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A performance prediction method for solar energy systems
Solar Energy Vol. 48, No. 3, pp. 169-175. 1992
0038-092X/92 $5.00 + .00 Copyright © 1992 Pergamon Press Ltd.
Printed in the U.S.A.
A PERFORMANCE PREDICTION METHOD FOR SOLAR ENERGY SYSTEMS H. SUEHRCKEt and P. G. MCCORMICKt tDepartment of Mechanical Engineering,James Cook University of North Queensland, Townsville, QId 4811, Australia, t Department of Mechanical Engineering,University of Western Australia, Nedlands, WA 6009, Australia Abstract--Recent studies of solar radiation have shown that instantaneous solar radiation exhibits a distinct bimodal character associated with clear and cloudy states. This suggeststhat many solar systems may simply be modelled to operate in an on/off fashion corresponding to clear/cloudy time intervals. It is shown that the average-dailysolar system performancemay be calculated from the product of clear-skysolar performance and the average time fraction of clear sky. This approximationgreatly simplifiesthe solar system performance prediction. The proposed method is verifiedwith a solar hot water system. The predicted results compare favourablywith resultspredicted by the ~f chart method and givea conservativeestimateof the experimentally measured performance.
!. I N T R O D U C T I O N
The prediction of the performance of solar energy systems is closely related to the prediction of solar radiation that drives the solar system. Monthly averages of daily solar radiation are available for many locations and allow the estimation of future insolation energy quantities reaching the earth's surface. However, average insolation values alone are insuificient information for most performance predictions since solar systems generally respond nonlinearly to the incident solar radiation. The radiation variation due to intermittent clouds and changes in the angular position of the sun must also be known. Liu and Jordan [ 1] showed that the distribution of the individual daily insolation values of a month essentially depends only on the average clearness index. Later Liu and Jordan[2] showed that this result also holds for the fluctuation statistics of hourly insolation values. Many solar performance prediction methods have subsequently been developed on the basis of hourly and daily insolation where the effect of radiation fluctuations is considered using the concept of utilisability [ 3- I l ]. The most prominent example is the ~.f chart method [ 3,4]. The utilisability based methods free the user from the need for detailed meteorological data as is necessary for detailed simulations such as TRNSYS [ 12 ], hut often [ 3,4 ] use correlations determined from simulations that incorporate specific load patterns. Utilisability calculations assume that collector heat capacity effects can be neglected [13 ]. A new approach that addresses the need for simplified performance prediction is the input-output performance prediction method for domestic hot water (DHW) systems by Bourges et al. [14]. The method is based on the observation that the daily output of many DHW systems stands in approximately linear relationship to the daily radiation received by the collector. As a result the solar system performance may be predicted directly in terms of average meteorological
quantities provided that the system input-output relationship is known. This paper proposes a performance prediction method for both thermal and photovoltaic solar systems. The raison d~tre for the proposed method is that solar systems are driven by instantaneous radiation. The statistical behaviour of instantaneous solar radiation has not been investigated until recently. The fact that hourly and daily insolation values follow the same frequency distribution lead most investigators to assume that instantaneous radiation would not behave differently. However, recent experimental investigations of instantaneous solar radiation at several locations [ 15-17 ] have found a distinct bimodal character associated with clear and cloudy states. An opportunity exists to simplify solar performance prediction for locations with similar bimodal character of instantaneous solar radiation. Typical cumulative probability distribution curves of instantaneous radiation for Perth, Australia are shown in Fig. 1. The instantaneous cumulative probability curves represent ascending order plots of instantaneous radiation measurements at a particular air mass (incidence angle). It can be seen from Fig. 1 that the probability curves exhibit a step that separates the radiation values; the radiation values of lower magnitude (clearness index) are associated with moments in time when there is a cloud in front of the sun whereas the higher (clear sky) values correspond to moments in time when there is no cloud in front of the sun. Such bimodal behaviour is not observed for hourly or daily insolation values because in the course of an hour or day many clear and cloudy time intervals are integrated together to form a random spectrum of intermediate radiation values. It has also been shown that when a bimodal character of instantaneous solar radiation exists for the horizontal surface, it further increases for the tilted surface [ 18 ]. This is due to the fact that tilting a solar energy receiving surface from the horizontal position
169
170
H. SUEHRCKEand P. G. MCCORMICK
I.O
m = 1.5
= ®
F, = 0
.
1
}~
/I
~
_= ~ F ©
_~
// 0.5
kT = 0.56
p=o'//
~ 0,0
I~= 45" i
L
0.0
i
i
I 0.5
,
,
*
,
I 1.0
f - Fraction of Time
Fig. 1. A cumulative probability distribution curve of instantaneous solar radiation for horizontal and tilted surface radiation [ 18 ].
toward the equator generally increases the beam, but not the diffuse radiation. However, as shown in Fig. 2, most beam radiation reaches the earth's surface when no cloud is in front of the sun. Thus, the effect of surface tilt is to increase the clear (beam) radiation values, but not the cloudy (diffuse) radiation values so that the step in the probability distribution becomes more pronounced. An example of this increase in bimodal radiation character due to surface tilt is shown in Fig. I. it is the increased bimodal radiation character on the tilted solar collector surface that provides the basis of the performance prediction method to be developed.
average fraction of clear sky at the particular point in time. A recent study of the diffuse fraction of instantaneous solar radiation showed that once a cloud is in front of the sun virtually all the radiation reaching the earth's surface is diffuse [ 20 ]. This means that the beam radiation reaches a point on earth's surface essentially only during the time when there is no cloud in front of the sun. Thus, measurements of beam radiation may be used to determine the average time fraction of clear sky. This is illustrated in Fig. 2 which shows the respective distribution of beam and diffuse radiation. The area labelled beam under the curve in Fig. 2 represents the average beam radiation, Gb/Go, at this particular air mass. Since this area is approximately equivalent to a rectangle of constant height, the average time fraction of clear sky may be obtained by relating the average beam radiation, GblGo, to the m a x i m u m possible rectangle area, the clear sky beam radiation, G~,ear/ Go. Hence, the average fraction of clear sky can be expressed asJ~l~a, = dblG~, . . . . Monthly-average daily values of fdc~, can be expressed in a similar manner. Studies of the statistics of instantaneous solar radiation indicate that the average fraction of clear sky is approximately independent of air mass [ 16,21 ]. This result is obtained when the time intervals of clear sky are more or less randomly distributed over the day. Assuming the clear sky fraction to be independent of air mass, the average time fraction of clear sky may simply be expressed as
f~l~a, = where/lh and t t ~
nb
(1)
Hbclear
denote the monthly-average daily
2. PROPOSED METHOD
2.1 Basic concept Many solar energy systems exhibit a threshold to solar radiation, i.e., the radiation level must exceed a critical value before useful output is produced. For example, for a photovoltaic water pumping system the threshold is determined by the pressure head that the p u m p has to overcome before a net flow results. For a solar thermal collector the threshold is determined by the collector heat loss that has to be offset before a positive output can be produced. A threshold causes the instantaneous input to output relationship of the solar system to become nonlinear. When the bimodal radiation character of instantaneous tilted surface radiation is considered in conjunction with the nonlinear solar system performance characteristic, m a n y solar systems may be modelled to operate in an o n / o f f fashion corresponding to clear/ cloudy time intervals [ 19 ]. The average instantaneous solar system performance at a particular point in time may be obtained from the average fraction of time that the solar system operates in the " o n " mode, i.e., the
1,0
m = 1.5 /
=
t/
k = 0.63
_=
E o.5 ©
Beam
i
.
