Solar irradiation inside a single span greenhouse

J. agric. Engng Rex (1989) 43, 221-229

Solar Irradiation Inside a Single Span Greenhouse Rur ROSA*; ANA MARIA SILVA*; ANT~NIO MIGUEL* A model is offered through which the solar irradiation can be computed inside a single span hemicylindrical tunnel greenhouse. The model is mathematically exact. From the physical point of view it is founded on the assumption that the cladding surface is fully diffusive and that the radiation diffused through the atmosphere and by the ground is fully isotropic. It is found that the solar irradiation inside the greenhouse depends upon its orientation. A greenhouse with its longitudinal axis aligned along the north-south direction collects more radiation in summer and less in winter as compared with a greenhouse with its axis aligned along the east-west direction. On the other hand the east-west oriented greenhouse is more efficient in collecting solar radiation in winter than in summer, while the north-south oriented greenhouse exhibits a constant efficiency the year round. Finally, it is shown that the collected radiation depends upon the optical properties of the cladding surface being mainly determined by its transmittance.

1. Introduction A number of authors have studied the influence of the shape and orientation of single span greenhouses and of the properties of their cladding surfaces upon the solar irradiation observed inside Edwards and Lake,’ Smith and Kingham,’ Basiaux et al.,’ Silva and Carvalho,4 Critten i ). Such studies require data about the optical properties of the cladding materials and these have been the subject of a number of studies (Blaga,6 Godbey eC a1.,7 Nijskens et al. “).

In this paper a model is explained which offers the possibility of computing the solar irradiation inside a single span greenhouse, provided the solar irradiation observed outside the orientation of the greenhouse and the optical properties of its cladding surface are known. The validity of this model is based on the assumptions that the radiation is transferred between diffusely reflecting and diffusely transmitting surfaces and that the optical properties of the surfaces do not depend upon the direction of the incident radiation. These assumptions are not entirely realistic because a direct solar beam is present whenever the sun shines and actual surfaces exhibit to some extent both specular and diffuse transmittance and reflectance. But many materials, such as fibreglass, and plastic corrugated and honeycomb structures, diffuse most of the radiation or become diffusive as a result of dust or dew deposition. The foregoing assumptions are expected to hold true when diffuse radiation predominates in the greenhouse environment, that is to say when shadows are not observed inside, even when the sun shines outside. 2. Theory Consider a single span greenhouse. Although the model applies to an arbitrary shape, the computation is limited to an hemicylindrical greenhouse, Assume also a fully diffusive * Departamento de Fisica, Universidade de gvora, Largo dos Colegiais 2, 7000 &ora, Portugal Received 29 September 1988; accepted in revised form 12 March 1989

221 @X21-8634/89/07@221

+ 09 $03.00/O

@ 1989

The

British

Society

for Research

in Aaricultural

EneincerinP

222

SOLAR

IRRADIATION

INSIDE

A

GREENHOUSE

Notation

azimuth of the sun relative to the south angular height of the sun above the horizontal Cartesian coordinates surface area, mz angle or shape factor for direct exchange of diffuse radiation between surfaces n and m daily sum of solar irradiation measured on the ground outside, WhmP2 daily sum of solar irradiation measured inside the greenhouse, WhmU2 total flux of solar radiation that reaches the envelope of the greenhouse, W direct beam solar irradiation flux density, on the horizontal plane, outside the greenhouse, Wmm2 diffuse solar irradiation flux density, on the horizontal plane, outside the greenhouse, Wmm2 hemispherical solar irradiation flux density, on the horizontal plane, outside the greenhouse, Wme2 solar irradiation flux density measured inside the greenhouse, Wmm2 length of the greenhouse, m

P

R (Y Y

P 8

t trel

fraction of diffuse radiation emanating from the underside of the cladding surface that reaches the inside ground, either directly or indirectly, after an arbitrary number of reflections on the cladding surface radius of the envelope of the greenhouse, m absorptance of a surface for solar radiation angle between the longitudinal axis of the greenhouse and the east-west direction angle of the solar beam with its projection on the plane of the greenhouse cross-section reflectance of a surface for solar radiation angle of the projection of the solar beam on the cross-sectional plane of the greenhouse with the horizontal plane transmittance of a surface for solar radiation relative transmittance of the envelope of the greenhouse Subscripts

a atmosphere C cladding surface 0 outside ground g inside ground

cladding surface and that the radiation diffused by the sky and by the ground is fully isotropic. Let &, 1, and 1, denote the direct beam, the diffuse and the hemispherical solar irradiation flux density on an horizontal plane as measured outside the greenhouse (Fig. 1). Let h denote the angular height of the sun above the horizon (that is, the angle between the solar beam and the horizontal plane) and a its azimuth relative to the south (that is, the angle between the projection of the solar beam on the horizontal plane and the south direction), as shown in Fig. 2. Let L, R and y stand for the length of the

R.

