British Journal of Renewable Energy 01-01, 0032-0037, (2016) An Open Access Journal
British Journal of Renewable Energy www.measpublishing.co.uk/BJRE
General Algorithm for Optimum Tilt Angles Determination at Tropical Region S. Soulayman a, *, W. Sabbagh b a b
Department of Applied Physics, Higher Institute for Applied Sciences and Technology, Damascus, Syria. 2National Energy Research Center, NERC, Damascus, Syria. *
Corresponding author at: S. Soulayman, Department of Applied Physics, Higher Institute for Applied Sciences and Technology, Damascus, Syria, Tel: 00000000000000000
E-mail address:
[email protected] (S. Soulayman)
Abstract:
Article Information:
Is the rule of thumb, which says that solar collector should be orientated towards Equator, is valid for tropical region? The present work focuses on determining the optimum tilt in the tropical region as for Equator facing and for Pole oriented collectors. Moreover, two simple equations are proposed: one is for predicting daily optimum tilt angle while the other is for predicting optimum tilt angle for any number of consecutive days which could be used for calculating weekly, fortnightly and monthly optimum tilt angles. The yearly possible energy gain in relation to horizontal surface was also calculated on the basis of daily and monthly optimum tilts. It was found that the rule of thumb, which says that solar collector should be orientated towards the Equator, is not applicable for a number of consecutive days in the year. The value of this number depends on the latitude value but it reaches its maximum value (6 months) for Equator (latitude = 0o). A comparison with available data is provided.
Keywords: Optimum tilt General formulae Pole facing Equator facing Energy gain Submitted: 01 Nov 2015 Revised form: 17 Dec 2015 Accepted: 21 Dec 2015 Available Online: 00 Dec 2015
1. Introduction The majority of installations of solar collectors are with fixed tilt angles. Therefore, it is often practicable to orient the solar collector at an optimum tilt angle, Bopt, and to correct the tilt from time to time. For this purpose, one should be able to determine the optimum slope of the collector at any latitude, for any surface azimuth angle, and on any day or any number of days during the year. As the goal of this work is to treat this question regarding the tropical region, it is reasonable to restrict ourselves to main available literature concerning directly or indirectly this zone. In this context, Bari [1] proposed a procedure for calculating daily optimum tilt angle, Bopt,d, and optimum tilt angle, Bopt,p, for any number of days for different latitudes (from L=5o to L=19o) of the Philippines. Stanciu and Stanciu [2] proposed a simple formula for determining the optimum tilt of solar collector at latitudes from 0o to 80o. Soulayman and Sabbagh [3] proposed an algorithm for determining Bopt,d and Bopt,p at any latitude, L, and for any direction (surface azimuth angle, G). They applied this algorithm for determining daily Bopt,d, monthly, Bopt,m and for any number of days Bopt,p, in tropical region [4] and introduced new idea regarding Equator facing, North Pole facing and South Pole facing conditions. Nigegorodov et al. [5] presented 12 equations (one for each month) for determining optimum tilt angle for any location that lies between latitude 60° south to 60° north. Oko and Nnamchi [6] studied theoretically the optimum tilt angles for the territory of Nigeria (L=4.86- 13.02oN) and provided also expressions for different optimum tilt angles with respect to the low latitudes. The main objective of the present work is to propose a simple and easy procedure for finding Bopt,d and Bopt,p at tropical region and to review the most relevant available literature results related to this region..