0.0 0,0
!
'rr'l''J'''l'''l'
"'l
"''l''
0,5
'v " 1 1,0
f - FracUon of Time Fig. 2. The distribution of beam and diffuse solar radiation[18]. The diffuse and the beam component of the cumulative probability distribution has been plotted using the equations kd = kGJG and kb = k( l - Gd/G), where Gd/G = Gd/G(k, m) is the instantaneous diffuse fraction[20].
A performance prediction method beam radiation and average daily clear-sky beam radiation, respectively. It should be noted that the assumption of the fraction of clear sky being independent of air mass does not necessarily mean that the time intervals of clear and cloudy radiation are uniformly distributed over the day. Even for asymmetric days that are clear in the morning and cloudy in the afternoon, the definition of the monthly average fraction of clear sky is valid (fc~a~ = 0.5 in this case). The advantage of expressing the fraction of clear sky in this manner is that it may then be determined from readily available monthly averages of daily radiation. This suggests that the monthly-average daily solar performance can be estimated from the monthly-average daily clear-sky performance multiplied by the average time fraction of clear sky as shown schematically in Fig. 3. Significant simplification is achieved through the replacement of the fluctuating solar radiation on the input side by clear sky radiation. The radiation fluctuation due to intermittent clouds is subsequently considered using only the parameter f¢,~. The main feature of the proposed method is its conceptual simplicity and directness to the solar system operation. 2.2 Determination of radiation quantities The proposed method requires 12 monthly averages of either daily horizontal surface radiation or the average number of sunshine hours. The monthly averages of daily radiation allow the monthly-average clearness index to be determined whereas the average number of sunshine hours provides a direct measure for the fraction of clear sky. Using monthly-average daily radiation,/~, as the input, the average beam radiation quantities Hb and H ~ , ~ and the tilted surface radiation Gxct~a~may be obtained in the following description. The average-monthly beam radiation can be found from measurements (if available) or from the monthlyaverage daily diffuse fraction, Ho/H, using the equation /lb = H[I - (Ild/ll)]
(2)
where / I is the monthly-average daily insolation. A simple straight line relationship of the form l~d/l~ = 1.0 - c/~, where g ( = H / H o ) is the monthly-average clearness index, may be used to determine the monthlyaverage daily diffuse fraction [ 22 ]. Page [ 23 ] originally proposed this relationship and suggested the value c = 1.13 for the slope. Recently, this relationship has been evaluated on the basis of instantaneous solar radiation data [ 20,21 ]. The value of the constant found was 1. I 1, almost identical to the value of Page. It was also shown that the monthly-average fraction relation-
171
ship is only weakly dependent on air mass of the location [ 20 ]. The average daily clear-sky radiation, H ~ , on the other hand may be found from experimental measurements (if available ) or from the integration of the instantaneous beam radiation, G~,o~, over the average day of the month. This can be accomplished by using the simple broad-band beam radiation model, G~,~, = G o r s e x p ( - B , m ) , proposed by [21]. This beam radiation model is based on the observation that at certain wavelengths the transmittance of solar radiation is reduced to near zero due to spectral atmospheric absorption while at other wavelengths the spectral transmittance is dependent on air mass[24,25 ]. The factor rs takes into account the decrease in broad-band beam solar radiation due to spectral absorption and #, combines the various spectral beam extinction coefficients. For Perth, Western Australia, good agreement with experimental data was achieved for rs = 0.85 and /~, = 0.15 [ 21 ]. Hence, the monthly-average daily clearsky beam radiation may be expressed as Hu.j~, = rs f Goexp(-~¢m)dt
(3)
where the above integral extends over the average day of the month. The determination of H~,~r has the potential to be simplified. The tilted-surface clear-sky radiation as a function of time of day can be expressed as
GTd~
= RG¢,¢~,
(4)
where R denotes the ratio between instantaneous tilted and horizontal surface radiation. The ratio R may be determined using Liu and Jordan's model[26] that considers the tilted surface radiation to be composed of beam radiation, isotropic sky diffuse radiation, and ground reflected isotropic radiation. However, the clear-sky radiation mainly consists of beam radiation so that R may simply be approximated by cos 0/cos 0,, where 0 and 0, are the beam incident angles for tilted and horizontal surface radiation, respectively [ 27 ]. The horizontal-surface clear-sky radiation, Gd~r (=Gbc.eaf + Gdc,~af), on the other hand may be found from the distribution of radiation over the day[21]. According to this distribution Go,,,, = GoKc,¢~,[X( 1 - / l a / / 7 ) e x p ( - f l ~ m ) + tIa/trl], where
X = rsllo/ll~j~ar.
Output
The clearness index for clear sky may be obtained from the monthly-average diffuse-fraction straight-line relationship as
Fig. 3. Performance prediction schematic for t h e f ~ method.
Kclea, = (H~l~r/(CHo)) t/2
.ClearSky~ Radiation
CleerSky~
Output
172
H. SUEHRCKEandP. G. MCCORMICK
2.3 Assumptions, errors, and heat capacitance effects There are three assumptions implicit in the statement that a solar system operates in an on/off fashion corresponding to clear/cloudy. First, it is assumed that the location where the solar system is situated exhibits bimodal instantaneous radiation statistics, i.e., it is assumed that there is a predominance of two significantly different radiation levels, clear and cloudy. The clear radiation level is associated with moments in time when there is no cloud in front of the sun while the cloudy level corresponds to moments in time when there is a (significant) cloud in front of the sun. The degree of the bimodal character of the instantaneous solar radiation will depend on location. However, if there is a distinct separation between clear and cloudy insolation values, the effect of collector tilt will in general strongly amplify this separation (see Fig. 1). Second, it is assumed that when a cloud is in front of the sun, the output of the solar process is zero due to the threshold of the solar process. Clearly, this is not always the case. For example, a high volume, low pressure head photovoltaic pumping system or an evacuated tube thermal collector may still produce output even when there is a cloud in front of the sun. However, a calculation based on the assumption that output is produced only when there is no cloud in front of the sun will yield a conservative estimate of performance. The reason is that, although the solar process may still be operating during cloudy conditions, its output may be small compared to the clear sky output. This point is illustrated in Fig. 4, which shows the error incurred for low thresholds when only clear sky output is considered (dashed area). Finally, the assumption that the solar process will operate in a strict on/off fashion is certainly not true for solar thermal collectors that have a variety of time constants due to collector heat capacitance (e.g., the time constant for thermal fiat-plate collectors may
1.0
-
rn
=
1.5 t
J 0.5
/ 0.0
threshold i
i
i
0.0 f-
o
I 0.5
Fraction
a
t
i
~
I 1.0
of Time
Fig. 4. The error associated with the assumptionthat the solar system produces no output under cloudy conditions for low thresholds.
