A Mathematical Model of Temperature Regimes in Mulched Greenhouse Soil

Copyright © IFAC Mathematical and Control Applications in Agriculture and Horticulture, Hannover, Gennany, 1997

A MATHEMATICAL MODEL OF TEMPERATURE REGIMES IN MULCHED GREENHOUSE SOIL C. Arcidiacono1, G. Cascone1 and D. Gutkowski2

(l) Istituto di Costruzioni Rurali, Universitiz di Catania, Via Valdisavoia 5, 95123 Catania, Italy; (2) Dipartimento di Fisica, Universitiz di Catania, Corso Italia 57, 95129 Catania, Italy.

Abstract: A model for the analysis of the thennal regimes in mulched soils under greenhouse has been developed on the basis of the measured temperature at two depths. The model is based on the expression of a measured soil temperature function as a Fourier series of terms representing damped monochromatic plane waves in the soil. For such waves the phase shift and the damping factor can be determined from experimental data taken at the second depth. The self
measured quantities, since the theory of the basic equations is not well known and consequently it is difficult to obtain results of general validity. A different approach has been used since the twenties of last century (Fourier, 1822) till recent years (Cenis, 1989; HOTton, et al., 1983; HoTton and Wierenga, 1983; Van Wijk, et aI., 1963) in order to get an approximate description of soil thennal regimes or to deal with related problems. This model is essentially based on the assumptions that, at any depth, temperature is a periodic function of time and that heat diffuses by conduction in a homogeneous medium. Then it can be shown by the Fourier theorem (Titchmarsh, 1960) that the knowledge of temperature at two arbitrary depths during a time period allows one to determine the temperature at any depth for any time. The model is quite simple, very well founded from a mathematical point of view due to the work of eminent mathematicians for more than a century (Titchmarsh, 1960). Important general results are known and a useful picture of the process in term of propagating thermal waves is

I. INTRODUCTION The knowledge of soil thennal regimes for a specified set of relevant conditions is important to many purposes: soil temperature influences plant growth rates, soil physical properties and biochemical reaction rates. Furthermore, when soil solarization is applied for partial soil disinfestation, the control of soilborne plant pathogens depends on soil thermal regimes (Pullman, et al., 1981). Several models provide a theoretical description of soil temperature regimes. Some of them involve the soil energy balance and the solution of coupled partial non-linear differential equation for heat and moisture diffusion (Cascone and Arcidiacono, 1994; Gutkowski and Terranova, 1991; Mahrer, et al., 1984). The main results of these models are numerical solutions to be compared with the The authors' contributions towards the present work are to

be considered equal in every way.

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obtained. A version of this model, which is an essential part of the present paper, is described in the next Section. The proposed model, on the basis of measured soil temperatures at two different depths however fixed., allows to estimate soil temperatures at any depth including soil surface.

P

r ll (t) = T(zll,t)-! pJor T(zlI,t) dt, (GII ,GIt )= 211: p

P

r r;(t) dt . Jo

Hypothesis d) and Fourier theorem imply:

linlN....... TN(Zh,t) = T(z",t).

Though there are some reasons, which will be discussed below, for which such a model is not expected to be suitable for the description of soil temperature regimes, there are cases in which the

The Fourier series in the left hand side of Equation (4) converges in L2(O,p) (fitchmarsh, 1960). It can be shown that the Fourier coefficients defined by Equations (3a) and (3b), but not in general the Fourier coefficients of every function f E L 2(0,p), satisfy the equation:

calculated quantities satisfactorily approximate the observed ones. Therefore, it seems worthwhile to find the limits and to explain the reasons both of the successes and of the failures of the method and not only to notice them on the basis of the experimental evidence. Some of these topics are discussed below, with reference to examples drawn from experimental observations and partial conclusions are reported.

ao

1I:L(a;" +b!,) = 1, 11=1

therefore '1& that

E

]O,I[ a natural number M exists such M

1-& ::;7!L(a~+b~)::;l.

