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The applicability of diffuse solar radiation models to Huntsville, Alabama
Solar Energy Vol. 38, No. 1, pp. 55-57, 1987
0038-092X/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.
Printed in the U.S.A.
THE APPLICABILITY OF DIFFUSE SOLAR RADIATION MODELS TO HUNTSVILLE, ALABAMA GLADIUSLEWJS Department of Mechanical Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, U.S.A.
(Received29 May 1986; accepted31 July 1986) Raleigh, NC) to derive the following season-dependent correlation:
INTRODUCTION
For many solar designs and applications it is necessary to have knowledge of both the direct and diffuse components of the incident solar radiation. Whereas considerable information exists for global solar radiation on horizontal surfaces, only few data are available for diffuse solar radiation. The purpose of the present work is to test the applicability of correlations for the estimations of diffuse radiation against measured values in one location: Huntsville, AL (latitude: 34°39'N; longitude: 86°46'W; altitude: 190 m). The correlations for estimating horizontal diffuse radiation belong to two types. The first type expresses the monthly average daily diffuse radiation fraction, Qo/Qr, as a function of the monthly average daily clearness index, QT/Qo. The second type expresses the fraction Qo/Qr or QD/Qo as a function of the monthly averaged daily values of bright sunshine hours, n, and maximum possible sunshine hours, N.
Q---P' QrD= 0"775 + 0"347 (1-'~0)(°~"- 90°) - [0.505 + 0.261 (1Y0)(~os- 90°) ] x cos[2(~oor -
0.9)]
(3)
for tos ~ 90° during February-April and AugustOctober; cos ~ 100° during May-July; o~ ~ 80° during November-January. The quantity 2(Qr/Qo 0.9) is in radians. TYPE II C O R R E L A T I O N S
Iqbal[5] used data obtained from three locations in Canada (Toronto, Goose Bay and Montreal) to propose the correlation 079,
TYPE 1 C O R R E L A T I O N S
Liu and Jordan[l] developed a statisticallybased correlation from results obtained from one station, Blue Hill, MA. Klein[2] developed the following mathematical relation for the correlation:
0635
By eliminating Qr, he then used the same body of data to obtain Q--P-D 0 ' 1 6=3 Q 0
+ 0"478 ( N ) -
0"655 ( N ) 2"
(4)
Qo 1.390 _ 4.027 ( Q r ) (Qr~ 2 O--~ = Qoo + 5.531 \Qoo] -
3.108 \Q'o/
for 0.4 -< Kr-< 0.5.
Hay[6] considered the relationship between radiation before and after multiple reflections between the earth and the cloud cover. These relationships are (a) for global radiation
(1)
Page[3] developed correlations between daily total and diffuse radiation for ten widely-spread sites in the 40°N to 40°S latitude belt and obtained the following relationship: and (b) for diffuse radiation Q--£ =
Qr
1.00-
1.13 Q"---~ Qr .
(2)
Qo-Qb=Qr,[pa(N )
Collares-Pereira and Rabl[4] used data obtained from five locations in the US (Albuquerque, NM; Fort Hood, TX; Livermore, CA; Maynard, MA and
+pc(I-N)
] .
