Mathematical links between optimum solar collector tilts in isotropic sky for intercepting maximum solar irradiance

Journal of Atmospheric and Solar-Terrestrial Physics 137 (2016) 58–65

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Research paper

Mathematical links between optimum solar collector tilts in isotropic sky for intercepting maximum solar irradiance Dorin Stanciu a, Camelia Stanciu a,n, Ioana Paraschiv b a

University Politehnica of Bucharest, Faculty of Mechanical Engineering and Mechatronics, Engineering Thermodynamics Department, Romania University Politehnica of Bucharest, Faculty of Engineering and Management of Technological Systems, Machines and Production Systems Department, Splaiul Independentei, 313, Sector 6, 060042 Bucharest, Romania

b

art ic l e i nf o

a b s t r a c t

Article history: Received 12 September 2015 Received in revised form 25 November 2015 Accepted 27 November 2015 Available online 28 November 2015

The paper presents a mathematical modeling of the optimum tilt for solar collectors for intercepting maximum solar irradiance (power density), at different geographical locations, periods of time and different base-ground types. The solar irradiance received by the collector is estimated based on isotropic sky analysis models, namely Hottel & Woertz model and Liu & Jordan model. The optimum value for the tilt is considered for maximum hourly and respectively daily noon incident solar irradiance. This paper emphasizes the mathematical link between the optima computed under the two considered models assumptions. Also the ground reflectance factor influence on the optimum tilt difference between considered models is presented related to latitude. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Solar irradiance Optimum tilt Isotropic sky Ground reflectance

1. Introduction Solar collectors are widely used for different applications, from ordinary water heating during summer in household use (achieved through non-concentrating collectors) to solar power systems (using concentrating collectors). For both types of solar collectors, the target is to catch either maximum possible solar irradiance or solar energy on the aperture area of the collector. Non-concentrating collectors are characterized by the same area for intercepting and absorbing solar radiation, equivalent to unity concentration ratio. Contrary, concentrating collectors intercept solar radiation by a larger aperture area in comparison to the absorbed (receiver) one, thus concentration ratios are higher than unity. In the first case, the collectors are usually fixed mounted all along a day, often tilted on North–South direction, facing southwards in the Northern hemisphere. Maximum intercepted solar energy is targeted, for which optimum tilt is then determined. They could be daily, monthly or seasonally adjusted or even fixed over the entire year of operation at a specified tilt. In the second case, Sun tracking mechanisms are usually used for moving and tilting the concentrating collector so that the Sun rays fall perpendicular on the aperture area. In this regards, there n

Corresponding author. E-mail addresses: [email protected] (D. Stanciu), [email protected] (C. Stanciu), [email protected] (I. Paraschiv). http://dx.doi.org/10.1016/j.jastp.2015.11.020 1364-6826/& 2015 Elsevier Ltd. All rights reserved.

are two kinds of tracking mechanisms: single-axis and two-axis one. For example, parabolic trough collectors, characterized by 10 to 40 concentration ratios (Kalogirou, 2004), are usually designed with a single-axis tracking mechanism, moving the collector either on East–West axis, or North–South one. For parabolic dish reflectors having concentration ratios higher than 100, two-axes tracking mechanisms are usually applied, so that the Sun rays are intercepted perpendicular to the aperture area all day long (Kalogirou, 2004). Their concentration ratios could arrive to more than 10000. Feidt et al. (2004) presented in a published work elements regarding optimum concentration ratios for different kind of parabolic solar concentrators. As a conclusion, different optimizing criteria might be sought when computing optimum tilt or path for the solar collector, mainly depending on the collector type and economic reasons. As non-concentrating collectors are intercepting both beam and diffuse components of solar radiation, maximizing received total solar energy is a common criterion used for determining the optimum tilt. In case of collectors having concentration ratios higher than 10, only beam component of the solar radiation is mainly used (Prapas et al., 1987) (as cited by Kalogirou (2004)) and thus targeted. In technical literature one may find different proposed methods for determining the optimum tilt, considering different criteria and expressing the tilt function on latitude or declination or other local constants. Maximizing total incident solar energy on the collector, either in atmospheric conditions or extraterrestrial ones,

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Nomenclature

Subscripts and superscripts

a0 , a1 C G k n

B CS D ET FPC g (HW) (LJ) max n (N) opt (S) SC t T 12 h

constants for standard atmosphere notation for term given by Eq. (28) solar irradiance, Wm  2 constant for standard atmosphere number of a day in an year

