sw—soil and water: a transient-spectral thermal model of soil under radiative-interfering cover

J. agric. Engng Res., 2000, 77 (1), 93}102

SW*Soil and Waterdoi:10.1006/jaer.2000.0565, available online at http://www.idealibrary.com on

A Transient-spectral Thermal Model of Soil under Radiative-interfering Cover

V. De Luca; G. Ruocco

DITEC, Universita della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy; e-mail of corresponding author:[email protected]

(Received 8 May 1999; accepted in revised form 31 March 2000; published online 25 May 2000)

In this paper, the soil temperature distribution under a plastic cover is determined, by devising a transient,one-dimensional model to describe the di!erent heat transfer interaction between soil and environment. Theinherent radiative heat exchange along a spectral range has been considered, including the variability ofincidence angle of solar radiation. Conduction through the soil layers, their volumetric heat capacity, convec-tion within the air gap and radiation through the cover has been taken into account, considering weather data,at the considered site, as boundary conditions. The simulation is carried out by using a numerical method, andresults have been favourably validated by experimental observation from "eld tests. The reported studycon"rms that, for the employed materials, the thermal regime within the soil depends on their long-waveradiative behaviour.

( 2000 Silsoe Research Institute

1. Introduction

Understanding soil temperature distribution is impor-tant when the ground is covered with a mulch, or witha small unheated greenhouse (sometimes called a tunnel),in order to suppress the growth of unwanted seedlings, orto promote the ripening of certain vegetables. Heat trans-fer control is also needed during drying or sterilization bysolar energy. The temperature under a mulch or in anunheated tunnel can be modelled in a similar fashion,although the geometries and the mass transport pro-cesses are di!erent.

The thermodynamics involved in such agricultural sys-tems has received considerable attention, particularly theassessment of the heat in#ux to the soil, and the relatedthermal regime. The study of the heat transfer, due to thesolar gain through a partially transparent cover (typi-cally made of plastic material), can generally be per-formed by introducing an appropriate simulation, suchthat the transient characterization of the heat #ux accu-rately predicts the time progress of the thermal regimeand ultimately the complete heat transfer of the system.

Many numerical models for simulating temperatureregimes of both the bare soil surface and the soil surfaceunder a plastic cover have been developed, but the in#u-ence of cover optical properties over their entire spectra

0021-8634/00/090093#10 $35.00/0 93

has not been speci"cally addressed. The "rst approacheswere performed by Mahrer (1979, 1980) and Mahrer andKatan (1981). Mahrer (1979, 1980) and Mahrer et al.(1984) introduced a numerical one-dimensional model tostudy soil temperature and moisture regimes of soil un-der a mulch of transparent polyethylene "lm. The model,which employed environmental conditions, soil physicalcharacteristics and cover transmittance to short- andlong-wave (SW and LW) optical properties of the cover,demonstrated that the soil temperature maxima increaseswith the soil moisture content. A two-dimensional exten-sion of that model in order to infer the lateral e$ciency ofa heating mulch was developed by Mahrer and Katan(1981). Avissar et al. (1986) also focused on the e!ect ofaging on the photometric properties of the cover, buttheir analysis was limited to SW and LW parameters.More recently, Chung and Horton (1987) focused on thecoupling of heat and water #ow in soil under the partialcover of a mulch by means of a "nite-di!erence two-dimensional model. They con"rmed that the mulchgreatly reduces evaporation and daily soil temperatureand moisture content variations. Sui et al. (1992) studiedthe e!ects of mulch and soil transmittance, in both theSW and LW bands, as inputs to their "nite-elementtwo-dimensional model, in order to determine the heatand moisture regimes. A di!erent approach by Ham and

( 2000 Silsoe Research Institute

V. DE LUCA; G. RUOCCO94

Notation

aj, b

jcoe$cients

cp

speci"c heat of air at constant pressure,J/kgK

C heat capacity, J/m3 KE emissive power, W/m2lmf shape factor, dimensionless

Fu

humidity factor of soil, dimensionlessG

dsolar di!use irradiance, W/m2

Gr

solar beam irradiance, W/m2

k thermal conductivity, W/mKK mass transfer coe$cient of soil, kg/m sh heat transfer coe$cient, W/m2K

hL

latent heat of vaporization of water, J/kg¸ latent heat #ux of water in air gap, W/m2

