Radiative heat loss inside a greenhouse

J. agric Engng Res. (1987) 37, 155-162

Radiative

Heat Loss inside a Greenhouse ANA MARIA SILVA*; RUI ROSA*

In this paper the exact solution of a realistic physical model of the thermal radiation exchange inside a single span greenhouse is offered which yields the net radiation flux density at ground level. A similar but approximate model is also offered for a multispan greenhouse.

The rate of radiative heat loss was computed for the particular situation when the inside ground, the cladding material and the outside air and ground are all at freezing temperature, and the sky is clear. It turns out that the radiative heat loss rate is nearly the same for the single and for the multispan greenhouse, and is mainly determined by the transmittance of the cladding material, being little dependent on the relative magnitudes of its emittance and reflectance. Finally, the coefficient of radiative heat transfer for the inside ground surface was obtained as well.

1. Introduction Greenhouses exhibit a greater thermal radiation interaction with their surroundings than most buildings. Thermal radiation loss can then become the dominant mechanism of total night-time heat loss. Thermal radiation exchange is therefore a very important factor in determining the thermal environment inside a greenhouse. A number of authors have recently worked on the problem of describing the thermal radiation environment in relation to the radiometric properties of the cladding material (Kindelan,’ Garzoli and Blackwell,’ Chandraj and Cooper and Fuller4). However, none have yet offered an exact solution of a realistic model for computing the rate of radiative heat loss at ground level inside the greenhouse. In a previous paper (Silva and Rosa’) a study was reported on the radiative environment inside a semi-cylindrical tunnel-type greenhouse. In particular, we reported the study of the net radiation during the night-time, in relation to the radiometric properties and temperature of the surfaces which form the envelope of the greenhouse and the external radiation fluxes (atmospheric and terrestrial thermal radiation). The radiation budget during the night-time was based on an approximate solution which holds true only when the material covering the greenhouse exhibits a fairly high emittance. The present paper establishes a more accurate model which is generally applicable to a single span greenhouse with any configuration and any cladding material. A similar model is then offered for the case of a multispan greenhouse as well. This particular model is approximate, for it neglects radiation paths involving multiple radiation exchanges. However, it is pointed out that its accuracy is very good in practical circumstances. The models offered in this paper may then be applied to estimating the radiative heat loss rate observed at ground level. This is of utmost importance in forecasting frost formation. A complete study of the heat balance of a greenhouse ought to take into account not only heat radiation but also sensible and latent heat fluxes. Such a complete study is outside the scope of the present work but it is hoped that this may nevertheless prove to be a useful contribution to such a global model. * Departamento

de Fisica, Universidade de fivora, Largo dos Colegiais, 7000 kvora, Portugal

Received 5 May 1986; accepted in revised form 28 September 1986

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Notation surface area, m2 F 12 geometrical shape factor for exchange of diffuse radiation between surfaces 1 and 2 h coefficient of radiative heat transfer, W mm2 K-’ Ld atmospheric radiation flux density, W mm2 LU terrestrial radiation flux density, Wme2 R thermal radiation net flux density, W mm2 T temperature, K or “C E emittance for thermal radiation A

4 p c r

component of thermal radiation flux density, W me2 reflectance for thermal radiation Stefan-Boltzmann constant, equal to 5.67 x lo-* W me2 KP4 transmittance for thermal radiation

Subscripts

a c o s

atmosphere cladding surface outside ground inside ground

2. The net radiation inside a greenhouse at night-time Let E, r and p stand for the emittance, transmittance and reflectance of the cladding material; Ld and L, are the atmospheric (downward) and terrestrial (upward) thermal radiation flux densities observed outside the greenhouse (W mm2); aT4 is the black-body radiant emissive power at absolute temperature T(K); 6, and R are radiation flux densities (Wm -?); A stands for surface area (m’); F,, is a geometrical shape factor for perfectly diffuse radiation (i.e. the fraction of diffuse radiation emanating from surface 1 that is received on surface 2). The subscripts s, c, a and o refer to the ground inside the greenhouse, the plastic cladding, the atmosphere and the outside ground respectively. It is assumed that the inside ground has unity emittance and zero reflectance. The greenhouse cladding may have arbitrary radiometric properties but is assumed to be grey and fully diffusive. Multiple reflections on the cladding surface are taken into account. Consider in the first place an infinitely long single span greenhouse. The radiation flux arriving at the ground that has been emitted by the greenhouse cladding surface (either directly or after one, two, or more reflections on the cladding surface) has a flux density $,, that is given by A,&,

= E~T,~A,F,,(~+~F,,+~~F,Z,+

. . .)

