Modelling of Greenhouse Temperatures Using Time-Series Analysis Techniques

cc,

CHP\IIg:h ! 11- .\(. ~'d! J lH"lIlll,d \\"" lId ( " , 1\.:ln __ Bud.lpt·. . 1. JI IIII~.II\ 1'1."'1

MODELLING OF GREENHOUSE TEMPERATURES USING TIME-SERIES ANALYSIS TECHNIQUES A. * /kIHlrlllloll III

J. u.

/'li nin flllr! ::::;;'u "\,,,, /"/"/t

ten Cate* and

J.

van de Vooren**

.\lI 'II 'IIIII!O,!..,'.'·. Agl"/nd/lIl"rI/ L·1II; 'I'I"\I(\". n 'fl,L!"I' IIIJlgl' l/, .\/0/11111 /111" F/orin"IIII"I'. AO!I/IU'f' I. TI/I' SI'f/t 1'1"/0 I/(/.'

'/1,1' SI'I/tI'rlUW/'

Abstract. An innovations approach in the modelling of the dynamics of the temperature in greenhouses is presented. Heating, ventilation and solar radiation are used as input variables. The parameters of the model are validated in terms of widely available heating-load coefficients, including the nonlinear characteristics of the heating system. Also, the working-point of the dynamic model is described using static or quasistationary relati ons . The proposed model is demonstrated to be accurate which - added to the level of detail - makes this way of modelling very suitable for design and analysis of greenhouse control systems. Eecause the dynamic model describes the relatively high frequency behaviour of the process and the quasi-stationary model the low frequency behaviour, time-series analysis techniques are essential in the modelling. Keywords. Identification, modelling, agriculture, air condition. are formulated along these lines. Typically in practice control procedures are improved by adding operator knowledge in terms of conditional compensations or logical decisions. The dyna~ic nature of the process under control is not analyzed generally. Also, the term control is used in a broad sense. In a perception of operator procedures, the drawing of a thermal screen or even chalking of the glasscover can be seen as control. When control of a dynamic process using a continuously operating actuator 1S meant, the term GCFC (greenhouse climate feedback/feedforward control) has been introduced (Udink ten Cate, 1982,

INTRODUCTION In the ~etherlands, the popularity of mini- and microcomputers for the control of greenhouses is steadily growing. At present about 3000 greenhouse computers are in operation at commercial hold1ngs. In th1S development the control algorithms in the computers usually perform the same functi ons as conventional analogue controllers. In a computer system, however, more sophisticated control methods could be applied, which has motivated research in this area. The computer fits well into the development of automation in greenhouses. Automation made a hesitating start in the late fifties (Vijverberg and Strijbosch, 196 8) by the introduction of thermostats for heating control. Later, analogue electronic controllers were introduced. At first performing similar functions as thermostats, but later with increasing capabil ities (\Jinspear , 1968), such as the automation of ventilation windows (Strijbosch, 1966 ).

The research on climate control reflects the operator 's attitude and little attention is paid to GCFC as such. As a result relatively few studies on GCFC have been published, and were restricted to the heating system control loop (O 'Flaherty et al., 1973, Tantau, 1979, Udink ten Cate and Van de Vooren, 19 77, 1981, Otto et al., 1982).

An interesting aspect of the deve l opments in the recent decade is that the underlying philosophy of control with greenhouse computers conforms to an operator manipulating various actuators. Using add1tlOnal measurements and time-clocks it is possible to transform this operator approach into procedures which are carried out by the controller. In these procedures actuator signals ( like heating system temperature, or ventilation window aperture) are directly related to the (assumed) reactions of the crop, so that in practice most control procedures

In contrast to the limited availability of GCFC models, models of the greenhouse cl imate that are based on physical phenomena of heat and mass tr~nsfc~ are widely used. Typically, in these models the outside weather and the actuator actions serve as input variables. The thermodynamic properties of the greenhouse structure, the soil and the crop enter into the mode l as parameters that can be calculated from known physical phenomena. The model can be of the steady-state type (e.g. 5usinger, 1963, Kimball, 1973, Garzol i

2033

1983).

