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Daily utilizability from hourly cumulative frequency curves
Renewable Ener,qy, Vol. 4, No. 8, pp. 891 895, 1994
~ Pergamon
Copyright t ' 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960 1481/9457.00+0.00
0960-1481(94)00052-2
DAILY UTILIZABILITY FROM HOURLY CUMULATIVE FREQUENCY CURVES J. C H A N D R A S E K A R A N a n d S. K U M A R School of Energy, Environment and Natural Resources, Madurai Kamaraj University, Madurai 625 021, India
(Received 28 March 1994 ; accepted 26 May 1994) Abstract~This paper describes a simple method of estimating daily utilizability (~) on a horizontal surf:ace at Madras and Kodaikanal (two locations in South India). The daily utilizability has been derived from the cumulative frequency curves of I/[m using five years of hourly global radiation data on a horizontal surface. The results of the analysis have been compared with data and the results of Klein at these two locations, in terms of standard deviation and relative standard deviation. The results show that the proposed method is easy to use and is more accurate than Klein's method.
N n
INTRODUCTION
2Z(l--lc) --
Utilizability is the fraction of the incident solar radiation that can be converted into useful heat by a collector having FR (z~) = 1 and the daily utilizability is the sum, over all hours and all days for a month, of the radiation on a surface (horizontal or tilted) that is above a critical level, divided by monthly average radiation falling on that surface. In the design of solar hot water systems, a knowledge of useful energy obtained from the incident solar radiation at the location of application is required. For calculating the amount of useful energy, it is necessary to determine daily utilizability. Klein [1] and Theilacker and Klein [2], making use of Liu and Jordan's generalized K curves [3], generated the daily utilizability curves for different values of K', lc and collector tilts, and proposed analytical equations for computing the daily utilizability function. A similar correlation has been proposed by Collares-Pereira and Rabl [4,5], which also includes concentrating solar collectors. Evans et al. [6], based on actual measured radiation data from several U.S. locations, have also developed an empirical correlation for the daily utilizability factor, wherein an explicit value has been taken, not of the critical radiation lc, but of the collector performance parameters FR (zc0 and FRUL. The utilizability for a horizontal surface can be calculated from long-term data by assuming a critical radiation Ic, using the following expression :
+
.~. ,,
(1)
Y~ZI where + indicates that the summation is extended over all hours in which ( I - I c ) is positive. F o r tilted surfaces the total radiation incident (IT) can be calculated using the isotropic sky model of Liu and Jordan [3] as
IT = (l-- ld)Rb + ld
(1 +cos//) 2
+ lp
(1 -cos//) 2
(2)
and the utilizability can then be estimated by substituting I = IT in eq. (1). The monthly average daily utilizability from the Klein [1] method is of the form
where A = 7 . 4 7 6 - 20.000/£+ 11.188/£ 2, B = -- 8.562 + 18.679/£-- 9.948£ 2, C = - 0.722 + 2.426K'+ 0.439/£ 2. The present analysis for the estimation of ~ proposes a much simpler method. The main object of this article is to describe the 891
892
J. CHANDRASEKARAN and S. KUMAR
methodology used to estimate the daily utilizability function directly from the cumulative frequency curves using long term data. This has been done by using measured data of hourly global radiation on a horizontal surface at two tropical locations in India (Madras and Kodaikanal). The results have been compared with data obtained by the summation using eq. (1), and also by the method proposed by Klein [1]. An illustrative example shows the procedure to calculate daily utilizability and the useful energy delivered by flat plate solar collectors placed on a horizontal surface.
DATA USED The data of hourly global solar radiation for the years 1983 to 1987 pertaining to Madras (latitude 13.00°N and longitude 18.18°E) and Kodaikanal (latitude 10.23°N and longitude 77.47°E) were obtained from Indian Meteorological Department. The data shows 17,076 h of global hourly radiation data for Madras, and 17,280 h for Kodaikanal.
METHODOLOGY The methodology that was followed to obtain the daily utilizability using cumulative frequency curve is as follows : • the ratio 1lira is calculated using the measured values of global horizontal radiation (/) at Madras and Kodaikanal ; • the cumulative frequency curves are drawn between I/[m and fractional t i m e f for all 12 months of the year for Madras and Kodaikanal. Liu and Jordan [3] have shown that the hourly utilizability factor can be obtained by graphically integrating the cumulative frequency curve for daily total radiation. A similar method is followed here to derive the expression for daily utilizability, except that the cumulative frequency curve has been drawn by arranging the ratio of the hourly global radiation (considering all the hours of the day for a month) to the monthly average hourly global radiation on horizontal surfaces. The daily utilizability function O is given by the portion of the distribution curve that is above a critical ratio, defined by I X = =-
Im
and
(4a)
Ic
Arc = [mm"
(4b)
The mathematical expression for • can be written as shown by Reddy [7] and Reddy et al. [8] as
I~
'max
4~ =
(l - f )
dX.