Greek symbols

α β δ φ γ ρ τ θ θz ω

solar absorptance slope of a tilted surface, deg solar declination geographical latitude, deg surface azimuth angle, deg diffuse reflectance atmosphere transmittance angle of incidence of beam radiation on a surface, deg zenith angle, deg hour angle, deg

is the main applied criterion. A brief summary of the published papers regarding flat plate collectors is presented by (Armstrong and Hurley, 2010) who deduced the optimum tilt particularly for climates susceptible to frequently overcast skies and validated for all other climate types. Some of the published methods calculate the optimum tilt by maximizing the extraterrestrial solar energy (Soulayman, 1991; Soulayman and Sabbagh, 2014; Gunerhan, 2005), others the direct solar radiation incident on the collector (Kern and Harris, 1975) and others maximize the total incident solar energy (Tamimi, 2011; Agarwal et al., 2012; Bakirci, 2012; Morcos, 1994). Particular correlations for optimum tilt have been derived at different latitudes. Based on the firstly derived isotropic sky model, Hottel and Woertz model, Tamimi (2011) proved a mathematical dependence of the optimum tilt on latitude, declination and length of the day and applied it for Amman, Jordan. The optimization criterion was the daily terrestrial solar energy, expressed as the product between average clearness index (constant value) and extraterrestrial solar energy. An optimum value is provided for each month and finally a yearly average tilt was proposed, correlated to NASA measurements. The proposed final correlation is βopt ¼ φ 7 (10°  15°). Agarwal et al. (2012) also maximized terrestrial solar energy for India and compared the tilts numerically derived by 4 isotropic and 4 anisotropic models. Numerical and graphical results are compared for Liu and Jordan, Reindl, Hay and Badescu models finding maximum differences of about 1.5° between models, reported in December, in India. Another numerical simulation of the optimum tilt for maximizing daily total solar energy received in Turkey, by applying Liu and Jordan model, is presented by Bakirci (2012). Considering a ground reflectance of 0.20 and correlating total terrestrial radiation to extraterrestrial one by the clearness index, the author presented numerical values for the optimum tilt and derived an expression function on solar declination δ proposing polynomial correlations valid for Turkey (e.g. 34.783  1.4317δ  0.0081δ2 þ0.0002δ3). An analytical expression of the optimum tilt for maximum total solar energy computed with Liu and Jordan model is derived by Morcos (1994). The tilt is expressed in terms of azimuth angle, declination, latitude, ground reflectance, zenith and hour angles. The author derived also the analytical expression of the azimuth

59

beam clear sky conditions diffuse extraterrestrial radiation flat plate collectors ground Hottel and Woertz model Liu and Jordan model maximum value normal to a plane North hemisphere optimum South hemisphere solar constant tilt total at 12 O'clock

angle for maximum total energy and applied the results for Assiut (Egypt). Hartley and Martinez-Lozano (1999) maximized solar irradiance on a horizontal plane, found a monthly average for this variation and computed the corresponding irradiance intercepted by a tilt collector using Liu and Jordan model, but without considering the view factor of the collector to the ground. The corresponding solar energy was obtained and numerical values for the optimum tilt were proposed. From the above discussions, one might separate two optimization criteria: maximum intercepted solar energy (radiation, in J/m2) and maximum intercepted solar irradiance (power, in W/m2). With respect to Hottel and Woertz model, Liu and Jordan forecast model takes into account the view factors to the sky and to the ground. In the current literature, the view factors calculation is of interest and proposed for application to solar radiation forecast (Sugden, 2004). In order to compute the radiation reflected by the ground or surface, ground albedo value is required. Psiloglou and Kambezidis (2009) reported several methods to calculate ground albedo underlining its important influence on the estimation of solar radiation incident on tilted surfaces. They have researched a methodology for deriving new site-specific groundalbedo expressions for locations without albedo measurements and compared the obtained reflected irradiance on a sloped surface to experimental measurements, emphasizing that a constant albedo of 0.20 is not convenient for Athens. Shamim et al. (2015) recently proposed a methodology for estimating clear sky global solar radiation, incorporating analytical tools for determining optical transmittance and surface albedo. We have previously studied collectors optimum tilt calculation (Stanciu and Stanciu, 2014) and after applying a numerical simulation for the tilt at different latitudes, the following result was revealed: the correlation βopt,noon ¼ φ  δ applies for the optimum tilt when using Hottel and Woertz model for intercepting maximum solar irradiance at noon, but some deviations from this formula along the year appeared when using other forecast models (Liu and Jordan and Hay–Davis–Klucker–Reindl models). Up to 10° difference in the tilt value was obtained between models. The purpose of this paper is to find analytical expressions for the optimum tilt, for further use in optimization procedures and

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also for explaining the source of those deviations between models, from mathematical point of view, by the influence of atmospheric transmittance and ground reflectance. A correlation between the optimum tilt computed by Hottel & Woertz and Liu & Jordan models is emphasized. The influence of the ground types characterized by a certain reflectivity related to latitude on the optimum collector tilt is emphasized. This procedure is applied in clear sky conditions for the case of non-concentrating collectors with a horizontal axis tracking mechanism. The maximization criterion for the optimum tilt is the total solar irradiance captured by the collector.

