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A model of humidity and its applications in a greenhouse
AGRICULTURAL AND FOREST METEOROLOGY ELSEVIER
Agricultural and Forest Meteorology 76 (1995) 129-148
A model of humidity and its applications in a greenhouse Cecilia Stanghellini *, Taeke de Jong DLO-lnstitute of Agricultural Engineering (IMAG-DLO), P.O. Box 43, 6700 AA Wageningen, The Netherlands
Received 2 March 1994; revision accepted 22 November 1994
Abstract A model of humidity within a greenhouse is developed and some new applications in greenhouse climate management are discussed. In the model, ambient vapour concentration results from the balance of three fluxes: crop transpiration, ventilation and condensation at the cover. Transpiration and ventilation rates are calculated by means of models developed earlier, described briefly here. Condensation is given by standard theory of mass transfer at plane surfaces, neglecting the small slope of a greenhouse cover. Through experimental verification in a greenhouse it was shown that the model, which takes into account the many feed-backs present in such a system, provides reliable estimates of both ambient humidity and crop transpiration. As an application of the model, the use of parameters such as relative humidity or saturation deficit in the climate control of modern greenhouses is discussed. The merit of each one is shown to be conditional upon the intended result. Roughly speaking, putting a ceiling on relative humidity is likely to reduce chances of dew formation, whereas a threshold on saturation deficit in general forces the transpiration rate above some level. However, there is no general rule linking a value of any humidity parameter to a level of the intended process, since the relationship in question is affected by prevailing weather, as well as present leaf area. With a humidity model such as the one described here, values of the regulated climate variables, like the temperature and ventilation set points, can be deduced from the desired level of a crop process (in this way eliminating the use of an intermediate humidity parameter in climate control). Accordingly, it is concluded that modern greenhouse climate management, which aims at steering crop processes should incorporate a similar humidity model.
* Corresponding author. 0168-1923/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0168-1923(95 )02220-8
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0. List of symbols a A b C Cf d D~ E F Gr g g h Ho I k l Lo LAI Le Nu r Rn
Sh T u V w X
ratio of row height to row midline distance area (m 2) ratio of row width to row midline distance condensation flux density (kg m 2 s 1) discharge coefficient typical dimension (m) molecular diffusion coefficient of water in air (m 2 s - 1) transpiration flux density (kg m 2 S 1) reduction of radiation absorption due to row shape Grashof number conductance to heat or mass transfer (m s - a ) acceleration due to gravity (m s - 2 ) ratio of greenhouse volume to ground area (m) height of opening radiation flux density (W m - 2 ) canopy extinction coefficient mean length of leaves (m) length of opening leaf area index ( m 2 m - 2 ) Lewis number Nusselt number resistance to heat or mass transfer (s m 1) net radiation available to the canopy (W m - 2 ) Sherwood number temperature (K) air velocity indoor (m s 1) ventilation flux density (kg m 2 s 1) wind speed outdoor (m s -1) maximum water vapour flux density indoor (kg m - 2 s 1)
G r e e k letters
a /3 E h u p r r q~ X
opening angle thermal expansion coefficient of air (K -1) rate of change of latent heat content with sensible heat content of saturated air latent heat of vaporization of water (J k g - 1 ) kinematic viscosity of air (m 2 s a) reflectance time constant (s) transmittance ventilation flux (m 3 s - 1) vapour concentration (kg m -3) slope of the cladding
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131
Subscripts a
b b C
cond d g 1
min 0 r S S
tot trans U
vent W oo
indoor air boundary layer owing to buoyancy canopy of condensation down, lower hemisphere ground, soil surface longwave minimal, specie-specific outdoor air roof, cladding surface stomata shortwave, solar total of transpiration upper hemishere of ventilation owing to wind of a dense canopy
Superscripts time derivative * at saturation per unit leaf area ~ virtual
1. I n t r o d u c t i o n
More advanced automatic climate control equipment and a better understanding of the processes behind greenhouse climate have greatly expanded the possibilities for climate control in greenhouses during the last decade. In the future 'optimal control strategies' will eventually lead to the economical optimisation of the greenhouse production system. Such strategies may involve various objectives, like amount and quality of yield, low production costs and little environmental burden. Obviously, the ultimate goal of 'optimal control' is still not achieved. More sophisticated models are required of economic, biological (growth and yield of the crop) and physical processes in the greenhouse. Moreover, the application of new control theory, required to find a solution of the multivariate problem such that the economic output is maximised, is still experimental in greenhouses. The prospect of using mathematical models has triggered off targeted research on the physical and dynamical behaviour of the greenhouse climate and on the development and production of the crop in relation to its microclimate. This,
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in turn, is stimulating the development of new control strategies, aimed at optimising specific processes or environmental factors. Bakker (1991) proved that growth and production of all major greenhouse vegetables are affected by ambient humidity. At present, however, the control of humidity in commercial greenhouses is restricted either to avoid dew forming on the crop, in order to prevent fungal diseases (Hand, 1988), or to prevent exceedingly small transpiration rates, often linked to calcium deficiency (Aikmar, and Houter, 1990) and to unsatisfactory crop development (Holder and Cockshull, 1990). In order to meet such a limited purpose, most climate control systems are made to restrict a humidity parameter (either relative humidity or saturation ,~eficit) within pre-set boundaries. Moreover, lacking direct means to control humidity, any humidity parameter is manipulated through some combination of heating and ventilation, something reckoned to add a large fraction to the heating costs of the Dutch glasshouse industry (Bakker, 1994). In fact, the large feed-back between humidity and transpiration, theoretically explored by Aubinet et al. (1989), ensures that attempting to depress ambient humidity is often a self-defeating exercise, as experienced already by Matthews and Saffell (1981). In a previous paper (Stanghellini and van Meurs, 1992) it was shown that environmental control of transpiration is possible with the systems now available, by reversing a model of the transpiration rate of a greenhouse crop, in order to derive the set point of air temperature and humidity that would ensure a desired transpiration rate. In this paper it is demonstrated that the current, relatively simple, humidity control operations can be refined considerably when the processes determining the water vapour balance of the greenhouse air are taken into account. This requires a mathematical description of the interaction between crop transpiration (the main source of vapour in the greenhouse) and the microclimate, as well as of the condensation and ventilation processes (which remove vapour from the air). Bakker (1986) demonstrated experimentally that crop transpiration could be determined as the flux balancing the main vapour sinks in a greenhouse. From the balance of these fluxes, a humidity model of the greenhouse air will be developed here and its experimental verification will be described. It will be further shown that such a model provides additional and sufficient knowledge for the control of humidity as a climatic variable, which makes a more efficient manipulation of crop processes such as dew forming or transpiration possible. This is of great importance for future greenhouse management since direct manipulation of crop growth could lead to better standards of quality and more efficient production (Cockshull, 1988).
2. Theory A model fit for the control of humidity-related crop processes has to account for the environmental effect on the vapour content of the air in such a way that the impact of regulated variables (like air temperature and ventilator opening) can be separated from the effect of weather. Such a model was developed based on the balance of transpiration, condensation and ventilation and their interaction. In the following it is shown that vapour concentration can be described with a good approximation by a first order differential equation, wherein the relevant vapour fluxes are described by transfer
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equations. By this means the time constant of the system can be determined, something imperative for climate control applications, which could not be obtained by linearisation of the vapour balance equation (Jollier, 1994). Crop evapotranspiration, E, is the main source of vapour in a greenhouse (fogging will not be considered here), whereas vapour removal takes place through both condensation, C, and ventilation, V, so that the following balance equation holds: ( k g m -2 s -1)
hj(a=E-C-V
(1)
where h (m) is the ratio of the greenhouse volume to its ground area, that is, the mean height of the greenhouse, and Xa is the mean vapour content of the air within the greenhouse (kg m-3), all the fluxes being referred to unit ground area and quantified as follows. 2.1. Transpiration
The transpiration ( E ) of a crop (evaporation from the soil surface, commonly mulched in Dutch greenhouses, will be neglected here) can be described by a PenmanMonteith (or big-leaf) equation: + 2LAI( Xa* - Xa) (1 + e ) r b + r s
• rbRn/h
E =
(kg m -2 s - ' )
(2)
where • is the rate of change of latent heat content with sensible heat content of air, were it saturated, r b and r~ are the boundary layer and canopy resistances, respectively (s m - l ) , R n is the net radiation available to the canopy (W m-2), A is the latent heat of vaporisation of water (J kg-1), LAI is leaf area index and the superscript* indicates saturation. In order to preserve the analogy with a transfer equation, Eq. (2) can be re-written as follows: E=gt
....
( Xc - Xa)
( k g m -z s -1 )
(3)
whereby a ' transpiration conductance': 2LAI gt . . . .
(m s -1 )
(4)
(1 + •)r b + r s
and an effective vapour concentration at the 'big leaf' surface: rb
Xc~Xa* + • 2 ~
R n
A
(kgm-3)
have been defined. Thus the effective vapour concentration of the canopy is two terms: the first one follows directly from ambient temperature, whereas one is proportional to the net radiation. The calculation of the parameters of Eq. (2), namely R., r b and rs, in case of a greenhouse tomato crop has been described earlier (Stanghellini, only the relevant equations of that work will be reported briefly hereafter.