2034

A. J. U. ten Cate and J . van de Vooren

and Bl ackwe ll , 1973) or can account for energy storage in the greenhouse (e.g. Takakura et al., 1971, Froeh li ch et a l., 1979, Bot et al., 1978). The models that incorporate energy storage el ements could be employed as GCFC mode l s, but on l y reasonab l e results are claimed when low-frequent (idea li zed) input signals are applied. No resu l ts are claimed for rapidly time-varying cond itons, which are of prime interest in GCFC.

9round area Aa[m 21 of the greenhous~, leading to C* ~ C /~ . q~ ~ q /r . k* ; 1/(R A )[W m- 2K-l] g g g' v v g' h h 9 and k; ;1/(n r Ag )[W m- 2K- l 1. Keeping the other variables constant linearizing yields for the re l ation between en and eh: de " C* _g = - (q* c . ;: , + k; + khJ9g (t) + 9 dt v p,a1r a1r +

The mode l s could be improved using detai l ed measure ments for val idation and employing parameter estimation of relevant clusters of physical procedures. This necessarily leads to a high degree of comp le sity whi ch is not su itable for GCFC app l ications. Quite another approach is to employ the black - box type of modelling that is familiar in time-series ana l ysis. This approach i s followed in this paper to formu l ate a model in terms of heating-load coefficients which have a physica l meaning. For the temperature in the greenhouse this is done for a linearized mode l , including heating, ventilation and the influence of so l ar radiation. The mode l is f ormu lated in terms of incremental variables around a s l ow ly time - varying workirg point . By separating the variables in highfrequency components, which define the linearized model, and low-frequency components which determine the working point, a high degree of accuracy is achieved . The I'esults are considered to be of key importance in GCFC and summarize and extend a part of the materia l reported in Udink ten Cate ( 1983) . TEMPERAT U~E

kh ~ h(t)

(2 )

In analogy with the other k- factors, a factor k* is introduced. With the average height of the v greenhouse ~g ~Vg/Ag , where Vn [m 31 is the green house volume k*v where S 1S the a1r change rate Sv(t) ;~v(t) . 3600/Vg [h - 1, so that 3 l 1; = c ,air.' air/3600 ~ 1/3 [W h m- K- 1. p The linearizations can be carried out for 2g ' 25 and al so for the disturbance :; " ; ; /A [W m- l~ This l eags to a combined lineafizedSeqgation C * ddStg - (k* + k* + k*) ; (t) + k* 9 (t) + 9 v r h g hh + r ~ (e; :;)5 (t) + n2s'( t) (4) ' ga _ gv GCFC transfer functions

MODEL

In fi9. 1 a qreenhouse is depicted with the actuators that are commonly in use in GCGC in the Netherlands. Restri cting ours elves to the climate factor air temperature inside the greenhouse 2 [ ' C], it is seen that this depends on the outsi8e weather (factors: ambient air temperature ea [OC], wind vel ocity Vw [m/s ] and direction, shortwave (solar) rad i ation ", s [Ill, l ongwave rad i ation 4>1 [W]) and on heating and vent il ation. Heating i s done by a heating system of S1 mm stee l pipes in which water i~ circulated 'lI ith inlet temperature e [DC]. The outlet (return) water with temperature e m[OC] is mixed wit~ feedwater from the main boi l er ofrtemperature 8f [DC] by a mi xing va l ve with positon r E [0, 100%]. Tne average temperature of the heatiWg system i s eh[OC ]. Ventilat i on is ac hieved by ventilation in the ~oof with aperture r t: [0, 1001; ] l eading to a vo l umetric exchange rate qW[m 3/s] .