range from 1-10 min). However, it is well known that the collector response lag causes the collector output to stay below the equilibrium output during a cloudyclear transition, but above the equilibrium output during a clear-cloudy transition. Hence, there is a strong tendency for the heat capacity effects to cancel. Klein et aL [28] found that the effects of intermittent sunshine can be neglected while Soltau [29] estimated the performance reducing effect of heat capacitance of unglazed swimming pool collectors to be less than 3%. Thus, for the long-term collector output calculation a zero collector time constant may be assumed even for thermal collectors since the net effect of heat capacitance may be neglected. All previous performance prediction methods implicitly made this assumption since none considers the effect of collector time constant. The fact that this assumption becomes apparent with the proposed method merely highlights the fact that some of the other methods have been quite removed from the actual solar system operation. For photovoltaic systems the assumption that the system operates in an on/off fashion is excellent, provided that the system threshold is large enough. A good example is a photovoltaic water pumping system where the threshold is given by the pressure head that the pump has to overcome before flow results. 3. EXPERIMENTAl,VERIFICATION The performance prediction method proposed has been experimentally verified for a 5.7 m 2 flat-plate collector hot water system with 440 litre storage tank. The experimental data are based on a one year performance monitoring carried out in Perth, Western Australia (q~ = - 32 o ) [ 30 ]. The system is not ideally suited to the proposed approach since the selective surface solar collectors exhibit a low threshold and the location experiences air mass values close to 1.0 in the summertime (this reduces the amplifying effect of collector tilt on the bimodal radiation character). Nevertheless, the system has been chosen because of the availability of detailed experimental data. The comparison of predicted and measured results for a thermal solar system do not imply that the use of the proposed performance prediction method is restricted to thermal systems. A schematic of the closed circuit flat-plate collector system is shown in Fig. 5. The flow in the collector loop was controlled by the temperature difference between the collector outlet and the bottom of the storage. Whenever this temperature difference was positive, the collector circulation pump was turned on (the specific controller threshold of the collector was estimated 60 W / m 2 [ 30]). No heat exchanger was employed in the collector loop. In the load delivery loop the flow was controlled by the load delivery temperature. When the load delivery temperature fell below a certain minimum value, Tmi,, the load pump was turned off. The load supply water was drawn offat the top of the storage tank and passed through a heat exchanger, where the energy was transferred to the load. The selective surface collectors were tilted 27 ° to-
173
A performance prediction method Collector
"%
~
OL
Fig. 5. Schematic ofthe flat-platecollector solar system. wards the equator. Prior to the commencement of the performance monitoring the collectors were tested to determine the collector etficiency and the incidence angle characteristic. Using the terminology of the standard fiat-plate collector equation [24], the collector performance parameters were found to be F R ( r a ) , = 0.705 and FRUL = 4.31 W / K m -2. For the incidence angle modifier the equation K,,, = i + bo( 1/cos 0 - 1 ) was employed, where bo was found to be approximately -0.075. Throughout the monitoring the specific collector flow rate was kept constant at 0.02 kg/s m -2. The storage tank was characterised by the product of heat loss coetficient and tank area ( UA)s. The average (UA)s product was measured as 4.26 W / K . Fluctuations of the (UA)s value (less than +0.4 W / K ) were observed because of leakages caused by an unreliable one way flow valve that was installed in the collector loop to prevent reverse thermosiphon flow and because of leakages of the storage tank overpressure relieve valve. Under the assumption of a fully mixed storage tank, the performance of the solar system shown in Fig. 5 can described by the heat balance equation dTs (MG)'-~=
Qu-Q,-QL
(5)
where M is the mass of the storage, Cp the specific heat of the storage fluid and Ts the storage tank temperature. Using the flat-plate collector equation and the concept of the fraction of clear sky, the useful energy transferred to the system can be expressed as O, =f~t~arAckR[(ra)GTd~af -- Um.(Ts - Ta)] ~ where ( r a ) = K , , ( r a ) , . The tank heat loss on the other hand can
be expressed as Qs = ( UA)s( Ts - T'~). Theheat transferred to the load (~L is subsequently denoted by L. Equation (5) has been solved numerically using Euler's method with five minute step size. The method adopted for simulation is the representative day method originally proposed by T. A. Reddy et al.[l 1,31]. In this approach the instantaneous energy balance equation [eqn (5)] is solved numerically for a representative day of the month. The representative day is based on the average day of the month where the radiation fluctuation is suitably superimposed (see also [30]). In this particular case the average day is characterised by the average daily radiation and the average time fraction of clear sky. The effect of fluctuation due to intermittent clouds is superimposed by multiplication of the clear sky performance with the fraction of clear sky. To ensure that the numerical solution corresponds to the equilibrium solution of the system, the storage tank temperature at the beginning of the day is compared with that at the end of the day, i.e., I Ts(t) - Ts(t + 24 h)l < ~. The simulation was repeated with the end temperature as the initial temperature until ~<0.1K. The numerical solution was carried out using the data shown in Table 1. The corrected value of FRU'L that is shown in Table 1 incorporates the pipe heat loss in the collector equation [ 32 ]. The reason that the product FRUL did not remain constant was that the heat loss was affected by wind. The value of FR(ra) remained unchanged. Also shown in Table I is the average fraction of clear sky that was calculated using eqns ( 2 ) and ( 3 ) and the monthly-average diffuse fraction relationship,/td//7 = I - 1.11 /(. For the calculation of the instantaneous horizontalsurface clear-sky radiation, Gc~.... the distribution of radiation over the day[21] was used as described in Section 3. The predicted solar fractions of the proposed method are compared in Table 2 with measured results and results of the ~f chart method. When the solar fractions of Table 2 are compared, it is noted that the measured solar fractions are around six percentage points above the predicted results. This discrepancy is due to stratification in the storage tank, which was not taken into account by either prediction method. It is well known that storage tank stratification increases the performance of thermal systems. For part
Table I. Monthly average values of the input data[30]
January February March April May June July August September October November December
n
/4 [MJ/day]
L [W]
Tm,~ [°C]
T, [°C]
b'RU~ [W/K m-21
f¢~,,
17 47 75 105 135 162 198 228 258 288 318 344
31.13 23.29 20.26 17.59 10.06 9.03 9.22 13.78 16.52 24.20 28.33 31.09
614 602 605 608 610 612 615 611 619 61 I 613 611
52.3 51.7 51.4 51.7 51.7 51.8 52.3 52.5 51.5 51.6 52.3 52.8
25.2 22.2 20.3 18.0 15.8 13.9 11.8 11.9 15.2 15.2 19.1 19.3
5.14 5.01 5.07 5.05 4.83 5.02 5.16 5.08 5.14 5.22 5.13 5.18
0.85 0.58 0.61 0.78 0.45 0.50 0.45 0.61 0.50 0.70 0.75 0.82
174
H. SUEHRCKEand P. G. MCCORMICK Table 2. Measured and predicted solar fractions Solar fractions in percent Measured
~fchart method
f¢~ method
January February March April May June July August September October November December
97 77 71 69 26 23 16 36 44 65 81 88
90 71 65 61 20 13 8 30 40 65 76 81
93 64 63 66 24 20 14 36 39 64 73 78
Average
58
52
53
of October the collector pump was inoperative, causing the measured solar fraction to be smaller than expected from the measured solar radiation. There are two more noticeable exceptions in the solar fractions of Table 2. First, for the month of January the solar fraction predicted byf¢~,, method comes close to the measured value. This however, cannot be credited to the f¢~,~ method, but is simply a consequence of the fact that for the fcaear method energy dumping was neglected. At solar fractions close to 1.0 neglect of energy dumping tends to offset the underestimation of the solar fraction because of the assumption of a fully mixed storage tank. Second, for the summer months December and February thef~ear-method significantly underpredicts the solar fraction. This may be explained from the fact that during the summer months the sun is directly overhead at the test site so that radiation levels below a cloud can still be high enough for a solar collector with low threshold to produce some output. This is particularly significant for February with a clear-sky time fractions of only 0.58, which means that during February, for more than 40% of the time, there were clouds in front of the sun. The ~ f chart method on the other hand, seems to underpredict at very low solar fractions during the months June and July. On an annual basis, however, the solar fractions predicted by the f~ar method and the ~ f chart method are very similar.