= EoT,~A~F_/(~-~F,,).

Given that A, F,, = A, F,, (Siegel and Howel16) we have A,F,, = A,F,, so that we obtain 4,s = F,,sc~.‘/(l-

PF,,).

(1)

- PF,,)

(2)

Similarly, we have 4,s = F,,F,, pG4/(l

for the flux density arriving at the ground that has been emitted by the inside ground itself. For the flux densities at ground level that are due to the external atmospheric and terrestrial radiation we have A, = F,,F,,rL,l(l - PF,,), (3) 60, = F,,F,,Gl(l

-

PF,,).

(4)

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Now, the net radiation flux density at ground level inside the greenhouse is R = A, + A, + 4,s + A, - G4.

(5)

Taking into account the flux densities given by Eqns (1) to (4) and bearing in mind that F,, = 1, we finally obtain (R+aK4)(1-pF,,)

= EoTP+F,,~~T,~+~(F,,L,+F,,L,).

(6)

Consider, next, a multispan greenhouse, infinitely long and infinitely wide. In this case it is not possible to offer an exact model, as before, as a result of the complexity of the radiation exchange paths among the roof panes. However, a very good approximation can be obtained by simply taking into account single exchange paths, that is to say, by retaining only terms of zero and first degree in F,, (the shape factor for exchange of radiation between roof panes). This approximation is very good because a path involving n radiation exchanges gives a contribution proportional to Fi, and, as a rule, F,, c 1, so that higher order terms can be neglected. With reference to Fig. 1 (fop), the radiation flux arriving at the ground that has been emitted by the greenhouse cladding has a flux density 4,, that is given by A,$,, = saT,4M’,,(1

+ PF,, + rF,J

This includes three contributions: (1) the direct flux; (2) the flux that arrives at the ground after one reflection 0n.a neighbouring roof pane (proportional to pF,,); (3) the flux which,

\

\ Fig. 1. Thermal radiation paths inside a multispan greenhouse, neglecting multiple exchange paths among the roof panes. (Top) Radiation emitted by the cladding surface which arrives at the ground. (Middle) Radiation emitted by the ground which is turned back by the cladding surface. (Bottom) Atmospheric radiation transmitted down to the ground through the cladding surface

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being emitted in the opposite direction, arrives at the ground after transmission through a nearby roof pane (proportional to zF,,). As we have A,F,, = A,F,,, it turns out that 4,s =

F,,s(l + PF,, + zF,,)oq4.

(7)

Similarly, with reference to Fig. 1 (middle), the radiation flux arriving at the ground that has been emitted by the inside ground itself has the following flux density: $,, = F,,F&

+ P’F,, + r2F,,)G4.

(8)

This also includes three contributions: (1) the flux arriving at the cladding surface from the inside ground that is reflected back directly to the ground (proportional to p); (2) the flux that is reflected back to the ground after one extra reflection on a nearby roof pane (proportional to p2F,,); (3) the flux that, having been transmitted through a first roof pane, is sent back to the ground after transmission through a neighbouring pane (proportional to r2FJ. Finally, the radiation flux arriving at the ground that originates in the sky has the following flux density (refer to Fig. I, bottom): 4,s = F,,F,,r(l+2pF&,.

(9)

This includes three contributions as well: (1) the flux emanating from the sky that is transmitted through the cladding directly to the ground; (2) the flux that is transmitted through a roof pane that arrives at the ground after reflection at a nearby pane (proportional to zpF,,); (3) the flux that is reflected by a first roof pane but which arrives at the ground after transmission through a neighbouring pane (also proportional to zpFcc). For the net radiation flux density R, being given by Eqn (5), and with c$,, = 0 and F,, = 1, we finally obtain R+aK4

= ~(l+pF,,+zF,,)rrT,~+F,,(p+p~F,,+~~F,,)a~~+F,,z(l+2pF,,)L,.

The shape factors can now be evaluated. For an infinitely long semi-cylindrical type greenhouse (Siegel and Howell6 and Kittas’) F,, = 0.364;

F,, = 0636;

F,, = 0.818;

(10) tunnel

F,, = 0.182.

