Calculating the solar contribution to solar assisted systems

Solar Energy VoL 28, No. 5, pp. 377-383, 1982 Printed in Great Britain.

0038-092X182/050377-07503.00/0 © 1982 Pergamon Press Ltd.

CALCULATING THE SOLAR CONTRIBUTION TO SOLAR ASSISTED SYSTEMS J. P. GEROFI, E. MANNIK and G. G. FENTON Solar Desalination Group, Department of Chemical Engineering, University of Sydney, Australia, 2006 (Received 24 March 1981; revision accepted 5 October 1981)

Abstract--After summarizing the methods for calculating the solar contribution for systems without thermal storage, this paper extends a previously proposed method which is based on using a frequency distribution of insolation data. This extension allows rapid hand calculation of solar contribution for most collector types and for any specified collector inlet and outlet temperatures. Typical results are shown to be accurate to within 1 per cent relative to dynamic computer simulation methods. The effect on the method of collector orientation and tilt is discussed, and a simple method of determining the maximum possible (i.e. infinite collector area) solar contribution for a given collector system is described. INTRODUCTION Hand calculation of the performance of solar collectors has traditionally been done using the utilizability concept[l,2]. The definitive work of Liu and Jordan[2] describes the principles of the method. Developed before detailed radiation data were recorded, the method relies on highly aggregated radiation information and certain assumptions about the nature of climates and the performance curves of collectors. Hourly radiation data is now becoming available for more and more locations and digital computer simulations such as TRNSYS[4] can be used to predict accurately the average performance of any type of collector. Their implementation requires a digital computer and large amounts of computer time. In this paper we examine a hand calculation method which uses local data and does not require any major assumptions about the climate or the form of the collector curve. It agrees with results from TRNSYS to -+1 per cent, and is applicable to all systems without thermal storage. A means is included of calculating the contribution even if excess energy is rejected by the system at times of high insolation. While such calculations can also be done with the utilizability method, they have not been described in the major papers on the subject[l, 2].

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0.7

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0.5

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0.3

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4

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2. METHODSOF CALCULATINGCOLLECTORPERFORMANCE Insolation data was first presented as a frequency distribution by Liu and Jordan, in their analysis of the inter-relationship of direct and diffuse radiation [5]. They plotted normalized daily radiation against the fraction of time that the radiation was less than or equal to the value concerned (see Fig. 1). A "clearness index" K,--which is related to the long term average insolation for the site concerned--is their only site dependent variable. In a later paper, Liu and Jordan quoted results from [1] which showed that for a particular location and a vertical south facing surface, the frequency distribution of hourly radiation was not very different from that of the daily radiation. They integrated the frequency distribution curves from a cut-off insolation (corresponding to the

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i

fi

0.2

4

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0.I

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i

_o "

i

O0

0.2

ii 0.4

0.6

0.8

LO

FRACTIONAL TIME, f , DURING WHICH DAILY TOTAL RADIATION ~ H

Fig. 1. Generalized insolation curves, from 13]. collector no-flow limits) up to the maximum available insolation, to generate a set of utilizability curves from which the performance of flat plate collectors could be calculated. Klein[3] introduced some simplifications to Liu and Jordan's computation method. Recently the method has been further refined and extended to cover other types of 377

J. P. GEROFIet aL

378

collector by Collares-Pereira and Rabl. This last refinement brings the accuracy to within 5 per cent [6]. For any given location and any given month, only three items of site specific data are required--the average insolation, K,, and the mean temperature. Reliance on such highly aggregated data is possible because it is assumed that a universal statistical distribution of insolation is valid. These methods also require the collector efficiency curve to be expressed in the form

n=no-L r°7- T~

(l)