ROSA

ET

223

AL.

Fig. 1. Diffuse solar irradiation The hemispherical irradiation

from the sky (a) and solar direct beam (b) on an horizontal surface. is the superposition of both the diffuse and the direct components

greenhouse, the radius of its envelope and the angle of its axis with the east-west direction. Let A,, A", A, and A, = 2RL stand for the area of the atmosphere (that is, the area of a virtual sky vault, whose exact shape and dimension do not affect the final results), the outside ground, the cladding surface and the inside ground. Let t, and pC represent the transmittance and the reflectance of the cladding surface and pO and pe the reflectance of the external and internal ground; all these are global quantities and therefore assumed independent of the direction and wavelength of the incident radiation. In what follows, F,, denotes the angle factor or shape factor between surfaces n and m, that is to say, the fraction of diffuse radiation emanating from surface n that reaches surface m. In the present circumstances one has F,, + F,, = 1, that is to say, the radiation from the upperside of the cladding surface is necessarily directed either to the outside

Fig. 2. Position

of the sun, S, relative to a single span hemicylindrical lying on the plane x-y with its axis aligned

tunnel greenhouse with the x-axis

located at 0,

224

SOLAR

IRRADIATION

Solar

Fig. 3. Cross-section

INSIDE

A

GREENHOUSE

beam

of a single span hemicylindrical tunnel greenhouse being intercepted by its envelope

and the

radiation

ground or to the atmosphere; on the other hand, the radiation from the underside of the cladding surface is necessarily directed either to another 1 ooint of the envelope or to the inside ground, that is to say, -FCC + F,, = 1. The solar radiation that reaches the greenhouse envelope is the sum of three contributions H=H’+H”+H”’ (1) The first contribution is the direct beam of solar radiation that is intercepted by the greenhouse envelope; it is given by H’

=

&

LR( 1 + sin f3) cos $

(24

where R(l + sin 0) represents the width of the intercepted beam (Fig. 3), 8 being the angle between the projection of the solar beam on the cross-sectional plane of the greenhouse and the horizontal plane, and C$the angle between the solar beam and its projection on the plane of the greenhouse cross-section (Fig. 2). The second contribution is the diffuse radiation from the sky that reaches the envelope which amounts to H”=I

d A.F‘I ‘1C

(2b)

The third contribution is the radiation reflected by the outside ground that reaches the envelope; it amounts to H”’ = h,~oAo&

(2c)

The solar radiation that finally reaches the ground inside the greenhouse is

HO?

1+ pspcP + p;pff2

+

. )=

Ht,P 1 - PgPP

where P stands for the fraction of the diffuse radiation emanating from the underside of the cladding that reaches the inside ground, either directly or indirectly, after an arbitrary number of reflections on the cladding surface (Fig. 4). The terms between brackets represent the contributions of the radiation paths that include zero, one, two, and so on, reflections on the inside ground.

R.

ROSA

225

ETAL.

Fig. 4. Radiation transfer paths from the underside of the cladding surface to the inside ground of the greenhouse. Paths a and b exemplify the angle or shape factor for direct exchange of diffuse radiation &. AN paths a to d contribute to the overall exchange parameter P defined in the text

The average inside flux density of solar radiation lint = HtcPIAg(l-

will therefore be

(4)

~g~cf’)

In order to obtain a practical formula one has to relate H and P to known quantities. With regard to H it will suffice to give explicit forms to angles 8 and I$. To that effect notice (Fig. 2) that sin 8 = *IjIm,

@a)

cos @ = V-/r,

(5b)

where x = r cos h sin (a - v),

(64

y = r cos h cos (a - y),

(6b)

z=rsinh,

(6~)

r=j/x*+y*+z*.