(5) the sum of the second the special 1987), and
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2.1.1. Net radiation
Net radiation of a homogeneous crop, with solar radiation I S (W m - 2 ) incident at the canopy top, was calculated by means of the theoretical equations for radiation exchange of a semi-transparent medium: R n = F{(1 + ~'s pg)[(1 - %) - (1 - ~'l)P~] Is + (1 - ~',)[ o ' ( T 4 + T4 -
2T~)]}
= F{(1 + C 1 ) [ C 2 - C 3 p~]I S +
(W m -2)
(6)
C3111] }
The coefficient groups can be explained as follows: C 3 is the soil cover, that also is the area exchanging longwave radiation ( I 0, C 3 p= is the reflected fraction of intercepted radiation, C 2 is the shortwave absorbance and C~ represents the transmitted radiation incoming at the bottom of the canopy owing to soil reflectance. Transmittance of the canopy for longwave and shortwave radiation, r~ and r S respectively, are given by: T I ~- e
klLAI
"rs =- e_k~LA1
(7)
with k J k S, the ratio of the canopy extinction coefficients for longwave and shortwave radiation, equal to 1.35 (e.g. Monteith and Unsworth, 1990). Values of k S (0.48), reflectance of an otherwise similar, dense canopy ( p~ = 0.12) as well as reflectance of the underlying white plastic mulch ( pg = 0.58) were determined experimentally. Longwave radiation exchange ( I l) is calculated through Tc, which is the mean temperature of the canopy, and Tu and Td, which are the apparent temperatures of the radiation reaching the upper and lower surfaces of the canopy, respectively. These are in practice given by a weighted mean of either the cover or the soil surface temperature and of the interposed heating elements. The factor F accounts for the reduction in absorption owing to the row shape of the canopy, according to a model developed by Stoffers (1975): 1 - 0.64~'~ l~h - 0.36'7"1-2"75b F = [1 - 0.07(1 - b ) ( 4 . 7 - a)] I - 0.64~'~ 18 - 0.36~'1275
(8)
where b and a are the ratio of the width and height of a row, respectively, to the distance between the midline of adjoining rows. The coefficient of absorption of solar radiation, that is the coefficient (1 + C1)(C 2 - C 3 p~) in Eq. (6), is illustrated by Fig. 1 in the case of a homogeneous crop (no rows), and for a few row configurations. 2.1.2. Boundary layer resistance
The boundary layer resistance ( r b) of single leaves within a greenhouse canopy can be described by a model for mixed-convection regimes (Stanghellini, 1993). With a leaf dimension d and air velocity u, the resistance is given by (Stanghellini, 1987): 1174d 0.5 rb
)( ' d ']T c - T a ' +IZ 0 7 u
2"°'25
where T. is ambient temperature.
(sm
1)
(9)
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135
q.0 C
._o ..Q L 0
m C3
o.8-
0.6-
0 CD
0> 0
c-
f.o
0.40.2-
0.0
0
I
I
I
J
~
I
1
2
3
4
5
6
leaf
areo
index
Fig. 1. The fraction of solar radiation absorbed by a canopy, that is the coefficient of I s in Eq. (6), in the case of a homogeneous crop with extinction and reflection coefficients as given in the text (thick line), and for the same in the following row configurations: O relative width b = 0.8 and height a = 0.1; • b = 0.8 and a = 1; [] b = 0.5 and a = 2. The thin decreasing line shows the fraction absorbed per unit leaf area in this latter case, that is the ratio between the absorption coefficient and 2LAI.
2.1.3. Stomatal resistance Jarvis's (1976) multiplicative model of the stomatal resistance (r s) was parametrised by best-fitting transpiration and microclimate data of a tomato crop. The equation derived by Stanghellini (1987) was slightly modified here, by ignoring the weak dependence on carbon dioxide concentration, by adapting the factor dependent on humidity in order to make its derivative continuous and by preventing stomatal resistance from increasing at large leaf temperatures (Stanghellini and Bunce, 1993). The resulting relationship is: rs
= rmi, f ( L ) f ( Tc) f ( Xc - Xa) = rmin
[ is + 4"30 ] [ e°3r~ + 258 ] iss q- 0 . 5 4 e0.---65~+~ [4
.q_e-O.73()cc-x,)]-l/4
(sm-1 )
10 -3
(10) with i s the mean shortwave radiation per unit leaf area (see Fig. 1), Tc in °C and X in g m 3. The minimal stomatal resistance, rmin, determined as a best-fit coefficient, was found to be 82 s m -1. The factors f of Eq. (10), by which the latter is increased owing to radiation, canopy temperature and vapour concentration difference, respectively, are plotted in Fig. 2.
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136
8
k_ O
i
i
i
I
I
T
I
I
6
1
4 {D
I
#
2- k._.__ 0
I
0
I
'
J
10 20 30
5
I (W m -2)
I
1
[
15
25
35
T (°C)
0
I
I
[
10
20
30
40
X¢-Xo (g m-S)
Fig. 2. The factors of Eq. (10), by which stomatal resistance is increased in dependence on radiation, canopy temperature and vapour concentration difference. The coefficients of the three preselected functions were determined by best-fitting experimental data of a tomato crop.
2.2. Condensation The flux density of water condensing at the cover surface (indicated by the subscript
r) can be written: C=gcond(Xa--Xr *)
( k g m 2 S 1)
(11)
And, of course, as no vapour can leave the dry cover: C=0
for
Xa
( k g m - 2 s -1)
(12)
The mass transfer conductance gco,~ can be calculated by (e.g. Monteith and Unsworth, 1990): Dv Sh gcond = T S h = 2.49 1 0 - s - -d
(m s -1)
(13)
where D v is the molecular diffusion coefficient of water vapour in air and d a typical dimension of the cover surface. The Nusselt number of a greenhouse cover of small slope was found by Papadakis et al. (1992) to be similar to that of a horizontal surface, given the poor accuracy associated with similarity numbers in practice. Accordingly, the Sherwood number of a horizontal surface will be applied here: Sh
= LeW3Nu
-~
0.96 • 0.13Gr 1/3
(14)
with: g / 37d- ' (27a-/~r)---1.47 108d3(7~a - ;rr) Gr = v---
(15)
where 7~ is the virtual temperature, in order to account for the effect of vapour concentration gradients on the density (and thus buoyancy) of air. The values of 15.5 1 0 - 6 m 2 s - 1 and 3.42 10 -3 K -1 have been used, respectively, for the kinematic viscosity ( v ) and the thermal expansion coefficient (/3) of air at 20°C, and g is 9.81 m s -2. As C is referred to unit ground area, one has to take into a c c o u n t A r / A g , the ratio
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
137
of cover area to ground area. With substitution of Eqs. (14) and (15) into Eq. (13) this leads to: gcond = mr 1.64 1 0 - 3 ( L -- Tr) 1/3 Ag
(m s -1)
(16)
A r / A g is about 1.1, for a typical Dutch Venlo-type glasshouse (slope 26°), and the rightmost factor is a weak function of the temperature difference, commonly (that is, when there is condensation at the surface) contained between 1 and 2. 2.3. Ventilation
Similarly, the vapour flux due to ventilation can be written as: V=gvent(Xa--Xo)
(17)
( k g m -2 s -1 )
with Xo the absolute humidity outdoor. Since the driving force for ventilation, i.e. the pressure difference over the window opening, is generated by wind as well as buoyancy effects, both effects have to be incorporated in the 'ventilation conductance', gvent" 2.3.1. Ventilation owing to buoyancy effects The effect of buoyancy on natural ventilation through a greenhouse roof window was calculated by Bot (1983), following the theoretical considerations for natural convection through openings by Brown and Solvason (1962), which are commonly applied for predicting ventilation owing to buoyancy effects in buildings. In this approach, the air interchange across the opening is driven by the variation of the hydrostatic pressure on both sides of the opening (caused by temperature and vapour concentration differences), taking into account a neutral pressure plane at the height of the opening, at which the thermally induced pressure difference is zero. The incoming and outgoing ventilation flux through the front area of the window (parallel to the ridge) is then given by:
~)b,front=Cf(g[j)l/2~H3/2(L-
L) 1/2
(m 3 s -1)
(18)
with Cf a discharge coefficient and the other symbols defined by Fig. 3. H, the height of the opening is: H=Ho[sin 0- sin(0- a)]
(m)
(19)
Fig. 3. Schematicrepresentationof a roof greenhouseventilator.The symbolsused in the text are definedhere.
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C. Stanghellini et al. / Agricultural and Forest Meteorology 76 (1995) 129-148
0.03 I
~3 (D &
f,,.
/..I " ' ~ "/'"
0.02
0
=1
0 C
0 © C >
0'01
~"
0.00 0
i
i
10
20
30
opening angle, a (o) Fig. 4. The ventilation function, G, describing the relationship between the ventilation flux per unit window area of the cover surface and the average outdoor wind speed, as a function of the opening angle of the ventilators, a. The two full lines were determined for simulated infinite covers with ventilators of aspect ratio as indicated, the dash-point line is for the greenhouse used for the experiment described later. The maximum opening angle in this latter case was 11°. After de Jong, 1990. The ventilation flux through the two side areas of the opening, on the other hand, is: sin a ~bb,sides = 2Cf0.146( g ~ )1/2 _sin_ S [ s i n ( S ×HSo/2(Ta - To) 1/2
c~)13/2
(m 3 s -1)
(20)
The discharge coefficient, Cf, was taken to be 0.6, as measured by de Jong (1990) and suggested by Zhang et al. (1989) for openings of other agricultural buildings. Obviously, the total flux by natural convection through the window is: 6 b = 6b,front -}- 6b,sides
(m 3 s -1)
(21)
2.3.2. Ventilation owing to wind effects The wind-driven flux through a roof ventilator can be expressed by (de Jong and Bat, 1992): d?w=wG(o~)LoH o
( m 3 s a)
(22)
with G ( a ) , the ventilation function, a saturating function of the opening angle, c~, which describes the relation between the ventilation flux per unit w i n d o w area of the cover surface and the average outdoor wind speed (w) at reference height, as a function of the opening angle of the windows. Fig. 4 displays the ventilation function of ventilators of two aspect ratio, in a quasi infinite Venlo-type cover, and the ventilation function of a
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139
real greenhouse. Examples for other aspect ratios can be found in Fernandez and Bailey (1992). 2.3.3. Combined buoyancy and wind effects De Jong (1990) showed that superposition of the respective pressure components over the window leads to an expression for the total ventilation, (])vent, that provides a good representation for practical purposes: ~vent = ((~2 _t_ ~2)1/2
(m 3 s -1)
(23)
Hence, the normalised 'ventilation conductance' is: ~vent gvent -----Z o a o
(m s -1)
(24)
2.4. Vapour concentration of ambient air Finally, the vapour balance of the greenhouse air, Eq. (1), can be re-written as follows: hj(a=gtrans(Xc--Xa)--gcond(Xa--Xr*)- gvent(Xa--Xo)
(kg m-2 s l a ) (25)
which can be rearranged, by grouping the coefficients of ambient absolute humidity, Xa, and the terms independent from the latter: X-gtotXa-h~(a=O
(26)
( k g m -2 s -1 )
where gtot, a cumulative transfer conductance, is defined by: gtot = gt .... + gcona + gvent
(ms-1 )
(27)
whereas X, the flux that would occur if the air in the greenhouse were completely dry, is: X=--gt.... Xcq-gcondXr* +gvent Xo
( k g m -2 s - ' )
(28)
In fact, an analytical solution to the differential Eq. (26) exists only if both X and gtot are either unaffected by Xa or meet some analytical requirements that will be discussed later. Under such conditions, ambient vapour concentration at any time, t, would be given by: X~(t) = Xa(O0) -- [ Xa(O0) -- Xa(0)] e -'/~
(kg m -3)
(29)
with the vapour concentration at equilibrium (t ~ ~) given by: )¢a(~) = X/gtot
(kg m -3 )
(30)
7, the time constant of the system, or the time ambient humidity would take to get through about two-thirds of a variation owing to changing conditions, is calculated using: 7"= h/gto t
(s)
(31)
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2.5. Analysis Let us consider, first of all, the conditions mentioned above for the existence of an analytical solution to Eq. (26). In fact, all components of gtot change following a variation in ambient humidity. Stomata react to it (Eq. (10) and Fig. 2), virtual temperature of air (and thus gco.d) is a function of it, and the resulting variation in buoyancy will affect gvent too. None of these, however, is the primary factor determining any of the conductances. Hence, gtot can be regarded as constant, provided that the other conditions do not change. With respect to X, both Xa* and Xr* (univocally related to Ta and Tr) are independent of condensation or ventilation only as far as the climate control system is able to deliver the desired temperature set point, which, indeed, modern systems manage rather well. Altogether, Eq. (29) is likely to provide good estimates of true vapour concentrations in a greenhouse, as will be shown hereafter. In order to quantify the time constant, the typical magnitude of the various conductances is required. This will be illustrated using data collected in the experiments
15 (-
I
I
]
10
E b-
5 0
16-
I
~J
g,0,
.~EE 12 -
8-
Y oo
40
I
I
I
6
12
18
time
24
(M.E.T.)