Expressing the curve in this form implies that collector losses are independent of insolation, because only a monthly mean ambient temperature is used. Many authors now prefer to use the average collector temperature instead of the inlet temperature, while more accurate collector models show different curves (not straight lines) for different values of I [7] (see Fig. 2). The accuracy of computer simulation is well established and storage can be included, however, the cost of implementing the simulation is often high. The f chart and g - f chart methods [8, 9] are based on extensive simulation studies. They are relatively simple to use and enable calculation of the solar contribution to systems with various amounts of thermal storage. Lunde[10] used the frequency distribution concept more directly. Using measured hourly data, he produced a table showing the distribution of insolation and associated a mean temperature with insolation class. The table is equivalent to an insolation histogram. Collector performance curves were assumed linear in (AT/I). From the histogram, by solving the collector performance equation for zero efficiency, he was able to determine the proportion of the total incident radiation for which useful heat was generated by the collector. For the utilizable proportion of the insolation, Lunde found the mean class temperature and the mean class insolation. He showed that, because of the linearity of the collector performance equation and the independence of losses and radiation, the average performance of the collector is correctly given by using ordinary time averages of collector temperature, ambient temperature 1"0

l

I

t

I

and incident radiation. These time averages are taken over the period of operation of the collector. Lunde's method thus provided a very simple method of calculating the output of the solar collectors, provided that radiation data in his "specially preprocessed" form was available. Lunde's method has three drawbacks. These are: (i) It has no provision for the effects of thermal storage. (ii) The form of the collector performance equation is only approximate, as shown in Fig. 2, which gives some actual collector curves for flat plate, evacuated tubular and parabolic trough tracking collectors [7]. The linear performance assumption becomes rapidly worse as AT increases [11]. (iii) It does not allow for the collectors' producing more energy than the process being supplied can use. When this happens, the excess energy must in some way be dissipated. We have extended Lunde's method to overcome the latter two restrictions. This has involved using the mean ambient temperature associated with each insolation class rather than an overall average. This extension makes the method especially attractive for an important class of solar energy projects as described below. 3. SOLAR-ASSISTEDSYSTEMS

A solar-assisted system is defined as having storage and two alternative sources of heat: a conventional boiler and solar collectors. The most probable applications of such systems are in industrial process plants, designed to operate at constant output and energy demand, with the boiler providing all the energy requirements when there is no insolation, and making up for any short-fall in the insolation during the day. Hence the investment in the solar collector system can be directly compared with the fuel savings resultant from the solar energy contribution. Desalination is an obvious use for solar-assisted systems, and several proposals are current. Fenton and Gerofi[12] analysed the cost effectiveness of two such cases which had been proposed elsewhere, while more general design considerations were discussed by Gerofi and Fenton[13]. The system consists of a bank of solar collectors, a I

"

Flat

_ ~

Evacuated ~

P

I

I

I

Plate Selective Surface Tubular

Paraboloidal

Trough

0"5

0.2 AT/I

O'4

(Km2W -1 )

Fig. 2. Collectorperformance curves.

Calculating the solar contribution to solar assisted systems boiler and a desalination plant. A heat transfer fluid, in a thermal loop, flows through the collectors and the boiler and then passes through a heat exchanger where it gives up energy to the brine in the desalination plant (see Fig. 3). These systems operate with constant collector inlet temperature, although the calculation method can cope with some variations in that quantity. Inability to utilize or store energy in excess of their (constant) demand is a particular feature of such designs, and some means of dissipating excess heat must be provided. An analogous situation exists with passively heated buildings [14]. The calculation procedure can be adapted to other systems without storage.

