Comput. & Indus. Engng Vol. 8, No. 3/4, pp. 181-191, 1984 0360-8352/84 $3.00 + .00 Printed in the U.S.A. C 1984 Pergamon Press Ltd.

A S I M U L A T I O N O F A S O L A R E N E R G Y S Y S T E M F O R H E A T I N G W A T E R

ZILLA SINUANY-STERN, ABRAHAM MEH~Z, DAVID CHI a n d NITZAN COP Department of Industrial Engineering and Management, Ben-Gurion Univesity of the Negev, Beer

Sheva, Israel

(Received in revised f o rm November 1983)

Almtract--This paper presents a simulation model for evaluating the economic viability of a solar system for heating water in an apartmeat building. The system consists of solar collectors and a back-up oil burner. The model determines the optimal number of collectors needed and the optimal method for connecting the collectors---in series or in parallel.

The simulation results show that the solar system for heating water is more economical as the life span of the system increases, as the prices of oil increase, and as the demand for hot water increases. Furthermore, connecting the collectors in series is more economical than connecting them in parallel.

I. I N T R O D U C T I O N

This paper presents a model for evaluating the economic viability of a solar system for heating water in an apartment building. The building in this study is located in the center of Israel; it is a new building and no heating was installed before this study took place. The system includes solar collectors and a back-up oil-burner. Oil burners are commonly used in apartment buildings in Israel, therefore, the cost of an autonomous oil-burner is used as a base for economic comparison here. Furthermore, the model determines the optimal configuration of collectors; namely, the optimal number of collectors needed, and the method for connecting the collectors--in series or in parallel. The optimal configuration of the collectors is an important issue in any research and development of solar energy systems[2, 8].

The authors dealt with another simulation model[8] for determining the optimal configuration of electrical power systems using solar energy. The focus in that model was the performance of a specific power generator, the solar component was of secondary interest, and the solar radiation was taken directly from historical data. This paper focuses on the performance of the solar collectors. The simulated radiation is generated by the model stochastically, and is based on regression equations, estimated from the historical data.

In this case study it was found that the solar system for heating water is more economical as the life span of the system increases, as the price of oil increases, and as the demand for hot water increases. Connecting the collectors in series was found to be more economical than connecting them in parallel. In addition to the determination of the optimal number of collectors, the economic viability range was determined in order to allow flexibility.

2. T H E S Y S T E M D E S C R I P T I O N

The solar system for heating water, in this study, is for a residential building with 32 identical apartments. It includes a storage tank for hot water located in the basement of the building. The volume of the water storage tank is 3000 liters according to the Israeli standard requirements. The flat plate collectors are located on the roof of the building. A circulation pump circulates the water from the storage tank to the collectors. As back-up, an oil burner is operated whenever needed. The collectors in the model varies from 0 to 40. This is an open tpjstem, when the water flowing through the collectors is used by the consumers and is constantly refilled by the network water. When the temperature of the water outflowing the collectors is at least 6°C higher than the temperature of the

181

182 Z. SINqJANY-STERN et al.

water in the storage tank, then the circulation pump operates, forcing a flow between the storage tank and the collectors. The circulation pump stops its work when the temperature of the water outflowing the collectors is not at least 6°C higher than the temperature of the storage water. The control of the circulation pump is achieved via temperature sensors. The back-up oil burner operates whenever the storage tank water temperature drops under 50°C, as required by the Israeli standards.

The problem addressed in this study is to find the most economical configuration of collectors; namely, the optimal number of collectors and their s tructure-- in series or in parallel.

