Applied Energy 87 (2010) 836842

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Applied Energy
journal homepage: www.elsevier.com/locate/apenergy

Solar irradiation in diffusely enclosures with partitions

a r t i c l e

i n f o

a b s t r a c t
This paper presents an approach to obtain the income of solar irradiation within partitioned enclosures partially transparent to solar radiation. This model is mathematically exact and it is function of the outside solar irradiation, the orientation of the enclosure, the properties of the enclosure envelope, the geometry and properties of partitions. From the physical point of view it is founded on the assumption that the envelope and partitions surfaces are fully diffusive and that the radiation diffused through the atmosphere is fully isotropic. The model was applied to assess the solar irradiation at the ground of a hemicylindrical tunnel tted with inclined partitions. The results of this study demonstrate, among other things, that: (i) the solar irradiation inside the enclosure, with any cladding material and with any orientation, is mainly determined by the transmittance of the partitions and to a lesser extent by its reectance, (ii) the solar radiation inside an enclosure with highly transparent partitions (i.e., transmittance > 0.5) is noticeable inuenced by its orientation, but for a lower transmittance the inuence of orientation becomes negligible; and (iii) to prevent the overheating within the enclosure it is advisable to install partitions with a low transmittance and a high absorptance on their surface facing downward. 2009 Elsevier Ltd. All rights reserved.

Article history: Received 28 June 2009 Received in revised form 3 September 2009 Accepted 5 October 2009 Available online 1 November 2009 Keywords: Solar irradiation, Diffuse irradiation Direct beam Partitioned enclosure Modeling

1. Introduction Energy has become a top priority and concern in modern society. Solar energy has an important role to play in the energy policies because has advantages both from the point of view of ecology and economy [1]. It is an important free power source that can be used directly as a source of heat or to generate electricity [1 6]. Besides it does not contribute to increase greenhouse gases nor hazardous wastes, and reduce land use impacts from typical utility generation, transmission and distribution [79]. Therefore solar energy, together with other renewable sources, can be an important alternative to reduce emission of pollutants to the atmosphere. Solar energy is used in passive systems that are applied in different elds. It is a time-dependent energy source and of intermittent character. Therefore, there are several studies devoted to the solar radiation variations at different locations on earth (see for example Refs. [1016]). Surveys of the literature that discusses solar thermal technologies were also prepared by Meir et al. [17] and Thirugnanasambandam et al. [18]. As noted by theses authors there is a lack of studies concerned with the inuence of solar radiation on the thermal performance of partitioned enclosures. To overcome this problem, a realistic model for estimating the

radiative loss inside a partitioned enclosure (with given radiometric properties and conguration) is required [17,18]. This will constitute an important contribution for improving their performance. In reality, the system performance can be improved by a rational improvement of design and operation criteria [1,19,20]. The search for optimal design is considerably more challenging and effective than optimizing the operation of a system [19]. In this paper, a model is developed which offers the possibility of estimating the solar irradiation inside a partitioned enclosure with fully diffusive surfaces. Since this model is mathematically exact (although from the physical point of view is founded on some simplifying assumptions) it can be applied to a variety of different solar-energy based systems in order to optimize the energy collection efciency or their operation. 2. Theory 2.1. Solar irradiation reaching the enclosure cover Consider a hemi-cylindrical enclosure with a partially transparent envelope to solar radiation. Assume a fully diffusive envelope and that the radiation diffused by the sky and by the ground is fully isotropic [21,22]. The total solar radiation that reaches the enclosure envelope is the sum of the following contributions [1,2]

* Corresponding author. Address: Geophysics Centre of Evora, Rua Romao Ramalho 59, 7000-671 Evora, Portugal. E-mail address: afm@uevora.pt (A.F. Miguel). 0306-2619/$ - see front matter 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.10.003