379

which each range of insolation values occurs and the average ambient temperature associated with each class can then be computed. Table 1 is a typical table of insolations as required for calculation of collector performance, based on 1975 insolation data for Sydney converted to a collector tilt of 340 (the latitude angle), for a non-tracking collector. Figure 4 shows the same data graphically as a histogram. Table 1 can also be generated for tracking collectors in which case the insolation Ic would represent beam rather than total irradiance. Step 1 represents the computer-based aggregation of the insolation data for a particular orientation and tilt at the location concerned. The resulting table can then be used to determine--by hand calculation--the solar contributions for a variety of solar collectors and operating temperature ranges. Step 2 has two aims: firstly to determine the minimum insolation Imi, below which no useful output results; secondly to provide the basis for finding in Step 3 the insolation value (for any given collector area) above which energy must be dissipated. This is most conveniently done in table form (see Appendix 1), starting at the lowest insolation range, and using at first an average collector temperature (Ts) equal to T~. For each class c, ATdI~ is calculated, and the collector characteristic used to find "0Jc. I~,in can be found by linear interpolation within the first insolation band resulting in a positive value of "0tic. For higher insolation classes, a value of Ts equal to Tm should be used, to accomplish the second aim. In Step 3 a specific value of collector area A is a s s u m e d . Irain is already known, and the limiting upper insolation I, corresponding to A, above which energy must be dissipated, can be determined by linear interpolation from the table of Step 2. Below Irai, there is zero output; above I, the plant's total requirement will be provided: it remains to determine the collector output between /rain and I,. A table similar to that in Step 2 is constructed, but includes columns for frequency/c and total energy available Ec, where

4. CALCULATIONPROCEDURE

There are three main steps in the calculation procedure. These are: (1) Setting up a table of insolations for the location concerned corresponding to the required collector orientation and tilt. (2) For a given problem (i.e. with heat demand, fluid inlet, T~, plant top temperature and type of collector specified) calculating for each insolation class the collector area that would be needed to supply all the required demand and the insolation below which no output is available. For a given collector area, one can determine the limiting insolation levels corresponding to each of three insolation regimes: (a) zero output, i.e. the collector stagnation (no-flow) temperature does not reach ti; (b) full output, i.e. all the energy put out by the collectors is utilized by the plant; (c) excess output, i.e. the collectors are providing more energy than the plant can use: the excess energy must be dissipated somehow. (3) For a given collector area, calculating the average energy available for each insolation class. In Step 1, the table is generated from hourly recordings of insolation on the horizontal plane and ambient temperature. Insolation data for the particular orientation and tilt required is calculated by first separating the beam and diffuse components of the radiation, transforming each to the required plane and adding the results; several accepted methods for doing this are included in simulation programs such as TRNSYS[4]. The frequency with

Ec =

f~I~.

In this calculation Ts will vary between T~ and Tin, and must be estimated for each insolation class. The estimate Plant

Collectors I Flash I Stage ~lOptional

~ ( ~

(2)

Purnp i_

i

Fig. 3. Solar-assisted desalination--series arrangement.

Feedwater

380

J.P. GEROFIet al.

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(

):Average

Ambient

Temperature (*C)

0"10 U ¢D (3" 18'7)

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0

I

I

IIIll 0

200

600

400

I

800

Insolation

II(23"7~ 1000

1200

(W/m2)

Fig. 4. Typical insolation frequency histogram. Table 1. Frequency and mean temperature by insolation band Insolation

Frequency*

Av. Temp* oC

Temp. S.D.

0- 100 100- 200 200- 300

.1453 .0607 .0438

17.6 18.7 19.0

4.2 4.1 4.1

300- 400 400- 500 500- 600

.0418 .0417 .0371

19.0 19.6 20.2

4.3 4.2 4.2

600- 700 700- 800

.0352 .0377

19.7 20.5

4.1 4.2

800- 900 900-1000

.0373 .0316

20.8 23.2

4.2 4.8

1000-1100

.0175

23.7

3.7

1100-1200 1200-1300

.0010 .0003

20.7 18.5

3.1 3.4

5. MAXIMUMPOSSIBLESOLARCONTRIBUTION In a solar assisted system, the total solar contribution is not proportional to the area of collector available, but varies as shown in Fig. 5. As the collector area is increased from zero there is a region of proportionality which corresponds to those areas for which IA is never greater than P/~ (the excess output regime is never entered). Once this limit is exceeded there are diminishing returns from increases in collector area, and the curve is asymptotic to some value usually much less than 50 per cent (which is the absolute upper limit). It is often interesting to know the value of this asymptote for different collector types. Table 1 allows this to be calculated very rapidly. One simply adds all the frequencies of classes not in regime 1 and expresses the result as a percentage.

can be considered correct when T s - T, _ n c l c T,, - T, - VIA'

below about 25 per cent of nominal output. When the sun is supplying almost all the plant's energy the low boiler efficiency needs to be included in the calculation of actual fuel savings.