3. T H E M O D E L F O R M U L A T I O N

In order to formulate the model the parameters are first defined:

A, B = the parameters of the collector's efficiency curve Ct = oil price, ($/1.) C2 = electricity price, (S/KWh) C3 = the investment cost of a collector, $ C4 = the investment cost of the oil burner, $ C5 = the investment cost of the circulation pump, $ d, = the demand for water l./hr f = the reduction in the collector efficiency over time

f0 if the collectors are connected in series

i = ~ (1 if the collectors are connected in parallel j = the number of collectors in the system r = the interest rate

RT, = the total radiation, kcal/hr/m 2 S = the surface area of a collector, m 2 t = time index, t = 1 . . . . . T

T = life span of the system I * = /'=

To= TA,

TC( i , j )

TO/

TO, °m

the time needed to raise the storage tank water temperature level to 50°C the actual time the burner operates during the interval At. the system's guarantee period

= the atmospheric (ambient) temperature = the future value of the total cost o f a system with j collectors and collector

connection of type i = the temperature of the network water, °C = the temperature of the water outflowing the collectors, °C = the temperature of the water outflowing the m-th collector in series (i = 0),

m = l , . . . , j , °C TSt = the temperature of the storage tank water, °C

TS, I = the preheated water temperature, °C u = the annual increase of the insurance rates V = the volume of the storage tank, 1.

X~, = the future value of the oil used for a system with j collectors and connection of type i $

2 X#, = the future value of the electricity used for a system with j collectors and connection of type i, $

c~ = the caloric value of oil, kcal/l. /~ = the burner's oil consumption, 1.

= the burner's efficiency 3 = the water flow rate in the collector, l./hr

At = the simulation time increment = the electricity consumption of the circulation pump, K W h

As shown in Section 4.3, if the collectors are connected in series (i = 0) then the

A simulation of a solar energy system for heating water

temperature of the water outflowing the m-th collector is: 183

where

TO, ,. = A . f . R T , + TO,~'-'(6 .S - B.f[2) + B . f . TA,

h.S + B.f/2 m = ! . . . . . j (1)

TO:* = TS,'. (2)

Let us denote

TO,°= TOil. (3)

If the collectors are connected in parallel (i = 1) then the temperature of the water outflowing each collector is:

TO,' = A . f . RT, + TS,~(6 • Sj - B . f [2) + B . f . TAt (4) f . S .j + B .f[2

The temperature of the preheated water as affected by the demand for hot water is:

TS; = TS,_ ,( V - d,) + Tlt.d, (5) V

The oil burner operates if TSt '> TO, i - 6 and TS, '< 50. In order to raise the water temperature to 50°C, the burner must operate during a period,

t*= V(50- TS,') (6)

Actually, during period t the oil burner will operate for a time period:

.At ift*>>.At

~ = { 0 if t * < A t if the oil burner does not operate

(7)

There are five formulas to update the temperature of the water in the storage tank for five different situations:

(ia) if TS,' < TOt ° - 6 then the burner does not operate, / ' = 0, if the collectors are connected in series then the updated temperature is:

[V. TS,' + 6 .At(TO, ° - TS,')](V - d~)/V + dr" TI, TS, = (8)

V

(ib) but if the collectors are connected in parallel then the updated temperature is:

[V. TS, t + t~ . j .d t (TO, ' - TS,~)](V - d,)/V + d," TI, TS, =

V (9)

(ii) if TS, t > TO, z - 6 and TS: >1 50, then the oil burner does not operate, / '= 0, and the updated temperature is:

TS, = TS: (10)

(iiia) if TS,_I > T O / - 6, TS, ~ < 50 and 7= t*, then the burner operates partially, and

184 Z. SINUANY-SI'F~N et al. the updated temperature is:

TS, = 50 (1 1)

(iiib) but i f / -= At then the oil burner operates fully, and the updated temperature is:

V. TS,~ + i-.,.fl.~ TS, = (12)

V

A schematic description of the system is given in Figs. 4 and 5. For economic comparison the future value of the costs was compared for all the

possible configurations. In a system with j collectors ordered in method i, the future value of the cost of oil used by the burner is:

Xb, = Clfli'e '~r-'~, j = 1 . . . . . J (13)

and the future value of the cost of electricity used for the circulation pump is:

X~,= fC2"At.qe'tr-') if TS,~ <TO,I+6 (14)

otherwise

The objective is to minimize the total future value of the system's costs:

Min ,~=, [Xb, + X~,] + (C3"j +(74 + C5)[(1 + r)r + t ~

= To (1 + r)r-I( l + u) I- r01 . (15)

a b

cl

c2

d el, e2

e3-e5

4. THE SIMULATION MODEL

The model simulates the physical system over its life span in hourly increments (At = 1). The simulation procedure can best be understood by examining the simulation flow chart shown in Fig. 1. The step-by-step procedure is summarized as follows:

The hourly demand for hot water is generated as detailed in Section 4.1. The hourly radiation parameters are sampled, and the total radiation is calculated as detailed in Section 4.2. If the collectors are connected in series, then the water temperature increases as it flows from one collector to the next. If the collectors are connected in parallel then the water temperature increases as it flows through one collector. The temperature of the preheated water is calculated. If the temperature of the water outflowing the collectors is at least 6°C higher than the temperature of the storage water, then the water is heated by solar energy (el, e2), and the circulation pump operates by electricity. If the temperature of the water outflowing the collectors is not at least 6°C higher than the temperature of the storage water then there are two possibilities: (e3) if the preheated storage temperature is at least 50°C, then no heating is used; (e4) if the preheated storage temperature is less than 50°C, the oil burner is used

partially; (es) or fully in order to increase the water temperature to 50°C.

f The simulation program accumulates over the system's life span the following quan- tities: 1. the electricity costs for the circulation pump frO, and

A simulation of a solar energy system for heating water

(lp:t|,ra te " ' ~

~ollector$ ordercd in series

starting ~ith t collector

I~j-O X I j - O initiatinl operating coats

# l hot water demand

Q dt a # ,, i|

~ xxple radiation paraseters [b calculate total radiation rrt[

#

:::::::'; I Temperature of water outfl~ing the collectors ~ collectors in parallel: L

calculate TO1 ,, F

Tank t elperature compared / I-~I .Toi a~¥es --| Solar eeerEy uso~ to collectors ~ 7 -~ calculate Tst 1

¥ ,~ ! ~lcal¢olete ,~e -.,er I I ~ t ~ preheated temperature [J Calculat~ the ij I / 131 I electr ic costs XP~..t

t S a accumulate f l

[hesting is neededL~ Yesj~- . . . . . . ~ zj zj Li t

burner is fully used

t=6t

Calculate '~t part Jelly t=te

Calculate burner's ois cost x~ l t

accumulate oi l costs ! ! t

Xij-XJJ ÷ X~j

updat e number of collectors j-j*l

test parallel collectors i - I

® ,es

Calculate~ive ijNin ~t°tal(ij)c°sts TC(ij) I f3

4>

185

Fig, 1. The simulation flow chart.

186 Z. SINUANY-STERN et al.

2. the oil costs for the back-up burner (1"2). At the end of the simulation runs the total future value of the system's costs is calculated for various configurations of the collectors, and the optimal configuration is found by simple enumeration (f3).

The demand for hot water is discussed in Section 4.1; the radiation factors and their estimation are given in Section 4.2; an analysis of the collector performance is given in Section 4.3; and a listing of the values of the simulation parameters is given in Table 3.

4.1 Demand for hot water The demand is assumed to vary hourly and to be deterministic; and the data for each

apartment is given in Table 1. Sensitivity analysis was performed for three possible levels of consumption: high, medium and low. There are 32 identical apartments in the building. Since the building was not occupied at the time of the study, the demand probability distribution was not available. Therefore, the demand was considered deterministic; however, the three levels of consumption compensate somewhat for this lack.

4.2 The radiation factors In the real world, the radiation is stochastic in nature. Deif[2] has used a stochastic

approach for simulating a solar powered cooling system. He provides a review of methods for deriving weather input to simulation, either by using existing weather data bases or by using a statistical approach. In this paper a combination of the two approaches is used.

The total radiation is calculated as follows:

R T = R . R D + RI

where R T is the total radiation (kcal/hr/m2), R is the conversion factor of the direct radiation (0 ~< R ~< 1), RD is the direct radiation (kcal/hr/m 2) and R/ is the indirect radiation (spread by the clouds) (kcal/hr/m2).