4. CONCLUSIONS A simple method for the performance prediction of solar energy systems has been proposed. Using the bimodal character of instantaneous solar radiation on tilted surfaces, many solar systems may be modelled to operate in an o n / o f f fashion corresponding to the clear/cloudy time intervals. The predictions by this new method compare favourably with predictions by the ~ f chart method. However, the f~,~, method is conceptually more simple than the ~ f chart method and may be more realistic since solar collectors see instantaneous solar radiation, not integrated hourly or
daily values. The f~t,,, method has the potential to be simplified.
Acknowledgment--H.S. wishes to acknowledge the financial assistance given by the National Energy Research, Development, and Demonstration Council of Australia. NOMENCLATURE he
collector area [ m 2]
bo constant for incidence angle modifier constant in/~d/H correlation specific heat [ J / K kg- t ] fraction of time, solar fraction average time fraction of clear sky, eqn ( 1) collector heat removal factor collector performance parameter [ W / K m-2 ] FRU;. collector performance parameter incorporating pipe heat loss [W/K m -2] FR(Tah collector performance parameter G instantaneous total (beam + diffuse) horizontal surface solar radiation [W/m 2] average instantaneous total (beam + diffuse) horizontal surface solar radiation [ W / m 2] ab average instantaneous horizontal surface beam radiation [W/m 2] Gbciear average clear sky instantaneous beam horizontal surface radiation [W/m z] Gclcar instantaneous clear sky horizontal surface radiation [W/m 2] Go extraterrestrial horizontal surface solar radiation [W/m 2] GT instantaneous tilted total surface (beam + diffuse) radiation [W/m 2] aTcle~u- clear sky instantaneous radiation incident on tilted collector surface [ W / m 2] average daily total ( beam + diffuse) horizontal surface radiation [MJ/m 2] ~b average daily beam horizontal surface solar radiation [MJ/m 2] nbc~Lr average daily beam clear sky horizontal surface solar radiation [MJ/m 2] Hd/H monthly average daily diffuse fraction /40 daily extraterrestrial horizontal surface radiation [MJ/m z] k clearness index for instantaneous insolation (atmospheric transmittance), k = G~Go k average instantaneous clearness index, k = G/Go kb instantaneous atmospheric transmittance for beam radiation, kb = G~/Go C
G f f~r FR FRU~.
A performance prediction method kd diffuse component of the instantaneous clearness index,/q = Gd/Go kr instantaneous clearness index for tilted surface radiation, kx = GT/Go J~T average instantaneous clearness index for tilted surface radiation, J~r = Gr/Go average daily clearness index,/~ = 14/11o average daily clearness index for clear sky, K~=,
= H~.~a,/Ho KTa incidence angle modifier L load [Wl m air Mass, m = 1/cos 0~, where 0~is the zenith angle, see [27] for details M storage tank fluid mass [kg] (MCv) storage tank heat capacity [ kJ / K ] n day number storage tank heat loss rate [W] Qo instantaneous collector output [W] q~ instantaneous thermal heat load [ W ] R ratio between total radiation on tilted and horizontal surface, R = GT/G[-] Rb ratio between beam radiation on tilted and horizontal surface, Rb = Gt.b/G~ t time ro ambient temperature [°C] r'. ambient temperature of storage tank [°C] ri collector fluid inlet temperature [°C] Tmi~ minimum useful load temperature [°C] rs storage tank temperature [ °C] Ut. total collector heat loss coefficient [ W / K m -2] ( UA)s storage tank heat loss product [ W / K ] O. extinction coefficient for clear sky beam radiation constant 0 incidence angle for beam radiation measured from the surface normal [radian ] O, zenith angle, cos 0~ = cos 6 cos ¢ cos w + sin 6 sin [ radian ] transmittance-absorptance product (~a). transmittance-absorptance product for normal incident solar radiation latitude angle [ radian ] x constant in radiation distribution × = Ho fGoexp( - ~ m ) d t
Os
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1. 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). 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, 53-70 ( 1963 ). 3. S.A. Klein and W. A. Beckman, A general design method for closed-loop solar energy systems, Solar Energy 22, 269-282 (1979). 4. J. E. Braun, S. A. Klein, and K. A. Pearson, An improved design method for solar water heating systems, Solar Energy 31,597-604 (1983). 5. J. M. Gordon and A. Rabl, Design, analysis and optimization of solar industrial processes without storage, Solar Energy 28, 519-530 (1982). 6. M. Collares-Pereira, J. M. Gordon, A. Rabl, and Y. Zarmi, Design and optimization of solar industrial hot water systems with storage, Solar Energy 32, 121-133 (1984). 7. D. R. Clark, S. A. Klein, and W. A. Beckman, A method for estimating the performance of photovoltaic systems, Solar Energy 33, 551-555 (1984). 8. D. Feuermann, J. M. Gordon. and Y. Zarmi, A typical meteorological day (TMD) approach for prediction the long-term performance of solar energy systems, Solar Energy 35, 63-69 ( 1985 ). 9. J. M. Gordon and Y. Zarmi, An analytical model for the long-term performance of solar thermal systems with wellmixed storage, Solar Energy 35, 55-61 ( 1985 ).
