A practical tracer gas method to determine ventilation in greenhouses

J. agric. Engng Res. (1985) 31,309-319

A Practical

Tracer

Gas Method to Determine Greenhouses

Ventilation

in

E. M. NEDERHOFF*;J. VAN DE Voowm’j’; A. J. UDINK TEN CATER

A practical method for measuring leakage and low level ventilation in greenhouses is presented that is based on the use of carbon dioxide (CO,) as tracer gas. When natural gas is used for heating, as is done in the Netherlands, carbon dioxide can be obtained from the exhaust gases of the heating system. A dynamic form of the tracer gas method is employed, known as the decay rate method. Six small greenhouse compartments were injected with CO, up to a level of 2000 v.p.m. (volume per million volume) and then the injection was stopped. The ventilation rate was calculated from the decay rate of the concentration. A relation was established that describes the ventilation rate as a function of wind speed and ventilator aperture. CO, can also be used as a tracer gas in greenhouses containing crops, provided the CO, exchange by the plants is included in the calculation. A test was made to compare a simple CO, measuring device, which is commonly used by growers, with the infra-red gas analyser that was used in the experiments. It was found that the simple device allowed the proposed method to be used in practice. 1.

Introduction

Ventilation is a very important tool for greenhouse climate control. Except for cooling in the summer, ventilation is important for transport of water vapour and other gases (e.g. carbon dioxide for photosynthesis). Many energy saving measures affect the ventilation or the leakage of the greenhouse, particularly the fact that greenhouses are now being built more airtight than hitherto. For the individual grower it would be useful to know the ventilation characteristics of his greenhouse in order to make decisions about investments in energy saving. When this determination becomes a common requirement, a practical and simple method for ventilation measurement will be a necessity. In the Netherlands, where natural gas is used for greenhouse heating, the exhaust gases from the central boiler (or from smaller heaters in the greenhouse) contain carbon dioxide (CO,) and (normally) no toxic components. The gases are led into the greenhouses in order to stimulate plant growth.’ For controlling this “CO,-enrichment” many greenhouses are equipped with a simple CO,-measuring device. Therefore, CO, might be a suitable tracer gas for determining the ventilation rate. This idea was tested in an experiment. Carbon dioxide was injected to a certain level, then the supply was stopped and the decay rate of the concentration was measured. From this decay rate the ventilation rate was calculated. The “decay rate method” or “dynamic tracer gas method” was selected because it is easily applied in practice. The results are compared with measurements obtained by an “equilibrium” or “static tracer gas method”.2 In order to test the method, measurements were carried out under conditions of varying wind velocity, greenhouses with and without a crop, with all ventilators closed (leakage ventilation only) or with a small ventilator aperture (both leakage and natural ventilation). These conditions frequently occur in winter in greenhouses. The primary objective was to test the method by checking whether a reproducible relation existed between measured ventilation rate, wind velocity and ventilator aperture. If such a ‘Glasshouse tResearch $Agncultural Recewed

Crops Station

Research

and Experiment

for Floriculture,

Umversity,

Department

4 July 1983; accepted

Station.

Linnaeuslaan of Computer

Zuidweg

38.2671

2A. 1431 JV Aalsmeer. Science,

in revised Form 2 January

Hollandseweg

MN Naaldwijk,

The Netherlands

The Netherlands I, 6706 KN Wageningen,

The Netherlands

1984

309 0021~X634185/040309+

1I $03.00/O

D 1985 The Bntish

Society

for Research

m Agricultural

Engmeenng

310

VENTILATION

IN GREENHOUSES

NOTATION a

C h IRGA k L

1 P R r

Pco, s t

u Subscripts a

e g

a conversion factor from photosynthesis rate to decay rate of the CO, concentration CO, concentration, (ml/m3) v.p.m. average height of the greenhouse, m infra-red gas analyser a conversion factor from photosynthesis rate to ventilation rate leaf area index, leaf area per ground area (m2/m2) leakage, expressed in units of window aperture, % plant effect on the CO, concentration, photosynthesis (+ ) or respiration (-), g CO, . h-i . me2 greenhouse area radiation, global, measured outside greenhouse, W/m2 aperture of ventilation windows as percentage of maximum aperture, % gas density of carbon dioxide at 293 K and atmospheric pressure, kg/m3 ventilation rate in air volume per greenhouse volume per hour, h- ’ time, s wind velocity, m/s apparent, without correction for the plant effect environment outside the greenhouse inside the greenhouse

relation can be established for any greenhouse, it can be used to predict the ventilation from the environmental conditions. Also a control algorithm for ventilator opening based on this relation may be implemented in a future computerized climate control system.” While testing the method, special attention was paid to the influence of the crop, since plants have a profound effect on the concentration of CO, in the greenhouse. 2.