(5)
By modifying the limits of integration, we can write ~) = 1 -
X df - (1 - f ) dX.
(6)
0
The relationship between f and X can be obtained by curve fitting techniques. For example, the linear equation for Madras in January is f = 0.0117+0.5102X.
(7)
By substituting eq. (6) in eq. (5), the daily utilizability function for horizontal surfaces is easily obtained. Thus, q) = I-0.9883Xc+O.2551X~2.
(8)
The same procedure is also followed for other months. The general equation can be represented in a simple form : ep = 1 - L X c + M X ~ ,
(9)
where L and M are constants for each month. Once the daily utilizability values are known, the monthly useful energy delivered by a collector per unit area can be obtained from the following equation : Q = FR(V~)H~PN.
(10)
By using the above procedure, the cumulative distribution curves for each of the 12 months for both locations are drawn. As these curves showed large differences, it was decided to use one curve for each month to estimate L and M. For Madras and Kodaikanal the 12 sets of cumulative distribution curves were drawn and the daily utilizability fraction was estimated. The constants L and M for different months are given in Table 1 for these two locations. Reddy et al. [8] have also studied the daily utilizability fraction for a tropical location (Bangkok) by using monthly cumulative frequency curves. At Bangkok the cumulative frequency curves of hourly total radiation are linear and also close to each other. The same trend has also been observed for Chiang Mai [9], another tropical location in northern Thailand. Equation (9) is valid only for horizontal surfaces.
893
Hourly cumulative frequency curves t'O
Table 1. Constants L and M for Madras and Kodaikanal
O0
Madras Month
L
Kodaikanal L M
M
u ~LEI~
08 07
January February March April May June July August September October November December
0.9883 1.0006 1.0393 1.0480 1.0202 1.0022 0.9930 1.0000 0.9929 0.9819 0.9842 0.9892
0.2551 0.2569 0.2692 0.2796 0.2732 0.2573 0.2517 0.2521 0.2502 0.2470 0.2687 0.2697
0.9555 0.9206 0.9982 0.9649 0.9649 0.9757 0.9661 0.9711 0.9547 0.9522 0.9608 0.9625
0.2302 0.2122 0.2499 0.2355 0.2455 0.2490 0.2457 0.2752 0.2569 0.2587 0.2767 0.2548
Asokan [9] and Reddy and Asokan [10], basing their findings on the Bangkok solar radiation model of Exell [11], have generated monthly cumulative frequency curves of hourly total radiation on Southfacing surfaces of various tilts, and have observed that these curves exhibit the same properties as those on horizontal surfaces, in the sense that the frequency curves can still be assumed linear and are very close to each other. However, the slopes of these curves differ with surface tilt.
IC 0. c I}E
OE 0.5 0-4
0 o.I
,'.o
4°
02 02 O; 0"£ O0
kO
20
3%
2_
/c
Fig. 1. Comparison of utilizability between actual data, Klein's results and the present study, for Madras (March).
5o
Zc Fig. 2. Comparison of utilizability between actual data, Klein's results and the present study, for Kodaikanal (March).