In the models of eqns. (1 - 4) an "average" va lu e of the variab l es is used. In GCFC usually single point measurements are employed for ~ , and for the inlet water temperature em is tak~n. This leads to a time-delay when transfer functions are considered. Taking the Laplace transform l eads to the transfer functions that are depicted in fig. ~ Here ' d,h' Td s, I d are delay times (transport times).' For the sno~n parameters the following relations hold

Idealized mode l

The extra term for T various de l ays repreg~R~ will be shown that ' d hs equivalent l y , s + T d,~ ~

v

An idealized model can be co nstit ued by considering the green house as a perfectly stirred tank, assuming un1form values of the variables. When the temperature eq is of intere st , summing the (sensib le) heat fluxes l eads to d

Cg d(

1

-

~ (~

+

J-h ( ~ h

r

g

(t) -

~

a

(t))

'g

" "

...-

v

's

"

' ~ s (t)

+

+

k* r

+

Td,hs

kh

"

Cg/kt

(Sa)

Kg

kh/ kt

( Sb)

Kv Ks

r; ~g( ~a

;; )/ k* 'g t

n/kt

(5c) (5d)

is added because the small time constants; it = r d,h - ~ d,s or r h + Td,h·

Working point

-(k*v,ss

+

(t) - " (t)) + 9

k* v

The working point is calculated from the timeinvariant parts of eqn. (1)

- air( ~ g(t) - ~ a(t)) -

- qv(t) cp,air

k* t

( 1)

Here C is the greenho~se heat capacity [J/K], c p airg~ 10 3 [J kg- 1K- ] is the specific heat of dry air at constant pressure, and - a ir ~ 1.2[kg/r;)3] is the dens i ty; Rh' Rr [K/fJ] a re the therma l re sistances of the heating system and roof plus sidewalls respectively, and ' i s a fraction '( [0,1]. The influence of l ongwave radiation is not considered. Eqn. ( 1) is linearized around a working-point and fOt'olulated in terms of incremental variables defined as '- (t) ~ ~ (t) - ~ . In order to normalize the equations, the re lati ons are expressed in units of

+

k*r,ss )(~ g -

o

~

a)

+

k*h,ss (~Om - ~0g )

+

(6)

where the suffix ss denotes the steady - state (static) va l ues of the parameters which follow from steady-state relations. It is remarked here that the shape of <;: " is usually such that using ~" in eqn. (6) is n~t meaningful. When the workingp~int is slowly time - varying, a quasi - static model can be formulated.

Hodelling of Greenhouse Temperatures

d -'

Cg,ss

3t SS

kh,ss( ~ m,ss(t - " d,h) -g, ss(t)) -

=

- (k~ ,ss x( -~

+

g,ss

+

~ 'ss\, ss(t - ' d,))

(t) - 2

a,ss

X

(t))+

" ss '; "s,ss(t- ' d,s)

(7)

where the suffix ss for the variables denotes the slowly time-varying compo nents (low frequencies) of the signals, The time-delays are introduced to account for the time shift, Note that k* is not used here, but ~ 'sS (t) in order to alYd~sfor slowly timevarying vXla~s of the ventilation rate, The storage term C* s d S /dt is present to facilitate the model ~d aam~~Assudden peaks (due to numerical inaccuracies) that may occur in the generation of the time responses of the quasi -static variables, The effect is mainly cosmetic and for its value C*g,ss = Cg is selected, Actua tors The GCFC models in fig, 2 are not formulated in terms of the actuator positions rand r w' This is done because of the nonlineariti~s introduced by the actuators and because the actuator processes depend on the characteristics on the actuators themselves and as a result lack generality, For the usual mixing valve a relation holds

(8)

The value of Pr depends on the s urface conditions of the heat1ng p1pe network and the circulation rate of the water, An approximation can be made as e- Cj,m s (

8 r (S)

=

\ s:r-+l \ ~ m(s) - Km(in(S) - 8 g (S)))

(9)

m

wi th T ~ 2 mi nutes, T d depends on the flow rate of the he~ting water and ~~€[0,05, 0,2] depends on the temperature decrease of the water in the heating pipe network , It is assumed that Oh - "g >. ~\ m - er' For the ventilation Bot (1983) and Nederhoff et al, (1983) established an emperical relation for the particular greenhouse in which the experiments described in this paper were perfo rmed,