2. Prerequisites – solar irradiance in isotropic sky In order to closely simulate the operation of a solar collector, one needs a precise set of data regarding the solar radiation density (solar irradiance) incident on the collector surface. The most common available data are for total irradiance on horizontal surfaces, GT , which is the result of two components:

(1)

G T = GB + GD

where GB is the direct (or beam) solar irradiance component, received from the Sun without having been scattered by the atmosphere; GD is the diffuse (or sky) solar irradiance component, received from the Sun after a modified direction due to scattering by the atmosphere. In technical literature, there have been published many analytical models to estimate each component, the complexity increasing with their accuracy. In the present paper, prediction methods for isotropic sky conditions are applied. By isotropic sky conditions one should understand that all diffuse radiation incident on a surface is received uniformly from the entire sky dome (Duffie and Beckman, 2006), while the circumsolar diffuse and horizon brightening diffuse radiation components are neglected. The beam radiation transmitted through clear atmosphere is calculated based on Hottel’s method (Hottel, 1976), while for diffuse radiation for clear days Liu and Jordan’s empirical relation (Liu and Jordan, 1960) is applied. As these formulae are valid for standard ‘clear sky’, note the superscript ‘CS’ to refer to this. Complete details regarding the mathematical procedure are provided in (Duffie and Beckman, 2006). The solar radiation travels from the Sun to the Earth surface a mean distance of about 1.5  1011m. Outside the Earth's atmosphere, a constant intensity (named solar constant) is reported W equal toGSC = 1367 2 as received per unit area of a surface normal m

to its propagation direction at this mean distance. However, as the Sun–Earth distance varies due to Earth motion, the extraterrestrial radiation incident on a plane normal to the radiation, GETn, is computed as (Spencer, 1971):

⎛ 360n ⎞ ⎟ GETn = GSC ⎜ 1 + 0.033 cos ⎝ 365 ⎠

(2)

Further, this radiation enters the Earth atmosphere striking the air layers. A parameter called atmospheric transmittance, τB, is defined as the ratio between the direct radiation flux received on a surface on the Earth, GBn, and the extraterrestrial one, both of them incident on a plane normal to the radiation: CS GBn = τB GETn

(3)

Hottel (1976) introduced an empirical relation for computing the atmospheric transmittance for beam radiation for any zenith angle θz:

τB = a 0 + a1e

−

k cos θ z

(4)

where a0 , a1, k are constants for standard atmosphere with 23 km visibility, given as function of altitude and for four climate types (Duffie and Beckman, 2006; Hottel, 1976): mid latitude summer climate, tropical, subarctic summer and mid latitude winter climate. For example, for Bucharest located in a mid latitude summer climate and 80 m altitude, the computed values of these constants are: a0 = 0.131889; a1 = 0.743; k = 0.387 (Stanciu and Stanciu, 2014). The zenith angle θz in Eq. (4) represents the angle between the beam radiation and the normal to a horizontal surface. In order to determine this angle, one should know the Sun relative position to the Earth. This is described by the aid of a set of angles, linked by the following relation (Duffie and Beckman, 2006):

cos θ = sin δ sin ϕ cos β − sin δ cos ϕ sin β cos γ + cos δ cos ϕ cos β cos ω + + cos δ sin ϕ sin β cos γ cos ω + cos δ sin β sin γ sin ω

(5)

where: θ ¼angle of incidence of beam radiation on a surface (the angle between the beam radiation on a surface and the normal to that surface); δ ¼ declination, estimated by Cooper equation (Cooper, 1969) for the nth day of the year

⎛ 284 + n ⎞ ⎟ δ = 23.45 sin ⎜ 360 ⎝ 365 ⎠

(6)

 φ ¼latitude of the considered location on the Globe;  β ¼ the slope of a tilted surface (angle between the plane of the surface and the horizontal);

 ω ¼hour angle, ω = 150 × (time − 12);  γ ¼surface azimuth angle (for fixed flat plate collectors, it is commonly 0° in the Northern hemisphere and 180° in the Southern one). The zenith angle θz in Eq. (4) is computed for the particular case of a horizontal surface ( β = 0):

cos θz = cos ϕ cos δ cos ω + sin ϕ sin δ

(7)