After testing the radiation data collected in Israel in recent years[9] it was found that the radiation distribution varies during the day. After processing the radiation data by using a logarithmic transformation[l], the function describing the accumulative relative frequency of the radiation was estimated by fitting a least squared straight line to logarithmic data. The result is as follows:

In (P) = - a , + b,-RD

where P is the accumulative probability of the radiation (0 ~< P ~< 1), a, is the intercept of the line and b, is the slope of the line.

In the simulation, P is sampled randomly between zero to one, and the inverse

Table 1. The hourly demand per apartment (in liters)

hours

0-6

7-9

21-:; l

! o-i !

high ! medium

i } 0.5

8 b

6 3

10 ~ 10

I { 0 .3

124 f 9S

0.5

4

3

8

0.5

A simulation of a solar energy system for heating water 187

c ~p

~E

X {°C/Kcol lhr Irn z ]

Fig. 2. The efficiency curve.

transformation is used to produce an appropriate sample of the random variable RD:

In (P) + a, R D =

bl

The parameters a~ and b~ were calculated for each hour in two typical days in the year, a day in November and a day in May. For example, the direct radiation at noon in November is described by the function:

RD = (In P + 2.874)/0.039.

Similarly, the indirect radiation was expressed:

1 - e - e = a 2 + b 2 . R l R I = 1 - a 2 - e -e b2

The coefficient R is affected by the angle of the collector with respect to the sun hourly and daily:

where

R = sin ~ [cos (p - E) cos to + sin (p - ~)]

= 23.45-sin [360. (284 + 1)/365]

to = 1 5 ( 1 2 - k )

where E is the slope of the collector with respect to the horizon, p is the north latitude, to is the angle of the hour, l is the day in the year and k is the hour in the day.

4.3 The collector performance The efficiency curve (Fig. 2) describes the collector's efficiency E as a function of

temperature of the following parameters:

1. TE is the temperature of the water entering the collector (°C). 2. TO is the temperature of the water leaving the collector (°C). 3. TA is the atmospheric temperature (°C). 4. R T is the total radiation affecting the collectors (kcal/m2/hr).

The efficiency curve, E, is linear in X.

where

E = A - B ' X

188 Z. S n , ~ u A ~ ' - S ~ et aL

0 212

T1 124

I I j 18 7O

TI { *C )

Fig. 3. The collectors performance for RT = 700 kcal /m 2.

Two thermal equations[3] were used:

E = S .6[TO - TE]

RT

where S is the collector area (m2), 6 is the water flow rate in the collector (1./hr), f is the reduction in the efficiency of the collector as a result of detexioration over I yea r s , f = F ~- ~, F is the annual reduction of the collectors' efficiency, A is the intercept of the efficiency curve and B is the slope of the efficiency curve.

Solving TO from the above equations yields:

T O = A . f . R T + TE(6 . S - B . f /2) + B . f . TA

a. S + B . f /2

Figure 3 and Table 2 illustrate the relationship between TO and TE for radiation of 700 kcal/hx/m 2. It is evident that as the entering temperature increases, the collector's efficiency decreases.

5. NUMERICAL RESULTS

Two major configurations were tested with two different life spans: collectors ordered in series vs collectors ordered in parallel. The life spans considered are 10 and 13 years. The simulation was run for varying numbers of collectors: from zero to 40 collectors in increments of two. The zero number of collectors refers to an autonomous oil burner which is used as the base for comparison.

The system's parameters are given in Table 3. It was assumed that the oil prices will increase 5% every year. The system has an 8 year guarantee. Beyond the guarantee period the system can be insured for a fee equivalent to 2% of the system's worth per year. This cost was considered in the model.