(3)

Usually only one iteration is required. Appendix 2 gives an example calculation of Steps 2 and 3. The above procedure would be repeated for a number of different values of A, to obtain a plot of solar contribution vs collector area (see Fig. 5). The effect of auxiliary heat source efficiency characteristics may be a further complication which can be incorporated in this method of analysis [13]. In particular, most fuel oil boilers suffer a steep drop in efficiency

6. SYSTEMSWITHSTORAGE Application of the method as described for a given collector area yields the contribution with zero storage. If for the same area one-treats all insolation in regime 3 as if it were in regime 2, one can find the solar contribution for a very large storage volume. 7. ACCURACYOF THEMETHOD Like all methods which allow hand calculations, a considerable amount of data reduction and aggregation is

Calculatingthe solar contributionto solar assisted systems

381

25

2O

o> 15 LL

t_ 5

I 500

O O

I t000

I

1 1500

Collector

2000

250O

A r e o (rn 2)

Fig. 5. Solar contributionvs collector area (Ref, [12]). summers, the use of a single temperature to describe an insolation band for the whole year may be invalid. Places where this may be a problem can be identified by looking at the temperature standard deviations. If necessary seasonal tables can be produced and an annual result obtained by averaging. Note that "accuracy" is used here in connection with comparing results from a TRNSYS calculation for a particular period with those from a calculation based on the histogram approach using data from the same period. Because statistical fluctuations affect the weather from year to year, even the best long term averages cannot accurately predict the system performance in any particular year.

involved in producing the histogram of insolations. However, unlike other methods, the present procedure does not involve any assumptions about the weather pattern at the location concerned (beyond those made by a simulation using hourly data), other than that an average temperature can be associated with each insolation class. The accuracy can be improved if necessary by narrowing the insolation bands. The method outlined has been compared with the results of TRNSYS simulations for four locations with latitudes between 12 and 35°. Results cover a range of collector inlet temperatures from 30°C to 70°C, and are for flat plate (black paint), flat plate (selective surface), and evacuated tubular collectors. With a class interval of 100 W/m 2 on the histogram of insolation, the method presented here agrees with TRNSYS results to within 1 per cent. Table 2 gives some typical results. If the class interval is increased to 200 W/m2 one can Still get results to better than 5 per cent accuracy. In locations with very cold clear winters and hot

8. COLLECTORonmcrA~oN ANDTILT In designing a solar-assisted system, one would normally face the collectors either North or South and tilt them at the latitude angle. However, such systems are often retro-fitted to existing facilities and other orientations and tilts may be physically more convenient.

Table 2. Accuracy of method--typical results Location

Latitude

Alice Springs

Darwin

Sydney

*A

=

B = C =

Time period

24Os

1979

12°S

3~°S

Collector Type *

1979

1975 January 1975 July 1975

f l a t plate, black paint; f l a t plate selective surface; concentrating collector

**

as computed by the method

***

worst case examined.

B B B

Orientation

InlEt Temp (~C)

240

North

60

540

200 W of N

60

97.5

140

400 W of N

40

144.1

.01

440

200 E of N

40

131.8

.i

120

North

Tilt

Av. Rate of Energy Collection*" W/m2 121.5

Error (%1 .2 .I

60

120.3

.2

( f u l l y tracking)

85

102.0

.I

34o

60

86.0

60

124.8

60

73.0

North

.5 1.2"** ,2

J. P. GEROFIet aL

382

A simple program can be used to produce a table of insolations for any desired orientation and tilt. However, an alternative procedure is to produce a grid of tables for a range of orientations and tilts, and interpolate between them. We have produced frequency versus insolation tables for both tilts varying by intervals of 10°, and orientation varying by 200, and have found that for other angles linear interpolation between these tables produces answers to within 2 per cent accuracy.