Hc Hbc Hdc Hhc

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Nomenclature a A c C E Fi h H Ib Id Ih L m n N Pj azimuth of the sun, area, m2 segment of the enclosure envelope width of the beam intercepted by a segment c of the enclosure envelope, m ux density of the radiation input (transmitted) from surface k into the enclosure, W/m2 shape factor for exchange of radiation between surface i and k angular height of the sun above the horizon, ux of solar radiation that reaches the envelope of the enclosure, W/m2 direct beam solar irradiation ux density (horizontal plane), W/m2 diffuse solar irradiation ux density (horizontal plane), W/m2 hemispherical solar irradiation ux density (horizontal plane), W/m2 length of the enclosure, m number of segments of the enclosure envelope that are reached by the direct beam total number of segments of the enclosure envelope number of surfaces of an enclosure fraction of diffuse radiation emanating from the inside of surface i that reaches surface k either directly or indirectly after an arbitrary number of reections on the surfaces radius of the envelope of the enclosure, m Cartesian coordinates coordinates of any point j of the enclosure envelope coordinates of any point j resulting from the interception of the line, obtained by the projection of the solar beam on the cross-section plane passing through the point (Yj, Zj), with its perpendicular passing through the origin of the coordinate system Greek symbols a absorptance of a surface for solar radiation b angle between the line passing through any point where the partition is xed to the enclosure envelope and the origin with the z-axis, c angle between the longitudinal axis of the enclosure and the eastwest direction, / angle of the solar beam with its projection on the plane of the enclosure cross-section, U solar irradiation ux density, W/m2 q reectance of a surface for solar radiation h angle of the projection of the solar beam on the crosssection of the enclosure with the horizontal plane, s transmittance of a surface for solar radiation w ratio between daily total irradiation inside the enclosure and the same quantity observed on a horizontal plane outside the enclosure Subscripts a atmosphere b bottom side c cover (envelope) of the enclosure d downward ef effective h radiation reected by the outside ground i surface i of the enclosure j surface j of the enclosure k surface k of the enclosure I inside ground o outside ground s partition t top side u upward

R x, y, z Yj, Zj Y ; Z j j

with

Hbc

Ib LC cos / sin h

2a 2b 2c

where m stands for the number of segments that are reached by the direct beam of solar radiation (in the example depicted in Fig. 1, m = 5).

Hdc Id Aa F ac Hhc Ih Al ql F lc

where Hc is the total solar radiation that reaches the enclosure envelope, Hbc is the direct beam of solar radiation that is intercepted by a segment c of the enclosure envelope, Hdc is the diffuse radiation from the sky that reaches the segment c of the envelope, and Hhc is the radiation reected by the outside ground that reaches the same segment of the envelope. Consider that the enclosure has semi-transparent partitions (Fig. 1). The radiation that reaches the cover is equal to the summation of the radiation intercepted by each segment of the enclosure envelope

n X k1

Hck

Here n represents the total of segments of the enclosure envelope, Hbc, Hdc and Hhc are given by Eqs. (2a), (2b), (2c), respectively, and C (Eq. (2a)) is related to Ck by

m X k1

C k R1 sin h

Fig. 1. Cross section of a hemi-cylindrical enclosure with partitions. The solar radiation beam is intercepted by ve segments of the enclosure envelope (m = 5).

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In general, n P m the equality occurring when h = 90. For h < 90, Hbck = 0 whenever m < k 6 n. According to Fig. 2, the angles h and / can be given explicit forms

Z Z j cos2 h Y j sin h cos h j

10

h sin

z p 2 y2 z

! 5a ! 5b 5c

y h cos1 p 2 y2 z p! z2 y 2 / cos1 r
where

with j = 0, 1, 2, . . . , t. The last point given by the foregoing equations is the one where the solar beam is tangent to the enclosure envelope (Fig. 1), whose coordinates are Yt = R sin h and Zt = R cos h. The width of the beam intercepted by any segment of the enclosure envelope, Ck, between coordinates (Y0 = R, Z0 = 0) and (Yt = R sin h, Zt = R cos h) can be then evaluated by means of Eqs. (9) and (10), leading to