¸e Lewis number, dimensionlessn soil porosity, dimensionless

Nu Nusselt number, dimensionlessP parameter

Pr Prandtl number, dimensionlessr thermal contact resistance at the cover}

soil interface, m2K/WRa Rayleigh number, dimensionless

t time, s¹ temperature, K

;MF

uncertainty magni"cation factor, dimen-sionless

v wind speed, m/sw width of cover, mx soil fraction, m3/m3

xwr

residual soil moisture, m3/m3

X air humidity ratio, dimensionlessz spatial coordinate, m

a absorptance, dimensionless*t time step of discretization, s*z space width of soil discretized layer, m

e emittance, dimensionless/ net radiant #ux density, W/m2

j wavelength, lmh incidence, dego re#ectance, dimensionlessos

soil bulk density, kg/m3

q transmittance, dimensionless

Subscriptsa airb black bodyc clay fractiono outside environmentf soil surfacei air gapj discrete soil element

m coverp organic fractionq quartz fractions soil

isat air gap at saturationssat soil at saturation

v skyw soil moisturej monochromatic

Superscriptst time step0 initial

Kluitenberg (1994) started from optical measures in theSW and LW bands using a spectroradiometer anda Fourier-transform infrared spectrophotometer andcomputed transmittance and re#ectance as weighted in-tegrals over the wavebands. The results relative toa "nite-di!erence model showed that soil heat was sensi-tive to both SW and LW optical properties of the mulch.A di!erent representation of optical properties was intro-duced by Wu et al. (1996), who obtained the transmit-tance, re#ectance and emittance from published values.By studying model sensitivity to parameters, such assurface roughness length, soil bulk density, clay andquartz fractions, quartz shape factor, and mulch trans-mittance of long-wave radiation and solar radiation, theyshowed that their model was greatly a!ected by soil bulk

density, quartz fraction of soil and mulch transmittanceto long-wave radiation.

All these studies convey the importance of spectral en-ergy interaction, between sun and soil through the plasticcover: water droplets are formed on the inside cover surface,thus reducing cover transmittance to LW radiation, but notto SW radiation, and therefore increasing the greenhousee!ect. Yet, the band-averaged simpli"ed assumption hasnever been fully supported. The approach adopted insteadin the present work takes into account the spectral curves ofcover transmittance along a wide wavelength range(0)175}25)000 lm), in order to improve the modelling of thespectral interaction among all participating elements.

The model, consisting of governing energy balanceequations approximated by a "nite-di!erence scheme

Fig. 1. Model of soil cover: G, solar irradiance; Ev , emissivepower from sky; Em, emissive power of cover; Ef, emissive powerfrom soil surface; ho(Tm!To), convective heat yux between coverand outside environment; hi(Tm!Ti ), convective heat yux be-tween cover and air gap; L, latent heat yux of water in air gap;hf (Ts!Ti )#hLK(Xssat!Xisat ) , convective heat yux from soil sur-

face; !ksLTs/Lz, conductive heat yux from soil

95THERMAL MODEL OF SOIL

and a number of well-established empirical relations,has been validated against measured temperatures bya related experimental arrangement.

2. Analysis

2.1. Assumptions of the process

In the analytical procedure, it was assumed that:

(a) the considered mulch consists of cover, air enclosureand soil;

(b) the direct solar radiation #ux varies with the angle ofincidence;

(c) the cover is isothermal and airtight, the air within theenclosure being radiatively non-participating;

(d) all driving parameters are time-dependent, except theoptical properties of the cover; and

(e) the geometry is one-dimensional by neglecting theheat #uxes along the horizontal plane.