9. CONCLUSIONS The concept of analyzing the insolation by treating it as a frequency distribution can be used to calculate the performance of solar energy systems without storage. This presentation of insolation data has been used in the past, but its full potential has never been exploited. It is particularly useful for calculations of solar contribution to solar-assisted industrial plants and has the following advantages: (1) The method of data aggregation does not involve any major assumptions about the climate at the location concerned. (2) It allows reasonably fast and accurate hand calculation of the energy collected by any collector whose efficiency characteristics can be adequately described as any function of AT and I only. Linearity is not required. (3) Collector efficiency may be expressed in terms of collector inlet temperature or a mean collector temperature. (4) It copes easily with excess energy produced when the output of the collectors is greater than the plant's needs. (5) It immediately identifies the maximum possible solar contribution. The calculation is conceptually simple and is based on the reduction of real data. Provided the histograms or tables of data used are based on many years' data the results of this method provide a more accurate long term average than previous methods.

Acknowledgement--The Solar Desalination Group is supported by a grant from Saudi Arabian interests administered through the Science Foundation for Physics, University of Sydney. This financial assistance is gratefully acknowledged.

A E Ec fc I Ic train

I~

K, P T~ Tc T! T~

NOMENCLATURE collector area, m2 total energy available from collector, W/m2 energy available from collector for insolation class c, W/m2 fraction of 24 hr day for which insolation is in class c incident insolation, W/m2 average incident insolation for insolation class c, W/m2 minimum insolation required to generate useful output from the collector insolation above which excess energy is generated by the bank of collectors clearness index system energy requirement W ambient temperature, C° average ambient temperature for insolation class c, C° mean fluid temperature, CO(% + To)12 temperature of fluid inlet to the collector, C°

Tm the value of Tf when the collectors are supplying all the plant's requirement To temperature of fluid outlet from the collector, C° AT Tt-Ta r; collector efficiency "Oc collector efficiency for insolation class c REFERENCES 1, H. C. Hottel and A. Whillier, Evaluation of flat plate collector performance. Trans. Conf. on the Use of Solar Energy: The Scientific Basis II. Part l, Section A, 74-104 (1955). 2. B. Y. H. Liu and R. C. Jordan, A rational procedure for predicting the long term average performance of flat plate solar energy collectors. Solar Energy 7, 53-74 (1963). 3. S. A. Klein, Calculation of flat plate collector utilizability. Solar Energy 21,393--402 (1978). 4. S. A. Klein et al. A method of simulation of solar processes and its application. Solar Energy 17, 29-37 (1975). 5. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). 6. M. Collares-Pereira and A. Rabl, Simple procedure for predicting long term average performance on nonconcentrating and of concentrating solar collectors. Solar Energy 23, 235253 (1979). 7. D. Proctor, Performance characteristics of high temperature solar collectors. A Place .for the Sun. ISES, Australian and New Zealand Branch, (1980). 8. S. A. Klein, W. A. Beckman and J. A. Duffle, A design procedure for solar heating systems. Solar Energy 18, 113127 (1976). 9. S. A. Klein and W. A. Beckman, A general design method for closed loop solar energy systems. Solar Energy 22, 269-282 (1979). 10. P. J. Lunde, Prediction of average collector efficiency from climatic data. Solar Energy 19, 685-689 (1977). 11. J. M. Gordon, D. Govaer and Y. Zarmi, Temperaturedependent collector properties from stagnation measurements. Solar Energy 25, 465-466 (1980). 12. G. G. Fenton and J. P. Gerofi, Critical analysis of a proposal solar assisted HTE desalination system. Desalination 33, 211-219 (1980). 13. J. P. Gerofi and G. G. Fenton, Solar distillation--the solar assisted case. Desalination 36, 189-204 (1981). 14. W. A. Beckman, private communication (1980). APPENDIX l