For an infinitely long and infinitely wide multispan structure (Siegel and Howel16) F,, = 1 -sin

(u/2);

F,, = F,, = sin (a/2),

where c1stands for the roof angle. In the following numerical example we assume CI= 135”, in which case F,, = 0.076;

As F,, <<1, contributions

proportional

F,, = F,, = 0.924.

to F,” (n > 2) may be neglected, as assumed.

3. Application example The foregoing models are useful in predicting the rate of radiative heat loss at ground level inside the greenhouse, provided that the ground temperature, cladding surface temperature, atmospheric and terrestrial flux densities are known. For purposes of a numerical example it will be assumed that inside and outside ground temperature, outside air and cladding surface temperature are all at freezing point (T = 273 K); that the sky is clear, in which case its emissivity is close to 0.72 (Heitor and Rosa*); and that the emissivity of the outside ground is unity. This situation is not

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necessarily realistic, but rather a reference situation which is useful in evaluating behaviour of the greenhouse near freezing conditions. In these circumstances aK4 = oTc4 = L, = 316 W m-‘;

the

L, = E,oT,~ = 228 W mm2.

Notice that a clear sky exhibits a relatively low emissivity if its temperature is assumed (as is usual) equal to the air temperature at screen level. In other words, the apparent temperature of a clear sky (i.e. the temperature of a black-body emitting the same radiation flux density) is much lower than the air temperature at screen level (- 21°C against 0°C in our example). Eqns (6) and (10) can now be solved, yielding the radiative heat loss rate R as a function of the radiometric properties (a, r and p) of the cladding material. The results are plotted in Figs 2 and 3, the curves being labelled with the corresponding radiative heat loss rate R Pm -2) at ground level, inside a single span tunnel or a multispan greenhouse, as a function of the transmittance (z), emittance (E) and reflectance (p) of the cladding material. In these diagrams the representative point of a particular cladding material can be found with the help of any two of the three radiometric properties z, E and p, the third one being subjected to the condition r + E+ p = 1. Each co-ordinate on the diagram varies linearly with the distance measured perpendicularly to the side of the triangle, from 0 on the side to 1 on the opposite vertex. The value of the heat loss rate which corresponds to a particular representative point is estimated by interpolation between the two nearby lines of given heat loss rate. It should be appreciated, in view of the similarity between Figs 2 and 3, that the geometric configuration appears to be not very important in determining the radiative heat loss rate.

Fig. 2. Radiative heat loss rate R ( W me2). at ground level inside a single span semi-cylindrical tunneltype greenhouse as a function of the transmittance E, emittance t and reflectance p of the cladding material. Note that z +~+p = 1. Inside and outside ground, outside air and cladding surface are at freezing point and the sky is clear. Consider, for instance, a cladding material having T = 0.6, E = 0.3 and p = 0.1; the representative point P lies between the lines labelled -40 and -SO W me2; by interpolation R= -46 Wmm2

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p:l

Fig. 3. Radiative heat loss rate R ( W m- 2, at ground level inside a multispan greenhouse with roof angle u = 135” as a function of the transmittance 5, emittance E and reflectance p of the cladding material. Note that z + E+ p = 1. Inside and outside ground, outside air and cladding-surface are at freezing point and the sky is clear

This rate is found to be mainly determined by the transmittance of the cladding material, its emittance and reflectance playing a minor role. When the cladding material is transparent the heat loss rate is strongly correlated to the apparent temperature of the sky. However, it should be stressed that these results are obtained on the assumption of constant temperature over the cladding surface, which is merely an approximation; particularly in the case of a tunnel-type greenhouse, the angle of view of ground and sky varies rapidly along the circle of the envelope thereby producing a variation of its temperature. The radiometric properties of the cladding surface may change with time due to deposition of dust, ageing under the action of ultra-violet radiation from the sun or condensation. Condensation, in particular, can modify quickly and drastically the radiometric properties of the envelope. A nearly transparent polyethylene becomes nearly opaque to infrared radiation when film condensation takes place, thus suppressing the direct exchange of thermal radiation between the inside ground and the sky (Nijskens et cd.‘). We now consider in more detail the case of the single span semi-cylindrical type greenhouse. Fig. 2 gives the heat loss rate R (W m-‘), assuming all temperatures to be at freezing point and a clear sky. Next, assuming the same meteorological situation, we searched for the inside ground temperature T* at which the radiative heat loss rate becomes zero (R = 0); that would be the equilibrium ground temperature which would be attained if no other heat fluxes were involved and the meteorological situation persisted. The results of this calculation are shown in Fig. 4. It will be noted that the ground would freeze, except when T = 0. When T = 1 the ground would eventually attain the apparent sky temperature (-21°C in our example). For p = 1 the temperature is not defined, which is due to the fact that the radiation received by the inside ground is then exactly the radiation it has emitted, giving R = 0, whatever its temperature may be. Finally, working with the data of Figs 2 and 4 we were able to obtain the parameter h = -dR/dT (W m-’ K-‘) which can be interpreted as a coefficient of radiative heat