Other studies

In early research on ventilation in 19544 a method was used based on plant observations. The net amount of carbon assimilated was found from plant samples and a method was evolved to calculate the ventilation rate necessary for undiminished growth. Ventilation measurements were performed by Whittle and Lawrence5 using smoke, krypton and hydrogen as tracers. The results of the methods were similar, and preference was given to hydrogen as a tracer gas. A linear relation was found between ventilation and wind velocity. A method to perform photosynthesis experiments with simultaneous ventilation measurements was proposed by Lake6 and again described by Hand and Bowman.’ Ventilation was measured with N,O as tracer gas, which was kept at a fixed level (hence the “static tracer gas method”). Ventilation measurements with CO, as tracer gas are reported by Okada and Takakura.s An equation was derived for a relation between air infiltration, wind speed and temperature differences between inside and outside air. The distribution of CO, in greenhouses of different sizes was investigated by Gormley and Walshes and Van Berkel.‘O The latter also determined leakage ventilation in a greenhouse with a crop. During a night following a day when CO, enrichment had taken place, the decay rate of the CO, concentration was measured and from these data the ventilation rate was calculated.

E. M. NEDERHOFF

311

ET AL.

The respiration (CO, release) of the soil and the plants was taken into account by determining the ultimate CO, level, which was higher than the ambient CO, concentration. In the earlier work reported by Bat,” ventilation was measured at various window apertures in the same greenhouse complex in Naaldwijk as used in the experiments described in this paper. The greenhouses were free of crops and CO, was used as tracer gas with a static tracer gas method. 3. Methods 3.1. Measurements in greenhouses without a crop In a series of experiments, the tracer gas CO, was injected into an empty greenhouse until a concentration level of 2000v.p.m. was reached. As a result of exchange with outside air, the concentration in the greenhouse, decreased with a rate proportional to the CO, difference between inside and outside:” dC (t> A= dt

-S.

(C,(t)-C,).

l/3600

. ..(la)

where the factor l/3600 is introduced because S is defined in h- ’ and t in sec. Assuming that Sand C, are constant over the observed time interval, the CO, concentration in the greenhouse is C,(t)=C,+(C,(O)-C,). el-G . ..(lb) in which C,(O) is the initial CO, concentration measured at time t = t,, S can be found from:

s= y

.ln

at t = 0. With C,(tJ being the CO, concentration

[ ~~~t~)~~

]

. ..( lc)

S is the ventilation rate expressed as times per hour exchange of the greenhouse volume. The outside carbon dioxide concentration (C,) was assumed to be 330 v.p.m. Measurements of the environmental air (above the greenhouse) gave values of up to 380 v.p.m. but this (local) deviation was ignored. For practical reasons the time interval at which the CO, concentration was measured was set at 480 s. 3.2. Measurements in greenhouses with a crop Experiments have been carried out simultaneously in greenhouses with a fully grown crop and greenhouses without a crop. In both cases CO, was introduced up to a certain concentration level (2000 v.p.m.) and the concentration decay was measured. In compartments with a crop the CO, concentration is greatly influenced by the photosynthesis (CO* uptake in light) and respiration (CO, release in dark) of the plants. So the CO, concentration decay is the result of ventilation rate Sand the plant effect P (g CO, . m-’ . h-r): dC (t)

4 dt with

= -(S. (C,(t)-C,)-a.

.(2a)

P). &-

a=(h .pcO,. 1O-3)-’

. ..(2b)

For a value t= t,, Eqn (2) cannot be solved analytically as was the case with Eqn (1). With a linearization technique (see Appendix) and under the assumption that P, S and C, are constant over the observed time interval, a simple solution at time t = t,, including the effect of photosynthesis, is found as: S=S,-k P . .(3a) where

k=[C,(O)-C&t,)].