Sv

= (a

O

+

rw) a 1 vw

(10)

where a and a are constants (a = 1; a = 0.064}, r ....dO,38;, ] and l vw([l, 10 m/sJ. TRe equat10ns (8) and (10) can be 1 inearized for sma ll increments. However, no experimental data are available here. EXPERIi~ENTS

In order to determine the parameters of the models presented in the previous section, experiments have been carried out in the multifactoral glasshouse of the Naaldwijk Crops Research and Experiment Station, Here a process computer regulates 24 identical compartments independently (Van de Vooren and Koppe, 1975 ) . The g1asshouse is of the Venlo type and the size of each compartment is 56 m" (fig, 3). Relevant data are given in table 1. For the experiments the compartments 1-8 have been used. A Chrysanthemum crop was grown during the experiments, which were carried out in Winter/Spring 1982, For the experiments the increments of the variables were assumed to be small and relatively high frequent (periods of harmonics < 3 hours) compared to the quasi-stationary components which were assumed to be

20]5

large and low frequent (periods 5 hours), The parameters of the GCFC transfer functions of eqns. (5) were determined using a hi gh frequent test signal as input on cm and Sv or select a day with h1gh frequent disturbances :", while the other variables were kept constant ~t various wor king points, Disturbances and trend are filtered out in the frequency do~ain using a highly interactive time-series analysis package (Van Zee and Van den Akker, 1983), The K's and ' 's as Vlell as the time-delays are estimated using optimization techniques. The samp l ing time T is 1 minute, The estimation procedure is depicted in fig, 4 for the and heating transfer function. Fig , 4a shows before filtering, fig, 4b shows - after r~moval g of trend and obvious disturbances~ and fig, 4c gives the result of the optimization, More details are found in Udink ten Cate ( 1983), Estimation of kh From the experiments only the ratio's of the phYSical parameters C*, kh and k* can be found (k* follows directly ~rom eqns, (3) and (10) ) , Th¥refore, kh was determined from a separate exper1ment by cool1ng off the water in the heating pipe network with closed mixing valve and circula tion pomp running. Here the relation holds dOh * Ch dt = - ~ h( t) (11 )

= f( = - = , t). Since C~ is known, a relat10n for Q ~ cangbe found that is nonlinear with respect to eh - 3g . When 0m is taken for = a good flt4~as found w1th ~ ~(t) = 1 ,O( ~m(t- - d hr- \'g(t))' , leading to a Tinearized k~: '

where. ~ h

kh

=

1. 46 (8m - 9g ) 0 .46

(12 )

Estimation results With kh given according to eqn. (12), the results of several experiments are summarized in table 2. In experiment I (dur in g a night period) the windows were closed fw = 0, and various ~ h were kept 1n 7 compartments. In exp, 11 (night) ~ = 50"C and various fw are employed in 4 compartment~ , Exps. I and 11 were carried out to determine the the heating transfer function of eqn, (5b) , In exps. III (day) the transfer function of radiation (eqn, 5d) was established for f = 0 and various Sm in 4 compartments. In expo IYIb a high outside a1r temperature occurred , From table 2 it is seen that k* is fairly constant for the same type of experiments, The 10Vler values in exps, III can be explained because the corridors (fig, 3) are heated by radiation too. Note tha t C* / C* i r ~ 11 , where Cair ~ 3.9.90 3 YJ K- l m- 2 ] is the normalized heat capacity of a greenhouse with average height ng , This high ratio indicates that large paralle l heat capacities are due to the sidewalls, roof, irrigation system , crop, construction parts etc. No reliable results were obtained for the venti lation tra~sfer function which is explained by the fact that Sv could not be measured directly. Quasi -stationary results For exps, I and 11 the steady-state va lues of the parameters are calcu lated using eqn. (6) (night situation, ;5 = 0) and for exp, IlIa (day) using eqn, (7), The results are given in table 3. The data are calculated with kh s = : h/( ~ m - ~ ) accordi ng to eqn. ( 11 ) . As c6ul d be expecte§, the higher frequencies yield a higher k-factor. The k~ ss of exps. I and 11 complies with a value from literature k~ = 8. 3 for v = 4 m/s (von Zabeltitz, 191B}. The result~ of k~,ss in table 3 also 1nd1cate that the value of kh is realistic, A combined response of the GCFC mod~T and the