2.2. Heat -ow in the soil

The governing di!erential equations for transient heat#ow in the soil, and related boundary and initial condi-tions may be derived as follows:

LLz Aks

L¹s

Lz B"Cs

L¹s

Lt, z*0, t'0 (1a)

¹s(z, 0)"¹0

s, z*0 (1b)

L¹s(zPR, t)

Lz"0, t'0 (1c)

!ks

L¹s(0, t)

Lz"h

f[¹

s(0, t)!¹

i]

#hLK (X

ssat!X

isat)#/

f, t'0 (1d)

where z is the spatial coordinate, ks

the soil thermalconductivity, ¹

sthe soil temperature, C

sthe soil heat

capacity, t the time, ¹0s

the soil temperature at the initialtime, h

fthe heat transfer coe$cient at the soil surface,

¹ithe air gap temperature, h

Lthe latent heat vaporiza-

tion of water, K the mass transfer coe$cient of soil,X

ssatand X

isatare the air humidity ratio at saturation

near the soil surface and in the air gap, respectively, and/f

is the net radiant #ux density at the soil surface.Heat transfer between the cover and the enclosed soil

surface is estimated with reference to the rectangularcavities notation by Incropera and De Witt (1990), using

a thermal contact resistance de"ned as

r"w

mkaNu

(2)

where wm

is the width of cover, ka

the air thermal con-ductivity, and Nu the Nusselt number. In the following,the heat transfer coe$cients at the soil surface h

fand in

the air gap hiare assumed to be equal to the reciprocal of

the thermal contact resistance r.If the cover is colder than the soil, and the Rayleigh

number Ra is greater than 1708, then free convectiontakes place with Nu+1, otherwise, when cover iswarmer than soil or Ra(1708, conduction prevails andNu"(1/14)5)Ra1@3Pr0>074 where Pr is the Prandtlnumber.

The radiative term in Eqn (1d) is evaluated as

/f"P

2n

0P

=

0C

afj

(1!omjofj)

qmj(Grj#G

dj#Evj)

#

afj

(1!omjofj)

(Emj#o

mjEfj)!Efj#E

mj] d0 dj

(3)

where afj and o

fj are the monochromatic absorptanceand re#ectance of soil surface, respectively, o

mj andqmj the monochromatic re#ectance and transmittance of

cover, Grj and G

dj the solar beam and di!use irradiance,Evj, E

mj and Efj are the monochromatic emissive

powers, respectively, for sky, cover and soil surface, h isthe incidence, and j the wavelength.

In Eqn (3), the multiple inter-re#ections (Fig. 1) can becustomarily accounted for by considering the followingconverged term for the geometric series:

afj(1#o

mjolj#o2mjo2

lj#2):ofj

(1!omjofj)

(4)

Table 1Soil parameters used in the simulation

Parameters Symbol Value

Soil bulk density, kg/m3 os

1650Soil porosity, m3/m3 n 0)36Residual soil moisture, m3/m3 x

wr0)08

Fraction of clay, m3/m3 xc

0)23Fraction of organic matter, m3/m3 x

p0)06

Fraction of quartz, m3/m3 xq

0)35Shape factor of air, dimensionless f

a0)200

Shape factor of clay, dimensionless fc

0)010Shape factor of organic phase,

dimensionless fp

0)500Shape factor of quartz, dimensionless f

q0)140

Shape factor of water, dimensionless fw

0)140Thermal conductivity of air, W/mK k

a0)025

Thermal conductivity of clay, W/m K kc

2920Thermal conductivity of organic matter,

W/mK kp

0)250Thermal conductivity of quartz, W/mK k

q8800

Thermal conductivity of water, W/m K kw

0)570Heat capacity of clay, J/m3 K C

c2)39]106

Heat capacity of organic matter, J/m3 K Cp

2)50]106Heat capacity of quartz, J/m3 K C

q2)13]106

Heat capacity of water, J/m3 K Cw

4)20]106


The mass transfer coe$cient is de"ned after Takakura(1989):

K"

hf

¸ecp

(5)

where ¸e is the Lewis number and cp

is the speci"c heatof air at constant pressure.