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E

Fig. 4. Equilibrium inside ground temperature P (“C), that is, ground temperature at which the radiative heat loss rate becomes zero, for a semi-cylindrical tunnel-type greenhouse, for outside air, outside ground and cladding surface at freezing point and a clear sky. Consider, for instance, a cladding material having T = 06, p = 0.1 and E = 0.3; then T* = - 11.5”C

transfer for the ground surface and is shown in Fig. 5. For a relatively small temperature deviation from equilibrium, the radiative heat loss rate will then be given by -R

= h(T-

T*),

(11)

Fig. 5. Coeficient of radiative heat transfer h (W mm2 K-‘) for the inside ground surface, for a semicylindrical tunnel-type greenhouse. Outside air, outside ground and cladding surface are at freezing point and the sky is clear

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where T- T* is the temperature deviation from the equilibrium temperature T* (as given in Fig. 4). Notice that h = 0 for p = 1, that is to say, when the cladding material is a perfect reflector the net radiation is zero. 4. Conclusions This paper offers simple radiation models which relate the net radiation observed inside a greenhouse during the night-time to the radiometric properties and temperature of the cladding surface, ground temperature and external fluxes of thermal radiation (atmospheric and terrestrial). The geometries considered are the infinitely long single-span type and the infinitely long and wide multispan-type of greenhouse. The models were then applied in estimating the rate of radiative heat loss at ground level in the particular case where there are freezing temperatures everywhere and a clear sky. Heat loss rates range from zero (when the transmittance of the cladding material is zero) to about 80 W m-* (for a fully transparent material). The results are nearly the same for the single and for the multispan greenhouse, and are mainly determined by the transmittance of the cladding material, being little dependent on the relative magnitudes of its emittance and reflectance. Finally, a coefficient of radiative heat transfer for the ground surface was obtained. Acknowledgement This work was financed by JNICT (contract No. 604.84.49) and FundacPo C. Gulbenkian No. C/96/84).

(contract

References ’ Kindelan, M. Dynamic modeling of greenhouse environment. Transactions of the ASAE 1980, 23(5): 1232-1239 * Garzoli, K. V.; Blackwell, J. An analysis of the nocturnal heat loss from a single skin plastic greenhouse. Journal of Agricultural Engineering Research 198 1, 26: 203-214 3 Chandra, P. Thermal radiation exchange in a greenhouse with a transmitting cover. Journal of Agricultural Engineering Research 1982, 27: 261-265 4 Cooper, P. 1.; Fuller, R. J. A transient model of the interaction between crop, environment and greenhouse structure for predicting crop yield and energy consumption. Journal of Agricultural Engineering Research 1983, 28: 401-418 5 Silva, A. A.; Rosa, R. Radiative environment inside a greenhouse. Agricultural and Forest Meteorology 1985, 33: 339-346 6 Siegel, R.; Howell, J. R. Thermal Radiation Heat Transfer. McGraw-Hill, 1972 7 Kittas, C. Contribution theorique et exptrimentale a l’etude du bilan d’energie des serres; applications a l’analyse du determinisme des temperatures de la paroi et de fair inttrieur de la serre. (Theoretical and experimental contribution to the study of the energy budget of greenhouses; application to the analysis of the mechanisms determining the temperature of the cladding surface and inside air of the greenhouse.) These 3’“’ cycle. INRA-Centre de Recherches Agronomiques d’Avignon, France 198 1 8 Heitor, A.; Rosa, R. Variacao da emissividade atmosferica corn a nebulosidade e obtencao de normais climatologicas. (Variation of the atmospheric emissivity with cloudiness and its climatologic average.) Acta da 4”. Conferencia National de Fisica. Sociedade Portuguesa de Fisica, Lisboa 1984 s Nijskens, J.; de Halleux, D.; Deltour, J.; Co&se, S.; Nissen, A. Condensation effect on heat transfer through greenhouse claddings: glass and polyethylene. Acta Horticulturae 1985, 174: 135-138