2. Methodology/Algorithm Radiation data are the best source of information for estimating average incidence radiation. Lacking these data, it is possible to use empirical relationships to estimate radiation from hours of sunshine or cloudiness, relative humidity and ambiance temperature, which are widely available from many hundreds of stations in many countries. The main part of
empirical relationships is restricted to hours of sunshine or cloudiness. However, these relationships could be written as: (
)
(1)
where H = monthly average daily radiation on a horizontal surface, H0 = the monthly average daily extraterrestrial solar radiation on a horizontal surface, n = monthly average daily hours of bright sunshine, N = monthly average of the maximum possible daily hours of bright sunshine (i.e., the day length of the average day of the month), C = monthly average daily cloud cover, RH = relative humidity and T = ambiance temperature. Supposing Eq. (1) is applicable for daily values and differentiating it in relation to surface tilt angle B, a nonlinear algebraic equation will be obtained. By equating the left part of the derived equation to zero, the daily optimum tilt angle Bopt,d could be obtained. So, the daily optimum tilt angle, Bopt,d is the solution of the following nonlinear algebraic equation: ( ) ,( (
(
)(
)[
(
)[
( (
*
)+
)
( )(
(
)
(
)(
)] )
( (
)(
) )
)] (
)
*
)(
)+-
(2)
in relation to the surface tilt B, where C(N) is the Nth day correction factor for Sun-Earth average distance: ( )
(
)
(3)
is the solar declination angle which could be calculated using the equation of Cooper [7]: *
(
)
+
(4)
Wss (rad) is the sunset hour angle on tilted surface:
,
[
( )
( )]
(
)
( ) ) , (5)
Wsr (rad) is the sunrise hour angle on tilted surface:
32
[
,
( )
( )]
(
( ))
) (6)
A1, A2, A3 and A4 are functions of solar and collector angles: ( )[ ( ) ( ) ( ) ( ) ( )] (7) ( )[ ( ) ( ) ( ) ( ) ( )] (8) ( ) ( ) ( ) ( ) (9) The analytical solution of Eq. (2), in the case of Wss tilt angle independence, is: –
( )
*
(
)
+
(10)
Eq. (10), a) Eq. (13) gives the results of the algorithm with an absolute deviation < 2o. b) The equation in [1] overestimates slightly Bopt,d during the period starting from 22/3 to 21/9 and underestimates slightly Bopt,d during the period starting from 22/9 to 21/3 with an absolute deviation < 5o; c) The equation in [2] overestimates Bopt,d during the period starting from 22/3 to 21/9 and underestimates Bopt,d during the period starting from 22/9 to 21/3 with an absolute deviation < 20o. So, Eq. (13) could be applied with a very good accuracy with regard to the proposed algorithm while an acceptable agreement is observed between the results of equation in [1] and those of the proposed algorithm.
As for Equator facing (EF) and Pole facing (PF) surfaces in both Northern Hemisphere (NH) and Southern Hemisphere (SH) Wss = -Wsr and sin(G) = 0. So, A3 = 0. Wss is independent of tilt angle on PF surfaces at equator and for EF and PF surfaces for other latitudes in NH and SH for the period starting on 22/9 and ending on 21/3 in NH and for the period starting on 22/3 and ending on 21/9 in SH. For other periods Newton's iteration scheme could be applied for searching Bopt,d. The monthly optimum tilt, Bopt,m is the solution of the following nonlinear algebraic equation: ∑ ( ) ,( (
)(
)[
( *
)[
)
)+
( (
(
)(
( )
)]
)(
(
)
(
(
* )
)(
)
(
)
)] (
)(
)+-
(11)
where the summation covers the month in consideration. In the case of Wss tilt angle independence, the analytical solution of Eq. (11) is: – [∑ ( ) ( ) ∑ ( ) ( ) ( )], (12) This case takes place on Equator all over the year and in NH, for the period starting on 22/9 and ending on 21/3, and in SH, for the period starting on 22/3 and ending on 21/9. For other periods Newton's iteration scheme could be applied for searching Bopt,m.