of March at Madras and for Kodaikanal, respectively. The quantitative verification has been done by calculating the standard deviation (SD) and relative standard deviation (RSD), as shown below :
SD=Ilo~(7~d-7~)~I ''2
(lla)
and
RSD= no ~ cb~ ] 3
From the above analysis it is found that the daily utilizability function reduces to a very simple quadratic expression in terms of Ic only. The accuracy of the proposed correlation was compared by calculating the daily utilizability obtained from hour-to-hour summation of eq. (1), and also with the correlation proposed by Klein [eq. (3)]. The results are shown in Figs 1 and 2 for the month
~
0.5 0-4
l1
RESULTS AND DISCUSSION
°°
06
(llb)
The standard deviation values are given in Table 2 for both Madras and Kodaikanal. For Madras, the SD value obtained from the Klein method varies from 0.009 to 0.050, the average being 0.0237, while the RSD values vary from a minimum of 4.9% to a maximum of 17.9%, the average being 9.5%. For the proposed method the SD value varies from 0.008 to 0.031, the average being 0.018, while the RSD value varies from 1.8% to 14.2%, the average being 7.6%. Similarly, for Kodaikanal, a comparison between Klein and actual data shows the average SD to be 0.020 and the average RSD to be 8.3%, while the average SD and RSD values for the proposed correlation are 0.015 and 5.9%, respectively. The results indicate the improved prediction of daily utilizability by the proposed method, as compared to the conventional method of Klein [1]. However, it is important to stress two major factors : firstly, Klein's method uses data from temperate locations, whose cumulative frequency curves are different from those observed in the tropics [12] and, secondly, the proposed correlation has been obtained from the cumulative frequency curves of the location of interest, i.e. Madras and Kodaikanal.
894
J. CHANDRASEKARAN and S. KUMAR Table 2. Standard deviation (SD) and relative standard deviation (RSD) values at Madras and Kodaikanal Madras
Kodaikanal
Klein and data Month January February March April May June July August September October November December
Present study and data
Klein and data
Sd
RSD
SD
RSD
SD
0.009 0.010 0.017 0.022 0.028 0.038 0.050 0.044 0.031 0.016 0.010 0.010
0.059 0.049 0.057 0.059 0.081 0.127 0.179 0.160 0.127 0.081 0.074 0.091
0.026 0.018 0.014 0.008 0.010 0.008 0.016 0.014 0.019 0.019 0.027 0.031
0.113 0.060 0.037 0.018 0.042 0.034 0.120 0.081 0.088 0.066 0.108 0.142
0.043 0.037 0.022 0.017 0.012 0.035 0.029 0.028 0.017 0.012 0.032 0.038
A generalized method for the estimation of daily utilizability is possible for tropical locations when a large number of datasets is available for the tropical regions of the world. Then, based on the monthly average clearness index, cumulative frequency curves could be generated, from which simple equations could be constructed for the estimation of daily utilizability. The use of the proposed correlations is illustrated below by means of an example.
A fiat-plate collector placed horizontally at Madras has F R U L = 4.7 W m 2 °C 1 and FR(Z~) = 0.617. The fluid inlet temperature to the collector is 80°C for all hours of the day and all clays of the month. Calculate the useful energy delivered by the collector for the m o n t h of May ( H = 2 3 . 3 0 M J m -2 and T~mb= 27.5°C). Compare with actual data and the Klein method. The critical level of the collector can be calculated from the following equations : FR UL(T~ - Tamb) FR(Z~) 4.7(80--27.5) to0.617 = 400Wm
2
=
1.44MJm-2h-i
and [m = ] 2 = l ' 9 4 2 M J m - 2 h - j ' therefore, Xc = 0.742.
RSD 0.181 0.135 0.063 0.051 " 0.020 0.104 0.079 0.074 0.029 0.021 0.153 0.186
SD
RSD
0.028 0.023 0.011 0.012 0.006 0.014 0.014 0.009 0.015 0.008 0.009 0.028
0.108 0.073 0.029 0.044 0.020 0.083 0.071 0.042 0.065 0.029 0.033 0.109
The utilizability can be calculated from eq. (9) by substituting the values L and M from Table 1. Thus, = 1 - 1.0202 Xc + 0.2732 X~ = 0.394. The useful energy delivered by the collector for May is then calculated by using eq. (10). Thus, Q = 0.617* 23.30" 0.394* 31.0 Q=
ILLUSTRATIVE EXAMPLE
Present study and data
175.59MJm
2month-~.
The • obtained by using Klein's method [eq. (3)] is 0.416, and that calculated from the data [eq. (1)] is 0.390. Comparison
Present study Data Klein
(b
Q (MJ m -2 month -j)
0.394 0.390 0.416
175.59 173.81 185.39
CONCLUSION A simple method is described in this paper for the estimation of monthly average daily utilizability on a horizontal surface at Madras and Kodaikanal (two tropical locations in South India), using the monthly average hourly cumulative frequency curves. The results obtained have been compared by the hour-tohour summation of actual measured data, and also with the method given by Klein. The proposed methodology could, therefore, be extended for other trop-
895
Hourly cumulative frequency curves ical l o c a t i o n s for w h i c h l o n g - t e r m d a t a is available, a n d is s i m p l e r to use t h a n the K l e i n m e t h o d .