CS Since GBn is received on the Earth on a plane normal to the radiation, and the angle θz is between the Sun ray and the normal to the surface on the Earth, the beam radiation received normal on a horizontal surface is computed as:

CS GBCS = GBn cos θz

(8)

As mentioned before, in order to compute the diffuse radiation component, Liu and Jordan’s empirical relation (Liu and Jordan, 1960) is applied:

τD =

GDCS = 0.271 − 0.294τB GETn cos θz

(9)

wherefrom one gets the diffuse radiation component GDCS . When the collector is tilted at an angle β≠0 with respect to the horizontal, the angle between the Sun ray (or GBn) and the normal to the tilted surface (GBt) is θ so that:

GBt = GBn cos θ

(10)

As cos θ depends on the tilt angle β, the beam irradiance received by a tilted surface depends on β, too. As stated in the beginning of this paper, the total solar

D. Stanciu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 137 (2016) 58–65

irradiance incident on a tilted surface is estimated based on two isotropic sky analysis models, namely: - Hottel and Woertz model (Hottel and Woertz, 1942):

It assumes that both diffuse from the sky and ground-reflected radiations are isotropic and do not depend on the collector orientation. - Liu and Jordan model (Liu and Jordan, 1963):

1 + cos β 1 − cos β + ( GB + GD ) ρg 2 2

It takes into account the view factors to the sky

(12) 1 + cos β 2

for the

diffuse from the sky radiation and the view factor to the ground 1 − cos β respectively, for the solar radiation diffusely reflected from 2

the ground. The diffuse reflectance ρg depends on the base-ground type. Common values are (Markvart and CastaŁżer, 2003; Technical University of Crete): 0.60 for land and light building surfaces, 0.25 for green grass, 0.40 for sand, 0.13 for gravel roof. It should be mentioned that other isotropic correlations exist in the current literature (Badescu, 2002; Tian et al., 2001; Koronakis, 1986) to determine the diffuse component of solar radiation. The differences from mathematical point of view consist in replacing the view factor to the sky in Eq. (12) by other terms, function on β too.

3. Mathematical modeling of the optimum tilt for maximum total irradiance In the case of non-concentrating collectors with horizontal-axis tracking system, the collector is kept at fixed azimuthal orientation γ and the tilt β is optimized with respect to maximum total solar irradiance GTt. When one needs to compute the optimum value of a variable appearing in a complex mathematical expression, one should compute the derivative of this function with respect to this variable. 3.1. Northern Hemisphere Let compute the derivative of total solar irradiance on a tilt surface with respect to the tilt:

dG Tt =0 dβ

(13)

In the expression of GTt, all terms are variables depending on time, latitude and declination, as presented above. 3.1.1. Hottel and Woertz model When applying Hottel and Woertz model for computing the total solar irradiance, one may find the following mathematical condition:

dG Tt GB d (cos θ ) =0⇒ ⋅ =0 dβ cos θz dβ

tan βopt =

sin ϕ cos δ cos γ cos ω − cos ϕ sin δ cos γ + cos δ sin γ sin ω cos θ z

(16b)

Eq. (16b) provides the hourly optimum tilt for solar collectors intercepting maximum solar irradiance for γ angle orientation of the collector with respect to East–West axis and facing South. Two particular cases are emphasized: a) The particular case of fixed collectors oriented parallel to East–West axis (γ ¼0) in northern hemisphere (common case of flat plate collectors), for which the above relations become:

d (cos θ ) = sin (ϕ − β ) cos δ cos ω − cos (ϕ − β ) sin δ dβ

(17)

According to Eq. (14), it results that:

(

)

(

)

sin ϕ − βopt cos δ cos ω = cos ϕ − βopt sin δ

(18)

or equivalent to:

(

)

tan ϕ − βopt =

tan δ cos ω

(19)

One may note the optimum value of the tilt for hourly maximum solar irradiance intercepted by a fixed flat plate collector (FPC) is:

⎛ tan δ ⎞ ⎟ βopt, FPC = ϕ − tan−1⎜ ⎝ cos ω ⎠

(20)

Tamimi derived a similar expression for maximizing terrestrial solar energy on FPC (Tamimi, 2011):

⎛ tan δ ( ωS π /180) ⎞ Tamimi ⎟⎟ = ϕ − tan−1⎜⎜ βopt , FPC sin ωS ⎠ ⎝

(21)

where ωS is the day length, in degrees. One may note the difference between the two Eqs. (20) and (21) in terms of optimization criterion. b) For the above mentioned case, as daily maximum solar irradiance is received at noon solar time, one might find the well known relation (22). Note that at noon the hour angle ω is zero. From Eq. (20), one gets:

βopt, FPC ,12h = ϕ − δ

(22)

3.1.2. Liu and Jordan model Let apply condition (13) to Eq. (12). Taking into account that cos θ GBt = GB cos θ and cos θ is a function of β, when deriving Eq. (12) z

with respect to

β it results:

GB [ − sin βopt (sin φ sin δ + cos φ cos δ cos ω) cos θz + cos βopt (sin φ cos δ cos γ cos ω − cos φ sin δ cos γ + cos δ sin γ sin ω)] − sin βopt

( GB + GD ) ρg GD + sin βopt = 0 2 2

(23)

One may find that:

d (cos θ ) = − sin β (sin ϕ sin δ + cos ϕ cos δ cos ω) dβ

tan βopt =

+ cos β (sin ϕ cos δ cos γ cos ω

According to Eq. (14), it results that:

(16a)

(14)

in which

− cos ϕ sin δ cos γ + cos δ sin γ sin ω)

sin ϕ cos δ cos γ cos ω − cos ϕ sin δ cos γ + cos δ sin γ sin ω sin ϕ sin δ + cos ϕ cos δ cos ω

or

(11)

G Tt = GBt + GD

G Tt = GBt + GD

tan βopt =

61

(15)

GB +

GD 2

GB ( G + G ) ρg − B 2D

sin ϕ cos δ cos γ cos ω − cos ϕ sin δ cos γ + cos δ sin γ sin ω (24) cos θz

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It is convenient to express the beam and diffuse radiation densities function on atmospheric transmittance:

tan βopt =

2τ B

( τB + τD ) ( 1 − ρg ) + τB

sin ϕ cos δ cos γ cos ω − cos ϕ sin δ cos γ + cos δ sin γ sin ω (25) cos θz or

tan βopt =

2τ B

( 0.271 + 0.706τB ) ( 1 − ρg ) + τB

sin ϕ cos δ cos γ cos ω − cos ϕ sin δ cos γ + cos δ sin γ sin ω (26) cos θz Comparing Eq. (26) to Eq. (16b), one notices that the optimum tilts derived under the two presented assumptions, Hottel & Woertz (HW) and Liu & Jordan (LJ), are similar and could be linked by: (LJ ) (HW ) tan βopt = C tan βopt

(27)

where the term C denotes:

C=

2τ B

( 0.271 + 0.706τB ) ( 1 − ρg ) + τB

(28)

One may analyze the influence of different assumptions of the applied forecast model on the optimum tilt. Also the base-ground type effect could be studied based on Eq. (28) and could be optimized, too. 3.2. Southern Hemisphere For the Southern hemisphere, the latitude and the declination are considered negative, while the azimuth is expressed away from North (or equivalent 2π minus the azimuth away from South). Applying trigonometric functions of negative argument in Eq. (16b), one might notice that it is exactly optimum tilt for Northern hemisphere with minus sign: (S ) (N ) βopt = − βopt

(29)

4. Results and discussions For numerical simulation, let consider the case of a solar collector operating at different latitudes in the Northern hemisphere, along the year. When applying Eq. (20), the optimum tilt is computed on an hourly basis, as revealed by Fig. 1a–c. The presented results have been plotted for the particular case of zero azimuth, i.e. south orientation of the collector on East–West axis. Three types of daily variations emerge along the year, whatever the latitude is: summer, winter and spring–autumn behaviors, respectively. For the one specific to summer periods, an increase in the tilt is foreseen as approaching noon. The daily variation in this period is more important at high latitudes (Fig. 1c) and more damped at lower latitudes (Fig. 1a). This behavior is explained by the altitude of the Sun on the celestial globe at noon. High altitude in the sky involves lower tilts and also damped variations along a day. As the Sun climbs on the sky dome, the collector should be tilted to catch the rays perpendicular on its surface at each hour, as this was the optimization criterion chosen for this research. It should be mentioned that the collector surface is tilted on North– South direction, facing South, as single tracking axis, in order to avoid the continuous daily rotation along East–West direction. Neither the temporary North orientation of the collector was

Fig. 1. a. Hourly optimum tilt along the year for maximum hourly irradiance, at 30° Northern latitude (Hottel and Woertz model). b. Hourly optimum tilt along the year for maximum hourly irradiance, at 44.25° Northern latitude (Hottel and Woertz model). c. Hourly optimum tilt along the year for maximum hourly irradiance, at 60° Northern latitude (Hottel and Woertz model).