2.1. General Formulae for Predicting Optimum Tilts When applying the above mentioned algorithm on the latitudes of the tropical region and analyzing the obtained results as a function of δ and L it was found that Bopt,d (o) can be calculated using the following equation: (13) Moreover, it was found that Eq. (10) could be applied with a high accuracy for determining Bopt,d all over the year. The absolute difference between the results of Eq. (10) and precise results does not exceed 0.5 o. Figure 1 shows the results of applying Eq.(10) and Eq. (13) in determining Bopt,d for three different latitudes. It is seen from Figure 1 that the differences between the results of these two equations are negligible. By integrating Eq. (13) over several consecutive days or any number of days starting from day number N1 and ending on day number N2 inclusively. Then, dividing the obtained result by the number of days included in the studied period, the optimum tilt over this period, Bopt,N1-N2, is determined. So, Bopt,N1-N2 could be determined using the following equation: *
(
)
+
*
(
)
+
(14)
3. Results and Discussions 3.1. Optimum Daily Tilt Bari [1] proposed a sixth order polynomial equation for calculating Bopt,d for different latitudes of the Philippines (L=5o, 7o, 9o, 11o, 13o, 15o, 17o, 19o) where the constants of this equation are latitude dependent. Bari [1] explained the procedure for calculating Bopt,d at intermediate latitudes on an example. Recently Stanciu & Stanciu [2] found that when applying Hottel & Woertz model for estimating the incident solar radiation for flat plate collector at latitudes from 0o to 80o, the Bopt,d should be computed as simply as Bopt,d = L − δ and is function of both latitude, L, and solar declination angle, . When applying Eq. (13), that in [1], that in [2] and the proposed algorithm (Eq. (10)) in calculating Bopt,d for L= 5o and L=15o the results given in Table 1 are obtained. It is seen from Table 1 that regarding the precise results of
Figure 1: Daily optimum tilt Bopt.d at different latitudes. Upper curve corresponds L = 15o, middle curve corresponds L = 0o and lower curve corresponds L = -15o.
3.2. Optimum Tilt Over a Period Bari [1] proposed a sixth order polynomial equation with latitude dependent constants for calculating Bopt,p for any number of consecutive days and for different latitudes of the Philippines (L= 5o, 7o, 9o, 11o, 13o, 15o, 17o, 19o) and gave two examples for determining Bopt for intermediate latitudes. As Bari [1] obtained his equation by integrating the other one proposed by him for calculating the daily optimum tilt over the required period, he should not change the constants of daily optimum tilt equation, for L= constant, contrary to his proposal. When applying Eq. (14), and Eq. (12) of the proposed algorithm in calculating Bopt, N1-N2 for different latitudes in the tropical zone, it was found that Eq. (14) gives the results of the algorithm with an absolute deviation < 2o. Therefore, the comparison between the results of the present work and those of Eq. (12) with regard to different periods will not be considered as the results are approximately the same. Eq. (14) is used to obtain the optimum tilt angles Bopt,p over different periods and different latitudes in the tropical zone. At any latitude of this zone, one can distinguish two main characteristic periods of solar collector orientation: the first corresponds to Equator facing period (EFP) and the second corresponds to Pole facing period (PFP). The method of determining these two periods is described in [8]. Bopt,N1-N2 values over these two periods are presented in Table 2 where Bopt,N1-N2 positive values are related to Equator facing situation while negative values are related to Pole facing case. N1 and N2 represent the day numbers of the beginning and the end of the characteristic periods, at which solar collectors should be orientated toward the Pole while (N2 +1) and (N1 1) represent the day numbers of the beginning and the end of the characteristic periods, at which solar collectors should be orientated toward the Equator. In this context, a comparison with results of [1] for different latitudes of the Philippines and for different periods was planned, but when applying the equation in [1] for calculating the Bopt,N1-N2, the obtained results were found to be inadequate. This could be explained by a typescript error in the table of coefficients of [1]. The above mentioned hypothesis was proved by comparing the values of Bopt,N1-N2 calculated by [1] in two examples [one for period (February 10 to March 10 at L=16o ) and the other for period
33
(December 10 to January 20 at L=13o)] with those obtained in the present work. The values of Bopt,N1-N2, given in [1] are 27o and 41o respectively. Table 1: A comparison between different approaches in calculating B opt,d( o) Eq(10) [1] [2] Date (o) L = 5o
When comparing these values with those of the present work, namely, 29.35º and 44.02º, a good agreement is observed.