(64

As to P notice that P = Fcg+ Fccp, f’,

(7)

F,,+F,,= 1,

(8)

and hence

(9)

p = (1 - F,c)l(l - F,,P,). In order to obtain a practical formula,

it should be noted that

F,, + F,, = 1

(10)

and, for any pair of surfaces of area A (Siegel and Howell’) AX,,

= AnL.

(11)

From Eqn (4), taking in consideration Eqns (l), (2), (5), (6), (9), (10) and (ll), the following formula is obtained for the solar radiation density flux incident at ground level inside a single span hemicylindrical tunnel type greenhouse: < = (1 - F,,p,) -r;.#

- F,,)p,

[(I

-

F,,)%

+

F,aId

+

(1

-

F,&oh,],

(124

226

SOLAR

IRRADIATION

INSIDE

A

GREENHOUSE

where a = t(l + vl + cot2 h cos2 (a - y)).

Wb)

3. Results

From Eqn (12), the solar irradiation inside the greenhouse depends on the solar irradiation outside and the position of the sun, on the optical properties of the cladding surface, on the angle factors that have to do with the shape of the envelope and on the orientation of the axis of the greenhouse. For a hemicylindrical greenhouse the angle factors have definite values: F,, = 0.364 and F,, = O-818 (Siegel and Howell’). To explore the influence of the orientation of the greenhouse upon the captured solar radiation a greenhouse located at Evora (38.6” north latitude, 7.9” west longitude, 309 m altitude) was considered, where the monthly averages of hourly sums of solar irradiation and hourly insolation fractions are known. A hypothetical day, when I,,, 1, and I,, exhibit their average values on a particular month of the year was considered. To discard the influence of the optical properties of the cladding surface, this is assumed to be fully transparent (rC = 1, pC = 0). The solar radiation flux density was computed hourly and the daily total irradiation inside the greenhouse (Gi”,) was obtained. The result is shown in Fig. 5 as a function of the orientation of the axis of the greenhouse, relative to the east-west direction, for different months of the year. Fig. 6 shows the variation of the daily sum of solar irradiation flux density observed inside the greenhouse against the same quantity observed outside (G,,,) for the east-west directed and the north-south directed greenhouses. To investigate the influence of the optical properties of the cladding surface upon the solar radiation flux density inside the greenhouse it is sufficient to analyse the behaviour

“E > 1 c 6

5-

May

4-

September

3-

March November

2% Jonuory I-

30”

60”

90”

120”

150’

Fig. 5. Daily sum of solar irradiation (Gin,) inside a single span hemicylindrical tunnel type greenhouse as a function of the angle (y) between the greenhouse axis and the east-west direction on an average day of different months of the year. The computation was carried out with data from Evora (38.6” north latitude, 7.9” west longitude, 309m altitude). The cladding surface is assumed fully diffusive and fully transparent

R.

ROSA

227

ET AL

2

4 G,x,,

8

6 kWh/m*

inside a single span hemicylindrical Fig. 6. Daily sum of solar irradiation observed on the outside ground the daily sum of solar irradiation (EW) and the north-south (NS) alignments

of the first factor in Eqn (12a) which may be interpreted t

rc’ = (1 - F,,Pc) T;gu

tunnel greenhouse (Gin,) (G,,..), for the east-west

as a relative transmittance

- E&c.

(13)

It is taken F,, = 0.364 (as appropriate for a hemicylindrical tunnel with a circular crosssection) and ps = 0.20 (as a typical value for local grounds in summer). Fig. 7 shows how this relative transmittance varies with the transmission and reflectance of the cladding

Fig. 7. Relative transmittance (tre,) of a single span hemicylindrical tunnel type greenhouse as a function of the transmittance z,, reflectance pC and absorptance CY,of the cladding surface which is assumed fully diffusive. Inside ground rejiectance ps = 0.20 and angle factor F, = 0.364. As an example, if t, = O-7, pC = O-1 and cu, = O-2, then t,,, = O-735