considered. Due to the fact that the Sun does not arrive to be exactly at 90° at noon (except the June 21st for the latitude of 23.27 N), the collector tilt should compensate the difference between the Sun position and the normal (the zenith). Oppositely, winter periods are characterized by a decrease in tilt values towards noon. As the Sun climbs the least on the sky dome, the collector should be deeper tilted to compensate the difference towards the zenith. Comparing again Fig. 1c to 1a, one might notice that for high latitudes locations, the day length is much shorter and the tilt should be more important as the Sun is lower in the sky; the collector is close to vertical position. The third period is in-between the first two presented, namely spring and autumn periods, when the optimum tilt is less sensitive to the day time. An important aspect to notice from practical considerations is the fact that during summer, there are periods of quasi-constant optimum tilt of about five hours in June (between

D. Stanciu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 137 (2016) 58–65

90.00

1.40 1.35

80.00

ρg=0.60 Jan,15th

70.00

23.27

1.30

60.00

30

1.25

23.27

50.00

40

1.20

30

1.15

40

1.10

44.25

1.05

50

1.00

60

0.95

66.33

40.00

44.25

30.00

50

20.00

60

10.00 0.00

C [-]

β opt,12h [deg]

63

66.33 1

2

3

4

5

6 7 Month

8

9

0.90

10 11 12

Fig. 2. Optimum noon tilt variation along the year, at different latitudes (Northern hemisphere) (Hottel and Woertz model).

(i) two different behaviors are clearly put into evidence, for low latitudes (Fig. 4a) and high latitudes (Fig. 4b), respectively. Thus, one may consider that the latitude dictates the shape of Δβ variation. This could be explained by the different Sun path and altitude in the Sky, respectively the day light duration and rays direction at different latitudes. These parameters directly influence the magnitude of solar irradiance components, and thus the optimum tilt value; (ii) a maximum difference of about 9° is reveled for Δβ . The

8

9

10

11 12 13 Solar time

14

15

16

17

1.38 ρg=0.60 Jun,15th

1.37

C [-]

1.36

30

1.35

40

1.34

44.25

1.33

50 60

1.32

66.33

1.31 1.30

80 7

8

9

10

11 12 13 Solar time

14

15

16

17

1.40

ρg=0.60 Sept,15th

1.35

C [-]

10 and 14 o'clock) to seven hours daily in August (between 9 and 15 o'clock solar time). From economical point of view it is important to maintain a constant tilt as long as possible and efficient. Fig. 1b emphasizes the particular case of Bucharest, at 44.25 N. When extracting the optimum values for the tilt at 12 o’clock when the solar irradiance reaches the daily maximum, similar yearly variations are obtained for all latitudes. They are presented in Fig. 2. One may notice that June is characterized by the minimum tilt value whatever the latitude is. Also a yearly variation of about 40° is revealed between January and June for all latitudes. As this optimum value is given by Eq. (22), it is normal to have the profile of solar declination δ. Eqs. (27)–(28) give the difference between optimum tilt computed by Liu and Jordan model with respect to Hottel and Woertz (LJ ) (HW ) model. The daily variation of C = tan βopt is emphasized / tan βopt by Fig. 3a–c for different latitudes, covering three months along the year. One may notice from Eq. (28) that C increases as either the transmittance or/and ground reflectance are increasing. The atmospheric transmittance is higher in the summertime, thus one might notice higher values of C in June (Fig. 3b) with respect to January (Fig. 3a) or September (Fig. 3c) for the same latitude. It is worthy to mention that for low latitude locations (o 40°N), the C values are higher than 1.00 all along the year, while for high latitudes, there are short periods for which C o1.00 (during winter mornings and evenings). A C value higher than 1.00 indicates that the optimum tilt computed by Liu and Jordan model is higher than the one computed by Hottel and Woertz model. This could be the effect of considering diffuse and ground reflected radiation terms by the corresponding view factors when computing the total irradiance. Also one might notice important daily variations of C for high latitudes and a more damped variation for low latitudes. This daily variation of C is more important in January when comparing to June. Since the tilt values differ along the year and with latitude, the variation of C translates into a different variation of the tilt difference. For the sake of an easier understanding, Fig. 4a–b could be plotted, revealing the difference expressed in degrees between (LJ ) (HW ) the two tilts Δβ = βopt at noon. The following conclusions − βopt may be emphasized:

7

1.30

23.27

1.25

30 40

1.20

44.25

1.15

50

1.10

60

1.05

66.33

1.00

80 8

9

10

11

12 13 Solar time

14

15

16

17

(LJ ) (HW ) Fig. 3. Daily variation of C = tan βopt for all latitudes on land basis / tan βopt (ρg ¼0.60) in January (a), June (b) and September (c).