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10. J. I. Ajona and J. M. Gordon, An analytical model for the long-term prediction of solar air heating systems, Solar Energy 38, 45-53 ( 1987 ). I I. T. A. Reddy, J. M. Gordon, and I. P. D. De Silva, MIRA: A one-repetitive day method for prediction the long-term performance of solar energy systems, Solar Energy 39, 123-133 (1987). 12. TRNSYS, A transient simulation program, Engineering Experiment Station Report 38-12, Solar Energy Laboratory, University of Wisconsin-Madison, Madison, WI ( 1983 ). 13. H. Suehrcke and P. G. McCormick, Solar radiation utilizability, Solar Energy 43, 339-345 (1989). 14. B. Bourges, A. Rabl, B. Leide, M. J. Carvalho, and M. Collares-Pereira, Long-term performance prediction for DHW solar systems based on input-output test method, Proc ISES Solar World Congress 1989. Paper No. 4.P4.04, Kobe, Japan (September 1989). 15. H. Suehrcke and P. G. McCormick, The frequency distribution of instantaneous insolation values. Solar Energy 40, 413-422 (1988). 16. P. H. Marwood, The performance of solar panels, Honours Thesis. Electrical & Computer Engineering Department, James Cook University of North Queensland, Townsville, Australia (1989). 17. A. M. Robinson and D. G. Howell, Analysis of the bimodal behaviour in solar radiation measurements, 16th Annual Conference of the Solar Energy, Society of Canada, Halifax, NS, Canada (June 1990). 18. H. Suehrcke and P. G. McCormick, Cumulative probability distribution curves of insolation values for tilted surfaces, Proc. ISES Solar World Congress 1989, Paper No. 12.4.04, Kobe, Japan (September 1989). 19. H. Suehrcke and P. G. McCormick, Performance prediction of solar systems using the time fraction of clear sky, Proc. ISES Solar World Congress 1989, Paper No. 4.9.02, Kobe, Japan (September 1989). 20. H. Suehrcke and P. G. McCormick, The diffuse fraction of instantaneous solar radiation, Solar Energy 40, 423430 ( 1988 ). 21. H. Suehrcke and P. G. McCormick, The distribution of average instantaneous terrestrial solar radiation over the day, Solar Energy 42, 303-309 (1989). 22. C. C. Y. Ma and M. lqbal, Statistical comparison of solar radiation correlations, Monthly average global and diffuse radiation on horizontal surfaces, Solar Energy 33, 143148 (1984). 23. J. K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40°N--40°S, Prtw. U.N. Conf. on New Sources of Energy. Paper No. $98, Vol. 4, 378-390 ( 1961 ). 24. J. A. Duffle and W. A. Beckman, Solar engineering ¢f solarpr~z'sses, Wiley, New York, p. 55 and Ch. 6, (1980). 25. M. lqbal, An introduction to solar radiation. Academic Press, New York, p. 138 (1983). 26. B. Y. H. Liu and R. C. Jordan, Daily insolation on surfaces tilted towards the equator, ASHRAE J., 3(10), 53-59 (1961). 27. A. Rabl, Active solar collectors and their applications. Oxford University Press, New York ( 1985 ). 28. S. A. Klein, J. A. Duffle, and W. A. Beckman, Transient considerations of flat-plate solar collectors. J. Eng. Power, 96A, 109 (1974). 29. H. Soltau, The thermal performance of uncovered collectors, Proc. ISES Solar World Congress 1987, Paper No. 2.5.14, Hamburg (1987). 30. H. Suehrcke, The performance prediction of solar thermal systems, Ph.D. Thesis. University of Western Australia, Perth, Australia ( 1988 ). 3 I. T.A. Reddy, The design and sizing ~factive solar systems. Oxford Press (1987). 32. W. A. Beckman, Duct and pipe losses in solar energy systems, Solar Energy 21, 531-532 ( 1978 ).
0038-092X/92 $5.00 + .00 Copyright © 1992 Pergamon Press Ltd.
Printed in the U.S.A.
A PERFORMANCE PREDICTION METHOD FOR SOLAR ENERGY SYSTEMS H. SUEHRCKEt and P. G. MCCORMICKt tDepartment of Mechanical Engineering,James Cook University of North Queensland, Townsville, QId 4811, Australia, t Department of Mechanical Engineering,University of Western Australia, Nedlands, WA 6009, Australia Abstract--Recent studies of solar radiation have shown that instantaneous solar radiation exhibits a distinct bimodal character associated with clear and cloudy states. This suggeststhat many solar systems may simply be modelled to operate in an on/off fashion corresponding to clear/cloudy time intervals. It is shown that the average-dailysolar system performancemay be calculated from the product of clear-skysolar performance and the average time fraction of clear sky. This approximationgreatly simplifiesthe solar system performance prediction. The proposed method is verifiedwith a solar hot water system. The predicted results compare favourablywith resultspredicted by the ~f chart method and givea conservativeestimateof the experimentally measured performance.
!. I N T R O D U C T I O N
The prediction of the performance of solar energy systems is closely related to the prediction of solar radiation that drives the solar system. Monthly averages of daily solar radiation are available for many locations and allow the estimation of future insolation energy quantities reaching the earth's surface. However, average insolation values alone are insuificient information for most performance predictions since solar systems generally respond nonlinearly to the incident solar radiation. The radiation variation due to intermittent clouds and changes in the angular position of the sun must also be known. Liu and Jordan [ 1] showed that the distribution of the individual daily insolation values of a month essentially depends only on the average clearness index. Later Liu and Jordan[2] showed that this result also holds for the fluctuation statistics of hourly insolation values. Many solar performance prediction methods have subsequently been developed on the basis of hourly and daily insolation where the effect of radiation fluctuations is considered using the concept of utilisability [ 3- I l ]. The most prominent example is the ~.f chart method [ 3,4]. The utilisability based methods free the user from the need for detailed meteorological data as is necessary for detailed simulations such as TRNSYS [ 12 ], hut often [ 3,4 ] use correlations determined from simulations that incorporate specific load patterns. Utilisability calculations assume that collector heat capacity effects can be neglected [13 ]. A new approach that addresses the need for simplified performance prediction is the input-output performance prediction method for domestic hot water (DHW) systems by Bourges et al. [14]. The method is based on the observation that the daily output of many DHW systems stands in approximately linear relationship to the daily radiation received by the collector. As a result the solar system performance may be predicted directly in terms of average meteorological
quantities provided that the system input-output relationship is known. This paper proposes a performance prediction method for both thermal and photovoltaic solar systems. The raison d~tre for the proposed method is that solar systems are driven by instantaneous radiation. The statistical behaviour of instantaneous solar radiation has not been investigated until recently. The fact that hourly and daily insolation values follow the same frequency distribution lead most investigators to assume that instantaneous radiation would not behave differently. However, recent experimental investigations of instantaneous solar radiation at several locations [ 15-17 ] have found a distinct bimodal character associated with clear and cloudy states. An opportunity exists to simplify solar performance prediction for locations with similar bimodal character of instantaneous solar radiation. Typical cumulative probability distribution curves of instantaneous radiation for Perth, Australia are shown in Fig. 1. The instantaneous cumulative probability curves represent ascending order plots of instantaneous radiation measurements at a particular air mass (incidence angle). It can be seen from Fig. 1 that the probability curves exhibit a step that separates the radiation values; the radiation values of lower magnitude (clearness index) are associated with moments in time when there is a cloud in front of the sun whereas the higher (clear sky) values correspond to moments in time when there is no cloud in front of the sun. Such bimodal behaviour is not observed for hourly or daily insolation values because in the course of an hour or day many clear and cloudy time intervals are integrated together to form a random spectrum of intermediate radiation values. It has also been shown that when a bimodal character of instantaneous solar radiation exists for the horizontal surface, it further increases for the tilted surface [ 18 ]. This is due to the fact that tilting a solar energy receiving surface from the horizontal position
169
170
H. SUEHRCKEand P. G. MCCORMICK
I.O
m = 1.5
= ®
F, = 0
.