(6)
Assuming pa = 0.25, Pc = 0.60 and using data from stations in Canada, Hay proposed the following 1o55
56
G. LEWZS Table 1. Relevant meteorological and solar radiation data for Huntsville, AL (latitude: 34°35'N; longitude: 86°46'W; altitude: 190 m) Month
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
Q (a) T (MJm- 2day" I )
(MJm- 2day- i )
(h)
18.58 23.64 30.45 36.27 40.04 41.26 39.78 35.94 31.86 23.60 18.43 16.57
5.60 5.40 7.80 8.60 9.30 9.60 9.80 8.60 8.21 7.00 5.60 5.40
7.49 10.92 13.93 18.00 20.90 22.43 21.38 20.34 15.19 13.82 9.18 7.92
~(c)
Qo (c)
n(b)
Measured
values;
Source:
reference
[14]
(b)
Measured
values;
Source:
reference
[15]
(c)
Calculated
using
relationships
(h) lO.01 10.83 11.93 13.01 13.92 14.32 13.95 13.06 12.26 10.89 I0.01 9.67
(a)
given
(c)
in r e f e r e n c e
75.04 81.24 89.43 97.54 104.39 107.42 104.68 97.97 91.93 81.68 75.09 72.49
[16]
Table 2. Measured and estimated values of diffuse solar radiation for Huntsville, AL (latitude: 34°39'N; longitude: 86°46'W; altitude: 190 m) Month
Q ca) (MJm D2 day -i )
(i)
QDeCb) (2)
using equation (3)jm_ j M (z (4) I day- I)
Ic567
I Jan Feb Mar iApr LMay June July Aug Sept Oct Nov Dec
3.46 4.41 5.69 6.73 7.38 7.53 7.29 6.49 5.93 4.21 3.42 3.08
3.83 5.44 6.13 8.03 10.53 11.58 11.39 10.79 7.68 5.08 3.33 3.54
4.08 5.22 6.74 7.91 8.57 8.64 8.41 7.33 7.00 4.68 4.01 3.64
3.83 5.44 6.13 8.03 10.63 11.58 11.39 10.79 7.68 5.08 3.33 3.54
4.18 5.63 6.10 7.01 7.58 7.81 7.03 6.94 6.03 4.73 4.15 3.73
4.05 5.25 6.52 7.75 8.56 8.86 8.46 7.70 6.79 5.18 4.05 3.63
RMSE c¢)
2.40
1.71
3.32
2.27
1.62
(a) M e a s u r e d data [corrected shadow b a n d m e a s u r e m e n t s , with respect to the total diffuse r a d i a t i o n on an even cloudy day]; Source: reference [14] (b) E s t i m a t e d
(C)
data
RMSE = I ~(Qge12- QD)z
Applicability of diffuse solar radiation models c a t i o n - i n d e p e n d e n t relationship: Ob
0.9702 + 1.6688 ( Q r ~
Q--~r =
- 21.3030 {Q'r~ 2
\ O o/ + 51.2880 \Qoo,]
\ Qoo]
- 50.0810 \Q--0/
+ 17.5510 \Q-oo/ "
(7)
The solution procedure is, for a given Qr, obtain Q~ from eqn (5), Q b from eqn (7) and finally Qo from eqn (6). In the present work, p was put equal to 0.3. The procedure was to obtain estimates for Qo using eqns (1)-(7) and test their accuracy with respect to the measured data for Huntsville, AL. All the relevant data for this location are given in Table 1 while the results of the statistical method of cornparison (computation of the root mean squared error, RMSE) are presented in Table 2. It is seen that the measured data agree best with the correlations due to Page and Hay. (An arbitrary but plausible index of acceptability of M S E < 2.00 was adopted.) The correlation due to Hay, however, suffers from being too complex for routine design calculation purposes. The correlation due to Page, on the other hand, is very simple to use. It is as well to note, in addition, that the Page correlation has found widespread applicability in a n u m b e r of locations in Canada[7], India[8, 9], Israel[10], Z i m b a b w e [ l l ] , Australia[12] and the U n i t e d States[l 3].
Qo monthly average extraterrestrial daily radiation incident on a horizontal surface, MJ m -2 day-1 O,, monthly average daily diffuse radiation incident on a horizontal surface, MJ m-2 day-1 Q ~. monthly average dally global radiation emerging from the atmosphere before striking the ground, MJ m-2 Qb day - ' monthly average daily diffuse radiation emerging from the atmosphere before striking the ground, MJ m-2 day-1 ~o, sunset hour angle for a horizontal surface, degrees p ground albedo 00 cloudless sky albedo pc cloud albedo REFERENCES
1. B. Y. H. Liu and R. C. Jordan, The interrelationship 2. 3.
4.
5. 6. 7. 8.