228

SOLAR

IRRADIATION

INSIDE

A

GREENHOUSE

surface. Note that tc+pc+

a,=

1,

(14)

o, being its absorptance. 4. Discussion

Fig. 5 exhibits a remarkable influence of the orientation of the greenhouse upon the collection of solar radiation. The greenhouse with its axis directed north-south collects rather more radiation in summer and slightly less in winter as compared to the greenhouse with its axis directed east-west. The collection of radiation is more uniform the year round in a greenhouse with its axis aligned along the east-west direction. In the neighbourhood of the equinoxes the radiation collection is practically insensitive to the greenhouse orientation. According to Fig. 6 the greenhouse with its axis on the east-west direction is more efficient in collecting solar radiation in winter than in summer, while the greenhouse with its axis on the north-south direction exhibits a nearly uniform efficiency the year round. The fact that the greenhouse with its axis in the north-south direction collects more solar radiation in summer (less in winter) than the greenhouse with its axis in the east-west direction finds support in the observations of Edwards and Lake’ and in the computations of Smith and Kingham.’ Contrary to these authors, however, who found the collecting efficiency of the greenhouse for solar radiation (i.e. the ratio of the daily sum of radiation observed inside to that observed outside) to be nearly uniform the year round, when the greenhouse had its axis in the east-west direction, we find that, for such an alignment, the collecting efficiency is higher in winter (and lower in summer). The difference (apart from the influence of shading by structural elements which we did not consider) is due to ours being a greenhouse with a fully diffusive envelope while theirs had a clear glass envelope. In fact, according to Basiaux et al.’ the east-west aligned tunnel covered with a diffusing material collects more direct radiation when the sun height is below 35” (less than when the height is above 35”) than the same tunnel covered with a clear material. Our results also agree with the recent work of Critten” which offers a model for the multispan glasshouse under direct sun radiation at UK latitude; according to that author, the east-west alignment is best for high light input in winter (while the north-south alignment is best in summer) and the light loss is independent of season for the north-south aligned glasshouse. In spite of the difference of circumstances his conclusions and ours coincide as far as they are comparable. Fig. 7shows that the relative transmittance of the greenhouse is mainly determined by the transmittance of the cladding surface but that it also improves with its reflectance and is therefore reduced by its absorptance. 5. Conclusions

A model is offered for computing exactly the solar irradiation inside a single span tunnel greenhouse with a fully diffusive cladding surface. The results are compared and agree with those reported by other authors. This leads to the conclusion that among hemicylindrical tunnel greenhouses, with different orientations and cladding surfaces with different optical properties, the one aligned along the east-west direction, having a diffusive envelope is the only one which exhibits a modulating capacity upon the capture of solar radiation, having a larger collecting efficiency in winter than in summer.

R.

ROSA

ET

229

AL.

These findings are of interest in orienting a greenhouse according to the purpose in view. Different orientations are advisable for different purposes, such as growing short life-cycle or long life-cycle plants or drying crops. It also emphasizes the interest in employing diffusive envelopes when the collection of solar radiation is to be enhanced in winter. References ’ Edwards, R. I.; Lake, J. V. Transmittance of solar radiation in a large-span east-west glasshouse. Journal of Agricultural Engineering Research 1965, 10: 125-131 * Smith, C. V.; Kingham, H. G. A contribution to glasshouse design. Agricultural Meteorology 1971, 8: 447-468 ’ Basiaux, P.; Deltour, J.; Nisen, A. Effect of diffusion properties of greenhouse covers on light balance in the shelters. Agricultural and Forest Meteorology 1973, 11: 357-372 ’ Silva, A. M.; Carvalho, M. J. CBlculo da radia@o interior em estufas n,a regigo de kvora (Computation of the irradiation inside greenhouses in the region of Evora). Revista de Electricidade (Lisbon) 1983, 185: 122-126 ’ Critten, D. L. A general analysis of light transmission in greenhouses. Journal of Agricultural Engineering Research 1986, 33: 289-302 ’ Blaga, A. Use of plastics in solar energy applications. Solar Energy 1978, 21(4): 331-338 ’ Godhey, L. C.; Bond, T. E.; Zomig, H. F. Transmission of solar and long-wave-length energy by materials used as covers for solar collectors and greenhouses. Transactions of the ASAE 1979, 22(5): 1137-1144 * Nijskens, J.; Deltour, J.; Coutisse, S.; Nisen, A. Radiation transfer through covering materials, solar and thermal screens of greenhouses. Agricultural and Forest Meteorology 1985, 35: 229-242 ’ Siegel, R.; Howell, J. R. Thermal Radiation Heat Transfer. McGraw-Hill, 1972 lo Critten, D. L. The transmission of direct light by structureless symmetric-roofed multispan greenhouses with non-absorbing cladding. Journal of Agricultural Engineering Research 1988, 48: 225-232