Fig. 5. The trend of the total conductance, gtot and its components, for one of the experimental runs (9 May, LAI = 3), with fairly uninterrupted condensation and some afternoon ventilation, lower panel. The upper panel shows the corresponding time constant of humidity in minutes, calculated by means of Eq. (31), the height of the experimental glasshouse being 2.75 m.
C. Stanghellini et al. //Agricultural and Forest Meteorology 76 (1995) 129-148
141
described in the next section. An example is given in Fig. 5, for a day (9 May, LAI = 3) with fairly uninterrupted condensation and some afternoon ventilation. The upper panel of the same figure shows the corresponding time constant. In the daytime the maximum time constant is set by the transpiration conductance and will not normally be much larger than 5 min, except with an underdeveloped canopy. At night the maximum is apparently set by the condensation term, and this is likely to be much more variable. Fig. 5 shows that it can take several minutes for any change in conditions to show through ambient humidity. In taller houses, that are more common than the experimental greenhouse (gutter height 2.4 m), response time would be proportionally longer. Given the response time of a heating system and of air temperature, in order to avoid overshooting any climate control algorithm should take into account the time-dependent part of Eq. (29). Reduced transpiration rates enlarge the time constant of the system, as
15 c
10-
E F-
5-
Oi
E on c o
20-
© k_ C © 0 C 0 0
•
'~
10-
\J~
~ ~
u 'v: , w ~
..
/
0 C)_ (:3 >
/~,Eq(29)
0
0
I 6
I 12
time
I 18
24
(M.E.T.)
Fig. 6. Upper panel: the time constant of humidity concentration in the same glasshouse as in Fig. 5, with a smaller crop (LAI = 1.5) on a summer day (29 June). The lower panel shows the relevant vapour concentrations for the same day. )Ca and Xo were measured; Xa* and X,* were determined from measured temperatures; Xc was calculated by means of Eq. (5), with inclusion of Eqs. (6) and (9). Ambient vapour concentration, calculated by means of the model, Eq. (29), as explained in Fig. 7, is shown here for comparison.
142
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is made clear by the difference between day and night values, and by comparison with the upper panel of Fig. 6, that refers to a smaller crop (29 June, LAI = 1.5). The lower panel of Fig. 6 depicts the daily pattern of the various vapour concentrations, for the same fair, though not cloudless, summer day. The shaded area indicates the contribution of radiation to the effective vapour concentration of the canopy (see Eq. (5)). R n, however, is a saturating function of LAI and, as is shown in Fig. 1 (thin line), the ratio RJ2LAI is largest for small plants. Hence, it may be inferred that large vapour concentration differences ( X c - Xa) are more probable with small plants and incomplete soil cover. That is, under a given sunshine, water stress is more likely for a young crop.
3. Materials and methods
The model described so far was verified with data collected in a greenhouse where a tomato crop was grown. The experiment (16 selected days in April and May with a mature crop and 12 days in June and July with a young one) took place in a single-glass, Venlo-type, eight span, E - W oriented house, gutter height being only 2.4 m. Heating was provided by hot water pipes lying a few centimetres above the soil surface. The crop (cv. Sonatine) was grown on rockwool mats, 0.3 m wide and 1.6 m apart. It was trained in a V-shape, the upper width of rows (at height 2 m) being some 50 cm; overall plant density was 2 m 2. Both soil and rockwool were covered with white plastic sheets in order to increase the shortwave radiation available for the crop (a fairly common practice in The Netherlands). Incoming solar radiation at the top of the canopy was estimated by applying the measured mean transmissivity of the house (65%) to the global radiation measured above the roof. The temperature of the heating pipes and of the plastic sheets on the ground (needed for the thermal radiation balance) was measured by thermocouples glued to the surfaces, while thinner (0.1 mm), gold-plated thermocouples glued to the inner side of the glass cover were used to determine its mean temperature. Ambient dry and wet bulb temperatures were given by Assman aspirated psychrometers within the canopy and outdoors, with thin thermocouples glued to the bulbs. In addition, profiles of leaf temperature were provided by six series of five thermocouples each, held touching the lower surface of leaves, at three levels in the canopy and with various orientations. Air movement within the crop was measured by four hot-bulb anemometers, whereas external wind speed was measured at an height of some 3 m above the cover; opening angle of the roof ventilators (present only on the South-facing slope) was determined by a potentiometer that was calibrated daily. Actual transpiration rate was determined by continuously weighing two replicate portions of a crop row, each of four to six plants, grown on a tray, held at ground level. Two electronic scales were used for this purpose. In the present set-up (described by van Meurs and Stanghellini, 1992) they attained a nominal accuracy of 0.1 g on a full scale of 60 kg, i.e. roughly 1% and 10% of day- and night-time transpiration, respectively, of a mature crop. Water was supplied in the whole house by a trickle irrigation system, triggered either by a clock or by the amount of drainage. The plants grown on the scales
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143
received the same treatment as the others, the only difference being that during the experimental runs the drained water was not allowed out of the trays, which were provided with a reservoir for this purpose. Leaf area was estimated from the mean length of leaves, according to an empirical relationship determined by van der Varst and Postel (1972), that was shown here (by destructive measures on samples) to provide a reasonable estimate of the leaf area for the Sonatine variety: area
0.25• 2 1 - 1.4812
(m2)
(32)
Eq. (32) gives the area of one side of a leaf, when 1 is its length (m). Finally, the ventilation function of the glasshouse (shown in Fig. 4) was determined in situ, as described by de Jong (1990), by means of a tracer gas technique (decay rate method).
4. Results and discussion
Both gtrans and Xc depend on Xa directly as well as indirectly through leaf temperature. The latter was calculated iteratively, prior to the calculation of ambient vapour concentration by Eq. (29). The algorithm is described in Fig. 7. A number of geometrical parameters of the htmse and the crop need to be known, as well as leaf area and the constants of the function describing stomatal resistance (Eq. (10), Fig. 2). Required inputs are radiation, wind speed, temperature and humidity outdoors; window opening angle and temperatures of air indoors and of the cover. The model requires soil surface temperature as well as air speed within the house. However, Stanghellini and van Meurs (1992) demonstrated that these could be replaced by estimates without much loss of accuracy. Vapour concentration in the glasshouse as calculated by the model is shown, for instance, in Fig. 8 (upper panel) for a 48-hour experiment (another example is given in Fig. 6). The lower panel of Fig. 8 gives measured and calculated transpiration for the same period. The correlation coefficients between measured and estimated values for all experimental runs were all (except two) above 0.8, both for humidity and transpiration. It may be concluded, therefore, that the present model predicts accurately both ambient humidity and transpiration under various weather conditions and climate control measures. Since the required input variables (with the exception of cover temperature) are routinely recorded by current climate control systems, the explicit manipulation of water-related crop processes is within the reach of existing commercial hardware. Although a discussion of various applications is beyond the scope of this paper, it is worthwhile discussing an example here. It has been said above that most modem greenhouse climate control systems include a control of humidity (however indirect) with the dual purpose of sustaining a 'minimal' transpiration rate and of avoiding dew forming on parts of the canopy. The humidity parameter most commonly used is relative humidity (that is prevented from exceeding a preselected value), whereas some systems aim at constraining the saturation deficit above a fixed threshold. The present model, applied to a few simple instances, can give insight into the value
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C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
data: greenhouse and crop parameters~ Lreod:
To,T .....'Ts°i"Is'x°°"~'u'w
I
L calculate: ~'r,~*J [ Xo,o,d = Xo( t-1)J
l ini iol T I calculate: r ,rb,R° i no ~ ' ~ T.:,d]< ~ ~
yes
F cOIoulate: )('c'gt.... ,gcond,gvent LI"
1001 u 0re
5
Xo(oo)l
~.(t)=x~(°°)--[X~(°°)--X.,,old]e -,/'r ,k
t colco ote E(t) I Fig. 7. The algorithm used for calculating ambient vapour concentration and crop transpiration rate from measured climate data, as explained in the text.
of either parameter. It can also predict the true consequence of climate manipulation means such as set point air temperature and ventilator opening angle, on the processes one attempts to control, in this case temperature and/or transpiration rate of the crop. In order to calculate the condensation rates in the following example, roof temperature will be estimated as a 'weighted mean' (two-thirds outdoor and one-third indoor temperature), an approximation roughly valid for single-glass covers, and a linear relationship will be assumed to exist between set point air temperature and the temperature of the heating pipes.