X~ critical level ratio I~/L,, (dimensionless) X~,,~ maximum critical level ratio (dimensionless).
Greek svmbo& Acknowledgements
The authors are grateful to the Indian Meteorological Department, Pune for having supplied the hourly global radiation data for Madras and Kodaikanal. One of the authors (SK) is grateful to the International Centre for Theoretical Physics, Trieste, Italy and the Dipartimento di Energetica, Universita di Ancona, Ancona, ltaly for providing the necessary facilities for completion of this work.
~) monthly average daily utilizability (dimensionless) • ~)~ monthly average daily utilizability using the present study and the Klein method (dimensionless) ~d monthly average daily utilizability using actual data (dimensionless) r~ transmittance absorptance product p ground reflectance [:t slope of the collector with respect to the horizontal
(). NOMENCLATURE A B C f Fr /t
coefficient in eq. (3) coefficient in eq. (3) coefficient in eq. (3) fractional time heat removal factor of the solar collector monthly average daily global radiation on a horizontal surface (MJ m 2) 1 hourly global radiation on a horizontal surface (MJ m 2) 1~ critical radiation level (W m -~) ld hourly diffuse radiation on a horizontal surface (MJ m 2) ~, monthly average global solar radiation on a horizontal surface between 6 a.m. and 6 p.m. (=/~/12)(MJ m :) 1,, hourly extraterrestrial radiation on a horizontal surface (MJ m -~) lr hourly global radiation on a tilted surface (MJ m ~) K daily clearness index (dimensionless) R monthly average daily clearness index (dimensionless) L constant in eq. (9) M constant in eq. (9) N number of days in the month n number of hours nf~ number of data Q monthly useful energy delivered by solar collector (MJ m 2) ,q ratio of monthly average dail}, total radiation on tilted surface to that on a horizontal surface (dimensionless) Rb ratio of the daily beam radiation on a tilted surface to that on a horizontal surface (dimensionless) R,, ratio of radiation on a tilted surface to that on a horizontal surface at noon (dimensionless) SD standard deviation as given by eq. ( l l a ) RSD relative standard deviation as given by eq. (1 lb) UL overall heat loss coefficient of the solar collector ( W m -~C ') T~u,~b ambient temperature ( C ) T, fluid inlet temperature to the collector ( C ) X radiation ratio defined as = I,/Im (dimensionless)
REFERENCES
1. S. A. Klein, Calculation of fiat-plate collector utilizability. Solar Energy 21,393 (1978). 2. J. C. Theilacker and S. A. Klein, Improvements in the utilizability relationships. American Section Of h~ter-
national Solar Energy Societ3 Meet&g Proceedings, Phoenix, AZ, p. 271 (1980). 3. B. Y. H Liu and R. C. Jordan, A rational procedure for predicting the long-term average performance of flatplate solar energy collectors. Sohtr Enerqy 7, 53 (19633. 4. M. Collares-Pereira and A. RaN, Derivation of method for predicting long term average energy delivery of solar collectors. Solar Energy 23, 223 (t9793. 5. M. Collares-Pereira and A. Rabl, A simple procedure for predicting long term average performance of non concentrating and of concentrating solar collectors. Solar Energy 23, 235 (1979). 6. D. L. Evans, T. T. Rule and B. D. Wood, A new look at long term collector performance and utilizability. Solar Enerqy 28, 13 ( 19823. 7. T. A. Reddy, The Deson and Sizing ~?/ Actit:e Solar Thermal Systems. Clarendon Press, Oxford (1987). 8. T. A. Reddy, S. Kumar and G. Y. Saunier, Review o1" solar radiation analysis techniques for predicting longterm thermal collector performance applicability to Bangkok data. Reneuable Energy Rer. J. 7, 56 (1985). 9. C. Asokan, Analysis of the monthly cumulative frequency curves of solar radiation at Chiang Mai and Bangkok. Unpublished Research Study, Energy Technology Division, Asian Institute of Technology, Bangkok (1985). 10. T. A. Reddy and C. Asokan, Daily utilizability correlations for Bangkok and Chiang Mai. Paper presented at the International Conference on Solar and Wind Eneryy Applications, Beijing (1985). 11. R. H. B. Exell, Simulation of solar radiation in a tropical climate with data for Thailand. Asian Institute of Technology Research Report No. 115, RERIC, Bangkok (1980). 12. (3. Y. Saunier, T. A. Reddy and S. Kumar, A monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations. Solar Energy 38, 169 (1987).