A summary of the results is given in Table 4, whore the simulation output for each configuration is as follows: the optimal number of collectors, the investment costs, the total

Table 2. The collector performance for RT = 700 kc.~l/hx/m 2

TO/TI TO TI

2.12 38.26 18

1.65 49.61 30

1.47 59.07 40

1.37 68.55 50

1.30 77.99 60

I . 24 87.46 70

A ~ m u l a f i o n o f a s o l a r e n e r g y sys t em for he a t i ng w a t e r

Table 3. S~muhtf ion parameters

Input Parameters Value

V vo lume of the water tank

S s u r f a c e a rea o f • c o l l e c t o r

T l i f e - s p a n o f the system

A - i n t e r c e p t o f t h e c o l l e c t o r ' s e f f i c i e n c y curve

8 - s lope o f the c o l l e c t o r ' s e f f i c i e n c y curve

F - annual d e t e r i o r a t i o n o f the c o l l e c t o r ' s e f f i c i e n c y

3000 l i t e r

2 2.16 m

10 or 13 y r s .

0 .73

4.0816

0 .98

TI t - tempera ture o f the network water du r ing : Winter

Summer

a - c a l o r i c v a l u e o f o i l

- requirement o f o i l to the burner

y - b u r n e r ' s e f f i c i e n c y

6 supply o f water to the c o l l e c t o r s

n supply o f e l e c t r i c i t y t o the pump

r i n t e r e s t r a t e

u - annual i n c r e a s e o f i n s u r a n c e r a t e s

T O length o f guaran tee p e r i o d s

10°C

18"C

8640 g c a l / l i t e r

11.4 l i t e r / h r

0 .85

73.4 l i t e r / h r

0 .33 (KWh)

0.025

0 . 0 2

8 y e a r s

189

costs, the payback period, the amount of oil consumed, the percentage of oil saved, and the future value of the money saved. Actually the future value criterion is equivalent to the present value criterion for purposes of comparison.

From Table 4, it is evident that a system with collectors connected in series is more efficient. This result can be explained by the fact that when the water runs through the collectors in series each collector raises the temperature of the water beyond its prede- cessor. Therefore, there is a chance that the collectors will raise the water temperature by more than 6°C. But when the collectors are ordered in parallel, the water passes only

Tab le 4, S u m m a r y o f the s imu la t ion resul ts

13 YIL5 l i f e - s p a n

s e r i e s ) a r a l l e l

number o f c o l l e c t o r s 20 22

investment c o s t s ($) 32,000 35,200

iTotal costs ($) 93,000 100,000

Pay-back pe r iod 10 12

o i l consumed ( l i t e r s ) 2176 2321.4

o i l saved (~) 61 58

sav ings ' f u t u r e value ($) 12,800 6,300

10 YRS l i f e - s p a n

s e r i e s ) a r a l l e l

12 10

19,200 16,000

70,560 71,900

10 10

3220.4 3732.0

42 32

2 , 1 8 0 2500

190 Z. StNUANY-STERN e! al.

COLLECTORS IN SERIES

[SUm.y~ DB~TS, . T0~

Fig. 4. Schematic description of a system with collectors in series.

through one collector. The operation of the circulation pump validates this reasoning, as the pump operates only 22.4% of the time when the collectors arc ordered in parallel, vs 31.3% of the time when the collectors are in series.

Furthermore, as the life span of the system increases, the series system becomes more economical. The two economic criteria used here, which validate these results, are the payback period and the future value of the savings.

6. SENSITIVITY ANALYSIS Sensitivity analyses arc given for two parameters: (a) varying the demand from medium

level to low and high levels (Table 1); (b) increasing oil prices by 33%. It was found that thcse parameters are dominant relative to other parameters.

From Table 5 it is evident that as the demand for hot water increases the solar system

Table 5. Sensitivity analysis results

~ em configuration

)utput paramet0r~~

humber of collectors

col lectors' economic range

pay back per iod

D11 consumed ( l i t e r s )

o i l saved (%)

s a v i n g s ' f u t u r e v a l u e ($)

lumber of collectors

collectors' economic range

pay back period

o i l consumed ( l i t e r s )

o i l saved (%)

s av~ngs ' f u t u r e v a l u e ($)

~ b e r o f collectors

collectors' economic range

pay back period

o l l consumed ( l i t e r s )

o l l sa~ed (%)

s a v i n g s ' f u t u r e v a l u e ($)

number of collectors

collectors' economic range

pay back period

oil consumed (liters)

o i l saved (%)

s a v i n g s ' f u t u r e v a l u e ($)