The in#uence of water vapour exchanged between theair gap and the soil surface is taken into account bya humidity factor F

u, de"ned as follows:

Fu(t)"

Xssat

!Xisat

Xssat

if Xssat

!Xisat

'0 (6a)

Fu(t)"0 if X

ssat!X

isat)0 (6b)

where Xssat

and Xisat

are the air humidity ratio at satura-tion near the soil surface and in the air gap, respectively.Then the volumetric fraction of moisture x

wcan be ob-

tained by the product of the humidity factor with the soilporosity n:

xw"F

u(t)n (7)

and therefore, the volumetric fraction of air in the soilxais

xa"(1!F

u(t))n

According to de Vries (1963), the heat capacity of thesoil can be found by considering the heat capacities of thevarious phases:

Cs"x

cC

c#x

pC

p#x

qC

q#x

wC

w#x

aC

a(8a)

where x and C denote, respectively, the volumetric frac-tion and the heat capacity of a phase, i.e. clay (subscriptc), organic matter (subscript p), quartz (subscript q),liquid water (subscript w), and air (subscript a).

The overall thermal conductivity of the soil can beexpressed as

ks"

fcxckc#f

pxpkp#f

qxqkq#x

wkw#f

axaka

fcxc#f

pxp#f

qxq#x

w#f

axa

(8b)

where f denotes the shape factor of a phase and k itsthermal conductivity.

2.3. Heat -ow in the cover

The governing di!erential equation for transient heat#ow in the cover, and related boundary and initial condi-tions may be derived as follows:

ho(¹

m!¹

o)#h

i(¹

m!¹

i)#/

m#¸"w

mC

m

L¹m

Lt, t'0

(9a)

¹m(0)"¹0

m, t'0 (9b)

where hois the heat transfer coe$cient between the plas-

tic mulch and the atmosphere, ¹m

the temperature of thecover, ¹

othe outside temperature, h

ithe heat transfer

coe$cient in air gap, ¹ithe air gap temperature, /

mthe

net radiant #ux density of the cover, ¸ the latent heat #uxof water in air gap, C

mthe cover heat capacity, and

¹0m

the start time temperature of cover.The convective heat transfer coe$cient between the

plastic mulch and atmosphere is evaluated according toFuller et al. (1987):

ho"2)8#3)8v(t) (10)

v being the wind speed.The radiative term in Eqn (9a) is evaluated as

/f"P

2n

0P

=

0C

amj

(1!omjofj)

(1#qmjofj) (Grj#G

dj#Evj)

#

amj

(1!omjofj)

(ofjEmj#E

fj)!2Emj#E

fj] d0 dj

(11)

where amj is the monochromatic absorptance of cover,

while the converged term of the related geometric series isderived as in Eqn (4).

The absorptance of soil surface is conventionally ap-proximated into the following:

afj"const"0)70 if 0)1(j)3)2 (12a)

afj"const"0)75 if j'3)2 (12b)


and, therefore,

ofj (j)"1!a

fj (j) (12c)

j being the wavelength.From Kircho!'s law (Incropera & De Witt, 1990) the

following statements are derived:

amj(j, h)"e

mj(j, h) (12d)

afj (j)"e

fj(j) (12e)

where emj is the monochromatic emittance of cover and

efj the monochromatic emittance of soil surface, h being

the incidence.The apparent temperature of the sky is evaluated ac-

cording to Campbell (1977):

¹v(t)"0)0552¹

o(t) (13)

Fig. 2. Spectral transmittance, at 03 incidence angle, for three xlmlong-wave (LDPE-LW); and (c) p

and related emittance is computed, from Swinbank(1963), as follows:

ev(t)"!0)6458#0)005¹

o(t) (14)

Spectral emissive power of sky, cover and soil surface,considered as radiant grey bodies are expressed as usualas

Evj(j, ¹

v)"e

vEbj(j, ¹

v) (15a)

Emj(j, ¹

m)"e

mEbj(j, ¹

m) (15b)

Efj(j, ¹

f)"e

fEbj(j, ¹

f) (15c)

where Ebj is the spectral emissive power of the black

body, according to the Planck's law.The results presented in Section 3 are validated against

a number of "eld tests which have been carried out by

s: (a) low-density polyethylene (LDPE); (b) modixed LDPE forolyethylene terephthalate (PET)


using a low-density polyethylene (LDPE) "lm. Also,these results are compared with two simulation testsrelative to a modi"ed LDPE (LDPE-LW) "lm, whichfeatures an increased absorptance in the LW range, anda polyethylene terephthalate (PET) "lm. Upon experi-mental evaluation of transmittance spectral distributionof the "lms, a custom correction has been adopted, as inFuller (1987), to take into account the reduction of trans-mittance as a consequence of the variation of incidentangle of solar radiation.