Ck

q 2 2 Y Y Zk Z k k1 k1

11 12

C k Y k1 Y k sin h Z k Z k1 cos h 6a 6b 6c 6d

x r cos hsina c y r cos hcosa c z r sin h p r x2 y2 z2

At this point we need to evaluate the width of the intercepted beam by a segment ck of the enclosure envelope, Ck. Let (Yj, Zj) be the coordinates of any point j of the enclosure envelope (Fig. 1), such as those where the partition is xed to the envelope and the one where the solar beam is tangent to the envelope. The equation of the line obtained by the projection of the solar beam on the cross-section plane of the enclosure, having a slope h with the horizontal plane and passing through the point (Yj, Zj), and the equation of its perpendicular passing through the origin of the coordinate system are given, respectively, by

with k = 1, 2, . . . , t and bearing in mind Eq. (4). Eq. (2a) together with Eqs. (5a)(5c), (6a)(6d), (11), (12) offer the possibility of calculating the direction beam of solar irradiation that is intercepted by any segment of the enclosure envelope, provided the length of the enclosure (L), the coordinates of the points where the partition is xed to the envelope (Yk, Zk), the position of the sun (h, a), the orientation of the axis of the enclosure, c and the direct beam solar irradiation ux density on the horizontal plane outside the enclosure, Ib. The total solar irradiation that reaches any longitudinal segment ck of the envelope is then given by Eq. (1) together with Eqs. (2b) and (2c), provided the diffused, Id, and the hemispherical solar irradiation ux density on a horizontal plane outside the enclosure, Ih, the angle factors (Fack, Flck) and the reectance of the external ground, ql. 2.2. Solar irradiation reaching the inside ground of an enclosure with partitions Consider that the solar radiation will be transferred between diffusely transmitting and diffusely reecting surfaces, and the optical properties of the surfaces do not depend upon the direction of the incident solar radiation. According to [22], the radiation ux that reaches the inner side of the surface j, Uj, after an arbitrary number of reections on surface j and on all the other surfaces of an enclosure composed of N fully diffusive, grey, partially transparent surfaces that diffusely reect, transmit and absorb radiation, is given by

Z Zj tan h Y Yj Z cot h Y

Combining Eqs. (7) and (8), we obtain the coordinates of the point (Y ; Z ) resulting from the interception of the line, obtained j j by the projection of the solar beam on the cross-section plane passing through the point (Yj, Zj), with its perpendicular passing through the origin of the coordinated system, as shown in Fig. 1

Y Y j sin h Z j sin h cos h j

Fig. 2. Position of the sun, P, relative to the enclosure, lying on the plane xy, with its axis passing through 0 and aligned with x-axis, making an angle c with east direction.

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Uj

kj Ak Ek P kj

Aj Ej Pij Aj 1 qj Pjj

13

The downward solar radiation ux densities are given by

Ust1;d

Here Ek is the ux density of radiation input from surface k into the enclosure [22]. This quantity can be related to the radiation incident upon surface k from the outside of the enclosure, Uko, as follows

Ac Ec1 Pc1s AS Es1;u P s2s1t As 1 qst Psst Ac Ec2 Pc2s AS Es2;u P s1s2t As 1 qst Psst !

22a

Ust2;d
where

22b

Ek Uko sk

14

where sk is the total transmittance of surface k for solar radiation which is related with the absorptance, ak, and the reectance, qk, by

Ec1

sc
Ac

Ib LRsin h cos h cos / Id Ac1 F c1a Ih q1 Ac1 F c1l sinh !

23a Ec2

sk ak qk 1

15

The fraction of diffuse radiation emanating from the inside of surface k that eventually reaches surface j (for the rst time) either directly or indirectly (after an arbitrary number of reections on the surfaces) is dened by the parameter Pkj. This parameter can be related to the angle factor Fkj [1,22] as follows

sc I b LR1 cos h cos / Id Ac2 F c2a Ih q1 Ac2 F c2l Ac sinh

23b 23c 23d

Es1;u Usb1;u ss Es1;u Usb2;u ss

Pkj F kj

X
ij

F kj qkj Pij

16

P Here N F kj 1, i.e., the radiation from the inside of the surface k j1 will necessary impinge upon one of the different N surfaces of the enclosure and for any pair of surfaces of area A we have the reciprocity relationship [23]

Usb1;u

As Es1;d Ps2s1b As Es2;d Ps2s1b As 1 qsb Pssb As Es2;d Ps1s2b As Es1;d Ps1s2b As 1 qsb Pssb