Life

Span

10

YRS

13

YRS

Connectiol o i l p r i c e

Method i n c r e a s e

P I0

A 2-32 R A 7

L 3732.3 L

32 E L 8 ,100

S 12

E 4-36

R 7

I 3220.4

£ 42

S 12,275

P 22

A 2-40 R 8 A L 3221.4

L 58 E 26,625 L

S 2O

E 4-40

R 7

I 2176.3

E 61

S 34,135

High Medium Low Water Water Water DeaandiDemand Demand

I0 i0 I0

6-18 6-10 6-10

10 10 10

5013.3 3732.6 2716.8

27 32 40

1,500 225 275

12 12 12

8-24 4-24 4-16

9 I0 i0

4536.3: 2798.a 2209.6

55 45 51

4,120 2,180 2,150

22 22 22

4-36 4-2b 4-28

10 12 12

3502.1 2521.4 1523.5

50 58 71

11,575 6,215 6000

20 20 20

4-40 4-36 4-32

9 I0 i0

3552.2 2176.3 1195.

52 61 75

18,265 12,785 12,1b

A simulation of a solar energy system for heating water COLLECTORS IN PARALLEL

Fig. 5. Sch~at ie description of a system with collectors in parallel.

191

becomes more economical in all the possible configurations. The sensitivity of the solution to the oil price was tested for a medium demand level. For a higher oil price the system is more economical, for each one of the four configurations.

In addition to the optimal number of collectors, Table 5 provides the range of collectors for which the solar system is more economical than an autonomous oil burner. As a result, for example, the consumer can decide to install a larger number of collectors if he anticipates an increase in the price of oil. The cost of electricity for the pump use is negligible compared to the cost of oil for the burner.

7. C O N C L U S I O N

From the consumer's point of view, it was found in this study that combining a solar system with a back-up oil burner is more economical than an autonomous oil burner, which was commonly used in apartment buildings in Israel in the past. This result is valid for various configurations of the solar system. Furthermore, the optimal configuration of collectors was found. Finally, the sensitivity of the results was tested for changes in the system's life span, changes in the demand for hot water, and changes in the prices of oil.

From the national point of view, in order to save oil, the government should encourage consumers, either by advertisements or by subsidies, to install solar systems in larger numbers within the economical range. Since oil is mainly imported by Israel and solar collectors are manufactured in Israel, the above policy will reduce the dependence on imports on the one hand, and will encourage the local industry of sun collector manufacture, on the other.

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Jcrusalcn~, 1967 (In Hebrew). [2] I. N. Deif, D. K. Anand & R. W. Allen, Solar powered cooling performance using a stochastic approach.

College Park, Maryland, Sept. 1978. [3] J. Dul~e & W. Beckman, Solar Energy Thermal Processes. Wiley, New York (1974). [4] P. B. Howells & R. H. Marshall, An improved computer code for the simulation of solar heating systems.

Solar Energy 30(2), 89-108 (1983). [5] A. Johnson & G. Grosmum, Performance simulation of regenerating type solar collectors, Solar Energy

30{2), 87-92 (1983). [6] T. M. Khalil; Comparative analysis of energy resources, J. Prod. Res. 19(4), 401--409 (1981). [7] R. R. Levary & B. V. Dean, A natural gas flow model under uncertainty in demand. Ops Res. 28(6),

1360-1374 (1980). [8] A. Mehrez & Z. Sinuany-Stern, Simulating an integrated energy system. Simulation 131-139 (1982). [9] A. Mens, A. Titclman & y. Filling, Sun radiation and radiation balance. The Israeli Weather Service Center,

Beit-Dagan, 1970 (In Hebrew). [10] A. L. Walton, Variations in the substitutability of energy and non-energy inputs: the case of the middle

Atlantic region. J. Reg/ona/Sci. 21, 411-420 (1981). [1 I] T. A. Williams, Comparative economics of small solar thermal electric power systems. Pacific Northwest

Laboratory, Riehland, Washington. [12] J. Zahavi, Planning new electric generating capacity--the short term probability. Ops. Res. 31(5), 367-387

(1980).