4.1. Control of humidity vs control of transpiration Fig. 9 refers to a relatively warm autumn night. Outdoor temperature and relative humidity are assumed to be 10°C and 88% respectively, and wind speed 1 m s 1. Air
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
145
"A--" 2 5
20- ~
6
'
~
:
c o
15-
#:
1 .
u
,f
k-
o 10 EL ~D > 5
T
,'i
E
60
'
4o-;i
~ ,D
.,.
o
'
I
'
Jul 24
I
'
I
'
Jul 25
Jul 26
Fig. 8, Upper panel: measured and estimated vapour concentration in the glasshouse, for a 48-h experiment; LAI = 2. The lower panel gives measured and calculated transpiration for the same run. The solid line shows measured values and the broken line shows calculated values.
E (mg m -2 s-') 100
~
80' .
o
F
Tc - T,.. (°C)
I
6oZ
1
40-
~
o) :> 0
relative humidit
,
'
I
'\ '
~ ~o~
'
/
\,
I'~
10 12 14 16 18
'
10 12 14 16 18
I
\,\.
'
0 12 14 16 18 20
air temperature (°C) Fig. 9. Leftmost panel: the transpiration rate, following from a given combination of air temperature and ventilators opening (as a percentage of the largest angle, here 11 °) during a relatively warm autumn night. Outdoor temperature and relative humidity are assumed to be 10°C and 88% respectively, and wind speed 1 m s - 1. Air movement within the house is taken to be 6 cm s - 1 with a crop of LAI ~ 3. Corresponding relative humidity within the glasshouse is drawn in the middle. The rightmost panel shows the extent by which calculated canopy temperature would exceed the modelled dew point, under the same conditions,
146
C. Stanghellini et al. / Agricultural and Forest Meteorology 76 (1995) 129-148
movement within the house is taken to be 6 cm s-~ with a crop of LAI of 3. The transpiration rate, following from a given combination of air temperature and ventilators opening, is shown in the leftmost panel, whereas the central one displays corresponding relative humidity. Attempting to contain relative humidity below a preset ceiling (something commonly done in order to ensure a minimal transpiration rate) would often require ventilation. Ventilating, however, could fail to stimulate the transpiration rate, since the latter is a weak function of ventilation (leftmost panel). On the other hand, given the equivalence between X~ and X~* at night (Fig. 6), the transpiration rate would be almost proportional to the saturation deficit (see Eq. (3)), which would be then a far better parameter to use. However, the scaling factor, gtrans, would change with varying climate and crop conditions, as Eq. (4) attests, which makes it unlikely that a given threshold of saturation deficit would yield a uniform result. This is even more doubtful in day-time, when sunshine induces a difference between Xc and X,* (Fig. 6).
4.2. Prevention of dew forming In order to assess the relationship between dew forming and humidity, one has to realize that it is quite unlikely for the average temperature of a greenhouse canopy to be colder than the dew point of the air. That may happen, however, to some parts of the foliage, with consequent local dew forming. Controlling the climate in such a way that this should not happen requires a knowledge of the temperature distribution within the canopy, something a 'big-leaf' model can hardly provide. Ruling out the development of a multi-layer model accurate enough for this purpose, one has to rely on statistics. That is to say that, having an estimate of the distribution of temperature, one has to choose a threshold value for the difference between the 'big-leaf' temperature and the dew point. For instance, the standard deviation of leaf temperature proved rather constant at night, during our experiments, 0.75°C being roughly the upper limit. One can stipulate, then, that in order to ensure that there is less than 1% chance of dew forming on parts of the canopy, the 'big-leaf' temperature has to exceed dew point by at least 2.2°C (three standard deviations). The rightmost panel of Fig. 9 shows that this is equivalent to restricting relative humidity to values lower than 85%. Once such an equivalence has been established, relative humidity could well be an apt parameter for this purpose. However, a given chance of dew does not imply a fixed ceiling on relative humidity, as the relationship between the two depends on crop and weather conditions.
5. C o n c l u s i o n
Vapour content of air within a greenhouse is determined by many processes, of which crop transpiration, condensation at the cover surface and ventilation are by far the most important. In turn, transpiration and photosynthesis (the latter through stomatal reaction) are affected by ambient humidity. Humidity is also the single most important factor causing outbreak of some diseases, therefore modern climate control systems include some means of humidity control, the 'controlled' parameter being either relative humidity or saturation deficit, both with loyal advocates.
c. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
147
It has b e e n s h o w n here that relative h u m i d i t y is a good index of the chance of dew, whereas transpiration is better related to saturation deficit, although weather and crop conditions affects the relationship b e t w e e n the parameter and the process. Accordingly, a set point value o f h u m i d i t y fixed beforehand (whatever the parameter), is unlikely always to ensure the intended result. In addition, even given a target value of either h u m i d i t y parameter, available m e a n s of climate m a n i p u l a t i o n are, most c o m m o n l y , only (natural) ventilation and heating. O w i n g to the m a n y feed-back effects that a change in conditions w o u l d set in motion, the use of a h u m i d i t y model of the kind developed here w o u l d allow climate settings, such as air temperature or w i n d o w o p e n i n g angle, to be derived directly from ' p r o c e s s ' set points, thereby disposing o f intermediate h u m i d i t y parameters.
Acknowledgements W e are indebted to the personnel of the vegetable gardens of I M A G - D L O and in particular to F.J.M. Corver for helping with the transpiration measurements, besides skillfully taking care of the crop, and to our colleague Dr. H.J. H o l t e r m a n for revising the manuscript.
References Aikman, D.P. and Houter, G., 1990. Influence of radiation and humidity on transpiration: implications for calcium levels in tomato leaves. J. Hort. Science, 65: 31-39. Aubinet, M., Deltour, J., De Halleux, D. and Nijskens, J., 1989. Stomatal regulation in greenhouse crops: analysis and simulation. Agric. For. Meteorol., 48: 21-44. Bakker, J.C., 1986. Measurement of canopy transpiration or evapotranspiration in greenhouses by means of a simple vapour balance model. Agric. For. Meteorol., 37: 133-141. Bakker, J.C., 1991. Analysis of humidity effects on growth and production of glasshouse fruit vegetables. PhD dissertation, Agricultural University, Wageningen, 155 pp. Bakker, J.C., 1994. Vraagtekens bij doel minimum buis. Groenten + Fruit/Glasgroenten, 4(2): 10-11. Bot, G.P.A. 1983. Greenhouse climate: from physical processes to a dynamic model. Ph.D. dissertation, Agricultural University, Wageningen, 240 pp. Brown, W.G. and Solvason, K.R., 1962. Natural convection through rectangular openings in partitions. 1. Int. J. Heat and Mass Transfer, 5: 859-868. Cockshull, K.E., 1988. Integration of plant physiology and physical aspects of the greenhouse climate. Acta Hortic., 229: 113-123. De Jong, T., 1990. Natural ventilation of large multi-span greenhouses. Ph.D. dissertation, Agricultural University, Wageningen, 116 pp. De Jong, T. and Bot, G.P.A., 1992. Air exchange caused by wind effects through (window) openings distributed evenly on a quasi-infinite surface. Energy Build., 19: 93-103. Fernandez, J.E. and Bailey, B.J., 1992. Measurement and prediction of greenhouse ventilation rates. Agric. For. Meteorol., 58: 229-245. Hand, D., 1988. Effects of atmospheric humidity on greenhouse crops. Acta Hortic., 229: 143-158. Holder, R. and Cockshull, K.E., 1990. Effects of humidity on the growth and yield of glasshouse tomatoes. J. Hort. Sci., 65: 31-39. Jarvis, P.G., 1976. The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field. Phil. Trans. R. Soc. Lond., B 273: 593-610.
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Jolliet, O., 1994. HORTITRANS, a model for predicting and optimizing humidity and transpiration in greenhouses. J. Agric. Eng. Res., 57: 23-37. Matthews, R.B. and Saffell, R.A., 1981. Computer control of humidity in experimental glasshouses. J. Agric. Eng. Res., 33: 213-221. Monteith, J.L. and Unsworth, M.H., 1990. Principles of Environmental Physics. Edward Arnold, London, 291 Pp. Papadakis, G., Frangoudakis, A. and Kyritsis, S., 1992. Mixed, forced and free convection heat transfer at the greenhouse cover. J. Agric. Eng. Res., 51:191-205. Stanghellini, C., 1987. Transpiration of greenhouse crops: an aid to climate management. Ph.D. dissertation, Agricultural University, Wageningen, 150 pp. StangheUini, C., 1993. Mixed convection above greenhouse crop canopies. Agric. For. Meteorol., 66: 111-117. Stanghellini, C. and van Meurs, W.T.M., 1992. Environmental control of greenhouse crop transpiration. J. Agric. Eng. Res. 51: 297-311. Stanghellini, C. and Bunce, J.A., 1993. Response of photosynthesis and conductance to light, CO2, temperatore and humidity in tomato plants acclimated to ambient and elevated CO 2. Photosynthetica, 29: 487-497. Stoffers, J.A., 1975. Radiation absorption of canopy rows. Acta Hortic., 46: 91-95. Van der Varst, P.G.I. and Postel, J.D.G., 1972. Bepaling bladoppervlak van tomatenplanten. IT]?, Wageningen, Report 46, 36 pp. Van Meurs, W.T.M. and Stanghellini, C., 1992. Use of an off-the-shelf electronic balance for monitoring crop transpiration in greenhouses. Acta Hortic., 304: 219-225. Zhang, J.S., Janni, K.A. and Jacobson, L.D., 1989. Modelling natural ventilation induced by combined thermal buoyancy and wind. Trans. Am. Soc. Agric. Eng., 32: 2165-2174.