numerical results also show that this difference remains below 9° all day long (not only at noon). These results complete the authors' previous study (Stanciu and Stanciu, 2014) in which a roughly estimation was proposed, showing that sometimes the difference between the two tilts is negligible and sometimes the difference is in the limit of 10°. For two of the above studied latitudes, 40.00°N and 60.00°N, respectively, we have analyzed the influence of the ground reflectance ρg on Δβ in Fig. 5. Three ground types were selected for this plot: sand basis (ρg ¼0.40), land or asphalt basis (ρg ¼0.60), and green grass land (ρg ¼0.25). As one may observe, the shape of Δβ is preserved for each latitude (as shown by Fig. 4), but its variation amplitude directly depends on ρg. A higher ground reflectance value increases the corresponding component of the (LJ ) and Δβ , accordtotal irradiance and influences the value of βopt ingly. Among the three considered basis, the land one has the most

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D. Stanciu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 137 (2016) 58–65

(LJ ) (HW ) Fig. 4. Yearly variation of noon βopt for different latitudes on land basis (ρg − βopt ¼ 0.60).

Fig. 6. Azimuth angle effect on the daily difference between the two optima, for ρg ¼0.60, at 44.25N latitude, at 9 AM (a) and 12 PM (b).

(as for example different ground type considered for the study). When non-zero values of the azimuth angle were considered, the same behavior was observed, as C does not depend on γ angle (Eq. (28)). Nevertheless, it is important to mention that the numerical values of the two computed tilts and accordingly their difference vary as γ angle is varied (see Fig. 6a–b). At noon, optimum value of the azimuth is zero as the Sun is perfectly on North– South direction in any location on the Globe. In Fig. 6a the tilt difference corresponding to optimum azimuth is the one represented by a black continuous line. This optimum azimuth was computed according to the relation derived by Morcos (1994). In the presented case (44.25N latitude at 9 am), the optimum azimuth varies between  41.55° on January the 1st to  75.51° on June 22nd. The minus sign refers to the fact that the Sun is located towards East from South. As one might notice, the influence of the azimuth is more significant in the morning and evening periods, especially during summer (Fig. 6a) and less important around noon (Fig. 6b). This is easily explained by the magnitude of the solar irradiance at noon in comparison to morning and evening periods.

5. Conclusions

Fig. 5. Ground reflectance effect on the yearly variation of tilt difference for 44.25°N (a) and 60°N latitude (b) at noon.

important weight on the tilt difference between models. These results explain the source of different values reported for the optimum tilt, either when comparing the two models (Hottel and Woertz to Liu and Jordan), or even when comparing the same Liu and Jordan model results but in different operating conditions

Analytical expressions for the optimum tilt corresponding to maximum total irradiance are provided for use in mathematical optimization procedures, under the assumptions of two isotropic clear sky models. The novelty with respect to the published literature is the link between the optima found under different solar radiation forecast models. The optimum tilt that is to be used all day long is provided by Eq. (16b) if Hottel and Woertz forecast model is used. If Liu and Jordan assumptions are made, the tilt values provided by Eq. (16b) should be altered by the C expression given in Eq. (28). The effect of ground reflectivity (albedo) on the

D. Stanciu et al. / Journal of Atmospheric and Solar-Terrestrial Physics 137 (2016) 58–65

optimum tilt was presented. This result might be helpful in explaining the differences of few degrees in the optimum tilt reported by different papers. When comparing reported results, it is important to specify the model assumptions and maximization criterion. The work presented is thus potentially interesting for explaining small discrepancies obtained when model validations scenario are applied. When computed results are compared to experimental data, some small differences might be explained by consideration of different values for the constant parameters appearing in the model, as for example ground albedo. Psiloglou and Kambezidis (2009) explained deviations in solar radiation results when comparing to experimental data in Athens by the effect of considering a constant 0.20 value for the ground albedo in the model, which proved to be too high for Athens. As a consequence of differences in the magnitude of ground reflected radiation with respect to total radiation, the optimum tilt depends also on the right selection of ground type basis on which a collector would be mounted.

Acknowledgment The work has been partially funded by the Sectorial Operational Programme Human Resources Development 2007–2013 of the Ministry of European Funds through the Financial Agreement POSDRU/159/1.5/S/134398 (co-author Ioana Paraschiv). This work has been partially supported by a Grant of the Romanian National Authority for Scientific Research and Innovation, CNCS – UEFISCDI, Project number PN-II-RU-TE-2014-4-0846 (coauthor Camelia Stanciu).