2036

A. J. U.

ten Cate and J. van de Vooren

quasi - stationary working-point of expo IlIa is depicted in fig. 5, c l early demonstrating that a good fit can be obtained. Because the estimated resu l ts are based on the va l ues of kh ss and kh it is important also to check the values of kh in another way . This has been done by calculating a ; k'/{k* + k*) and c = C*/{k* + k*) from expo I i ~ sev~ra I Vways~ Becaus~ in ~xp.vI r several values of ~ were realized 2 a check can be made on the i ndepen~ency Of c _on Bm4 It was also found that a g = 4 . 41· 1 0 - 2(~ - g§g) ~'4~' With k~ + kF = 30.1, kb = 1.32{ e ~ - eg) . ,wh i ch conforms to eqn. \ 12). The dependency of k* on v and on the longwave radiative heat em i s~ion t~ the sky cou l d not be investigated in the experiments. From the experiments, indicative va l ues can be established for arbitrary greenhouses (table 4) . The valueof k; ss is usually known from literature for a specific'type of greenhouse. The time - delay ' d h is also dependent on the heating system l ay - out. ' With eqns. (5) GCFC mode l s can be establ i shed that can be used for analysis and simu l ation. The working point can al so be computed using eqns. (6) and (10) . TEMPERATURE MODE L INCLUDING HEATING SYSTEM In the GCFC models of eqns. (5) and fig . 2, the input signal is not the position of the actuator itse l f, but merely the actuated variable. For the ventilation windows th i s is a necessary restriction since the ventilation process in greenhouses is only partly known, with only steady- state data for particular greenhouses avai l able as yet. For the mixing va l ve position, the characteristics of the heating system are cance ll ed out by US i ng eh in the GCFC mode l . The actual heati ng system behaviour cou l d be inc l uded us i ng eqns. (8) and (9). Another approach is a I so possib l e which is more c l osely related to a physica l model. Now the heatin~ system is considered as a perfectly stirred tank with uniform temperature e ~ and the influence of the mixing va l ve is the supply of heat. This heat is measurab l e in a practical greenhouse by measuring Bm and 9 r . This l eads to the mode l

r::" h

k*/C*h

- p

k~/Ch

l/C h

Bh

' ~h{t- ' d,P)

+

k*/C* P g

0

g

0

(13 )

In this mode l k* is related to eh and is likely to have another fo~m than kh. It might be anticipated that k~ wi l l correspond more to physica l phenomena . The factor kt = k* + k* + k* . The delay- time Td p ' wil l be somewhat ¥mall~r th~n Td h' The heat input ~ h It) = qh{t) c at r ~ wat' ( ~ It) - Sr{t))/A where qh is the vo l ume~r1~ flow 5f t~e heating wate7. This ~ h can . be controlled in a separate loop, or regulated d1rectly Slnce from eqn . (8) Sm- 2r={ 0f - er)rm. As is typica l in heeting systems ' h ~ 0 and on l y sma l l negative values of ~ h can be achieved when Sr ~ S as is the case in most greenhouses. This, combine~ wi th the l arge storage capacity of the water f or heat, severely l imits the controllability of the greenhouse air temperature . CONCLUSIONS In this paper a GCFC model of the greenhouse temperature is presented. The parameters of the model are estimated using time- series analysis techniques. The paramete .·s are re l ated to heating l oad coefficients (k - values) which are wide l y available in the l iterature . This enables the formulation of GCFC models for arbitrary greenhouses, wh is of much importance for application of models in the analysis and