Eq (13)
Eq(10)
[1]
[2]
Eq (13)
L = 15o
1/1
-23.01
38.1
33.65
28.01
38.68
46.9
42.72
33.01
48.04
15/1
-21.27
35.8
32.16
26.27
36.11
44.8
41.15
31.27
45.47
1/2
-17.52
30.9
27.60
22.52
30.57
40.1
36.73
27.52
39.93
15/2
-13.29
25
22.10
18.29
24.34
34.5
31.41
23.29
33.70
1/3
-8.29
17.7
15.54
13.29
16.97
27.5
24.98
18.29
26.33
15/3
-6.76
15.3
8.41
11.76
14.71
23.2
17.91
16.76
24.07
1/4
4.02
0
-0.36
0.98
-1.19
8.5
9.10
5.98
8.17
15/4
9.41
-9.9
-7.16
-4.41
-9.15
0
2.18
0.59
0.21
1/5
14.90
-18.1
-13.95
-9.90
-17.2
-7
-4.80
-4.9
-7.88
15/5
18.79
-23.7
-18.69
-13.79
-22.98
-14.7
-9.72
-8.79
-13.62
1/6
22.04
-28.1
-22.58
-17.04
-27.77
-19.4
-13.80
-12.04
-18.41
15/6
23.31
-29.8
-24.06
-18.31
-29.65
-21.2
-15.38
-13.31
-20.29
1/7
23.12
-29.5
-23.76
-18.12
-29.37
-20.9
-15.13
-13.12
-20.01
15/7
22.80
-27.3
-21.78
-17.80
-28.89
-18.6
-13.11
-12.80
-19.53
1/8
17.91
-22.3
-17.38
-12.91
-21.69
-13.2
-8.59
-7.91
-12.33
15/8
13.78
-16.2
-12.33
-8.78
-15.60
-5
-3.407
-3.78
-6.245
1/9
7.72
-7
-4.87
-2.72
-6.66
2.9
4.25
2.28
2.70
15/9
2.22
1.7
1.979
2.78
1.47
11.7
11.23
7.78
10.83
1/10
-4.22
11.7
10.08
9.22
10.95
21.7
19.41
14.22
20.31
15/10
-9.60
19.9
16.95
14.60
18.89
29.6
26.28
19.60
28.25
1/11
-15.36
28.1
24.40
20.36
27.40
37.5
33.61
25.36
36.76
15/11
-19.15
33.2
29.32
24.15
32.98
42.3
38.36
29.15
42.34
1/12
-22.11
37
33.09
27.11
37.34
45.9
41.96
32.11
46.70
15/12
-23.34
38.5
34.43
28.34
39.15
47.3
43.28
33.34
48.51
Table 2: Optimum tilt over characteristic periods L (o) N1 N2 Bopt,N1-N2 (o)
Bopt,(N2+1)-(N1-1) (o)
23
161
183
-11.30
23.89
20
141
204
-13.63
25.75
15
122
223
-16.00
25.80
10
107
237
-18.18
24.84
5
94
251
-20.09
23.59
0
81
263
-21.90
21.95
-5
69
275
-23.64
19.98
-10
56
289
-24.88
18.03
-15
41
303
-25.83
15.78
-20
22
322
-25.75
13.21
-23
2
349
-23.25
8.93
23
161
183
-11.30
23.89
3.3. Monthly Optimum Tilt When trying to divide the characteristic periods with regard to calendar months, it was found that parts of the calendar months at the beginning and the end of each characteristic period are included (see some cases in Table 2). So, when calculating the monthly optimum tilt, Bopt,m, for Equator facing, NP facing and SP facing solar collectors, the results for these months should reflect this fact and indicate the number of days corresponding to each of the suitable orientations. Here it should be
mentioned that this fact is ignored by all other authors, except [4], when treating this subject. Nijegorodov et al. [5], Idowu et al. [9] and Oko & Nnamchi [6] proposed sets of equations for calculating Bopt,m. When comparing their results with the results of the present work at different latitudes it was found that the results based on the set of equations in [6] are not compatible with the results of set of equations in [5] and [9] and those of the present work and they could not be justified. Therefore, the results of set of equations in [6] are not included in Table 3 which contains the results of set of equations in [5] and [9] and those of the present work. From Table 3 it is seen that the results of set of equations in [9] are too approximate in relation to those of [5] and those of the present work. A relatively very good agreement is observed between the results of the present work and those of [5]. Here it should be mentioned that, for each of the studied latitudes, the present work gives two values of different signs for optimum tilt angle for two months (see Table 3). For L=10o these months are April and September while for L=20o these months are April and August. In addition, each tilt angle value is accompanied by a number in brackets. This should be read as follows: first value relates to the number of days in brackets for which the orientation of solar collector coincides with the anterior period while the second value relates to the number of days in brackets for which the orientation of solar collector coincides with the posterior period.