23e

Usb2;u

23f

Ak F kj Aj F jk

17

3. Solar irradiation incident upon the ground of a partitioned enclosure The foregoing equations presented in Section 2 are next applied to the study of enclosures with and without partitions. 3.1. Hemi-cylindrical enclosure without partitions Consider a hemi-cylindrical enclosure. According to Eq. (13) the solar radiation at the ground level of the enclosure is given by

where Es1,d and Es2,d are given by Eqs. (21a) and (21b), respectively. Substituting Eqs. (21a), (21b), (22a), (22b), (23a), (23b), (23c), (23d), (23e), (23f) into (20) yields

Ug sef
with

Ac Uc1;o Uc2;o Ag !

24

sef

sc ss Pcs Psg W 1 Q 1 qg Pgg 1 Psst Psst G

25a

& q! Ug sef 0:51 F cc 1 1 cot2 h cos2 a c Ib F ca Id ' 1 F ca qo Ih

25b

18

G1

s2 Psst Pssb s
1 qsb P ssb 1 qst Psst

25c

sef

sc
1 F cc qc 1 F cc qc qg

Q 19
and

s2 Psst Ps1s2b s 1 qsb Pssb 1 qst Psst 2

25d

where sef may be interpreted as the effective transmittance of the enclosure. It depends not just on the transmittance of the enclosure but also on the reectance of both envelope and bottom surfaces, and angle factor Fcc. 3.2. Hemi-cylindrical partitioned enclosure Consider next an enclosure with partitions as shown in Fig. 3. According to Eq. (13), the ux of solar radiation that reaches the surface g is given by

Ac Uc1;o Ib LR1 cot hj cosa cj Id Ac1 F c1a Ih q1 Ac1 F c1l Ac Uc2;o Ib LR ! q 1 cot2 h cos2 a c cot h j cosa cj 25e

Id Ac2 F c2a Ih q1 Ac2 F c2l 25f

A E P As Es2;d Psg Ug s s1;d sg Ag 1 qg Pgg

20

Es1;d Ust1;d ss Es2;d Ust2;d ss

21a 21b

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Fig. 3. Cross-section of a hemi-cylindrical partitioned enclosure.

cylindrical enclosure (radius 4 m) located at Evora (38.6 north latitude, 7.9 west longitude, 309 m altitude) was considered. The solar irradiation uxes were computed according to the formulation presented by [1] for both low and high insolation conditions. For sake of simplicity, we consider January and July as months of low and high insolation, respectively. The daily total irradiation inside the partitioned enclosure (Fig. 3) was obtained for fully transparent (sc = 1, ac = qc = 0) and partially transparent (sc = 0.8, ac = 0.1, qc = 0.1) enclosure envelope. In both situations, the total hemispherical reectance of both the inside and the outside ground is assumed to be 0.20. The angle factors were evaluated according to [23] and are presented in Table 1. The daily total irradiation inside the enclosure at ground level was then computed with Eqs. (18) and (24). The results were presented in terms of a dimensionless quantity, w, dened as the ratio between daily total irradiation inside the enclosure and the same quantity observed on a horizontal plane outside the enclosure. Fig. 4 shows how the effective transmittance, sef, of an enclosure without partitions varies with the transmittance and the reectance of the envelope. This plot shows that sef is mainly inuenced by the transmittance of the enclosure but it also increases with its reectance, and is therefore reduced by the absorptance. For a fully transparent enclosure envelope with partitions (Fig. 3), the effect of the orientation of the enclosure and the optical properties of the partition on w, is presented in Figs. 5 and 6 for conditions of low insolation (January, winter) and high insolation (July, summer), respectively. Fig. 7 shows w versus the orientation of the enclosure and the optical properties of the partition, for a partially transparent enclosure envelope. According to the previous gures there is a noticeable inuence of the orientation of

Fig. 4. Effective transmittance of an enclosure without partitions versus the transmittance, reectance and absorptance of the envelope (qg = 0.20).