1
}~
/I
~
_= ~ F ©
_~
// 0.5
kT = 0.56
p=o'//
~ 0,0
I~= 45" i
L
0.0
i
i
I 0.5
,
,
*
,
I 1.0
f - Fraction of Time
Fig. 1. A cumulative probability distribution curve of instantaneous solar radiation for horizontal and tilted surface radiation [ 18 ].
toward the equator generally increases the beam, but not the diffuse radiation. However, as shown in Fig. 2, most beam radiation reaches the earth's surface when no cloud is in front of the sun. Thus, the effect of surface tilt is to increase the clear (beam) radiation values, but not the cloudy (diffuse) radiation values so that the step in the probability distribution becomes more pronounced. An example of this increase in bimodal radiation character due to surface tilt is shown in Fig. I. it is the increased bimodal radiation character on the tilted solar collector surface that provides the basis of the performance prediction method to be developed.
average fraction of clear sky at the particular point in time. A recent study of the diffuse fraction of instantaneous solar radiation showed that once a cloud is in front of the sun virtually all the radiation reaching the earth's surface is diffuse [ 20 ]. This means that the beam radiation reaches a point on earth's surface essentially only during the time when there is no cloud in front of the sun. Thus, measurements of beam radiation may be used to determine the average time fraction of clear sky. This is illustrated in Fig. 2 which shows the respective distribution of beam and diffuse radiation. The area labelled beam under the curve in Fig. 2 represents the average beam radiation, Gb/Go, at this particular air mass. Since this area is approximately equivalent to a rectangle of constant height, the average time fraction of clear sky may be obtained by relating the average beam radiation, GblGo, to the m a x i m u m possible rectangle area, the clear sky beam radiation, G~,ear/ Go. Hence, the average fraction of clear sky can be expressed asJ~l~a, = dblG~, . . . . Monthly-average daily values of fdc~, can be expressed in a similar manner. Studies of the statistics of instantaneous solar radiation indicate that the average fraction of clear sky is approximately independent of air mass [ 16,21 ]. This result is obtained when the time intervals of clear sky are more or less randomly distributed over the day. Assuming the clear sky fraction to be independent of air mass, the average time fraction of clear sky may simply be expressed as
f~l~a, = where/lh and t t ~
nb
(1)
Hbclear
denote the monthly-average daily
2. PROPOSED METHOD
2.1 Basic concept Many solar energy systems exhibit a threshold to solar radiation, i.e., the radiation level must exceed a critical value before useful output is produced. For example, for a photovoltaic water pumping system the threshold is determined by the pressure head that the p u m p has to overcome before a net flow results. For a solar thermal collector the threshold is determined by the collector heat loss that has to be offset before a positive output can be produced. A threshold causes the instantaneous input to output relationship of the solar system to become nonlinear. When the bimodal radiation character of instantaneous tilted surface radiation is considered in conjunction with the nonlinear solar system performance characteristic, m a n y solar systems may be modelled to operate in an o n / o f f fashion corresponding to clear/ cloudy time intervals [ 19 ]. The average instantaneous solar system performance at a particular point in time may be obtained from the average fraction of time that the solar system operates in the " o n " mode, i.e., the
1,0
m = 1.5 /
=
t/
k = 0.63
_=
E o.5 ©
Beam
i
.
0.0 0,0
!
'rr'l''J'''l'''l'
"'l
"''l''
0,5
'v " 1 1,0
f - FracUon of Time Fig. 2. The distribution of beam and diffuse solar radiation[18]. The diffuse and the beam component of the cumulative probability distribution has been plotted using the equations kd = kGJG and kb = k( l - Gd/G), where Gd/G = Gd/G(k, m) is the instantaneous diffuse fraction[20].
A performance prediction method beam radiation and average daily clear-sky beam radiation, respectively. It should be noted that the assumption of the fraction of clear sky being independent of air mass does not necessarily mean that the time intervals of clear and cloudy radiation are uniformly distributed over the day. Even for asymmetric days that are clear in the morning and cloudy in the afternoon, the definition of the monthly average fraction of clear sky is valid (fc~a~ = 0.5 in this case). The advantage of expressing the fraction of clear sky in this manner is that it may then be determined from readily available monthly averages of daily radiation. This suggests that the monthly-average daily solar performance can be estimated from the monthly-average daily clear-sky performance multiplied by the average time fraction of clear sky as shown schematically in Fig. 3. Significant simplification is achieved through the replacement of the fluctuating solar radiation on the input side by clear sky radiation. The radiation fluctuation due to intermittent clouds is subsequently considered using only the parameter f¢,~. The main feature of the proposed method is its conceptual simplicity and directness to the solar system operation. 2.2 Determination of radiation quantities The proposed method requires 12 monthly averages of either daily horizontal surface radiation or the average number of sunshine hours. The monthly averages of daily radiation allow the monthly-average clearness index to be determined whereas the average number of sunshine hours provides a direct measure for the fraction of clear sky. Using monthly-average daily radiation,/~, as the input, the average beam radiation quantities Hb and H ~ , ~ and the tilted surface radiation Gxct~a~may be obtained in the following description. The average-monthly beam radiation can be found from measurements (if available) or from the monthlyaverage daily diffuse fraction, Ho/H, using the equation /lb = H[I - (Ild/ll)]
(2)
where / I is the monthly-average daily insolation. A simple straight line relationship of the form l~d/l~ = 1.0 - c/~, where g ( = H / H o ) is the monthly-average clearness index, may be used to determine the monthlyaverage daily diffuse fraction [ 22 ]. Page [ 23 ] originally proposed this relationship and suggested the value c = 1.13 for the slope. Recently, this relationship has been evaluated on the basis of instantaneous solar radiation data [ 20,21 ]. The value of the constant found was 1. I 1, almost identical to the value of Page. It was also shown that the monthly-average fraction relation-
171
ship is only weakly dependent on air mass of the location [ 20 ]. The average daily clear-sky radiation, H ~ , on the other hand may be found from experimental measurements (if available ) or from the integration of the instantaneous beam radiation, G~,o~, over the average day of the month. This can be accomplished by using the simple broad-band beam radiation model, G~,~, = G o r s e x p ( - B , m ) , proposed by [21]. This beam radiation model is based on the observation that at certain wavelengths the transmittance of solar radiation is reduced to near zero due to spectral atmospheric absorption while at other wavelengths the spectral transmittance is dependent on air mass[24,25 ]. The factor rs takes into account the decrease in broad-band beam solar radiation due to spectral absorption and #, combines the various spectral beam extinction coefficients. For Perth, Western Australia, good agreement with experimental data was achieved for rs = 0.85 and /~, = 0.15 [ 21 ]. Hence, the monthly-average daily clearsky beam radiation may be expressed as Hu.j~, = rs f Goexp(-~¢m)dt
(3)
where the above integral extends over the average day of the month. The determination of H~,~r has the potential to be simplified. The tilted-surface clear-sky radiation as a function of time of day can be expressed as
GTd~
= RG¢,¢~,
(4)
where R denotes the ratio between instantaneous tilted and horizontal surface radiation. The ratio R may be determined using Liu and Jordan's model[26] that considers the tilted surface radiation to be composed of beam radiation, isotropic sky diffuse radiation, and ground reflected isotropic radiation. However, the clear-sky radiation mainly consists of beam radiation so that R may simply be approximated by cos 0/cos 0,, where 0 and 0, are the beam incident angles for tilted and horizontal surface radiation, respectively [ 27 ]. The horizontal-surface clear-sky radiation, Gd~r (=Gbc.eaf + Gdc,~af), on the other hand may be found from the distribution of radiation over the day[21]. According to this distribution Go,,,, = GoKc,¢~,[X( 1 - / l a / / 7 ) e x p ( - f l ~ m ) + tIa/trl], where
X = rsllo/ll~j~ar.