~ Pergamon
Copyright t ' 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960 1481/9457.00+0.00
0960-1481(94)00052-2
DAILY UTILIZABILITY FROM HOURLY CUMULATIVE FREQUENCY CURVES J. C H A N D R A S E K A R A N a n d S. K U M A R School of Energy, Environment and Natural Resources, Madurai Kamaraj University, Madurai 625 021, India
(Received 28 March 1994 ; accepted 26 May 1994) Abstract~This paper describes a simple method of estimating daily utilizability (~) on a horizontal surf:ace at Madras and Kodaikanal (two locations in South India). The daily utilizability has been derived from the cumulative frequency curves of I/[m using five years of hourly global radiation data on a horizontal surface. The results of the analysis have been compared with data and the results of Klein at these two locations, in terms of standard deviation and relative standard deviation. The results show that the proposed method is easy to use and is more accurate than Klein's method.
N n
INTRODUCTION
2Z(l--lc) --
Utilizability is the fraction of the incident solar radiation that can be converted into useful heat by a collector having FR (z~) = 1 and the daily utilizability is the sum, over all hours and all days for a month, of the radiation on a surface (horizontal or tilted) that is above a critical level, divided by monthly average radiation falling on that surface. In the design of solar hot water systems, a knowledge of useful energy obtained from the incident solar radiation at the location of application is required. For calculating the amount of useful energy, it is necessary to determine daily utilizability. Klein [1] and Theilacker and Klein [2], making use of Liu and Jordan's generalized K curves [3], generated the daily utilizability curves for different values of K', lc and collector tilts, and proposed analytical equations for computing the daily utilizability function. A similar correlation has been proposed by Collares-Pereira and Rabl [4,5], which also includes concentrating solar collectors. Evans et al. [6], based on actual measured radiation data from several U.S. locations, have also developed an empirical correlation for the daily utilizability factor, wherein an explicit value has been taken, not of the critical radiation lc, but of the collector performance parameters FR (zc0 and FRUL. The utilizability for a horizontal surface can be calculated from long-term data by assuming a critical radiation Ic, using the following expression :
+
.~. ,,
(1)
Y~ZI where + indicates that the summation is extended over all hours in which ( I - I c ) is positive. F o r tilted surfaces the total radiation incident (IT) can be calculated using the isotropic sky model of Liu and Jordan [3] as
IT = (l-- ld)Rb + ld
(1 +cos//) 2
+ lp
(1 -cos//) 2
(2)
and the utilizability can then be estimated by substituting I = IT in eq. (1). The monthly average daily utilizability from the Klein [1] method is of the form
where A = 7 . 4 7 6 - 20.000/£+ 11.188/£ 2, B = -- 8.562 + 18.679/£-- 9.948£ 2, C = - 0.722 + 2.426K'+ 0.439/£ 2. The present analysis for the estimation of ~ proposes a much simpler method. The main object of this article is to describe the 891
892
J. CHANDRASEKARAN and S. KUMAR
methodology used to estimate the daily utilizability function directly from the cumulative frequency curves using long term data. This has been done by using measured data of hourly global radiation on a horizontal surface at two tropical locations in India (Madras and Kodaikanal). The results have been compared with data obtained by the summation using eq. (1), and also by the method proposed by Klein [1]. An illustrative example shows the procedure to calculate daily utilizability and the useful energy delivered by flat plate solar collectors placed on a horizontal surface.
DATA USED The data of hourly global solar radiation for the years 1983 to 1987 pertaining to Madras (latitude 13.00°N and longitude 18.18°E) and Kodaikanal (latitude 10.23°N and longitude 77.47°E) were obtained from Indian Meteorological Department. The data shows 17,076 h of global hourly radiation data for Madras, and 17,280 h for Kodaikanal.
METHODOLOGY The methodology that was followed to obtain the daily utilizability using cumulative frequency curve is as follows : • the ratio 1lira is calculated using the measured values of global horizontal radiation (/) at Madras and Kodaikanal ; • the cumulative frequency curves are drawn between I/[m and fractional t i m e f for all 12 months of the year for Madras and Kodaikanal. Liu and Jordan [3] have shown that the hourly utilizability factor can be obtained by graphically integrating the cumulative frequency curve for daily total radiation. A similar method is followed here to derive the expression for daily utilizability, except that the cumulative frequency curve has been drawn by arranging the ratio of the hourly global radiation (considering all the hours of the day for a month) to the monthly average hourly global radiation on horizontal surfaces. The daily utilizability function O is given by the portion of the distribution curve that is above a critical ratio, defined by I X = =-
Im
and
(4a)
Ic
Arc = [mm"
(4b)
The mathematical expression for • can be written as shown by Reddy [7] and Reddy et al. [8] as
I~
'max
4~ =
(l - f )
dX.