Agricultural and Forest Meteorology 76 (1995) 129-148
A model of humidity and its applications in a greenhouse Cecilia Stanghellini *, Taeke de Jong DLO-lnstitute of Agricultural Engineering (IMAG-DLO), P.O. Box 43, 6700 AA Wageningen, The Netherlands
Received 2 March 1994; revision accepted 22 November 1994
Abstract A model of humidity within a greenhouse is developed and some new applications in greenhouse climate management are discussed. In the model, ambient vapour concentration results from the balance of three fluxes: crop transpiration, ventilation and condensation at the cover. Transpiration and ventilation rates are calculated by means of models developed earlier, described briefly here. Condensation is given by standard theory of mass transfer at plane surfaces, neglecting the small slope of a greenhouse cover. Through experimental verification in a greenhouse it was shown that the model, which takes into account the many feed-backs present in such a system, provides reliable estimates of both ambient humidity and crop transpiration. As an application of the model, the use of parameters such as relative humidity or saturation deficit in the climate control of modern greenhouses is discussed. The merit of each one is shown to be conditional upon the intended result. Roughly speaking, putting a ceiling on relative humidity is likely to reduce chances of dew formation, whereas a threshold on saturation deficit in general forces the transpiration rate above some level. However, there is no general rule linking a value of any humidity parameter to a level of the intended process, since the relationship in question is affected by prevailing weather, as well as present leaf area. With a humidity model such as the one described here, values of the regulated climate variables, like the temperature and ventilation set points, can be deduced from the desired level of a crop process (in this way eliminating the use of an intermediate humidity parameter in climate control). Accordingly, it is concluded that modern greenhouse climate management, which aims at steering crop processes should incorporate a similar humidity model.
* Corresponding author. 0168-1923/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0168-1923(95 )02220-8
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0. List of symbols a A b C Cf d D~ E F Gr g g h Ho I k l Lo LAI Le Nu r Rn
Sh T u V w X
ratio of row height to row midline distance area (m 2) ratio of row width to row midline distance condensation flux density (kg m 2 s 1) discharge coefficient typical dimension (m) molecular diffusion coefficient of water in air (m 2 s - 1) transpiration flux density (kg m 2 S 1) reduction of radiation absorption due to row shape Grashof number conductance to heat or mass transfer (m s - a ) acceleration due to gravity (m s - 2 ) ratio of greenhouse volume to ground area (m) height of opening radiation flux density (W m - 2 ) canopy extinction coefficient mean length of leaves (m) length of opening leaf area index ( m 2 m - 2 ) Lewis number Nusselt number resistance to heat or mass transfer (s m 1) net radiation available to the canopy (W m - 2 ) Sherwood number temperature (K) air velocity indoor (m s 1) ventilation flux density (kg m 2 s 1) wind speed outdoor (m s -1) maximum water vapour flux density indoor (kg m - 2 s 1)
G r e e k letters
a /3 E h u p r r q~ X
opening angle thermal expansion coefficient of air (K -1) rate of change of latent heat content with sensible heat content of saturated air latent heat of vaporization of water (J k g - 1 ) kinematic viscosity of air (m 2 s a) reflectance time constant (s) transmittance ventilation flux (m 3 s - 1) vapour concentration (kg m -3) slope of the cladding
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
131
Subscripts a
b b C
cond d g 1
min 0 r S S
tot trans U
vent W oo
indoor air boundary layer owing to buoyancy canopy of condensation down, lower hemisphere ground, soil surface longwave minimal, specie-specific outdoor air roof, cladding surface stomata shortwave, solar total of transpiration upper hemishere of ventilation owing to wind of a dense canopy
Superscripts time derivative * at saturation per unit leaf area ~ virtual
1. I n t r o d u c t i o n
More advanced automatic climate control equipment and a better understanding of the processes behind greenhouse climate have greatly expanded the possibilities for climate control in greenhouses during the last decade. In the future 'optimal control strategies' will eventually lead to the economical optimisation of the greenhouse production system. Such strategies may involve various objectives, like amount and quality of yield, low production costs and little environmental burden. Obviously, the ultimate goal of 'optimal control' is still not achieved. More sophisticated models are required of economic, biological (growth and yield of the crop) and physical processes in the greenhouse. Moreover, the application of new control theory, required to find a solution of the multivariate problem such that the economic output is maximised, is still experimental in greenhouses. The prospect of using mathematical models has triggered off targeted research on the physical and dynamical behaviour of the greenhouse climate and on the development and production of the crop in relation to its microclimate. This,
132
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
in turn, is stimulating the development of new control strategies, aimed at optimising specific processes or environmental factors. Bakker (1991) proved that growth and production of all major greenhouse vegetables are affected by ambient humidity. At present, however, the control of humidity in commercial greenhouses is restricted either to avoid dew forming on the crop, in order to prevent fungal diseases (Hand, 1988), or to prevent exceedingly small transpiration rates, often linked to calcium deficiency (Aikmar, and Houter, 1990) and to unsatisfactory crop development (Holder and Cockshull, 1990). In order to meet such a limited purpose, most climate control systems are made to restrict a humidity parameter (either relative humidity or saturation ,~eficit) within pre-set boundaries. Moreover, lacking direct means to control humidity, any humidity parameter is manipulated through some combination of heating and ventilation, something reckoned to add a large fraction to the heating costs of the Dutch glasshouse industry (Bakker, 1994). In fact, the large feed-back between humidity and transpiration, theoretically explored by Aubinet et al. (1989), ensures that attempting to depress ambient humidity is often a self-defeating exercise, as experienced already by Matthews and Saffell (1981). In a previous paper (Stanghellini and van Meurs, 1992) it was shown that environmental control of transpiration is possible with the systems now available, by reversing a model of the transpiration rate of a greenhouse crop, in order to derive the set point of air temperature and humidity that would ensure a desired transpiration rate. In this paper it is demonstrated that the current, relatively simple, humidity control operations can be refined considerably when the processes determining the water vapour balance of the greenhouse air are taken into account. This requires a mathematical description of the interaction between crop transpiration (the main source of vapour in the greenhouse) and the microclimate, as well as of the condensation and ventilation processes (which remove vapour from the air). Bakker (1986) demonstrated experimentally that crop transpiration could be determined as the flux balancing the main vapour sinks in a greenhouse. From the balance of these fluxes, a humidity model of the greenhouse air will be developed here and its experimental verification will be described. It will be further shown that such a model provides additional and sufficient knowledge for the control of humidity as a climatic variable, which makes a more efficient manipulation of crop processes such as dew forming or transpiration possible. This is of great importance for future greenhouse management since direct manipulation of crop growth could lead to better standards of quality and more efficient production (Cockshull, 1988).
2. Theory A model fit for the control of humidity-related crop processes has to account for the environmental effect on the vapour content of the air in such a way that the impact of regulated variables (like air temperature and ventilator opening) can be separated from the effect of weather. Such a model was developed based on the balance of transpiration, condensation and ventilation and their interaction. In the following it is shown that vapour concentration can be described with a good approximation by a first order differential equation, wherein the relevant vapour fluxes are described by transfer
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
133
equations. By this means the time constant of the system can be determined, something imperative for climate control applications, which could not be obtained by linearisation of the vapour balance equation (Jollier, 1994). Crop evapotranspiration, E, is the main source of vapour in a greenhouse (fogging will not be considered here), whereas vapour removal takes place through both condensation, C, and ventilation, V, so that the following balance equation holds: ( k g m -2 s -1)
hj(a=E-C-V
(1)
where h (m) is the ratio of the greenhouse volume to its ground area, that is, the mean height of the greenhouse, and Xa is the mean vapour content of the air within the greenhouse (kg m-3), all the fluxes being referred to unit ground area and quantified as follows. 2.1. Transpiration
The transpiration ( E ) of a crop (evaporation from the soil surface, commonly mulched in Dutch greenhouses, will be neglected here) can be described by a PenmanMonteith (or big-leaf) equation: + 2LAI( Xa* - Xa) (1 + e ) r b + r s
• rbRn/h
E =
(kg m -2 s - ' )
(2)
where • is the rate of change of latent heat content with sensible heat content of air, were it saturated, r b and r~ are the boundary layer and canopy resistances, respectively (s m - l ) , R n is the net radiation available to the canopy (W m-2), A is the latent heat of vaporisation of water (J kg-1), LAI is leaf area index and the superscript* indicates saturation. In order to preserve the analogy with a transfer equation, Eq. (2) can be re-written as follows: E=gt
....
( Xc - Xa)
( k g m -z s -1 )
(3)
whereby a ' transpiration conductance': 2LAI gt . . . .
(m s -1 )
(4)
(1 + •)r b + r s
and an effective vapour concentration at the 'big leaf' surface: rb
Xc~Xa* + • 2 ~
R n
A
(kgm-3)
have been defined. Thus the effective vapour concentration of the canopy is two terms: the first one follows directly from ambient temperature, whereas one is proportional to the net radiation. The calculation of the parameters of Eq. (2), namely R., r b and rs, in case of a greenhouse tomato crop has been described earlier (Stanghellini, only the relevant equations of that work will be reported briefly hereafter.