Table 1 The angle factors evaluated according to [23]. Enclosure without partition Fcc = 0.364 Fca = 0.818 Partitioned enclosure (Fig. 3) Fs1s2 = Fs2s1 = Fss = 0.293 Fs1g = Fs2g = Fsg = 0.707 Fgs1 = Fgs2 = Fgs = 0.500 Fc1s = Fc2s = 0.900 Fc1l = Fc2l = 0.182 Fc1a = Fc2a = 0.818

the enclosure on the collection of solar radiation. This inuence depends also upon the optical properties of the partition. For the enclosure with partitions, the orientation northsouth collects more radiation in summer (July) and slightly less in winter (January) as compared with the enclosure with its axis directed eastwest. However, as the transmittance of the partition decreases, the radiation collected both in winter and summer becomes practically insensitive to the enclosure orientation. As we expected, Figs. 5 and 6 show that the radiation that reaches the bottom surface (ground) of the enclosure is, in both cases, mainly determined by the transmittance of the partition and the reectance and absorptance play a minor role. The plots also show that the radiation that reaches the enclosure ground decreases when the absorptance of the partition surface facing down increases (i.e., the reectance decreases). Fig. 7 shows that for a non-opaque partition (ss > 0) the higher the reectance of either the partition surface facing down or facing

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Fig. 5. w versus the angle c between the enclosure axis and the eastwest direction on an average day of winter (January) within the partitioned enclosure represented in Fig. 3 (fully transparent enclosure sc = 1.0). Fig. 7. w versus the angle c between the enclosure axis and the eastwest direction on an average day of summer (July) within a partitioned enclosure represented in Fig. 3 (partially transparent enclosure sc = 0.8 and qc = 0.1).

enclosure provided the outside solar irradiation, the position of the sun, the optical properties of both the enclosure envelope and partitions, the geometry of the partitions and the orientation of the axis of the enclosure. The model was applied to the study an enclosure where partitions form a closed cavity with the bottom surface (ground) of the enclosure, but can be also applied to another different conguration. The impact of the orientation of the enclosure, as well as the importance of the geometry and the optical properties of both the enclosure envelope and partitions on the income of solar irradiation are indicated.

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Fig. 6. w versus the angle c between the enclosure axis and the eastwest direction on an average day of summer (July) within the partitioned enclosure represented in Fig. 3 (fully transparent enclosure sc = 1.0).

up, the larger is the admission of solar radiation. However, a given increase in the reectance of the partition surface facing down produces a larger admission of radiation than the same increase on the reectance of the surface facing up. Another interesting feature is observed in Fig. 7: keeping the reectance of the partition surface facing down xed, the lower the transmittance of the partition the smaller the collection of solar radiation irrespective of the optical properties of the partition surface facing up. 5. Final remarks In this paper a model has been developed which offers the possibility of computing the solar irradiation inside a partitioned

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A.F. Miguel, A. Silva / Applied Energy 87 (2010) 836842 [19] Bejan A, Tsatsaronis G, Moran M. Thermal design and optimization. New York: Wiley; 1996. [20] Enibe SO, Iloeje OC. Design optimization of the at plate collector for a solid absorption solar refrigerator. Sol Energy 1997;60:7787. [21] Muneer T, Gueymard C, Kambezidis H. Solar radiation and daylight models. Oxford: Elsevier Butterworth-Heinemann; 2004. [22] Silva A, Miguel AF, Rosa R. Thermal radiation inside a single-span greenhouse with a thermal screen. J Eng Res 1991;49:28598. [23] Modest M. Radiative heat transfer. New York: Academic Press; 2003.

[14] Senkal O, Kuleli T. Estimation of solar radiation over Turkey using articial neural network and satellite data. Appl Energy 2009;86:12228. [15] Munawwar S, Muneer T. Statistical approach to the proposition and validation of daily diffuse irradiation models. Appl Energy 2007;84:45575. [16] Muneer T, Younes S. The all-sky meteorological radiation model: proposed improvements. Appl Energy 2006;83:43650. [17] Meir MG, Rekstad JB, Lovvik OM. A study of a polymer-based radiative cooling system. Sol Energy 2002;73:40317. [18] Thirugnanasambandam M, Iniyan S, Goic R. A review of solar thermal technologies. Renew Sustain Energy Rev 2009; in press.