2.4. Numerical procedure

The soil is discretized into a set of elements. Theproblem is solved by using a central di!erenced, "nitescheme, explicit in time. Local average temperature, ther-mal conductivity and heat capacity are attributed to eachelement. Soil parameters are explicitly linearized, as they

Fig. 3. Hourly solar global and diwuse irradiance, air temperatuE. 163

are evaluated at each time step using their values of theprevious time step.

Therefore, Eqn (1a) is discretized for the jth element, bythe Euler method of the solution

C tsj¹t`1sj

"atj¹tsj~1

#(Ctsj!at

j!bt

j)¹t

sj#bt

j¹tsj`1

(16a)

where the superscript t denotes the time step and thecoe$cients a

jand b

jare de"ned as follows:

atj"

*t

*zj

2ktsj

*zj~1

#*zj

, btj"

*t

*zj

2ktsj

*zj`1

#*zj(16b)

*z represents the thickness of the soil discretized layer.The value of the time step *t has to be chosen to satisfy

the following stability criterion:

1

Csj

(atj#bt

j))1 (16c)

re and wind speed recorded at test site (lat. N. 40323@, long.47@)


Calculations carried out on a Pentium II workstationrunning at 300 MHz typically took 4 min to simulate anexperimental session of 3 days, considering a "ve-nodediscrete scheme.

3. Results and discussion

Field experiments have been carried out in a sandylean clay soil in a site of the Basilicata region, Italy. Theassumed soil parameters are listed in Table 1. The soilhas been covered by a 5 m by 5 m of LDPE sheet, with50 lm thickness. Its heat capacity is assumed as50 k J/m3K. The sides of the "lm were covered at theedge by the surrounding soil. Soil temperatures at 0)01,0)05, 0)15, 0)30 and 0)50 m depths have been measuredhourly using thermistors linked to an electronic datalogger. Weather data over the entire observation period,from 28 July to 20 August, were collected hourly. The

Fig. 4. Measured (2) and simulated ( - - - ) soil temperatures,

transmittance spectral distribution curves of testedmaterials (Fig. 2) have been measured by a ultraviolet}visible (UV}VIS) Spectracomb 601 (Carlo Erba Instru-ments) in the 0)175}0)900 lm band, and a Fouriertransform infrared (FTIR) 2000 (Perkin-Elmer Instru-ments) in the remaining band. Further details on theexperimental set-up are given in De Luca et al. (1996).Figure 3 shows the hourly meteorological data which areused as boundary conditions in this validation, togetherwith the correspondent recorded soil initial temperatures.

Figure 4 shows the time evolution of the simulated andmeasured soil temperatures for four depths. A good con-sistency between the simulated soil thermal regime andthe available "eld measurements can be noticed. How-ever, the comparison is not as good at the deepest loca-tion, due to the evident scattering of experimental data.The di!erences in absolute maximum temperaturebetween the simulation and the experiment are 3)23C at0)01 m, while it is 3)0, 2)7 and 3)73C at 0)05, 0)15 and

at soil depths of: (a) 0)01; (b) 0)05; (c) 0)15 and (d) 0)30 m

Fig. 5. Soil temperature simulations carried out for three diwerent xlms at soil depths of: (a) 0)01 m; (b) 0)05 m; (c) 0)15 m;(d) 0)30 m; , low-density polyethylene (LDPE) xlm; - - -, modixed LDPE xlm for long-wave (LDPE-LW); , polyethylene

terephthalate (PET) xlm


0)30 m depths, respectively. Related mean square errorsare 1)4, 1)4, 1)2 and 1)23C, respectively.

