAUSHIK:

COLLECTOR/STORAGE SOLAR WATER HEATERS

teristics of industrial applications where the hot water is withdrawn in the afternoon hours at constant flow rate. The variation of the temperature of water corresponding to this demand pattern may be obtained by the combined use of expressions (4), (5), (8) and (9), keeping in mind that AB is the interval of nonwithdrawal of hot water in sunshine hours, BD is the interval of withdrawal of hot water at constant flow rate during sunshine hours and finally D F is the interval of nonwithdrawal of hot water during off-sunshine hours. The results of the computation of water temperatures for this demand pattern are exhibited in Fig. 5(b) and Fig. 5(c) for built-in storage solar water heater and shallow solar pond water heater, respectively. Another typical demand pattern which is the characteristic of hot water demand in rural hospitals of India, is depicted in Fig. 6(a). Calculations corresponding to this demand pattern were also made in the similar manner and are depicted in Fig. 6(b) and Fig. 6(c) for built-in storage solar water heater and shallow solar pond water heater, respectively. Thus, we conclude that the model presented in this

paper, will serve the needs of the designers of built-in storage solar water heater and shallow solar pond water heater for a variety of domestic and industrial applications. Acknowledgements--The authors are grateful to Dr S. C. Kaushik and Dr J. K. Nayak for their cooperation in the present work.

REFERENCES [1] R. S. Chauhan and V. Kadambi Solar Energy 18, 327 (1976). [2] H. P. Garg Solar Energy 17, 167 (1975). [3] W. H. Gopfforth, R. L. Davidson, W. B. Harris and M. J. Baird, Solar Energy 21, 317 (1968). [4] M. L. Khanna Solar Energy 15, 269 (1973). [5] A. I. Kudish and D. Wolf Solar Energy 21, 317 (1979). [6] M. S. Sodha, J. K. Nayak, S. C. Kaushik, S. P. Sabberwal and M. A. S. Malik Energy Cony. 19, 41 (1979). I-7] M. S. Sodha, J. K. Nayak and S. C. Kaushik Int. J. Energy Res. 4, 323 (1980). I-8] M. S. Sodha, P. K. Bansal and S. C. Kaushik lnt. J. Energy Res. 5, 95 (1981).