(5)
By modifying the limits of integration, we can write ~) = 1 -
X df - (1 - f ) dX.
(6)
0
The relationship between f and X can be obtained by curve fitting techniques. For example, the linear equation for Madras in January is f = 0.0117+0.5102X.
(7)
By substituting eq. (6) in eq. (5), the daily utilizability function for horizontal surfaces is easily obtained. Thus, q) = I-0.9883Xc+O.2551X~2.
(8)
The same procedure is also followed for other months. The general equation can be represented in a simple form : ep = 1 - L X c + M X ~ ,
(9)
where L and M are constants for each month. Once the daily utilizability values are known, the monthly useful energy delivered by a collector per unit area can be obtained from the following equation : Q = FR(V~)H~PN.
(10)
By using the above procedure, the cumulative distribution curves for each of the 12 months for both locations are drawn. As these curves showed large differences, it was decided to use one curve for each month to estimate L and M. For Madras and Kodaikanal the 12 sets of cumulative distribution curves were drawn and the daily utilizability fraction was estimated. The constants L and M for different months are given in Table 1 for these two locations. Reddy et al. [8] have also studied the daily utilizability fraction for a tropical location (Bangkok) by using monthly cumulative frequency curves. At Bangkok the cumulative frequency curves of hourly total radiation are linear and also close to each other. The same trend has also been observed for Chiang Mai [9], another tropical location in northern Thailand. Equation (9) is valid only for horizontal surfaces.
893
Hourly cumulative frequency curves t'O
Table 1. Constants L and M for Madras and Kodaikanal
O0
Madras Month
L
Kodaikanal L M
M
u ~LEI~
08 07
January February March April May June July August September October November December
0.9883 1.0006 1.0393 1.0480 1.0202 1.0022 0.9930 1.0000 0.9929 0.9819 0.9842 0.9892
0.2551 0.2569 0.2692 0.2796 0.2732 0.2573 0.2517 0.2521 0.2502 0.2470 0.2687 0.2697
0.9555 0.9206 0.9982 0.9649 0.9649 0.9757 0.9661 0.9711 0.9547 0.9522 0.9608 0.9625
0.2302 0.2122 0.2499 0.2355 0.2455 0.2490 0.2457 0.2752 0.2569 0.2587 0.2767 0.2548
Asokan [9] and Reddy and Asokan [10], basing their findings on the Bangkok solar radiation model of Exell [11], have generated monthly cumulative frequency curves of hourly total radiation on Southfacing surfaces of various tilts, and have observed that these curves exhibit the same properties as those on horizontal surfaces, in the sense that the frequency curves can still be assumed linear and are very close to each other. However, the slopes of these curves differ with surface tilt.
IC 0. c I}E
OE 0.5 0-4
0 o.I
,'.o
4°
02 02 O; 0"£ O0
kO
20
3%
2_
/c
Fig. 1. Comparison of utilizability between actual data, Klein's results and the present study, for Madras (March).
5o
Zc Fig. 2. Comparison of utilizability between actual data, Klein's results and the present study, for Kodaikanal (March).
of March at Madras and for Kodaikanal, respectively. The quantitative verification has been done by calculating the standard deviation (SD) and relative standard deviation (RSD), as shown below :
SD=Ilo~(7~d-7~)~I ''2
(lla)
and
RSD= no ~ cb~ ] 3
From the above analysis it is found that the daily utilizability function reduces to a very simple quadratic expression in terms of Ic only. The accuracy of the proposed correlation was compared by calculating the daily utilizability obtained from hour-to-hour summation of eq. (1), and also with the correlation proposed by Klein [eq. (3)]. The results are shown in Figs 1 and 2 for the month
~
0.5 0-4
l1
RESULTS AND DISCUSSION
°°
06
(llb)
The standard deviation values are given in Table 2 for both Madras and Kodaikanal. For Madras, the SD value obtained from the Klein method varies from 0.009 to 0.050, the average being 0.0237, while the RSD values vary from a minimum of 4.9% to a maximum of 17.9%, the average being 9.5%. For the proposed method the SD value varies from 0.008 to 0.031, the average being 0.018, while the RSD value varies from 1.8% to 14.2%, the average being 7.6%. Similarly, for Kodaikanal, a comparison between Klein and actual data shows the average SD to be 0.020 and the average RSD to be 8.3%, while the average SD and RSD values for the proposed correlation are 0.015 and 5.9%, respectively. The results indicate the improved prediction of daily utilizability by the proposed method, as compared to the conventional method of Klein [1]. However, it is important to stress two major factors : firstly, Klein's method uses data from temperate locations, whose cumulative frequency curves are different from those observed in the tropics [12] and, secondly, the proposed correlation has been obtained from the cumulative frequency curves of the location of interest, i.e. Madras and Kodaikanal.