(5) the sum of the second the special 1987), and
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C. StangheUini et al. / Agricultural and Forest Meteorology 76 (1995) 129-148
2.1.1. Net radiation
Net radiation of a homogeneous crop, with solar radiation I S (W m - 2 ) incident at the canopy top, was calculated by means of the theoretical equations for radiation exchange of a semi-transparent medium: R n = F{(1 + ~'s pg)[(1 - %) - (1 - ~'l)P~] Is + (1 - ~',)[ o ' ( T 4 + T4 -
2T~)]}
= F{(1 + C 1 ) [ C 2 - C 3 p~]I S +
(W m -2)
(6)
C3111] }
The coefficient groups can be explained as follows: C 3 is the soil cover, that also is the area exchanging longwave radiation ( I 0, C 3 p= is the reflected fraction of intercepted radiation, C 2 is the shortwave absorbance and C~ represents the transmitted radiation incoming at the bottom of the canopy owing to soil reflectance. Transmittance of the canopy for longwave and shortwave radiation, r~ and r S respectively, are given by: T I ~- e
klLAI
"rs =- e_k~LA1
(7)
with k J k S, the ratio of the canopy extinction coefficients for longwave and shortwave radiation, equal to 1.35 (e.g. Monteith and Unsworth, 1990). Values of k S (0.48), reflectance of an otherwise similar, dense canopy ( p~ = 0.12) as well as reflectance of the underlying white plastic mulch ( pg = 0.58) were determined experimentally. Longwave radiation exchange ( I l) is calculated through Tc, which is the mean temperature of the canopy, and Tu and Td, which are the apparent temperatures of the radiation reaching the upper and lower surfaces of the canopy, respectively. These are in practice given by a weighted mean of either the cover or the soil surface temperature and of the interposed heating elements. The factor F accounts for the reduction in absorption owing to the row shape of the canopy, according to a model developed by Stoffers (1975): 1 - 0.64~'~ l~h - 0.36'7"1-2"75b F = [1 - 0.07(1 - b ) ( 4 . 7 - a)] I - 0.64~'~ 18 - 0.36~'1275
(8)
where b and a are the ratio of the width and height of a row, respectively, to the distance between the midline of adjoining rows. The coefficient of absorption of solar radiation, that is the coefficient (1 + C1)(C 2 - C 3 p~) in Eq. (6), is illustrated by Fig. 1 in the case of a homogeneous crop (no rows), and for a few row configurations. 2.1.2. Boundary layer resistance
The boundary layer resistance ( r b) of single leaves within a greenhouse canopy can be described by a model for mixed-convection regimes (Stanghellini, 1993). With a leaf dimension d and air velocity u, the resistance is given by (Stanghellini, 1987): 1174d 0.5 rb
)( ' d ']T c - T a ' +IZ 0 7 u
2"°'25
where T. is ambient temperature.
(sm
1)
(9)
C. StangheUini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
135
q.0 C
._o ..Q L 0
m C3
o.8-
0.6-
0 CD
0> 0
c-
f.o
0.40.2-
0.0
0
I
I
I
J
~
I
1
2
3
4
5
6
leaf
areo
index
Fig. 1. The fraction of solar radiation absorbed by a canopy, that is the coefficient of I s in Eq. (6), in the case of a homogeneous crop with extinction and reflection coefficients as given in the text (thick line), and for the same in the following row configurations: O relative width b = 0.8 and height a = 0.1; • b = 0.8 and a = 1; [] b = 0.5 and a = 2. The thin decreasing line shows the fraction absorbed per unit leaf area in this latter case, that is the ratio between the absorption coefficient and 2LAI.
2.1.3. Stomatal resistance Jarvis's (1976) multiplicative model of the stomatal resistance (r s) was parametrised by best-fitting transpiration and microclimate data of a tomato crop. The equation derived by Stanghellini (1987) was slightly modified here, by ignoring the weak dependence on carbon dioxide concentration, by adapting the factor dependent on humidity in order to make its derivative continuous and by preventing stomatal resistance from increasing at large leaf temperatures (Stanghellini and Bunce, 1993). The resulting relationship is: rs
= rmi, f ( L ) f ( Tc) f ( Xc - Xa) = rmin
[ is + 4"30 ] [ e°3r~ + 258 ] iss q- 0 . 5 4 e0.---65~+~ [4
.q_e-O.73()cc-x,)]-l/4
(sm-1 )
10 -3
(10) with i s the mean shortwave radiation per unit leaf area (see Fig. 1), Tc in °C and X in g m 3. The minimal stomatal resistance, rmin, determined as a best-fit coefficient, was found to be 82 s m -1. The factors f of Eq. (10), by which the latter is increased owing to radiation, canopy temperature and vapour concentration difference, respectively, are plotted in Fig. 2.
C. Stanghellini et al. / Agricultural and Forest Meteorology 76 (1995) 129-148
136
8
k_ O
i
i
i
I
I
T
I
I
6
1
4 {D
I
#
2- k._.__ 0
I
0
I
'
J
10 20 30
5
I (W m -2)
I
1
[
15
25
35
T (°C)
0
I
I
[
10
20
30
40
X¢-Xo (g m-S)
Fig. 2. The factors of Eq. (10), by which stomatal resistance is increased in dependence on radiation, canopy temperature and vapour concentration difference. The coefficients of the three preselected functions were determined by best-fitting experimental data of a tomato crop.
2.2. Condensation The flux density of water condensing at the cover surface (indicated by the subscript
r) can be written: C=gcond(Xa--Xr *)
( k g m 2 S 1)
(11)
And, of course, as no vapour can leave the dry cover: C=0
for
Xa
( k g m - 2 s -1)
(12)
The mass transfer conductance gco,~ can be calculated by (e.g. Monteith and Unsworth, 1990): Dv Sh gcond = T S h = 2.49 1 0 - s - -d
(m s -1)
(13)
where D v is the molecular diffusion coefficient of water vapour in air and d a typical dimension of the cover surface. The Nusselt number of a greenhouse cover of small slope was found by Papadakis et al. (1992) to be similar to that of a horizontal surface, given the poor accuracy associated with similarity numbers in practice. Accordingly, the Sherwood number of a horizontal surface will be applied here: Sh
= LeW3Nu
-~
0.96 • 0.13Gr 1/3
(14)
with: g / 37d- ' (27a-/~r)---1.47 108d3(7~a - ;rr) Gr = v---
(15)
where 7~ is the virtual temperature, in order to account for the effect of vapour concentration gradients on the density (and thus buoyancy) of air. The values of 15.5 1 0 - 6 m 2 s - 1 and 3.42 10 -3 K -1 have been used, respectively, for the kinematic viscosity ( v ) and the thermal expansion coefficient (/3) of air at 20°C, and g is 9.81 m s -2. As C is referred to unit ground area, one has to take into a c c o u n t A r / A g , the ratio
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
137
of cover area to ground area. With substitution of Eqs. (14) and (15) into Eq. (13) this leads to: gcond = mr 1.64 1 0 - 3 ( L -- Tr) 1/3 Ag
(m s -1)
(16)
A r / A g is about 1.1, for a typical Dutch Venlo-type glasshouse (slope 26°), and the rightmost factor is a weak function of the temperature difference, commonly (that is, when there is condensation at the surface) contained between 1 and 2. 2.3. Ventilation
Similarly, the vapour flux due to ventilation can be written as: V=gvent(Xa--Xo)
(17)
( k g m -2 s -1 )
with Xo the absolute humidity outdoor. Since the driving force for ventilation, i.e. the pressure difference over the window opening, is generated by wind as well as buoyancy effects, both effects have to be incorporated in the 'ventilation conductance', gvent" 2.3.1. Ventilation owing to buoyancy effects The effect of buoyancy on natural ventilation through a greenhouse roof window was calculated by Bot (1983), following the theoretical considerations for natural convection through openings by Brown and Solvason (1962), which are commonly applied for predicting ventilation owing to buoyancy effects in buildings. In this approach, the air interchange across the opening is driven by the variation of the hydrostatic pressure on both sides of the opening (caused by temperature and vapour concentration differences), taking into account a neutral pressure plane at the height of the opening, at which the thermally induced pressure difference is zero. The incoming and outgoing ventilation flux through the front area of the window (parallel to the ridge) is then given by:
~)b,front=Cf(g[j)l/2~H3/2(L-
L) 1/2
(m 3 s -1)
(18)
with Cf a discharge coefficient and the other symbols defined by Fig. 3. H, the height of the opening is: H=Ho[sin 0- sin(0- a)]
(m)
(19)
Fig. 3. Schematicrepresentationof a roof greenhouseventilator.The symbolsused in the text are definedhere.
138
C. Stanghellini et al. / Agricultural and Forest Meteorology 76 (1995) 129-148
0.03 I
~3 (D &
f,,.
/..I " ' ~ "/'"
0.02
0
=1
0 C
0 © C >
0'01
~"
0.00 0
i
i
10
20
30
opening angle, a (o) Fig. 4. The ventilation function, G, describing the relationship between the ventilation flux per unit window area of the cover surface and the average outdoor wind speed, as a function of the opening angle of the ventilators, a. The two full lines were determined for simulated infinite covers with ventilators of aspect ratio as indicated, the dash-point line is for the greenhouse used for the experiment described later. The maximum opening angle in this latter case was 11°. After de Jong, 1990. The ventilation flux through the two side areas of the opening, on the other hand, is: sin a ~bb,sides = 2Cf0.146( g ~ )1/2 _sin_ S [ s i n ( S ×HSo/2(Ta - To) 1/2
c~)13/2
(m 3 s -1)
(20)
The discharge coefficient, Cf, was taken to be 0.6, as measured by de Jong (1990) and suggested by Zhang et al. (1989) for openings of other agricultural buildings. Obviously, the total flux by natural convection through the window is: 6 b = 6b,front -}- 6b,sides
(m 3 s -1)
(21)
2.3.2. Ventilation owing to wind effects The wind-driven flux through a roof ventilator can be expressed by (de Jong and Bat, 1992): d?w=wG(o~)LoH o
( m 3 s a)
(22)
with G ( a ) , the ventilation function, a saturating function of the opening angle, c~, which describes the relation between the ventilation flux per unit w i n d o w area of the cover surface and the average outdoor wind speed (w) at reference height, as a function of the opening angle of the windows. Fig. 4 displays the ventilation function of ventilators of two aspect ratio, in a quasi infinite Venlo-type cover, and the ventilation function of a
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
139
real greenhouse. Examples for other aspect ratios can be found in Fernandez and Bailey (1992). 2.3.3. Combined buoyancy and wind effects De Jong (1990) showed that superposition of the respective pressure components over the window leads to an expression for the total ventilation, (])vent, that provides a good representation for practical purposes: ~vent = ((~2 _t_ ~2)1/2
(m 3 s -1)
(23)
Hence, the normalised 'ventilation conductance' is: ~vent gvent -----Z o a o
(m s -1)
(24)
2.4. Vapour concentration of ambient air Finally, the vapour balance of the greenhouse air, Eq. (1), can be re-written as follows: hj(a=gtrans(Xc--Xa)--gcond(Xa--Xr*)- gvent(Xa--Xo)
(kg m-2 s l a ) (25)
which can be rearranged, by grouping the coefficients of ambient absolute humidity, Xa, and the terms independent from the latter: X-gtotXa-h~(a=O
(26)
( k g m -2 s -1 )
where gtot, a cumulative transfer conductance, is defined by: gtot = gt .... + gcona + gvent
(ms-1 )
(27)
whereas X, the flux that would occur if the air in the greenhouse were completely dry, is: X=--gt.... Xcq-gcondXr* +gvent Xo
( k g m -2 s - ' )
(28)
In fact, an analytical solution to the differential Eq. (26) exists only if both X and gtot are either unaffected by Xa or meet some analytical requirements that will be discussed later. Under such conditions, ambient vapour concentration at any time, t, would be given by: X~(t) = Xa(O0) -- [ Xa(O0) -- Xa(0)] e -'/~
(kg m -3)
(29)
with the vapour concentration at equilibrium (t ~ ~) given by: )¢a(~) = X/gtot
(kg m -3 )
(30)
7, the time constant of the system, or the time ambient humidity would take to get through about two-thirds of a variation owing to changing conditions, is calculated using: 7"= h/gto t
(s)
(31)
C. Stanghellini et al. / Agricultural and Forest Meteorology 76 (1995) 129-148
140
2.5. Analysis Let us consider, first of all, the conditions mentioned above for the existence of an analytical solution to Eq. (26). In fact, all components of gtot change following a variation in ambient humidity. Stomata react to it (Eq. (10) and Fig. 2), virtual temperature of air (and thus gco.d) is a function of it, and the resulting variation in buoyancy will affect gvent too. None of these, however, is the primary factor determining any of the conductances. Hence, gtot can be regarded as constant, provided that the other conditions do not change. With respect to X, both Xa* and Xr* (univocally related to Ta and Tr) are independent of condensation or ventilation only as far as the climate control system is able to deliver the desired temperature set point, which, indeed, modern systems manage rather well. Altogether, Eq. (29) is likely to provide good estimates of true vapour concentrations in a greenhouse, as will be shown hereafter. In order to quantify the time constant, the typical magnitude of the various conductances is required. This will be illustrated using data collected in the experiments
15 (-
I
I
]
10
E b-
5 0
16-
I
~J
g,0,
.~EE 12 -
8-
Y oo
40
I
I
I
6
12
18
time
24
(M.E.T.)