As mentioned in Section 2, further numerical tests havebeen conducted on di!erent "lms, to infer the dependenceof thermal regime on transmittance spectral distribution.These results prove that, with the adopted model, it isfeasible and economic to explore the behaviour of di!er-ent materials in mulching practice. Figure 5 shows thewarmer regime simulated for a PET-covered mulch thanwith LDPE or LDPE-LW "lms. Comparing PET andLDPE simulations, for example, con"rms that the LWband transmittance which for PET is consistently lowerat wavelength longer than about 7 lm, and signi"cantlycontributes to increase the underside thermal regime.The same may be inferred when comparing LDPE withits modi"ed form, but in this case a poorer performanceof the latter in the short-infrared band occurs.

Finally, a sensitivity analysis was carried out to investi-gate the e!ect of changes in some key parameters that arelisted in Table 2.

An uncertainty magni"cation factor (UMF) (Coleman& Steele, 1999), for each investigated parameters, wasintroduced to assess the model sensitivity. Each uncer-tainty magni"cation factor;

MFindicates the in#uence of

the uncertainty in that parameter on the uncertainty inthe result:

;MF

"

¹(P)!¹(P#*P)

*P

P

¹(P)(17)

where¹ represents soil temperature, P is each parameter,and *P the increase in the parameter. The UMF for eachparameter can be easily computed following the numer-ical simulation runs.

Table 2Input parameters considered in the sensitivity analysis

Parameters Symbol Value

Monochromatic absorptance of soil surface at 0)1(j)3)2, dimensionless afj 0)70

Monochromatic absorptance of soil surface at j'3)2, dimensionless afj 0)75

Humidity factor, dimensionless Fu

see Eqns(6a) and (6b)

Heat transfer coe$cient between the plastic mulch and the atmosphere,W/m2K

ho

see Eqn (10)

Space width of soil discretized layer at 0)01 m, m *z 0)01


The results of the above analysis, reporting UMFsversus time, in a 24 hour period, are shown in Fig. 6 fortwo di!erent depths 0)05 and 0)15 m.

The model showed considerable sensitivity to the ab-sorptance of soil surface and the humidity factor. In

Fig. 6. Model uncertainty magnixcation factor UMF versus time to(b) 0)15 m; - - -, monochromatic absorptance of soil surface (a

fj ) a(afj ) at j'3)2; , humidity factor (F

u); , heat transfer

, space width of soil discret

particular, the sensitivity is strongly negative with refer-ence to monochromatic absorptance of soil surfaceafj at j'3)2, even greater when the 0)15 m depth is

considered [Fig. 6(b)]. When the monochromatic absor-ptance of soil surface a

fj at 0)1(j)3)2 is monitored,

10% change in diwerent parameters, at soil depths of : (a) 0)05 andt 0)1(j)3)2; , monochromatic absorptance of soil surfacecoezcient between the plastic mulch and the atmosphere (h

o);

ized layer (*z) at 0)01 m depth


the sensitivity greater than zero both at 0)05 [Fig. 6(a)]and 0)15 m [Fig. 6(b)] depths. This indicates that thevariation in soil absorptance both at SW and LW bandsstrongly in#uences soil temperature simulations. For thehumidity factor F

ua low sensitivity was observed at

0)15 m depth, with a time evolution of alternate sign.There is no signi"cant sensitivity to other parameters atboth depths.

4. Conclusions

A transient, one-dimensional model is presented todescribe the di!erent heat transfer modes from the soil tothe environment. Spectral radiation and thermal interac-tion with the cover material, conduction within the soiland convection within the air gap has been analyticallydescribed and numerically solved.

The reported analysis shows that the soil regime undera radiative interfering cover can be simulated by employ-ing the transmittance spectral and directional data of theadopted "lm, in order to abandon the simpli"ed assump-tion of short-wave and long-wave bands averaged values.Di!erent materials can be studied by introducing theircorresponding radiative responses in the model, uponspectroradiometer measures.

The programme is therefore a suitable tool to study thesoil thermal regime under a cover "lm, in such agricul-tural system as tunnels and mulches.

References

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Coleman H W; Steele W G (1999). Experimentation and Uncer-tainty Analysis for Engineers. John Wiley and Sons,New York

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Mahrer Y; Naot O; Rawitz E; Katan J (1984). Temperatureand moisture regimes in soils mulched with transparent poly-ethylene. Soil Science Society of America Journal, 48(2),362}367

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sw—soil and water: a transient-spectral thermal model of soil under radiative-interfering cover

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