894
J. CHANDRASEKARAN and S. KUMAR Table 2. Standard deviation (SD) and relative standard deviation (RSD) values at Madras and Kodaikanal Madras
Kodaikanal
Klein and data Month January February March April May June July August September October November December
Present study and data
Klein and data
Sd
RSD
SD
RSD
SD
0.009 0.010 0.017 0.022 0.028 0.038 0.050 0.044 0.031 0.016 0.010 0.010
0.059 0.049 0.057 0.059 0.081 0.127 0.179 0.160 0.127 0.081 0.074 0.091
0.026 0.018 0.014 0.008 0.010 0.008 0.016 0.014 0.019 0.019 0.027 0.031
0.113 0.060 0.037 0.018 0.042 0.034 0.120 0.081 0.088 0.066 0.108 0.142
0.043 0.037 0.022 0.017 0.012 0.035 0.029 0.028 0.017 0.012 0.032 0.038
A generalized method for the estimation of daily utilizability is possible for tropical locations when a large number of datasets is available for the tropical regions of the world. Then, based on the monthly average clearness index, cumulative frequency curves could be generated, from which simple equations could be constructed for the estimation of daily utilizability. The use of the proposed correlations is illustrated below by means of an example.
A fiat-plate collector placed horizontally at Madras has F R U L = 4.7 W m 2 °C 1 and FR(Z~) = 0.617. The fluid inlet temperature to the collector is 80°C for all hours of the day and all clays of the month. Calculate the useful energy delivered by the collector for the m o n t h of May ( H = 2 3 . 3 0 M J m -2 and T~mb= 27.5°C). Compare with actual data and the Klein method. The critical level of the collector can be calculated from the following equations : FR UL(T~ - Tamb) FR(Z~) 4.7(80--27.5) to0.617 = 400Wm
2
=
1.44MJm-2h-i
and [m = ] 2 = l ' 9 4 2 M J m - 2 h - j ' therefore, Xc = 0.742.
RSD 0.181 0.135 0.063 0.051 " 0.020 0.104 0.079 0.074 0.029 0.021 0.153 0.186
SD
RSD
0.028 0.023 0.011 0.012 0.006 0.014 0.014 0.009 0.015 0.008 0.009 0.028
0.108 0.073 0.029 0.044 0.020 0.083 0.071 0.042 0.065 0.029 0.033 0.109
The utilizability can be calculated from eq. (9) by substituting the values L and M from Table 1. Thus, = 1 - 1.0202 Xc + 0.2732 X~ = 0.394. The useful energy delivered by the collector for May is then calculated by using eq. (10). Thus, Q = 0.617* 23.30" 0.394* 31.0 Q=
ILLUSTRATIVE EXAMPLE
Present study and data
175.59MJm
2month-~.
The • obtained by using Klein's method [eq. (3)] is 0.416, and that calculated from the data [eq. (1)] is 0.390. Comparison
Present study Data Klein
(b
Q (MJ m -2 month -j)
0.394 0.390 0.416
175.59 173.81 185.39
CONCLUSION A simple method is described in this paper for the estimation of monthly average daily utilizability on a horizontal surface at Madras and Kodaikanal (two tropical locations in South India), using the monthly average hourly cumulative frequency curves. The results obtained have been compared by the hour-tohour summation of actual measured data, and also with the method given by Klein. The proposed methodology could, therefore, be extended for other trop-
895
Hourly cumulative frequency curves ical l o c a t i o n s for w h i c h l o n g - t e r m d a t a is available, a n d is s i m p l e r to use t h a n the K l e i n m e t h o d .