4.
Comparison with Available Data
Wasiri et al. [10] stated that, for Kano, Nigeria L=12.1o N, Bopt,m for months of November to April were found to be 35o, 45o, 35o, 25o, 15o and 15o respectively. These results, except April, coincide well with the results of the present work (38.6o, 44.4o, 41.2o, 30.4o, 14.8o and 2.8°(10) -4.8° (8)).
34
Eke [11] found for Zaria, Kaduna State, Nigeria L= 11.13o N that Bopt,m values are 26.5o, 24.5o, 10.0o, 19.5o, 26.0o, 30.0o, 24.0o, 21.0o, 11.5o, 19.5o, 27.0o and 30.0o, for the months of January to December, respectively. When comparing Eke [10] results with those of Bopt,m obtained in the present work (40.4o, 29.6o, 14.1o, 4.5o (15) -4.3o (15), -16.2o, -22.3o, -19.5o, -8.4o, -0.5o (2) +8.3o (28), 24.7o, 37.8o, 43.6o ) and taking into consideration those of [9] three comments could be given: a) The results of [11] are too approximate regarding the period starting from 22/9 to 21/3; b) The sign from May to August, in results of [11], should be negative indicating that solar collector is orientated towards the North Pole during this period; c) The results of [11] for April and September could not be understood. Uba & Sarsah [12] found, for WA, Ghana (L=10.01o), that collector tilt should be changed three times a year: January – March (26.4o), April – August (29.7o) and September – December (25.9o). When calculating Bopt,p for these periods approximately using the proposed methodology of the present work one obtains: January 1st – April 7th (23.8o), April 8th – September 4th (-16.8o) and September 5th – December 31 (28.8o). So, the results of this work are in a good agreement with those of [12] for the first and third periods while the result of [12] should be corrected for the second period because 1) The solar collector should be orientated towards the North Pole during April – August period and 2) The proposed value by [12] is too high. Ng et al. [13] found, for Bangi, Malaysia (L = 3°), that Bopt,m = 22o, 16o, 5o, -8o, -18o, -22o, -21o, -11o, 0o, 11o, 19o, 24o for January to December respectively. When calculating Bopt,m for theses months using the proposed methodology of the present work one obtains that Bopt,m = 32.6°, 21.8°, 7.7°(26)-1.3°(5), -10.7°, -24.0°, -30.1°, -27.3°, -16.1°, -4.5°(15) +4.3° (15), 16.9°, 30.3°, 35.9° for January to December respectively. So, even the orientation is determined correctly in [13], Bopt,m is underestimated for October to March and overestimated for April to September. Table 3: A comparison between different approaches in calculating Bopt,m (o) [5] [9] Eq. (14) [5] [9] Eq. (14) Month L = 10o L = 20o Jan. Feb. March
37.9 26.7 14
35 25 10
April
0
10
May June July
-14.7 -25.3 -21.1
-15 -15 -15
39.2 28.4 12.8 1.7(7); -3.7(23) -17.4 -23.5 -20.7
Aug.
-7.3
-5
-9.6
Sep.
8
25
Oct. Nov. Dec.