C,(O) - C, C&,)-C,

1.

312

VENTILATION

IN GREENHOUSES -1

(C,(o)-C,).(C,(r,)-C,).~.~,ol.10-j

1

..

S is the true ventilation rate and S, the apparent ventilation rate. S, is calculated according to Eqn (lc), which does not account for the crop effect. To facilitate correction of the ventilation rate for the presence of the plants, the value of P has to be known. In the experiments described here, P was not measured but was calculated on the basis of related factors. These factors were either measured or estimated from data found in the literature. The measured factors are: radiation intensity (R), active leaf area (L), and CO, concentration in the greenhouse. For the latter the average CO, concentration (1500 v.p.m.) is taken. The estimated factors are: photosynthetic efficiency of cucumber leaf,” photosynthesis of a whole crop related to leaf photosynthesis, I3 transmission of radiation through the roof of the greenhouse used in these experimentslo and the fraction of radiation that is effective for photosynthesis.15 Estimates of the factors that influence the rate of the CO, uptake P by the crop lead to a very crude relation between P,the leaf area L and the global radiation R: P=0*0024R.L

[gC0,.m-2.h-‘]

. ..(4a)

It must be emphasized that Eqn (4a) is a simplified, linearized reproduction of the complex dependency of the photosynthesis process. The accuracy is, however, good enough for the purpose required. An estimate (based on Bierhuizenls and Challa’“) was also made of the amount of CO, that was released during the night from the soil, the roots and the leaves in the greenhouse used: P=-0.9gC0,.m-2.h-1 . ..(4b) The amount of CO, released from the soil and the roots during the daytime was small compared with CO, uptake by the leaves. Thus, the CO, release during the daytime was neglected. In greenhouses without a crop, CO, is produced only by organisms in the soil and this is neglected in these experiments. If necessary it is easy to correct for CO, from the soil by substituting for P in Eqn (3a) the amount of CO, with a negative sign, as in Eqn (4b). In order to check the validity of the linearized Eqn (3), a computer simulation has been carried out for a decay from C,(O) = 2000 v.p.m. to C,(tl) = 1000 v.p.m. Using Eqn (2) in small time steps (t), the time t, was calculated necessary for this CO, decay at various photosynthesis and respiration levels (P)and various true ventilation rates (S). Using Eqn (lc) and substituting t,, C,(tl), C,(O) and C,, an apparent ventilation rate S, was found. So S, is calculated from the decay rate of the CO, concentration, while the plant effect was not taken into consideration. The simulation results in Fig. 1 show at P = 0 the true ventilation rates S and at P # 0 the values of S,. By linear regression it follows that: s=s,-0.115

P

. ..(5)

This is in agreement with the factor k in Eqn (3b), where it is found that k =0.175 for C,(O)=2000v.p.m., C,(ti)= lOOOv.p.m., C,=330v.p.m. and h=2.9m. 4.

Experiments

Measurements were made in six small identical compartments of a Venlo type greenhouse at the Glasshouse Crops Research and Experiment Station at Naaldwijk. The dimensions of each house are 6 x 9.6m with a volume of 163 m 3. ‘O Ventilation was achieved with windows (1.45 x 0.78 m) situated in the roof, six on the east and six on the west side. Only windows on the leeside were opened. Aperture of the windows is expressed as % of maximum aperture, which is 0.6 m. When a window is opened 1% (= 0.006 m), the gaps at three sides of one window have a total surface of 0.0134 m2. The biggest aperture in the experiments was 15%. In four of the six compartments a cucumber crop (cultivar Corona) was present, planted on 10

E. M. NEDERHOFF

ET AL.

313

Fig. I. Calculated eflect of the CO, exchange of the plants P on the apparent ventilation rate S,. The value of S, was found according to Eqn. (1) for a decay of the CO, concentration from Zoo0 to 1000 vq.m. at various levels of P and real ventilation rates. P (+) is CO, uptake, P (-) is CO, release by theplants ing CO,. m-' hh’; Sands,, in h-t. ‘If P=O then S, = S.