design of control systems. Also, the working-point to which GCFC mode l s are related, can be calculated using steady- state or quaSi-stationary relations. It is demonstrated thatwhen a s i milar model structure is used as in the case of GCFC, the va l ues of the corresponding parameters differ. Since GCFC models are high frequent, the l ow frequent working - point models indicate a frequency dependency of the parameters of a type that is common in distributed parame t er systems. Because the nonlinearit i es of the k- va l ues are al so included in the models, and because a main disturbance (radiation from the sun) is model l ed too, good opportunities arise for gain- scheduling and feedforward compensation in industrial greenhouse computer contro l . It can be concluded that the accuracy and the level of detail makes the proposed GCFC models superior over other mode l s proposed in the literature. Recent research focuses on the model l ing of humidity, where the influence of the crop is more profound . More knowledge of the ventilation processes is a prerequisite here. ACKNOWLEDGEt~ENT

Prof . dr.ir . J. Schenk (Agricultural University at Wageningen) and Prof.ir. J.J. van Dixhoorn (Twente University of Technology) are acknow l edged for their stimulating criticism. REFERENCES Bot, G.P.A. , J . J. van Dixhoorn, and A.J. Udink ten Cate (1978). Dynamic modelling of glasshouse cl imate and the application to gl asshouse control. Phytotronic News l ett, 18, 70-80. Bot, G.P.A. (1983). Heat transfer and ve nti l ation in glasshouses. Thesis to be completed. Agric. Un1V. Wagen1ngen, The Netherlands. Businger, J.A. (1963). The glasshouse (greenhouse) climate . In: ~I.R. van vilJk (ed.) PhYS1CS of pl ant environment. North - Holland Publ. Co., Amsterdam. p. 277-318. Froehlich, D. P. , L. D. Al bright, N. R. Scott, and P. Chandra (1979) . Steady- periodic analysis of glasshouse thermal environment . Trans . ASAE, 22, 387 - 399. -Garzo l i, K. V., and J. Blackwell (1973). ThE: response of a glasshouse to high solar radiation and ambient temperature. J . Agric. Eng. Res., 18, 205 - 216. Kimball, B.A. 11973). Simulation of the energy balance of a greenhouse. Agric. Meteoro l ., 11 (2), 243 - 260 -Nederhoff, E.M . , J. van de Vooren, and A.J. Udink ten Cate (1983) . A practical method to determine the ventilation rate in greenhouses using a tracer gas method. Subm. to J. Agric. Eng. Res., Report Naa l dwijk Exp. Res . Sta. O'Flaherty, T., B.J. Gaffney, and J.A . Walsh 0973) Analysis of the temperature control characteristics of heated glasshouses using an analogue computer. J. Agric . Eng. Res. , 18, 117 - 132. --Otto, P. , K. Sokoll i k, J . Wernstedt, and M. Diezemann (1982) . Ein Innentemperaturmodel l zur Mi krorechnersteuerung der Heizungssysteme von Gewathsh~usern (in German, English summar~. Arch. Gartenbau, 30 (3), 139 - 146. Str1Jbosch, Th.(196~ Ventilatie en verwarming van tomatenkassen (in Dutch, English summary). Meded. Dir . Tuinb. 29 (9), 364 - 371. Takakur , T., k. A. Joraan, and L. L. Boyd (1971). Dynamic simulation of plant growth and environment in the greenhouse . Trans. ASAE, 14(5), 964-971. -- -Tantau, H. -J. (1979) . Analyse des Regelverhaltens kl imatisierter Gewa'chsha'user al s Grund l age zur Auswah l und Entw1ck l ung gee1gneter Regler ( I n