By choosing any fixed frequency v>O, and defining the phase

The model is based on the following assumptions: a) the soil temperature T(z,t) depends on one space coordinate z (depth) with z ~ 0 and on the time E }- 00,

CP(Z, t) = 2'KVt -

J(z,t) =exp (-)1CV1C.z) sincp(z,t)

T(z,t)

o2T(z,t) &2

satisfies equation (1). It can also be seen that j{z,t) propagates like a damped monochromatic plane wave with phase velocity

v = 2~7tV / 1C. and damping factor exp

oT(z,t) 1C.

Of;

(1)

(6) n=1

where

c. =exp [- ~: (2-2,)].

11=1

T*(Zh) =

!r

(-~ 'KV1C. Z) .

Let

d) '1z ~ 0, T(z,t) is periodic with respect to t with period P and continuous with respect to t in [0, P[. Let Z/r ~ 0 be any fixed value of z, for any natural numberN let

where

~1CV1C. z+ 'I' ,

where 'I' is an arbitrary constant, the function

+ 00[;

is twice continuously differentiable with respect to z and almost everywhere differentiable with respect to t ; c) T(z,t) satisfies the following heat conduction equation for a homogeneous medium with constant thermal diffusivity 1/" :

b)

(5)

11=1

2. MODEL DESCRIPTION

t

(4)

D"

=(a nh cos SIt +b nh sin sn)

with

T(Zh,t) dt ,

SII

Anh(t) = anh COSwn(t) + bnh sinw,,(t) with

= 2mt(t_ z-z,,) p VII

By Equation (6) TMC.z,t) is a linear combination of solutions of Equation (1) which are damped monochromatic waves propagating with phase velocity

(3a) (3b)

VII

and

= 2~1U'l / KP .

(7)

TJ.,z,t) for Z = Z/r becomes the function TJ.,z/t,t), approximating T(z/t,t), which is supposed to be known. Since Equation (1) is linear, TMC.z,t) is a solution of the same equation. It can be shown that if T(z/t,t) is known for two values of h, say h = 1, 2 and 2 (0 1 1

with

92

+ b121 ) (0~2 + b 12 ) 2

::#

0,

beginning of summer 1995 until the end of summer 1996 inside a greenhouse with plastic cover located in South-East of Sicily (360 48' N, 14 0 29' E). The soil was sand, composed by 96.0% sand, 2.7% silt and 1.3% clay. The soil surface was covered with a black polyethylene mulch 60 .,un thick while during summer soil was solarized. During the growing season an egg-plant crop has been cultivated in rows ofl m width.

=Jiiv

then VI can be detennined and v" l • In such a way Equation (6) provides an approximate solution of the problem of finding T(z,I). In practice soil temperatw'es T(z",I) for h = 1, 2 can be known not for any le [O,p] but only for a finite number of values of I. Consequently the integrals in Equations (3a) and (3b) cannot be exactly calculated but they can be calculated only numerically with a certain precision.

Soil temperatw'es and moisture contents of the mulched soil were measured every half-hour with thermal resistive sensors and with a time-domain retlectometry (TOR) system at 1 cm, 5 cm, 15 cm and 30 cm depth.