2000-

0

0 0

E 2 .-5 ;L

0

0

%

1500-

z z5

0 0

0 0

N s

Y

O

IOOO::

a

% 0

%

0 ,

I 20.00

,

% 4,

8

I

0

, 4, 22.00

\ I

I

@@%x3 ,4 I-7-‘8 I

Time

,

I

i 2.00

I

%P I

(h)

Fig. 2. A typical example of the course of the CO2 concentration (v.p.m.) in a greenhouse compartment measurement. An arrow (t) indicates injection of CO,.

during a ventilation

December 1980, and this was fully grown at the time the ventilation measurements were performed (June-July 1981). The plant spacing was 0.5 m (within rows) and 1.6m (between the rows). Leaf areas were measured on 18 June 198 1 and were found to be 140,160,170 and 125 m2 for compartments 1,3,4 and 6 respectively. Compartments 2 and 5 were empty. The tracer gas CO, (99.85% pure) was supplied from a tank by five tubes per greenhouse compartment. When a level of 2000 v.p.m. was exceeded, the supply was stopped and the CO, concentration started to decrease. When the concentration had fallen to 1000 v.p.m., CO, was injected again up to 2000 v.p.m.

314

VENTILATION

IN GREENHOUSES

It may seem that this arrangement was not the best, since the value of P is not exactly known. It would seem better to let the CO, concentration decrease to an equilibrium and to calculate the value of P from this equilibrium level (after Beck and Arnold15). However, at lower CO, concentrations P depends strongly on the concentration itself, whereas between 1000 and 2000 v.p.m. P is little affected by the concentration. Another reason for measuring the CO, decay from 2000 to 1000 v.p.m., is that for accurate measurements of the air exchange the concentration of the tracer gas must be distinctly higher than the concentration of this gas in the outside environment () 330 v.p.m.). A typical example of the CO, concentration response is given in Fig. 2. The homogeneity of the CO, distribution has been checked by Bat* in the same greenhouse compartments with the same distribution method, and it corresponds with the homogeneity found by Van Berkel’” for CO, injection tubes situated on the ground. The CO, concentration was measured with an infra-red gas analyser (IRGA, Hartman and Braun). The IRGA was calibrated twice a week with mixtures of carbon dioxide and nitrogen, provided by three mixing pumps (manufactured by Wiisthoff, West Germany). The CO, concentrations in the six greenhouse compartments were measured sequentially using the same IRGA. With a measuring cycle of 8 min each compartment was connected to the IRGA for 60 sec. One minute was found to be long enough to rinse the measuring cell and to obtain a steady response. The CO, concentration that was measured at the end of a minute was transmitted to a computer (datalogger and controller). In addition to the experiments with the IRGA, a test was done with a simple type of CO, measuring device (manufactured by Siemens), that is in common use on commercial buildings. This meter also measures” the absorption of infra-red radiation of CO, gas. Simultaneous measurements with the two devices were compared. The temperature inside the greenhouse, the outside global radiation, wind velocity and wind direction were measured every 60 s. Observations with wind velocity lower than 2.5 m/s have been omitted because of inaccuracy of the sensor. 5.

Results and discussion

5.1. Measurements in greenhouses without a crop Fig. 3 shows the relation between the ventilation rate S, the hourly averaged wind velocity U, and the ventilator aperture r. These measurements were done in compartments without a crop (2 and 5). The ventilation rate was calculated using Eqn. (lc), from the decay of the CO, concentration in time intervals of 8 min. Hourly values of S were found by taking the medians. During the measurements the windows were either closed or opened to a certain, fixed aperture, not more than 15% of the maximum aperture. Linear regression fits have been calculated and are shown in Fig. 3. In Fig. 4 the ventilation rates S are divided by the hourly averaged windspeed U. All ratios Sb obtained at a given ventilator aperture are averaged and this value is plotted against the ventilator aperture. Each point in Fig. 4 represents from nine to 51 hourly values. By linear regression the following equation was found for the relation between ventilation, wind speed and window aperture in these greenhouse compartments: S=O.O73 u. r+0+097 u or

S=O.O73 u(r+l)

with I= 1.33

. ..(6a) . . .(6b)

The physical meaning of 1 is that it indicates the leakage of the greenhouse in units of ventilator aperture. Measurements in the greenhouses with closed ventilators gave values for 1 of 1.0 to 1.8, which indicates that the effect of leakage is equal to a window aperture of 1.0 to 1.8% in a completely airtight house. The fact that the data obtained from over 250 hourly observations show a very high correlation (Fig. 4), may be regarded as evidence of the validity of the method of measurement. For

E. M. NEDERHOFF

315

ET AL.