Fig. 5. The trend of the total conductance, gtot and its components, for one of the experimental runs (9 May, LAI = 3), with fairly uninterrupted condensation and some afternoon ventilation, lower panel. The upper panel shows the corresponding time constant of humidity in minutes, calculated by means of Eq. (31), the height of the experimental glasshouse being 2.75 m.
C. Stanghellini et al. //Agricultural and Forest Meteorology 76 (1995) 129-148
141
described in the next section. An example is given in Fig. 5, for a day (9 May, LAI = 3) with fairly uninterrupted condensation and some afternoon ventilation. The upper panel of the same figure shows the corresponding time constant. In the daytime the maximum time constant is set by the transpiration conductance and will not normally be much larger than 5 min, except with an underdeveloped canopy. At night the maximum is apparently set by the condensation term, and this is likely to be much more variable. Fig. 5 shows that it can take several minutes for any change in conditions to show through ambient humidity. In taller houses, that are more common than the experimental greenhouse (gutter height 2.4 m), response time would be proportionally longer. Given the response time of a heating system and of air temperature, in order to avoid overshooting any climate control algorithm should take into account the time-dependent part of Eq. (29). Reduced transpiration rates enlarge the time constant of the system, as
15 c
10-
E F-
5-
Oi
E on c o
20-
© k_ C © 0 C 0 0
•
'~
10-
\J~
~ ~
u 'v: , w ~
..
/
0 C)_ (:3 >
/~,Eq(29)
0
0
I 6
I 12
time
I 18
24
(M.E.T.)
Fig. 6. Upper panel: the time constant of humidity concentration in the same glasshouse as in Fig. 5, with a smaller crop (LAI = 1.5) on a summer day (29 June). The lower panel shows the relevant vapour concentrations for the same day. )Ca and Xo were measured; Xa* and X,* were determined from measured temperatures; Xc was calculated by means of Eq. (5), with inclusion of Eqs. (6) and (9). Ambient vapour concentration, calculated by means of the model, Eq. (29), as explained in Fig. 7, is shown here for comparison.
142
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
is made clear by the difference between day and night values, and by comparison with the upper panel of Fig. 6, that refers to a smaller crop (29 June, LAI = 1.5). The lower panel of Fig. 6 depicts the daily pattern of the various vapour concentrations, for the same fair, though not cloudless, summer day. The shaded area indicates the contribution of radiation to the effective vapour concentration of the canopy (see Eq. (5)). R n, however, is a saturating function of LAI and, as is shown in Fig. 1 (thin line), the ratio RJ2LAI is largest for small plants. Hence, it may be inferred that large vapour concentration differences ( X c - Xa) are more probable with small plants and incomplete soil cover. That is, under a given sunshine, water stress is more likely for a young crop.
3. Materials and methods
The model described so far was verified with data collected in a greenhouse where a tomato crop was grown. The experiment (16 selected days in April and May with a mature crop and 12 days in June and July with a young one) took place in a single-glass, Venlo-type, eight span, E - W oriented house, gutter height being only 2.4 m. Heating was provided by hot water pipes lying a few centimetres above the soil surface. The crop (cv. Sonatine) was grown on rockwool mats, 0.3 m wide and 1.6 m apart. It was trained in a V-shape, the upper width of rows (at height 2 m) being some 50 cm; overall plant density was 2 m 2. Both soil and rockwool were covered with white plastic sheets in order to increase the shortwave radiation available for the crop (a fairly common practice in The Netherlands). Incoming solar radiation at the top of the canopy was estimated by applying the measured mean transmissivity of the house (65%) to the global radiation measured above the roof. The temperature of the heating pipes and of the plastic sheets on the ground (needed for the thermal radiation balance) was measured by thermocouples glued to the surfaces, while thinner (0.1 mm), gold-plated thermocouples glued to the inner side of the glass cover were used to determine its mean temperature. Ambient dry and wet bulb temperatures were given by Assman aspirated psychrometers within the canopy and outdoors, with thin thermocouples glued to the bulbs. In addition, profiles of leaf temperature were provided by six series of five thermocouples each, held touching the lower surface of leaves, at three levels in the canopy and with various orientations. Air movement within the crop was measured by four hot-bulb anemometers, whereas external wind speed was measured at an height of some 3 m above the cover; opening angle of the roof ventilators (present only on the South-facing slope) was determined by a potentiometer that was calibrated daily. Actual transpiration rate was determined by continuously weighing two replicate portions of a crop row, each of four to six plants, grown on a tray, held at ground level. Two electronic scales were used for this purpose. In the present set-up (described by van Meurs and Stanghellini, 1992) they attained a nominal accuracy of 0.1 g on a full scale of 60 kg, i.e. roughly 1% and 10% of day- and night-time transpiration, respectively, of a mature crop. Water was supplied in the whole house by a trickle irrigation system, triggered either by a clock or by the amount of drainage. The plants grown on the scales
C. Stanghellini et aL /Agricultural and Forest Meteorology 76 (1995) 129-148
143
received the same treatment as the others, the only difference being that during the experimental runs the drained water was not allowed out of the trays, which were provided with a reservoir for this purpose. Leaf area was estimated from the mean length of leaves, according to an empirical relationship determined by van der Varst and Postel (1972), that was shown here (by destructive measures on samples) to provide a reasonable estimate of the leaf area for the Sonatine variety: area
0.25• 2 1 - 1.4812
(m2)
(32)
Eq. (32) gives the area of one side of a leaf, when 1 is its length (m). Finally, the ventilation function of the glasshouse (shown in Fig. 4) was determined in situ, as described by de Jong (1990), by means of a tracer gas technique (decay rate method).
4. Results and discussion
Both gtrans and Xc depend on Xa directly as well as indirectly through leaf temperature. The latter was calculated iteratively, prior to the calculation of ambient vapour concentration by Eq. (29). The algorithm is described in Fig. 7. A number of geometrical parameters of the htmse and the crop need to be known, as well as leaf area and the constants of the function describing stomatal resistance (Eq. (10), Fig. 2). Required inputs are radiation, wind speed, temperature and humidity outdoors; window opening angle and temperatures of air indoors and of the cover. The model requires soil surface temperature as well as air speed within the house. However, Stanghellini and van Meurs (1992) demonstrated that these could be replaced by estimates without much loss of accuracy. Vapour concentration in the glasshouse as calculated by the model is shown, for instance, in Fig. 8 (upper panel) for a 48-hour experiment (another example is given in Fig. 6). The lower panel of Fig. 8 gives measured and calculated transpiration for the same period. The correlation coefficients between measured and estimated values for all experimental runs were all (except two) above 0.8, both for humidity and transpiration. It may be concluded, therefore, that the present model predicts accurately both ambient humidity and transpiration under various weather conditions and climate control measures. Since the required input variables (with the exception of cover temperature) are routinely recorded by current climate control systems, the explicit manipulation of water-related crop processes is within the reach of existing commercial hardware. Although a discussion of various applications is beyond the scope of this paper, it is worthwhile discussing an example here. It has been said above that most modem greenhouse climate control systems include a control of humidity (however indirect) with the dual purpose of sustaining a 'minimal' transpiration rate and of avoiding dew forming on parts of the canopy. The humidity parameter most commonly used is relative humidity (that is prevented from exceeding a preselected value), whereas some systems aim at constraining the saturation deficit above a fixed threshold. The present model, applied to a few simple instances, can give insight into the value
144
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
data: greenhouse and crop parameters~ Lreod:
To,T .....'Ts°i"Is'x°°"~'u'w
I
L calculate: ~'r,~*J [ Xo,o,d = Xo( t-1)J
l ini iol T I calculate: r ,rb,R° i no ~ ' ~ T.:,d]< ~ ~
yes
F cOIoulate: )('c'gt.... ,gcond,gvent LI"
1001 u 0re
5
Xo(oo)l
~.(t)=x~(°°)--[X~(°°)--X.,,old]e -,/'r ,k
t colco ote E(t) I Fig. 7. The algorithm used for calculating ambient vapour concentration and crop transpiration rate from measured climate data, as explained in the text.
of either parameter. It can also predict the true consequence of climate manipulation means such as set point air temperature and ventilator opening angle, on the processes one attempts to control, in this case temperature and/or transpiration rate of the crop. In order to calculate the condensation rates in the following example, roof temperature will be estimated as a 'weighted mean' (two-thirds outdoor and one-third indoor temperature), an approximation roughly valid for single-glass covers, and a linear relationship will be assumed to exist between set point air temperature and the temperature of the heating pipes.