Two main reasons for which the above model seems DOt to be suitable for the description of soil temperature are to be discussed. The first reason concerns the periodicity condition for the soil temperature. Obviously even in principle, apart from factual considerations, a test of the periodicity for any time I E ]- 00 , + oo[ would be beyond the human p?ssibilities; furthermore this assumption is in disagreement both with accredited cosmological theories (peebles, 1971) and with faith believes. The second reason concerns the laws of heat transfer. There is theoretical and experimental evidence of effects which cannot be described within a theory of heat conduction (Philip and de Vries, 1957) and that the basic equations are non-linear. Due to both the above reasons, the Fourier theorem, which is in any case a well established mathematical result. would seem not to be useful to solve the problem, for even if Equations (2), (3a) and (3b) hold true and the monochromatic waves would be approximate ~lutions of the non-linear temperature equations, a linear combination of plane waves could not be a solution of this equation. Furthermore there is the problem of the choice of the v~ue of the ~riod P, since one would expect that different chOIces will give different results. Intuitively, apart from practical difficulties in ~onitoring data for too long periods, a good choice IS expected to be related to periodic processes affecting the soil energy balance such as the solar day, the tropic year, periods of variation of solar activity and so on. Yet. if the self consistence of the ~odel is required then the effects of periods of different order of magnitude have to be, in some sense, separated. Indeed nobody knows whether there are relevant periods of the order of 109 years. If so, and if the effects of different periods are not ~parat~ it wo~d be hopeless to get approximate mformatlon on sod temperature applying the method for P = 1 day or P = 1 year. In Section 4 calculations based on the choice P = 1 day and P = 1 year are reported and the separation of the effects pertaining to the two periods is shown. Moreover, examples are given both of good and of bad approximations and explanations are proposed for the obtained results.

The experimental results have been processed by the model both for the daily and the annual cycle with the following choices concerning the period P, the value of & appearing in Equation (5), the two reference depths z/ and Z2, the finite set of time values tlf and corresponding temperature T" at the reference depths necessary to provide the model with the known functions T(z,.,t), for h = 1, 2. For the daily cycle: P = 1 day, & = 0.001, either Zl = 5 cm and Z2 = 15 cm or ZI = 15 cm and Z2 = 30 cm, I,. ="..1t for" = 0, 1, 2, ... , 48 and ..1t = 0.5 h and T,. measured temperatures at the time I,. . For the annual cycle: P = 1 year, & = 0.003, Z/ = 15 cm and Z2 = 30 cm; the mean daily values of measured temperatw'es taken every seven days were interpolated to obtain a three days step.

4. RESULTS AND DISCUSSION Concerning the daily cycle, results are presented for two days selected among those for which the analysis have been performed: August 3, 1995 and December 17, 1995. August 3 represents the most common regular cases for which temperature periodicity conditions in near days are almost satisfied and then the difference between the last and the first temperatw'e value recorded in the day is small. December 17 represents exceptional irregular

cases. Figures 1, 2, 3 and 4 show the results obtained for August 3 with the choice ZI = 15 cm, Z2 = 30 cm. Two harmonics were sufficient to obtain the solution. According to previous results (Van Wijk, el al., 1963) usually two harmonics are sufficient to describe the temperature variation in homogeneous soils. The analysis of data showed that for regular days two to four harmonics provided the results within the chosen approximation. For regular days the second term in the right hand side of Equation (6), representing the oscillating part, was at all the depths in good agreement with the experimental data, while the constant term T" in Equation (6), representing the daily mean temperatw'e at the reference depth Z /0 may not agree with experimental

3. MATERIALS AND METHODS The experimental data were collected from the

93

..-. u 0

--

i8. e

58

58

56

56

54

54

52

52

50

50

48

48

G

46

--~

0

44

e 8. e

42

~

46 44

42

~

40

40

38

38

36

36

34

34

0

3

6

9

12

15

18

21

24

0

3

6

time (hours)

9

12

15

18

21

24

time (hours)

Fig. 1. Soil temperature of August 3, 1995 at 1 cm depth: -measured, -.- model, ... correction 1, - - - correction 2.

Fig. 2. Soil temperature of August 3, 1995 at 5 cm depth: -measured, _.- model, ... correction 1, - - - correction 2.

58

58

56

56

54

54

52

52

50

50

48

48

..-.

u e.....

e8.~ e

46

G

--~

46

0

44

e 8. e

42

~

40

~

44

----

40

38

38

36

36

34

-.-.-.- /

42

/'

./

./

,.-

,.-

34

0

3

6

9

12

15

18

21

0

24

3

6

9

12

15

18

21

24

time (hours)

time (hours)

Fig. 3. Soil temperature of August 3, 1995 at 15 cm depth: - measured, •. - model, ... correction 1, - - - correction 2.