CONCLUSION
The main conclusion of the present work is that the diffuse solar radiation incident upon horizontal surfaces in Huntsville, A L may be estimated from the following correlation, due to Page[3]: Q_._.~o= 1 . 0 0 Qr
1.13 Q__._r. Q0
NOMENCLATURE
C cloudcover or fraction of daytime sky covered by clouds n monthly average daily number of bright sunshine hours, hours N monthly average daily length of day, hours Qr monthly average daily global radiation incident on a horizontal surface, MJ m - 2 day - 1
57
9. 10. 11. 12. 13. 14. 15. 16.
and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). S. A. Klein, Calculation of monthly average insolation on tilted surfaces. Solar Energy 19, 325-329 (1977). 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 to 40°S. Proc. of UN Conf. on New Sources of Energy, Rome, Italy (1961). M. Collares-Pereira and A. Rabl, The average distribution of solar radiation. Correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22, 155-164 (1979). M. Iqbal, Correlation of average diffuse and beam radiation with hours of bright sunshine. Solar Energy 23, 169-173 (1979). J. E. Hay, Calculation of monthly mean solar radiation for horizontal and inclined surfaces. Solar Energy 23, 301-307 (1979). M. Iqbal, A study of Canadian diffuse and total solar radiation data. I: Monthly average daily horizontal radiation. Solar Energy 22, 81-86 (1979). N. K. O. Choudhury, Solar radiation at New Delhi. Solar Energy 7, 44-52 (1963). V. Modi and S. P. Sukhatme, Estimation of daily total and diffuse insolation in India from weather data. Solar Energy 22, 407-411 (1979). G. Stanhill, Diffuse, sky and cloud radiation in Israel. Solar Energy 10, 96-101 (1966). G. Lewis, Diffuse irradiation over Zimbabwe. Solar Energy 31, 125-128 (1983). D. J. Norris, Solar radiation on inclined surfaces. Solar Energy 10, 72-77 (1966). F. Vignola and D. K. McDaniels, Correlations between diffuse and total global insolation for the Pacific Northwest. Solar Energy 32, 161-168 (1984). Alabama Solar Energy Center, Solar Radiation in Alabama (unpublished); Alabama Solar Energy Center, private communication (1986). National Weather Service, Comparative Climatic Data for the US. National Weather Service, Asheville, NC (1985). J. A. Duffle and W. A. Beckman, Solar Thermal Processes. Wiley, New York (1974).
0038-092X/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.
Printed in the U.S.A.
THE APPLICABILITY OF DIFFUSE SOLAR RADIATION MODELS TO HUNTSVILLE, ALABAMA GLADIUSLEWJS Department of Mechanical Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, U.S.A.
(Received29 May 1986; accepted31 July 1986) Raleigh, NC) to derive the following season-dependent correlation:
INTRODUCTION
For many solar designs and applications it is necessary to have knowledge of both the direct and diffuse components of the incident solar radiation. Whereas considerable information exists for global solar radiation on horizontal surfaces, only few data are available for diffuse solar radiation. The purpose of the present work is to test the applicability of correlations for the estimations of diffuse radiation against measured values in one location: Huntsville, AL (latitude: 34°39'N; longitude: 86°46'W; altitude: 190 m). The correlations for estimating horizontal diffuse radiation belong to two types. The first type expresses the monthly average daily diffuse radiation fraction, Qo/Qr, as a function of the monthly average daily clearness index, QT/Qo. The second type expresses the fraction Qo/Qr or QD/Qo as a function of the monthly averaged daily values of bright sunshine hours, n, and maximum possible sunshine hours, N.
Q---P' QrD= 0"775 + 0"347 (1-'~0)(°~"- 90°) - [0.505 + 0.261 (1Y0)(~os- 90°) ] x cos[2(~oor -
0.9)]
(3)
for tos ~ 90° during February-April and AugustOctober; cos ~ 100° during May-July; o~ ~ 80° during November-January. The quantity 2(Qr/Qo 0.9) is in radians. TYPE II C O R R E L A T I O N S
Iqbal[5] used data obtained from three locations in Canada (Toronto, Goose Bay and Montreal) to propose the correlation 079,
TYPE 1 C O R R E L A T I O N S
Liu and Jordan[l] developed a statisticallybased correlation from results obtained from one station, Blue Hill, MA. Klein[2] developed the following mathematical relation for the correlation:
0635
By eliminating Qr, he then used the same body of data to obtain Q--P-D 0 ' 1 6=3 Q 0
+ 0"478 ( N ) -
0"655 ( N ) 2"
(4)
Qo 1.390 _ 4.027 ( Q r ) (Qr~ 2 O--~ = Qoo + 5.531 \Qoo] -
3.108 \Q'o/
for 0.4 -< Kr-< 0.5.