4.1. Control of humidity vs control of transpiration Fig. 9 refers to a relatively warm autumn night. Outdoor temperature and relative humidity are assumed to be 10°C and 88% respectively, and wind speed 1 m s 1. Air
C. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
145
"A--" 2 5
20- ~
6
'
~
:
c o
15-
#:
1 .
u
,f
k-
o 10 EL ~D > 5
T
,'i
E
60
'
4o-;i
~ ,D
.,.
o
'
I
'
Jul 24
I
'
I
'
Jul 25
Jul 26
Fig. 8, Upper panel: measured and estimated vapour concentration in the glasshouse, for a 48-h experiment; LAI = 2. The lower panel gives measured and calculated transpiration for the same run. The solid line shows measured values and the broken line shows calculated values.
E (mg m -2 s-') 100
~
80' .
o
F
Tc - T,.. (°C)
I
6oZ
1
40-
~
o) :> 0
relative humidit
,
'
I
'\ '
~ ~o~
'
/
\,
I'~
10 12 14 16 18
'
10 12 14 16 18
I
\,\.
'
0 12 14 16 18 20
air temperature (°C) Fig. 9. Leftmost panel: the transpiration rate, following from a given combination of air temperature and ventilators opening (as a percentage of the largest angle, here 11 °) during a relatively warm autumn night. Outdoor temperature and relative humidity are assumed to be 10°C and 88% respectively, and wind speed 1 m s - 1. Air movement within the house is taken to be 6 cm s - 1 with a crop of LAI ~ 3. Corresponding relative humidity within the glasshouse is drawn in the middle. The rightmost panel shows the extent by which calculated canopy temperature would exceed the modelled dew point, under the same conditions,
146
C. Stanghellini et al. / Agricultural and Forest Meteorology 76 (1995) 129-148
movement within the house is taken to be 6 cm s-~ with a crop of LAI of 3. The transpiration rate, following from a given combination of air temperature and ventilators opening, is shown in the leftmost panel, whereas the central one displays corresponding relative humidity. Attempting to contain relative humidity below a preset ceiling (something commonly done in order to ensure a minimal transpiration rate) would often require ventilation. Ventilating, however, could fail to stimulate the transpiration rate, since the latter is a weak function of ventilation (leftmost panel). On the other hand, given the equivalence between X~ and X~* at night (Fig. 6), the transpiration rate would be almost proportional to the saturation deficit (see Eq. (3)), which would be then a far better parameter to use. However, the scaling factor, gtrans, would change with varying climate and crop conditions, as Eq. (4) attests, which makes it unlikely that a given threshold of saturation deficit would yield a uniform result. This is even more doubtful in day-time, when sunshine induces a difference between Xc and X,* (Fig. 6).
4.2. Prevention of dew forming In order to assess the relationship between dew forming and humidity, one has to realize that it is quite unlikely for the average temperature of a greenhouse canopy to be colder than the dew point of the air. That may happen, however, to some parts of the foliage, with consequent local dew forming. Controlling the climate in such a way that this should not happen requires a knowledge of the temperature distribution within the canopy, something a 'big-leaf' model can hardly provide. Ruling out the development of a multi-layer model accurate enough for this purpose, one has to rely on statistics. That is to say that, having an estimate of the distribution of temperature, one has to choose a threshold value for the difference between the 'big-leaf' temperature and the dew point. For instance, the standard deviation of leaf temperature proved rather constant at night, during our experiments, 0.75°C being roughly the upper limit. One can stipulate, then, that in order to ensure that there is less than 1% chance of dew forming on parts of the canopy, the 'big-leaf' temperature has to exceed dew point by at least 2.2°C (three standard deviations). The rightmost panel of Fig. 9 shows that this is equivalent to restricting relative humidity to values lower than 85%. Once such an equivalence has been established, relative humidity could well be an apt parameter for this purpose. However, a given chance of dew does not imply a fixed ceiling on relative humidity, as the relationship between the two depends on crop and weather conditions.
5. C o n c l u s i o n
Vapour content of air within a greenhouse is determined by many processes, of which crop transpiration, condensation at the cover surface and ventilation are by far the most important. In turn, transpiration and photosynthesis (the latter through stomatal reaction) are affected by ambient humidity. Humidity is also the single most important factor causing outbreak of some diseases, therefore modern climate control systems include some means of humidity control, the 'controlled' parameter being either relative humidity or saturation deficit, both with loyal advocates.
c. Stanghellini et al. /Agricultural and Forest Meteorology 76 (1995) 129-148
147
It has b e e n s h o w n here that relative h u m i d i t y is a good index of the chance of dew, whereas transpiration is better related to saturation deficit, although weather and crop conditions affects the relationship b e t w e e n the parameter and the process. Accordingly, a set point value o f h u m i d i t y fixed beforehand (whatever the parameter), is unlikely always to ensure the intended result. In addition, even given a target value of either h u m i d i t y parameter, available m e a n s of climate m a n i p u l a t i o n are, most c o m m o n l y , only (natural) ventilation and heating. O w i n g to the m a n y feed-back effects that a change in conditions w o u l d set in motion, the use of a h u m i d i t y model of the kind developed here w o u l d allow climate settings, such as air temperature or w i n d o w o p e n i n g angle, to be derived directly from ' p r o c e s s ' set points, thereby disposing o f intermediate h u m i d i t y parameters.
Acknowledgements W e are indebted to the personnel of the vegetable gardens of I M A G - D L O and in particular to F.J.M. Corver for helping with the transpiration measurements, besides skillfully taking care of the crop, and to our colleague Dr. H.J. H o l t e r m a n for revising the manuscript.
References Aikman, D.P. and Houter, G., 1990. Influence of radiation and humidity on transpiration: implications for calcium levels in tomato leaves. J. Hort. Science, 65: 31-39. Aubinet, M., Deltour, J., De Halleux, D. and Nijskens, J., 1989. Stomatal regulation in greenhouse crops: analysis and simulation. Agric. For. Meteorol., 48: 21-44. Bakker, J.C., 1986. Measurement of canopy transpiration or evapotranspiration in greenhouses by means of a simple vapour balance model. Agric. For. Meteorol., 37: 133-141. Bakker, J.C., 1991. Analysis of humidity effects on growth and production of glasshouse fruit vegetables. PhD dissertation, Agricultural University, Wageningen, 155 pp. Bakker, J.C., 1994. Vraagtekens bij doel minimum buis. Groenten + Fruit/Glasgroenten, 4(2): 10-11. Bot, G.P.A. 1983. Greenhouse climate: from physical processes to a dynamic model. Ph.D. dissertation, Agricultural University, Wageningen, 240 pp. Brown, W.G. and Solvason, K.R., 1962. Natural convection through rectangular openings in partitions. 1. Int. J. Heat and Mass Transfer, 5: 859-868. Cockshull, K.E., 1988. Integration of plant physiology and physical aspects of the greenhouse climate. Acta Hortic., 229: 113-123. De Jong, T., 1990. Natural ventilation of large multi-span greenhouses. Ph.D. dissertation, Agricultural University, Wageningen, 116 pp. De Jong, T. and Bot, G.P.A., 1992. Air exchange caused by wind effects through (window) openings distributed evenly on a quasi-infinite surface. Energy Build., 19: 93-103. Fernandez, J.E. and Bailey, B.J., 1992. Measurement and prediction of greenhouse ventilation rates. Agric. For. Meteorol., 58: 229-245. Hand, D., 1988. Effects of atmospheric humidity on greenhouse crops. Acta Hortic., 229: 143-158. Holder, R. and Cockshull, K.E., 1990. Effects of humidity on the growth and yield of glasshouse tomatoes. J. Hort. Sci., 65: 31-39. Jarvis, P.G., 1976. The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field. Phil. Trans. R. Soc. Lond., B 273: 593-610.
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Jolliet, O., 1994. HORTITRANS, a model for predicting and optimizing humidity and transpiration in greenhouses. J. Agric. Eng. Res., 57: 23-37. Matthews, R.B. and Saffell, R.A., 1981. Computer control of humidity in experimental glasshouses. J. Agric. Eng. Res., 33: 213-221. Monteith, J.L. and Unsworth, M.H., 1990. Principles of Environmental Physics. Edward Arnold, London, 291 Pp. Papadakis, G., Frangoudakis, A. and Kyritsis, S., 1992. Mixed, forced and free convection heat transfer at the greenhouse cover. J. Agric. Eng. Res., 51:191-205. Stanghellini, C., 1987. Transpiration of greenhouse crops: an aid to climate management. Ph.D. dissertation, Agricultural University, Wageningen, 150 pp. StangheUini, C., 1993. Mixed convection above greenhouse crop canopies. Agric. For. Meteorol., 66: 111-117. Stanghellini, C. and van Meurs, W.T.M., 1992. Environmental control of greenhouse crop transpiration. J. Agric. Eng. Res. 51: 297-311. Stanghellini, C. and Bunce, J.A., 1993. Response of photosynthesis and conductance to light, CO2, temperatore and humidity in tomato plants acclimated to ambient and elevated CO 2. Photosynthetica, 29: 487-497. Stoffers, J.A., 1975. Radiation absorption of canopy rows. Acta Hortic., 46: 91-95. Van der Varst, P.G.I. and Postel, J.D.G., 1972. Bepaling bladoppervlak van tomatenplanten. IT]?, Wageningen, Report 46, 36 pp. Van Meurs, W.T.M. and Stanghellini, C., 1992. Use of an off-the-shelf electronic balance for monitoring crop transpiration in greenhouses. Acta Hortic., 304: 219-225. Zhang, J.S., Janni, K.A. and Jacobson, L.D., 1989. Modelling natural ventilation induced by combined thermal buoyancy and wind. Trans. Am. Soc. Agric. Eng., 32: 2165-2174.