Fig. 4. Soil temperature of August 3, 1995 at 30 cm depth: - measured, -.- model, .. . correction 1, - - - correction 2 .

94

n ,------------------------,

45 ,---------------------------,

21

40

Scm

20

,..., ....,

u

0

19

e

.B

,..., ....,

u 0

.~

i

8-

35

'~\...

30

i

.,...

8-

18

e

25

.B

17

16

20

c.................~....................~.........~.................J.~.....................;

o

3

6

9

12

15

18

21

15 ~--~~--~--~--~~~ 197 260 323 n 86 147 197

24

time (hours)

Giulian day of the year

Fig. S. Soil temperature at 5 cm and 30 cm depth of December 17, 1995: - measured., -- model, ... correction 1.

Fig. 6. Soil temperature at 15 cm and 30 cm depth from July 16, 1995 until July 15, 1996: - measured at 15 cm, -.- model at IS cm, ... measured at 30 cm, - - - model at 30 cm.

values at depths different from z1 • This is because 1" is constant in the model with respect to the depth z, while in reality, as the experimental data showed, it may be not constant The calculated curves correctly describe the pattern of temperature variations in soil both in summer and in winter conditions. The temperature variations versus depth consist in reduction of amplitudes of the oscillations and displacement of peaks as depth increases. The model simulates quite well the measured temperatures at the reference depth Zt. At shallower depths than Zt the model over-estimates the temperatures in winter while under-estimates them in summer. At deeper depths than Zt the model under-estimates the temperatures in winter while over-estimates them in summer. The temperature differences are higher if the reference depths Zt and Z2 are chosen deeper in the soil. In order to correct the tenn 1" two methods have been introduced. A first correction (dotted line in fig.s 1, 2, 3 and 4) has been applied to the constant tenn by substituting it with the measured mean temperature at the depth considered. Another correction has been obtained (dashed line in fig.s 1, 2, 3 and 4) substituting the constant tenn with the result obtained by the construction of a parabolic function which gives the daily mean temperature for each depth on the basis of the measured data available for the other three depths.

In the 5 cm curve it is possible to see a Gibbs' phenomenon (Mathews, 1. and R.L. Walker, 1965) due to the difference between the first and the last value of the observed temperature, consisting in a short range pronounced error of the approximate solution.

'The picture of thennal waves provides in this case the following explanation for the failure of the model. Since the phase velocity of the first harmonic is found to be about 4 cmh- t , what is observed in the first four hours of a day, for example at 30 cm depth, is the effect of thermal waves flowed through the soil at a depth of about IS cm in the last four hours of the preceding day. If the periodicity condition is approximately satisfied the temperature graph of the last four hours of a day is almost equal to the temperature graph of the last four hours of the preceding day. If the periodicity condition is far from being fulfilled the two graphs are very different and consequently calculated and observed values do not agree. Analogous considerations can be made for the other depths.

The model results with the previous choices only depend on the temperature of the considered day but the soil temperature is also affected by any periodical cycle with period greater than one day, as discussed in Section 2. Therefore the effects of the annual cycle on the day temperature have been calculated according to the choices given in the preceding Section 3 for the annual cycle. The results are shown

When the daily periodicity is far from being fulfilled, as happened on December 17, then the calculated results do not agree with the measured ones (fig. 5).

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another improvement of the model could be achieved by considering a piecewise constant thennal diffusivity, instead of a constant one. At last, further investigations on the anomaly of the dispersion of phase velocity, pointed out at the end of Section 4, seem to be interesting.

in fig. 6. Thirtyfive harmoBics were needed to provide the results within the chosen approximation. For this reason the precision of the calculation is less than in the case of the daily cycle. The phase velocity of the harmonic of frequency of lIday when I P = 1 year has been found to be equal to 1.08 cmh- . As the phase velocity of the harmonic of the same frequency with P = 1 day is about 4 cmh-I it should be remarked that the propagation of the thermal waves is slower in the annual cycle than in the diurnal cycle. Therefore by comparing the calculated results obtained for P = 1 day with those obtained for P = 1 year it is possible to observe that the condition given by Equation (7) is not satisfied. The differences of the phase velocity for thermal waves having the same frequencies, but pertaining to different cycles, can be probably explained by considering the numerous physical and biochemical processes influencing the thermal regimes of the considered mulched soil under greenhouse. It is known that when chemical and biological processes involving heat exchanges occur, Equation (7) is not necessarily satisfied (Freire, 1995).