Hay[6] considered the relationship between radiation before and after multiple reflections between the earth and the cloud cover. These relationships are (a) for global radiation
(1)
Page[3] developed correlations between daily total and diffuse radiation for ten widely-spread sites in the 40°N to 40°S latitude belt and obtained the following relationship: and (b) for diffuse radiation Q--£ =
Qr
1.00-
1.13 Q"---~ Qr .
(2)
Qo-Qb=Qr,[pa(N )
Collares-Pereira and Rabl[4] used data obtained from five locations in the US (Albuquerque, NM; Fort Hood, TX; Livermore, CA; Maynard, MA and
+pc(I-N)
] .
(6)
Assuming pa = 0.25, Pc = 0.60 and using data from stations in Canada, Hay proposed the following 1o55
56
G. LEWZS Table 1. Relevant meteorological and solar radiation data for Huntsville, AL (latitude: 34°35'N; longitude: 86°46'W; altitude: 190 m) Month
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
Q (a) T (MJm- 2day" I )
(MJm- 2day- i )
(h)
18.58 23.64 30.45 36.27 40.04 41.26 39.78 35.94 31.86 23.60 18.43 16.57
5.60 5.40 7.80 8.60 9.30 9.60 9.80 8.60 8.21 7.00 5.60 5.40
7.49 10.92 13.93 18.00 20.90 22.43 21.38 20.34 15.19 13.82 9.18 7.92
~(c)
Qo (c)
n(b)
Measured
values;
Source:
reference
[14]
(b)
Measured
values;
Source:
reference
[15]
(c)
Calculated
using
relationships
(h) lO.01 10.83 11.93 13.01 13.92 14.32 13.95 13.06 12.26 10.89 I0.01 9.67
(a)
given
(c)
in r e f e r e n c e
75.04 81.24 89.43 97.54 104.39 107.42 104.68 97.97 91.93 81.68 75.09 72.49
[16]
Table 2. Measured and estimated values of diffuse solar radiation for Huntsville, AL (latitude: 34°39'N; longitude: 86°46'W; altitude: 190 m) Month
Q ca) (MJm D2 day -i )
(i)
QDeCb) (2)
using equation (3)jm_ j M (z (4) I day- I)
Ic567
I Jan Feb Mar iApr LMay June July Aug Sept Oct Nov Dec
3.46 4.41 5.69 6.73 7.38 7.53 7.29 6.49 5.93 4.21 3.42 3.08
3.83 5.44 6.13 8.03 10.53 11.58 11.39 10.79 7.68 5.08 3.33 3.54
4.08 5.22 6.74 7.91 8.57 8.64 8.41 7.33 7.00 4.68 4.01 3.64
3.83 5.44 6.13 8.03 10.63 11.58 11.39 10.79 7.68 5.08 3.33 3.54
4.18 5.63 6.10 7.01 7.58 7.81 7.03 6.94 6.03 4.73 4.15 3.73
4.05 5.25 6.52 7.75 8.56 8.86 8.46 7.70 6.79 5.18 4.05 3.63
RMSE c¢)
2.40
1.71
3.32
2.27
1.62
(a) M e a s u r e d data [corrected shadow b a n d m e a s u r e m e n t s , with respect to the total diffuse r a d i a t i o n on an even cloudy day]; Source: reference [14] (b) E s t i m a t e d
(C)
data
RMSE = I ~(Qge12- QD)z
Applicability of diffuse solar radiation models c a t i o n - i n d e p e n d e n t relationship: Ob
0.9702 + 1.6688 ( Q r ~
Q--~r =
- 21.3030 {Q'r~ 2
\ O o/ + 51.2880 \Qoo,]
\ Qoo]
- 50.0810 \Q--0/
+ 17.5510 \Q-oo/ "
(7)
The solution procedure is, for a given Qr, obtain Q~ from eqn (5), Q b from eqn (7) and finally Qo from eqn (6). In the present work, p was put equal to 0.3. The procedure was to obtain estimates for Qo using eqns (1)-(7) and test their accuracy with respect to the measured data for Huntsville, AL. All the relevant data for this location are given in Table 1 while the results of the statistical method of cornparison (computation of the root mean squared error, RMSE) are presented in Table 2. It is seen that the measured data agree best with the correlations due to Page and Hay. (An arbitrary but plausible index of acceptability of M S E < 2.00 was adopted.) The correlation due to Hay, however, suffers from being too complex for routine design calculation purposes. The correlation due to Page, on the other hand, is very simple to use. It is as well to note, in addition, that the Page correlation has found widespread applicability in a n u m b e r of locations in Canada[7], India[8, 9], Israel[10], Z i m b a b w e [ l l ] , Australia[12] and the U n i t e d States[l 3].