Fig. 3. Average ventilation rates S (h-l) as a function of wind velocity (hourly averages) ii (m/s) at various ventilator apertures r (% ) Measurements were performed between 17 June and 19 July 1981, in greenhouses without a crop. The wind velocity was r 2.5 m/s, except for 0% window aperture, where it was > 1.5 m/s.

I

0

I

I4

2.5

1

5.0

i

I

7.5

I

I

10.0

I

I

12.5

I

I

I

15.0

r,%

Fig. 4. Ratio of average ventilation rate S (h-l) and wind velocity (hourly average) ii (m/s) as a function of ventilator aperture r (O/o). Same measurements as in Fig. 3.

a check on the method, the results of two other studies in the same greenhouse complex are reviewed. Bot’ performed ventilation measurements with a static tracer gas method with CO, as tracer in the same compartments without a crop. A linear relation was found between ventilation and wind velocity, and an exponential relation between ventilation and window aperture. For apertures up to 20% a simplified linear form was found to be valid:

316

VENTILATION

Results of ventilation measurements

Dale

IN GREENHOUSES

TABLE 1 in compartments without and with a crop and the effect of accounting for the CO, exchange of the plants

Time ih)

T

Ventilation rate

F

Compartments without a crop

I

s 3

Column no.

4

Comparrments with a crop

1

Neglecting plants

Takingplants inlo consideration

-L-

n

3

(5

5

6

7

n

s

0

8

9

10

25 June 1981

15.OOG15.30 15.30-16.00 16.00-16.30 16.3C-17.00 17.00-17.30

0.87 0.68 0.76 0.69 0.70

0.09 0.09 0.10 0.07 0.14

4 6 6 4 8

1.34 1.02 1.18 0.96 0.78

0.19 0.21 0.26 0.20 0.20

7 5 6 8 9

0.84 0.58 0.70 0.57 0.47

0.10 0.13 0.14 0.17 0.20

9 July 1981

00.00~1 .oo 01.00-02.00 02.00-03.00 03.00-04.00 04.0&05.00

0.25* 0.24* 0.26* 0.22* 0.24*

0.06 0.05 0.05 0.05 0.05

5 7 8 7 7

0.11* 0.08 0.08 0.06 0.04

0.03 0.05 0.04 0.03 0.03

6 10 17 13 19

0.24* 0.25 0.25 0.25 0.24

0.03 0.04 0.05 0.04 0.04

fMeasurements performed in only one compartment. S= average ventilation rate h- ‘, o = standard deviatmn of average, n = number of observations.

s=o-071 24(r+ 1.0)

. ..(7)

which is very similar to Eqn. (6b). In experiments by Bakker and Van de Vooren3 the above mentioned exponential curve of Beta is implemented in a computer algorithm with which the water vapour transport may be controlled. The results of these experiments established the reliability of the ventilation formula. 5.2. Measurements in greenhouses with a crop Ventilation rates have been calculated according to Eqn (lc) for compartments without a crop (2 and 5) and with a crop (1, 3,4 and 6). This was done for two periods, one during the day and one at night, when the ventilators were closed in all compartments. Since plants take up the CO, in daytime and release CO, in the dark, the ventilation rates calculated neglecting the presence of plants are too high during daytime and too low at night. The results can be seen in Table 1 (columns 3 and 6). The values of the ventilation rate S are an average of 4-19 measurements which were taken every 8 min and were performed simultaneously in 24 compartments (except the measurements marked with a “*” in Table 1). The ventilation rates of the compartments with a crop are recalculated with Eqns (3) and (4), to take account of CO, exchange of the plants. The results of these calculations are given in column 9 of Table 1, It is obvious that the recalculated ventilation rates in the compartments with a crop approach the rates measured at the same time in the compartments without a crop (column 3). The standard deviations g are decreased (column 10 compared to 7), which means that the ventilation rates calculated for different compartments, have come closer to each other. This can be explained by the fact that the crops in the four compartments had a different active leaf area. Therefore the disturbance of the calculated ventilation rate was different and CTwas larger when no correction for the presence of plants was made. As can be Seen from observations made on 25

E. M. NEDERHOFF

317

ET AL.