REFERENCES Cascone, G. and C. Arcidiacono (1994). Simulation of greenhouse soil temperature with different moisture contents. AgEng'94, Report n.94-C-059. Cenis, I.L. (1989). Temperature evaluation in solarized soils by Fourier analysis. Phytopathology, 79, 506-510. Fourier, I. (1822). Theorie Analytique de la Chaleur. Firmin Didot Pere & Fils, Paris. Freire, E. (1995). Thermodynamics of partly folded intermediates in proteins. Annual Review in Biophysics and Biomolecular Structure, 24, 141165. Horton, R and P.1. Wierenga (1983). Estimating the soil heat flux from observations of soil temperature near the surface. Soil Sci.Soc.Am.J., 47, 14-20. Horton, R, PJ. Wierenga and D.R Nielsen (1983). Evaluation of methods for determining the apparent thermal diffusivity of soil near the surface. Soil Sci.Soc.Am.J., 47,25-32. Gutkowski, D. and S. Terranova (1991). Physical aspect in soil solarization. FAO Plant Production and Protection, 109,48-68. Mahrer, Y., O. Naot, E. Rawitz and J. Katan (1984). Temperature and moisture regimes in soils mulched with transparent polyethylene. Soil Sci. Am. J., 48,362-367. Mathews, J. and R.L.Walker (1965). Mathematical Methods of Physics. Chapter 4., W.P. Benjamin Inc., New York. Peebles, P.J.E. (1971). Physical Cosmology. Princeton University Press, Priceton, NJ. Philip, J.R, and D.A. de Vries (1957). Moisture movemennt in porous materials under temperature gradients. Am. Geophys. Union, Trans., 38, 222-232. Pullman, G.S., J.E. de Vay and R.H. Garber (1981). Soil solarization an thennal death: a logarithmic relationship between time and temperature for four soilborne plant pathogens. Phitopathology, 71, 959-964. Titchmarsh, E.C. (1960). The Theory of Functions. Chapter XIII, Clarendon Press, Oxford. Van Wijk, W.R. and D.A. de Vries (1963). Periodic temperature variations in a homogeneous soil. In: Physics of Plant Environment. (W.R. Van Wijk, Ed.) pp. 102-140. North Holland, Amsterdam.

5. CONCLUSIONS When the periodicity assumption for the soil temperature is fulfilled, the proposed model quite correctly describes the soil temperature regimes at all the examined depths. The average half-hourly difference between measured and calculated temperatures does not exceed 0.5°C and it diminishes when the proposed correction methods are applied to the model. In particular, the absolute maximum difference between measured and calculated temperatures diminishes from about 2°C to about 1°C when the corrections are applied. The differences between measured and calculated temperatures can be explained by the error associated with the evaluation of the mean daily soil temperature at the different considered depths. In general, if the assumption of periodicity for the soil temperature is valid, both the methods proposed for the correction of the mean daily temperature lead to similar results. The model leads to errors not tolerable in the estimation of soil temperatures when the assumption of periodicity is not valid. This frequently happens in the open field where abrupt climatic changes may occur in short periods, while it is less probable in the mulched soil covered by a greenhouse where the soil temperature is less influenced by the climate conditions. Therefore the proposed model is particularly suitable for predicting temperatures of a mulched greenhouse soil with uniform properties. Further refinements of the model could be obtained by improving the estimate of the effects of the annual cycle in relation to soil properties, climate conditions and agricultural practices. Since the results obtained showed that the phase velocity is weakly dependent on the soil depth,

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