Qo monthly average extraterrestrial daily radiation incident on a horizontal surface, MJ m -2 day-1 O,, monthly average daily diffuse radiation incident on a horizontal surface, MJ m-2 day-1 Q ~. monthly average dally global radiation emerging from the atmosphere before striking the ground, MJ m-2 Qb day - ' monthly average daily diffuse radiation emerging from the atmosphere before striking the ground, MJ m-2 day-1 ~o, sunset hour angle for a horizontal surface, degrees p ground albedo 00 cloudless sky albedo pc cloud albedo REFERENCES
1. B. Y. H. Liu and R. C. Jordan, The interrelationship 2. 3.
4.
5. 6. 7. 8.
CONCLUSION
The main conclusion of the present work is that the diffuse solar radiation incident upon horizontal surfaces in Huntsville, A L may be estimated from the following correlation, due to Page[3]: Q_._.~o= 1 . 0 0 Qr
1.13 Q__._r. Q0
NOMENCLATURE
C cloudcover or fraction of daytime sky covered by clouds n monthly average daily number of bright sunshine hours, hours N monthly average daily length of day, hours Qr monthly average daily global radiation incident on a horizontal surface, MJ m - 2 day - 1
57
9. 10. 11. 12. 13. 14. 15. 16.
and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1-19 (1960). S. A. Klein, Calculation of monthly average insolation on tilted surfaces. Solar Energy 19, 325-329 (1977). 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 to 40°S. Proc. of UN Conf. on New Sources of Energy, Rome, Italy (1961). M. Collares-Pereira and A. Rabl, The average distribution of solar radiation. Correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22, 155-164 (1979). M. Iqbal, Correlation of average diffuse and beam radiation with hours of bright sunshine. Solar Energy 23, 169-173 (1979). J. E. Hay, Calculation of monthly mean solar radiation for horizontal and inclined surfaces. Solar Energy 23, 301-307 (1979). M. Iqbal, A study of Canadian diffuse and total solar radiation data. I: Monthly average daily horizontal radiation. Solar Energy 22, 81-86 (1979). N. K. O. Choudhury, Solar radiation at New Delhi. Solar Energy 7, 44-52 (1963). V. Modi and S. P. Sukhatme, Estimation of daily total and diffuse insolation in India from weather data. Solar Energy 22, 407-411 (1979). G. Stanhill, Diffuse, sky and cloud radiation in Israel. Solar Energy 10, 96-101 (1966). G. Lewis, Diffuse irradiation over Zimbabwe. Solar Energy 31, 125-128 (1983). D. J. Norris, Solar radiation on inclined surfaces. Solar Energy 10, 72-77 (1966). F. Vignola and D. K. McDaniels, Correlations between diffuse and total global insolation for the Pacific Northwest. Solar Energy 32, 161-168 (1984). Alabama Solar Energy Center, Solar Radiation in Alabama (unpublished); Alabama Solar Energy Center, private communication (1986). National Weather Service, Comparative Climatic Data for the US. National Weather Service, Asheville, NC (1985). J. A. Duffle and W. A. Beckman, Solar Thermal Processes. Wiley, New York (1974).