June 198 1, the difference in ventilation rates between compartments with and without a crop decreased later in the afternoon. This is probably explained by a decreasing CO, uptake by the plants at the end of the day.‘* Since the correction for the presence of plants is based on a fully active crop the correction is too large at the end of the afternoon (columns 3, 6 and 9 at 17.00-17.30 h). When the ventilation windows were opened, the difference in the ventilation rates between greenhouses with and without a crop was found to be insignificant. The explanation is that when the ventilation windows are opened, the amount of CO, lost by air exchange is much bigger than the amount taken up by the plants. This can be seen from Eqn. (3a) and from Fig. I. It is concluded that correcting for the presence of plants in ventilation measurements is possible; however, it is not recommended for practical use. The observations and calculations that are necessary for estimating the plant effect, make the method less accurate and less attractive than determining the ventilation characteristics of the greenhouse when it is free of plants. 5.3. Measurements with a simple CO2 analyser A test has been performed in order to compare the results obtained using a simple commercial device” with those using the IRGA from the experiments. When the analogue display of the simple device is read, the resolution is small. Nevertheless, ventilation rates calculated as the measured CO, concentration declined from 2000 to 1000 v.p.m., were almost equal (+ 1%) using both devices. Hence, the simple CO, analyser can be used for ventilation measurements with the proposed method.18 6.

Conclusions

The dynamic tracer gas method, also known as the decay rate method, can be employed to determine the leakage and low level ventilation in greenhouses. This method is based on measuring the rate of decay of a tracer gas concentration from an initial concentration. For practical use carbon dioxide (CO,) is a suitable tracer gas, since equipment for supplying and measuring CO, is available in an increasing number of greenhouses in the Netherlands. A disadvantage is that the CO, concentration is influenced by the CO, exchange of the plants. The results of the proposed dynamic tracer gas method were similar to those from earlier measurements made by means of a static tracer gas method. Both measurements were made in the same greenhouse compartments without a crop. For measuring ventilation in greenhouses with a crop, the same method can be used but it is necessary to correct for the effect of the crop on the CO, concentration. The correction has been validated in experiments in identical greenhouse compartments, some of which contained crops. The equipment commonly used by growers for measuring CO, in greenhouses gave satisfactory results in a small test on ventilation measurement. Thus the proposed method may be used by growers to determine the leakage and the ventilation rate at small ventilator apertures. It is anticipated that these ventilation measurements could be useful when energy saving measures in greenhouses are considered, and that in the future an established relation between ventilation, wind speed and ventilator aperture might be used in a computerized climate control system.

REFERENCES

Berkel, N. van CO, enrichment in the Netherlands. In Carbon Dioxide Enrichment of Greenhouse Crops. (Eds, H. Z. Enoch; B. A. Kimball). Boca Raton, U.S.A.: C.R.C. Press Inc. (in press) p Bot, G. P. A. Greenhouse climate: from physical processes to a dynamic model. Thesis, Department of Physics and Meteorology, Agricultural University of Wageningen, 1983 3 Bakker, J. C.; Vooren, J. van de Water vapour transport by means ofventilation. I. Efsects on climate. Acta Hort., 1984 148 535-542