2037

Modelling of Greenhouse Temperatures

German). Gartenbautechn. Inforr.l. Heft 7. Taspo Verlag, Hannover, Germany. 146 p. Udink ten Cate, A.J., and J. van de Vooren (1977). Digital adaptive control of a glasshouse heating system. In: H.R. Van Nauta Le~ke (ed.) Digital computer applications to process control. North-Holland Publ., Amsterdam. p:-5U2-512 Udink ten Cate, A.J., and J. van de Vooren (198 1). Adaptive systems in greenhouse climate control. In: Preprints 8th IFAC World Congress, Kyoto, Japan (Aug. 1981). Late papers. p. 9-15. Udink ten Cate, A.J. (1982). What is the problem in greenhouse climate control. Proc. Int. Tcch~ Kolloquium, Heft 2, Techn. Univ. rlmenau, DDR. p. 207-210. Udink ten Cate, A.J. (1983). Model ing and (adaptive) contro l of reenhouse cllmates. ' eS1S, gnc. nlV., 'iagemngen, e Netherlands. 159 p. Vooren, J. van de, and R. Koppe (1975). The climate qlasshouse at Naaldwijk. Neth. J. Agri~ Sci., 23, 238 -~47. VijYerber~ A.J., and Th. Strijbosch (1968). Ontwikkelingen in de klimaatregeling bij tomaat en sla (in Dutch). 11eded. Dir. Tuinb., 31 (12), -427-475. Winspear, K.W. (1968). Control of heating and ventilation in glasshouses. Acta Hort., 2., 62-78. Zabeltitz, C. von (1978) . Gewa'chshiiuser - Planung und Bau (in German). Verlag Eugen Ulmer. Stuttgart, Germany. 267 p. Zee, G.A. van, and D. van den Akker (1983). TSAPACK users guide . Dept. Physics and Meteorol. Agrlc. Onlv., Wageningen, The Netherlands. Rep. LH-NW-MRS-83-1. 55.7 m32 A ground surface vg m 163.0 air volume 2.93 m2 fi g Vg?Ag average height 85 . 2 m2 s idewalls surface As 63.0 m Ar roof surface heating pipes length 110.0 m lh (active) 0.051 m diam. 17.6 n132 heating pi pes surface Ah 0.216 m Vh heating pipes volume TI,LL[ 1 ::'li::ssl1oJse rlil:.cnsi ons k* r

expo no.

k* v

C*

-

10~

-

~'9 . 7

11

28 . 3 -

47.4

IlIa 21.8 0.3

0.71

II Ib 15.9 0.5

0.58

II Ic 19.6 0.3

0.72

TABLE

L

k* r

k~/k~,ss

9.9

29.7

3.0

11

9. 8

28. 3

2.9

IUal

7.7

21.8

2.8

TABLE 3

k*r,ss

10

k~/kr,ss

2- 3

C*/C* 9 air

10

n

0.7

m

1 ( ld,h

=

7 min

Td, s

=

1 min

j

Td,hs

=

nss

min Td,s

Quasi-stationary results

0.71

' d,hs

' d,h

10• 1\s r~

;vw

>(rw qv

Of Og

Fig. 1 The greenhouse climat control system

n/ nss

1.6

+

TABLE 4 Indicative values for GCFC models

Fig. 2 GCFC model s 0.45

8g )0.46

min

6 min

n

-

1.5

Experimental results

* kr,ss

expo no.

lft6( a

h

i)

9

0.4 43.7 10 3

I

k*

2038

A. J. U. ten Cate and J. van de Vooren

BIB BiB Effij

i N

Fi g. 3 The Naaldl/ijk J:1ultifacto ral glasshouse (after Van de Vooren and Koppe, 1975)

, . . . E f B73 5 f B151 E E23 9 5850

4

8

12

16

20

weather station

24

j. 2~0~.h~---:-c-,---'-_ _ _~_ _ _ _ _ _-"-'--_ _ _ _ _- ' . .32.0 0':.;. 91300 .; corridors compartments

computer

- - - - - -- -----------

~g -. DC

18

0

720

360

tIme mln

21 DC !

V\ .

·2' 0

, 360

llil

,

,

~L--'~ 720

time mln ,

2 -

·~~-~--~---L--~-~~-~---L---L--~-~350 time mln

Fig. 4 Filtering and data-fitting in compartment no. 1