References Armstrong, S., Hurley, W.G., 2010. A new methodology to optimize solar energy extraction under cloudy conditions. Renew. Energy 35, 780–787. Agarwal, A., Vashishtha, V.K., Mishra, S.N., 2012. Comparative approach for the optimization of tilt angle. Int. J. Eng. Res. Technol. 1 (5). Bakirci, K., 2012. General models for optimum tilt angles of solar panels: Turkey case study. Renew. Sustain. Energy Rev. 16 (8), 6149–6159. Badescu, V., 2002. A new kind of cloudy sky model to compute instantaneous values of diffuse and global irradiance. Theor. Appl. Clim. 72, 127–136. Cooper, P.I., 1969. The absorption of solar radiation in solar stills. Sol. Energy 12 (3),

65

333–346. Duffie, J., Beckman, W., 2006. Solar Engineering of Thermal Processes. United States of America, John Wiley and Sons Inc. Feidt, M., Costea, M., Petre, C., Boussehain, R., 2004. Genie Energetique Appliqué au Solaire – Energie Solaire Thermique. Printech Publication House, Bucharest, Romania. Gunerhan, H., 2005. Determination of the optimum tilt of solar collectors for building applications. Build. Environ. 42 (2), 779–783. Hartley, L.E., Martinez-Lozano, J.A., 1999. The optimization of the angle of inclination of a solar collector to maximize the incident solar radiation. Renew. Energy 17, 291–309. Hottel, H.C., 1976. A simple model for estimating the transmittance of direct solar radiation through clear atmospheres. Sol. Energy 18, 129. Hottel, H.C., Woertz, B.B., 1942. Performance of flat plate solar heat collectors. Trans. ASME 64, 91. Kalogirou, S.A., 2004. Solar thermal collectors and applications. Prog. Energy Combust. Sci. 30, 231–295. Kern, J., Harris, I., 1975. On the optimum tilt of a solar collector. Sol. Energy 17, 97–102. Koronakis, P.S., 1986. On the choice of the angle of tilt for south facing solar collectors in the Athens basin area. Sol. Energy 36, 217. Liu, B.H., Jordan, R.C., 1960. The interrelationship and characteristic distribution of direct, diffuse and total solar radiation. Sol. Energy 4 (3), 1. Liu, B.H., Jordan, R.C., 1963. The long-term average performance of flat plate solar energy collectors. Sol. Energy 7, 53. Morcos, V.H., 1994. Optimum tilt angle and orientation for solar collectors in Assiut Egypt. Renew. Energy 4 (3), 291–298. Markvart, T., CastaŁżer, L., 2003. Practical Handbook of Photovoltaics: Fundamentals and Applications. Elsevier, Amsterdam, Oxford, UK. Prapas, D.E., Norton, B., Probert, S.D., 1987. Optics of parabolic trough solar energy collectors possessing small concentration ratios. Sol. Energy 39, 541–550. Psiloglou, B.E., Kambezidis, H.D., 2009. Estimation of the ground albedo for the Athens area, Greece. J. Atmos. Sol. -Terr. Phys. 71 (8–9), 943–954. Soulayman, S., 1991. On the optimum tilt of solar absorber plates. Renew. Energy 1 (3–4), 551–554. Soulayman, S., Sabbagh, W., 2014. Comment on ‘Optimum Tilt Angle and Orientation for Solar Collectors in Syria’ by Skeiker, K. Energy Conversion and Management. Sugden, S., 2004. View factor for inclined plane with Gaussian source. Appl. Math. Model. 28, 1063–1082. Shamim, M.A., Remesan, R., Bray, M., Han, D., 2015. An improved technique for global solar radiation estimation using numerical weather prediction. J. Atmos. Sol. -Terr. Phys. 129, 13–22. Stanciu, C., Stanciu, D., 2014. Optimum tilt angle for flat plate collectors all over the World – A declination dependence formula and comparisons of three solar radiation models. Energy Convers. Manag. 81, 133–143. Spencer, J.W., 1971. Fourier series representation of the position of the sun. Search 2, 172–173. Tamimi, A., 2011. Modeling of optimum inclination angles of solar systems for Amman, Jordan. J. Renew. Sustain. Energy 3, 043109. Technical University of Crete, Intelligent Systems Laboratory. 〈http://www.in telligence.tuc.gr/renes/fixed/info/reflectanceinfo.html〉. Tian, Y.Q., Davies-Colley, R.J., Gong, P., Thorrold, B.W., 2001. Estimating solar radiation on slopes of arbitrary aspect. Agric. For. Meteorol. 109, 67–77.