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VENTILATION

IN GREENHOUSES

Morris, L. G.; Postlethwaite, J. D.; Edwards, R. I. Ventilation and the supply of carbon dioxide to a glasshouse tomato crop. Tech. memo 87, NIAE, Silsoe, 1954 ’ Whittle, R. M.; Lawrence, W. J. C. Climatology ofgreenhouses II. Ventilation. J. agric. Engng Res., 1950 5 3-1 ’ Lake, J. V. Measurement and control of the rate of carbon dioxide assimilation of glasshouse crops. Nature, 1966 209 97-98 ’ Hand, D. W.; Bowman, G. E. Carbon dioxide assimilation measurement in a controlled environmental glasshouse. J. agric. Engng Res., 1969 14(l) 92-99 a Okada, M.; Takakura, T. Guide and data for greenhouse air conditioning. 3. Heat loss due to air injiltraJ. agric. Met., 1973 28(4) 223-230 tion of heatedgreenhouses. s Gormley, T.R.; Waishe, P. E. Carbon dioxide distribution in glasshouses. Irish J. agric. Res., 1979 18 45-53 ” Berkel, N. van CO, from gas-jred heating boilers-its distribution and exchange rate. Neth. J. agric. Sci., 197523202-210 ‘I’ Businger, J. A. The glasshouse (greenhouse) climate. In Physics in the Plant Environment (Ed. W. R. van ’

Wijk). Amsterdam: North-Holland Publishing Company, 1963 ” Challa, H. An Analysis of the Diurnal Course of Growth, Carbon Dioxide Exchange and Carbon Hydrate Reserve Content of Cucumber. Wageningen: Pudoc, 1976

‘3 Acock, B.; Charles-Edwards, D. A.; Fitter, D. J.; Hand, D. W.; Ludwig, L. J.; Warren Wilson, J.; Withers, A. C. The contribution of leaves from difSerent levels within a tomato crop to canopy net photosynthesis: an experimental examination of two canopy models. J. exp. Bat., 1978 29( 111) 8 15-827 I4 Vooren, J. van de; Koppe, R. The climate glasshouse at Naaldwijk. Neth. J. agric. Sci., 1975 23 238-247 ” Monteith, J. L. Principles of Environmental Physics. London: Edward Arnold, 1973 ” Beck, J. V.; Arnold, K. J. Parameter Estimation in Engineering and Science. New York: John Wiley, 1977 ” Siemens CO, Controller Instruction Manual, No. TN59129. Den Haag: Siemens De Tuinderij, ‘I* Nederhoff, E. M. Meten van ventilatie in kassen [Measuring ventilation in greenhouses]. 1983103&31and1983113637 ” Bierhuizen, J. F. Carbon dioxide supply and netphotosynthesis. Acta Hort., 1971 32 119-126

Appendix Eqn. (2a) will be linearized,

which then results in Eqn. (3). Rewrite

dCg(O= -s. dt

Eqn. (2a) as:

. ..(Al)

(C,(t)-CT,)-U’P

where S’= S/3600 and a’= a/3600. Assuming that s’, P and C, are constant time interval and, for convenience, C(t) = C,(t) - C,, then:

(

C(t)= -q-+

C(O)+

over the observed

a’P 7

.

. . .(A2)

eBS”

>

An explicit solution for s’ cannot be obtained analytically, given the value C(tI) at t= t,. Therefore, a linearization is carried out. To do so, P is written explicitly:

p=--.

S

[C(tI)-

a’

A working

C(O)] . epS”l

point is defined for P= 0, and as a result ?? is found according C(t,)-

. ..(A3)

1 -e-S”1

C(0). e3”1 =o

to Eqn. (1 b): . . .(A4)

E. M. NEDERHOFF

319

ET AL.

For small variations p around the working point using the Taylor series approximation:

p=

s=s.

i:

a=-

L?‘=(usingEqn. (A4))= -a$

1

a'

Since

With a =

e-S’tl=

or

C(0) C(t,) . C(O)-C(t,)

from P”= - aS, it follows that

.(A5b)

1 -@l

.t,=ln[ *] 1 (I=- a’

e-S’tl.gl

St

C(O)

Jn[

*,

*]

. ..(A6)

~=-i~, a

(h . pco 2 . lO-3)-’ this results in: g=-k.ij

and

. .(A5a)

. .(A7a)

10e3. C(0). C(tr) . In

. .(A7b)

The relation_holds that x=x+ 3. ?? follows from Eqn. (A4). S- S, and Pr P. Since P= 0, it follows that in Eqn. (3a). This leads to S=S,-k.

which was to be demonstrated.

P

. ..(AS)