Output
The clearness index for clear sky may be obtained from the monthly-average diffuse-fraction straight-line relationship as
Fig. 3. Performance prediction schematic for t h e f ~ method.
Kclea, = (H~l~r/(CHo)) t/2
.ClearSky~ Radiation
CleerSky~
Output
172
H. SUEHRCKEandP. G. MCCORMICK
2.3 Assumptions, errors, and heat capacitance effects There are three assumptions implicit in the statement that a solar system operates in an on/off fashion corresponding to clear/cloudy. First, it is assumed that the location where the solar system is situated exhibits bimodal instantaneous radiation statistics, i.e., it is assumed that there is a predominance of two significantly different radiation levels, clear and cloudy. The clear radiation level is associated with moments in time when there is no cloud in front of the sun while the cloudy level corresponds to moments in time when there is a (significant) cloud in front of the sun. The degree of the bimodal character of the instantaneous solar radiation will depend on location. However, if there is a distinct separation between clear and cloudy insolation values, the effect of collector tilt will in general strongly amplify this separation (see Fig. 1). Second, it is assumed that when a cloud is in front of the sun, the output of the solar process is zero due to the threshold of the solar process. Clearly, this is not always the case. For example, a high volume, low pressure head photovoltaic pumping system or an evacuated tube thermal collector may still produce output even when there is a cloud in front of the sun. However, a calculation based on the assumption that output is produced only when there is no cloud in front of the sun will yield a conservative estimate of performance. The reason is that, although the solar process may still be operating during cloudy conditions, its output may be small compared to the clear sky output. This point is illustrated in Fig. 4, which shows the error incurred for low thresholds when only clear sky output is considered (dashed area). Finally, the assumption that the solar process will operate in a strict on/off fashion is certainly not true for solar thermal collectors that have a variety of time constants due to collector heat capacitance (e.g., the time constant for thermal fiat-plate collectors may
1.0
-
rn
=
1.5 t
J 0.5
/ 0.0
threshold i
i
i
0.0 f-
o
I 0.5
Fraction
a
t
i
~
I 1.0
of Time
Fig. 4. The error associated with the assumptionthat the solar system produces no output under cloudy conditions for low thresholds.
range from 1-10 min). However, it is well known that the collector response lag causes the collector output to stay below the equilibrium output during a cloudyclear transition, but above the equilibrium output during a clear-cloudy transition. Hence, there is a strong tendency for the heat capacity effects to cancel. Klein et aL [28] found that the effects of intermittent sunshine can be neglected while Soltau [29] estimated the performance reducing effect of heat capacitance of unglazed swimming pool collectors to be less than 3%. Thus, for the long-term collector output calculation a zero collector time constant may be assumed even for thermal collectors since the net effect of heat capacitance may be neglected. All previous performance prediction methods implicitly made this assumption since none considers the effect of collector time constant. The fact that this assumption becomes apparent with the proposed method merely highlights the fact that some of the other methods have been quite removed from the actual solar system operation. For photovoltaic systems the assumption that the system operates in an on/off fashion is excellent, provided that the system threshold is large enough. A good example is a photovoltaic water pumping system where the threshold is given by the pressure head that the pump has to overcome before flow results. 3. EXPERIMENTAl,VERIFICATION The performance prediction method proposed has been experimentally verified for a 5.7 m 2 flat-plate collector hot water system with 440 litre storage tank. The experimental data are based on a one year performance monitoring carried out in Perth, Western Australia (q~ = - 32 o ) [ 30 ]. The system is not ideally suited to the proposed approach since the selective surface solar collectors exhibit a low threshold and the location experiences air mass values close to 1.0 in the summertime (this reduces the amplifying effect of collector tilt on the bimodal radiation character). Nevertheless, the system has been chosen because of the availability of detailed experimental data. The comparison of predicted and measured results for a thermal solar system do not imply that the use of the proposed performance prediction method is restricted to thermal systems. A schematic of the closed circuit flat-plate collector system is shown in Fig. 5. The flow in the collector loop was controlled by the temperature difference between the collector outlet and the bottom of the storage. Whenever this temperature difference was positive, the collector circulation pump was turned on (the specific controller threshold of the collector was estimated 60 W / m 2 [ 30]). No heat exchanger was employed in the collector loop. In the load delivery loop the flow was controlled by the load delivery temperature. When the load delivery temperature fell below a certain minimum value, Tmi,, the load pump was turned off. The load supply water was drawn offat the top of the storage tank and passed through a heat exchanger, where the energy was transferred to the load. The selective surface collectors were tilted 27 ° to-
173
A performance prediction method Collector
"%
~
OL
Fig. 5. Schematic ofthe flat-platecollector solar system. wards the equator. Prior to the commencement of the performance monitoring the collectors were tested to determine the collector etficiency and the incidence angle characteristic. Using the terminology of the standard fiat-plate collector equation [24], the collector performance parameters were found to be F R ( r a ) , = 0.705 and FRUL = 4.31 W / K m -2. For the incidence angle modifier the equation K,,, = i + bo( 1/cos 0 - 1 ) was employed, where bo was found to be approximately -0.075. Throughout the monitoring the specific collector flow rate was kept constant at 0.02 kg/s m -2. The storage tank was characterised by the product of heat loss coetficient and tank area ( UA)s. The average (UA)s product was measured as 4.26 W / K . Fluctuations of the (UA)s value (less than +0.4 W / K ) were observed because of leakages caused by an unreliable one way flow valve that was installed in the collector loop to prevent reverse thermosiphon flow and because of leakages of the storage tank overpressure relieve valve. Under the assumption of a fully mixed storage tank, the performance of the solar system shown in Fig. 5 can described by the heat balance equation dTs (MG)'-~=
Qu-Q,-QL
(5)
where M is the mass of the storage, Cp the specific heat of the storage fluid and Ts the storage tank temperature. Using the flat-plate collector equation and the concept of the fraction of clear sky, the useful energy transferred to the system can be expressed as O, =f~t~arAckR[(ra)GTd~af -- Um.(Ts - Ta)] ~ where ( r a ) = K , , ( r a ) , . The tank heat loss on the other hand can