Example calculation Consider a solar assisted plant requiring 266 kW at 80°C. The plant heat exchanger rejects the heating water at 65°C. Beasley flat plate collectors (figure 2) will be used in series with the boiler. We wish to find the energy available to the plant from 590 m2 of collectors. Step 1. Location is Sydney, and insolation data in the form required is available from Table 1. Step 2. Considering each insolation class in turn, Table A1 is constructed. We start with Tl equal to T~ (65°C). For insolation up to 300 W/m2, calculated collector efficiency is zero. Iminis in the range 300-350 W/m2. For higher insolation classes, a Tf equal to T,, (here 72.5°C) is used. 2 For 590 m collector area, an output of 451 W/m2 is needed to fully supply the plant. This corresponds (from Table AI) to the centre of the 800-900 W/m2 band, i.e. It equals 850 W/m2 so half this band will be treated as being in the "proportional" regime, and half in the "excess power" regime. Step 3. Table A2 is now constructed over the insolation range covering Iminand It (300-900 W/m2). Columns 1, 2 and 7 are from Table 1. In setting column 3, the two extreme entries of 65 and 72.5°C are first inserted at the two ends of the operating range. Intermediate values are interpolated and can be checked with eqn (3) using column 6. Here our first approximations to It were correct to within ±I°C.

Calculating the solar contribution to solar assisted systems

383

Table A1. 1 I

C

2

3

T

Tf

C

4

5

6T -r c

r)C

50

17.6

65

0.95

0

150

18.7

65

0.31

0

6

ncl C 0 0 0

250

19.0

65

0.18

0

350

19.0

65

0.15

0.05

18

15200

450

19.6

72.5

0.12

0.25

113

2364

550

20.2

72.5

0.095

O. 37

204

1307

650

19.7

72.5

0.081

0.43

280

952

750

20.5

72.5

0.069

0.48

360

739

850

20.8

72.5

0.061

0.53

451

590

950

23.2

72.5

0.052

0.59

561

475

1050

23.7

72.5

0.046

0.61

641

415

1150

20.7

72.5

0.045

0.61

702

379

1250

18.5

72.5

0.043

0.61

763

349

Table A2. 1 Ic

2

3

4

5

6

7

8

TC

Tf

AT IC

~c

~Ic

fc

Ec

350

19.0

66

0.13

0.2

70

0 . 0 4 2 2.94

450

19.6

68

0.10

0.35

158

0 . 0 4 2 6.61

550

20.2

69

0.088

0.4

220

0 . 0 3 7 8.14

650

19.7

70

0.076

0.47 306

0 . 0 3 5 10.69

750

20.5

71

0.067

0.51 383

0 . 0 3 8 14.54

850

20.8

72.5

0.061

0.53 451

0 . 0 3 7 16.98

Column 5 can now be read from Fig. 2 using data from column 4. Column 8 corresponds to the value ~clJc. In the proportional band, the energy per unit area can be calculated from column 8 of Table A2 and is (0.5 x 16.98) + 14.54 + 10.69 + 8.14 + 6.61 + 2.94 = 51.4 W/m2. For 590m 2 of collector this yields 30.3kW average. In the "excess power" regime the plant's total requirement is always available. The proportion of time during which this occurs is

taken from Table 1 and is (0.5 × 0.037) + 0.032 + 0.018 + 0.001 + 0.0003 = 0.07. Hence the total energy from this regime is 0.07 × 266 = 18.6 kW average. .. The annual average energy available from the array is 30.3+18.6=48.9kW, which represents an overall solar contribution of 18.4%.