22 34.3 42.7
25 35 35
-1.1(4); 7.6(26) 23.5 36.6 42.5
46.8 36.4 24
45 35 20
10
20
-5.4 -16.6 -12.2
-5 -5 -5
48.6 37.7 22.2 6.4(24); -1.5(6) -8.1 -14.2 -11.4 -4.0(18); 3.6(13)
2.4
5
18
35
15.9
32 43.6 51.4
35 45 45
32.8 46.0 51.8
Kamanga et al. [14] proposed a study to determine the optimum tilt angle for installing photovoltaic solar panels in Zomba district, Malawi. The study was conducted at Chancellor College Meteorological Station in Zomba district, Malawi, located at latitude of -15.387°. The goal of [14] was to determine the optimum monthly tilt angles of PV solar panels and the seasonal adjustments needed for the panels in order to collect maximum solar radiation throughout the year. In [14], global solar radiation on four tilted Equator-facing surfaces of 0°, 15°, 20°, and 25°, was measured. Using obtained data, it was concluded that the optimum tilt angle is 0° from October to February and 25° from March to September and that two seasonal tilt adjustments were suggested: one is at the end of February and the other is at the end of September. The yearly optimum tilt angle for north facing surfaces was proposed to be 25° [14]. When providing Bopt,m and Bopt,p calculation for the same region in [14] using the procedure proposed in the present work, the following results were obtained:
1)
Bopt,m = 15.4°, 6.1° (25) -1.0° (3), -10.9°, -27.9°, -41.2°, -47.3°, -44.5°, -33.3°, -17.2°, -4.1° (15) +4.2° (16) , 12.8°, 18.6° for January to December respectively; 2) Bopt,57-288 = -30.23° and Bopt,289-56 = 7.7°. So, as [14] deals with Equator facing case only, the optimum tilt during the period starting from 16/10 to 25/2 should be zero. This result coincides well with the finding of [14] and it gives more precisely the starting and ending dates of period. On the other hand, as -25° is the maximum absolute tilt angle value used in [14] in measuring global solar radiation, the obtained result in this work coincides with the finding of [14]. Finally Diaz et al. [15] determined Bopt,m theoretically in Philippine (L from 4° to 21°) using the method of El-Kassaby [16] which suffers from uncertainty during period starting from 22/3 to 21/9 (see [17] for more details). Diaz et al. [15] verified his calculation at L = 14.56° experimentally during days of February repeating again the same error of El-Kassaby mentioned in [17] by verifying the applicability of the formula during the period of its validity which is starting from 22/9 to 21/3 [17].
5.
Tilt Factor
The geometric tilt factor Rb is the ratio of solar radiation on the tilted surface to that on a horizontal surface at any time or period of time. This factor can be calculated by appropriate use of solar incidence angle on tilted surface and on horizon. For daily, monthly, seasonally and yearly values of Rb, the following equation can be used: ∑ ( ) ∑ ( ) (15) where the collector azimuth angle, G, is 0o or -180o or 180o depending on EF case, North Pole facing (NPF) case or South Pole facing (SPF) case and the summation by N should cover the length of period in consideration. For daily values no summation by N is needed. When calculating the daily solar radiation at B = 0, B = Bopt,d for latitudes L=0o, 15o, -15o and taking the ratio between the value on a tilted surface to the value on a horizontal, according to Eq. (15), the obtained results are given in Figure 2 while Figure 3 gives the same calculation but at B = 0, B = Bopt,m. From Figures 2 & 3 one can conclude that monthly orientation is similar to daily orientation regarding energy gain. Moreover, according to Figures 2 & 3, Rb varies from 1 to 1.21 at L = 0o and from 1 to 1.48 at L = ±15o for daily and monthly orientation. The yearly energy gain is 1.10 and 1.15 for L = 0o and L = ±15o respectively. Here it should be mentioned that at L=0o it is reasonable to undertake orientation adjustment during May, June, July, November, December and January months as the energy gain will be 1.2. If the energy gain of 1.31 is required for L = ±15o, it is advised to provide orientation adjustment during months from October to February for L = 15o and from April to August for L = -15o inclusively.