be expressed as Qs = ( UA)s( Ts - T'~). Theheat transferred to the load (~L is subsequently denoted by L. Equation (5) has been solved numerically using Euler's method with five minute step size. The method adopted for simulation is the representative day method originally proposed by T. A. Reddy et al.[l 1,31]. In this approach the instantaneous energy balance equation [eqn (5)] is solved numerically for a representative day of the month. The representative day is based on the average day of the month where the radiation fluctuation is suitably superimposed (see also [30]). In this particular case the average day is characterised by the average daily radiation and the average time fraction of clear sky. The effect of fluctuation due to intermittent clouds is superimposed by multiplication of the clear sky performance with the fraction of clear sky. To ensure that the numerical solution corresponds to the equilibrium solution of the system, the storage tank temperature at the beginning of the day is compared with that at the end of the day, i.e., I Ts(t) - Ts(t + 24 h)l < ~. The simulation was repeated with the end temperature as the initial temperature until ~<0.1K. The numerical solution was carried out using the data shown in Table 1. The corrected value of FRU'L that is shown in Table 1 incorporates the pipe heat loss in the collector equation [ 32 ]. The reason that the product FRUL did not remain constant was that the heat loss was affected by wind. The value of FR(ra) remained unchanged. Also shown in Table I is the average fraction of clear sky that was calculated using eqns ( 2 ) and ( 3 ) and the monthly-average diffuse fraction relationship,/td//7 = I - 1.11 /(. For the calculation of the instantaneous horizontalsurface clear-sky radiation, Gc~.... the distribution of radiation over the day[21] was used as described in Section 3. The predicted solar fractions of the proposed method are compared in Table 2 with measured results and results of the ~f chart method. When the solar fractions of Table 2 are compared, it is noted that the measured solar fractions are around six percentage points above the predicted results. This discrepancy is due to stratification in the storage tank, which was not taken into account by either prediction method. It is well known that storage tank stratification increases the performance of thermal systems. For part
Table I. Monthly average values of the input data[30]
January February March April May June July August September October November December
n
/4 [MJ/day]
L [W]
Tm,~ [°C]
T, [°C]
b'RU~ [W/K m-21
f¢~,,
17 47 75 105 135 162 198 228 258 288 318 344
31.13 23.29 20.26 17.59 10.06 9.03 9.22 13.78 16.52 24.20 28.33 31.09
614 602 605 608 610 612 615 611 619 61 I 613 611
52.3 51.7 51.4 51.7 51.7 51.8 52.3 52.5 51.5 51.6 52.3 52.8
25.2 22.2 20.3 18.0 15.8 13.9 11.8 11.9 15.2 15.2 19.1 19.3
5.14 5.01 5.07 5.05 4.83 5.02 5.16 5.08 5.14 5.22 5.13 5.18
0.85 0.58 0.61 0.78 0.45 0.50 0.45 0.61 0.50 0.70 0.75 0.82
174
H. SUEHRCKEand P. G. MCCORMICK Table 2. Measured and predicted solar fractions Solar fractions in percent Measured
~fchart method
f¢~ method
January February March April May June July August September October November December
97 77 71 69 26 23 16 36 44 65 81 88
90 71 65 61 20 13 8 30 40 65 76 81
93 64 63 66 24 20 14 36 39 64 73 78
Average
58
52
53
of October the collector pump was inoperative, causing the measured solar fraction to be smaller than expected from the measured solar radiation. There are two more noticeable exceptions in the solar fractions of Table 2. First, for the month of January the solar fraction predicted byf¢~,, method comes close to the measured value. This however, cannot be credited to the f¢~,~ method, but is simply a consequence of the fact that for the fcaear method energy dumping was neglected. At solar fractions close to 1.0 neglect of energy dumping tends to offset the underestimation of the solar fraction because of the assumption of a fully mixed storage tank. Second, for the summer months December and February thef~ear-method significantly underpredicts the solar fraction. This may be explained from the fact that during the summer months the sun is directly overhead at the test site so that radiation levels below a cloud can still be high enough for a solar collector with low threshold to produce some output. This is particularly significant for February with a clear-sky time fractions of only 0.58, which means that during February, for more than 40% of the time, there were clouds in front of the sun. The ~ f chart method on the other hand, seems to underpredict at very low solar fractions during the months June and July. On an annual basis, however, the solar fractions predicted by the f~ar method and the ~ f chart method are very similar.
4. CONCLUSIONS A simple method for the performance prediction of solar energy systems has been proposed. Using the bimodal character of instantaneous solar radiation on tilted surfaces, many solar systems may be modelled to operate in an o n / o f f fashion corresponding to the clear/cloudy time intervals. The predictions by this new method compare favourably with predictions by the ~ f chart method. However, the f~,~, method is conceptually more simple than the ~ f chart method and may be more realistic since solar collectors see instantaneous solar radiation, not integrated hourly or
daily values. The f~t,,, method has the potential to be simplified.
Acknowledgment--H.S. wishes to acknowledge the financial assistance given by the National Energy Research, Development, and Demonstration Council of Australia. NOMENCLATURE he
collector area [ m 2]
bo constant for incidence angle modifier constant in/~d/H correlation specific heat [ J / K kg- t ] fraction of time, solar fraction average time fraction of clear sky, eqn ( 1) collector heat removal factor collector performance parameter [ W / K m-2 ] FRU;. collector performance parameter incorporating pipe heat loss [W/K m -2] FR(Tah collector performance parameter G instantaneous total (beam + diffuse) horizontal surface solar radiation [W/m 2] average instantaneous total (beam + diffuse) horizontal surface solar radiation [ W / m 2] ab average instantaneous horizontal surface beam radiation [W/m 2] Gbciear average clear sky instantaneous beam horizontal surface radiation [W/m z] Gclcar instantaneous clear sky horizontal surface radiation [W/m 2] Go extraterrestrial horizontal surface solar radiation [W/m 2] GT instantaneous tilted total surface (beam + diffuse) radiation [W/m 2] aTcle~u- clear sky instantaneous radiation incident on tilted collector surface [ W / m 2] average daily total ( beam + diffuse) horizontal surface radiation [MJ/m 2] ~b average daily beam horizontal surface solar radiation [MJ/m 2] nbc~Lr average daily beam clear sky horizontal surface solar radiation [MJ/m 2] Hd/H monthly average daily diffuse fraction /40 daily extraterrestrial horizontal surface radiation [MJ/m z] k clearness index for instantaneous insolation (atmospheric transmittance), k = G~Go k average instantaneous clearness index, k = G/Go kb instantaneous atmospheric transmittance for beam radiation, kb = G~/Go C
G f f~r FR FRU~.
A performance prediction method kd diffuse component of the instantaneous clearness index,/q = Gd/Go kr instantaneous clearness index for tilted surface radiation, kx = GT/Go J~T average instantaneous clearness index for tilted surface radiation, J~r = Gr/Go average daily clearness index,/~ = 14/11o average daily clearness index for clear sky, K~=,
= H~.~a,/Ho KTa incidence angle modifier L load [Wl m air Mass, m = 1/cos 0~, where 0~is the zenith angle, see [27] for details M storage tank fluid mass [kg] (MCv) storage tank heat capacity [ kJ / K ] n day number storage tank heat loss rate [W] Qo instantaneous collector output [W] q~ instantaneous thermal heat load [ W ] R ratio between total radiation on tilted and horizontal surface, R = GT/G[-] Rb ratio between beam radiation on tilted and horizontal surface, Rb = Gt.b/G~ t time ro ambient temperature [°C] r'. ambient temperature of storage tank [°C] ri collector fluid inlet temperature [°C] Tmi~ minimum useful load temperature [°C] rs storage tank temperature [ °C] Ut. total collector heat loss coefficient [ W / K m -2] ( UA)s storage tank heat loss product [ W / K ] O. extinction coefficient for clear sky beam radiation constant 0 incidence angle for beam radiation measured from the surface normal [radian ] O, zenith angle, cos 0~ = cos 6 cos ¢ cos w + sin 6 sin [ radian ] transmittance-absorptance product (~a). transmittance-absorptance product for normal incident solar radiation latitude angle [ radian ] x constant in radiation distribution × = Ho fGoexp( - ~ m ) d t
Os
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