X~ critical level ratio I~/L,, (dimensionless) X~,,~ maximum critical level ratio (dimensionless).
Greek svmbo& Acknowledgements
The authors are grateful to the Indian Meteorological Department, Pune for having supplied the hourly global radiation data for Madras and Kodaikanal. One of the authors (SK) is grateful to the International Centre for Theoretical Physics, Trieste, Italy and the Dipartimento di Energetica, Universita di Ancona, Ancona, ltaly for providing the necessary facilities for completion of this work.
~) monthly average daily utilizability (dimensionless) • ~)~ monthly average daily utilizability using the present study and the Klein method (dimensionless) ~d monthly average daily utilizability using actual data (dimensionless) r~ transmittance absorptance product p ground reflectance [:t slope of the collector with respect to the horizontal
(). NOMENCLATURE A B C f Fr /t
coefficient in eq. (3) coefficient in eq. (3) coefficient in eq. (3) fractional time heat removal factor of the solar collector monthly average daily global radiation on a horizontal surface (MJ m 2) 1 hourly global radiation on a horizontal surface (MJ m 2) 1~ critical radiation level (W m -~) ld hourly diffuse radiation on a horizontal surface (MJ m 2) ~, monthly average global solar radiation on a horizontal surface between 6 a.m. and 6 p.m. (=/~/12)(MJ m :) 1,, hourly extraterrestrial radiation on a horizontal surface (MJ m -~) lr hourly global radiation on a tilted surface (MJ m ~) K daily clearness index (dimensionless) R monthly average daily clearness index (dimensionless) L constant in eq. (9) M constant in eq. (9) N number of days in the month n number of hours nf~ number of data Q monthly useful energy delivered by solar collector (MJ m 2) ,q ratio of monthly average dail}, total radiation on tilted surface to that on a horizontal surface (dimensionless) Rb ratio of the daily beam radiation on a tilted surface to that on a horizontal surface (dimensionless) R,, ratio of radiation on a tilted surface to that on a horizontal surface at noon (dimensionless) SD standard deviation as given by eq. ( l l a ) RSD relative standard deviation as given by eq. (1 lb) UL overall heat loss coefficient of the solar collector ( W m -~C ') T~u,~b ambient temperature ( C ) T, fluid inlet temperature to the collector ( C ) X radiation ratio defined as = I,/Im (dimensionless)
REFERENCES
1. S. A. Klein, Calculation of fiat-plate collector utilizability. Solar Energy 21,393 (1978). 2. J. C. Theilacker and S. A. Klein, Improvements in the utilizability relationships. American Section Of h~ter-
national Solar Energy Societ3 Meet&g Proceedings, Phoenix, AZ, p. 271 (1980). 3. B. Y. H Liu and R. C. Jordan, A rational procedure for predicting the long-term average performance of flatplate solar energy collectors. Sohtr Enerqy 7, 53 (19633. 4. M. Collares-Pereira and A. RaN, Derivation of method for predicting long term average energy delivery of solar collectors. Solar Energy 23, 223 (t9793. 5. M. Collares-Pereira and A. Rabl, A simple procedure for predicting long term average performance of non concentrating and of concentrating solar collectors. Solar Energy 23, 235 (1979). 6. D. L. Evans, T. T. Rule and B. D. Wood, A new look at long term collector performance and utilizability. Solar Enerqy 28, 13 ( 19823. 7. T. A. Reddy, The Deson and Sizing ~?/ Actit:e Solar Thermal Systems. Clarendon Press, Oxford (1987). 8. T. A. Reddy, S. Kumar and G. Y. Saunier, Review o1" solar radiation analysis techniques for predicting longterm thermal collector performance applicability to Bangkok data. Reneuable Energy Rer. J. 7, 56 (1985). 9. C. Asokan, Analysis of the monthly cumulative frequency curves of solar radiation at Chiang Mai and Bangkok. Unpublished Research Study, Energy Technology Division, Asian Institute of Technology, Bangkok (1985). 10. T. A. Reddy and C. Asokan, Daily utilizability correlations for Bangkok and Chiang Mai. Paper presented at the International Conference on Solar and Wind Eneryy Applications, Beijing (1985). 11. R. H. B. Exell, Simulation of solar radiation in a tropical climate with data for Thailand. Asian Institute of Technology Research Report No. 115, RERIC, Bangkok (1980). 12. (3. Y. Saunier, T. A. Reddy and S. Kumar, A monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations. Solar Energy 38, 169 (1987).