Figure 2: Daily tilt factor calculated using Bopt,d. Upper curve on 146th day number corresponds L = -15o, middle curve corresponds L = 0o and lower curve corresponds L = 15o
35
6.
Energetic Evaluation
The rule of thumb says that solar collectors should be orientated towards Equator and latitude tilted. In the tropical region this rule is not valid as: a) There is a period during which solar collector should be orientated towards the Pole; b) The latitude is not the optimum daily or monthly tilt angles. In order to provide an energetic evaluation of daily optimum tilting solar radiation receiver in relation to latitude tilting case, solar radiation received in both cases should be measured or evaluated and then calculate the ratio of these values. Figure 4 demonstrate the energy gain of optimum tilting in relation to latitude tilting for L=15o, where it is seen that energy gain reaches 125% and it is greater than 115% during a long period of time. The average yearly gain is 1.112.
Figure 4: Daily energy gain of daily optimum tilting surface in relation to latitude tilted surface for L = 15o.
8.
Figure 3: Daily tilt factor calculated using Bopt,m. Upper curve on 146th day number corresponds L = -15o, middle curve corresponds L = 0o and lower curve corresponds L = 15o.
7.
Economic Evaluation
In order to evaluate preliminary the economic effect of daily optimum tilting in relation to latitude tilting case it is reasonable to do that on a practical example. Suppose that energy load of a small energy efficient solar home is required to be covered by solar panels. 4 kW is a good peak power target for small energy efficient solar home. Supposing that the solar system will be installed in a location of 15o latitude, the daily average of bright sunshine hours at latitude tilted solar panels is 6 hours and the stored energy by batteries covers three rainy or cloudy days. To close the system of variable the daily energy produced by latitude tilted solar panels is supposed to be equal to the daily energy used in the home. This assumption is required to determine the number of solar panels. Here it should be mentioned that there are two types of power requirements one needs to know when designing a solar system: The peak power delivered to the load, and the peak power produced by the solar panels by the system. The peak power delivered to the load is the total maximum power level one expects to be drawn by appliances in the home. So, in the provided example: The daily energy used is 4 kW x 6 hr = 24 kWhr. The required peak power of solar panels is 4 kW. The size of the system's inverter is 4 kW. The size of batteries is 72 kWhr. As the invertor cost is about 1000$/kW, the system's inverter cost is 4000$ and it is constant for latitude tilting and optimum tilting surfaces. So, it is out of comparison. The cost of batteries is about $100 per kWhr storage making the batteries cost about 7200$ and this value is out of comparison as it is constant for latitude tilting and optimum tilting surfaces. The cost of the charge controller is about 400$ and it is out of comparison also. As determined from a survey of current market prices, it costs about 3$/Watt-peak. Therefore, the upfront cost of the solar panels is 12000 $ for latitude tilted surface. For optimum tilted surface the daily average yearly energy gain is 1.112 (see figure 4). So, the upfront cost of the solar panels is 12000$/1.112 = 10791$. When considering the system life-cycle cost an additional benefit of optimum tilting case rises. The lifetimes of the batteries should be increased.
Conclusions
Finally, the following conclusions could be formulated: Formulae (13) and (14) are very effective in calculating daily optimum tilt and optimum tilt over a period at tropical zone. The available literature data regarding the tropical zone should be used carefully because some of these data should be corrected. It is sufficient to adjust solar collector tilt monthly. Adjusting solar collector tilt monthly or daily leads to solar energy gain that varies from 0% to 21% at L=0o and from 0% to 48% at L = ±15o. The yearly energy gain of 10% and 15% for L = 0o and L = ±15o respectively could be achieved by daily or monthly tilt adjustment. In northern part of the tropical zone, it is advised to adjust solar collector tilt angle during months from October to February only, while in southern part, it is advised to adjust solar collector tilt angle during months from April to August only.
Acknowledgements The authors thank Dr. R. Jabra for useful discussions and English reviewing. Thanks to reviewers for constructive comments.
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