Development and validation of model for estimating temperature
within maize ear
S. Khabba
a,∗, J.-F. Ledent
b, A. Lahrouni
aaDépartement de physique, Faculté des Sciences Semlalia, BP 2390, Marrakech, Morocco
bECOP Grandes Cultures, Université Catholique de Louvain, 2 pl. de la Croix du Sud, B-1348 Louvain-la-Neuve, Belgium
Received 21 January 2000; received in revised form 18 July 2000; accepted 31 July 2000
Abstract
We present a three-dimensional computer model that simulates ear temperatures under field conditions for both daytime and night-time. The meteorological data used are total and diffuse radiation, wind speed, air temperature and humidity (or wet bulb temperature). The model is based on the energy variation of volume elements on ear surface. It takes into account, net radiation, sensible and latent heat exchange and heat diffusion within the ear. The model performs a radiation balance that separates direct, diffuse and scattering components. The husk stomatal resistance was parameterised as a function of water vapour deficit and solar radiation deduced from our experimental data. The model was tested in two stages: first, the calculated flux of downward and upward all wave radiation, at ear level, was compared with real measurements. Second, the calculated grain temperatures were compared with air temperature, and with data collected, for different polar positions around the cob, in two experiments conducted in 1997, in Morocco, and 1998, in Belgium. The agreement was satisfactory; the average difference between the model estimates and measurements of grain temperature were 0.5◦C in Belgium and 0.6◦C
in Morocco, whereas using air temperature as the simplest estimate of the grain temperature gave average differences against the measured grain temperature of 1.1 and 1.8◦C, respectively. © 2001 Elsevier Science B.V. All rights reserved.
Keywords: Maize; Ear; Temperature; Model
1. Introduction
Temperature has a major influence on plant develop-ment, growth and yield of maize (Miedema, 1982). In maize, heat stress, around flowering, may be the cause of unsuccessful fertilisation with losses of 30–32% in grain yield (Saadia et al., 1996). During grain filling, low temperatures affect grain growth (Ledent, 1988) possibly due to effects on the transfer of assimilates through the cob to the grains. Grain temperature may
∗Corresponding author. Tel.:+212-4-43-46-49;
fax:+212-4-43-74-10.
E-mail address: [email protected] (S. Khabba).
also affect characteristics of seed quality (Rossman, 1949). These effects are determined by temperature within maize grains which may differ significantly from the surrounding air temperature especially when the latter changes rapidly. The ear has thermal iner-tia (Ledent, 1988; Khabba et al., 1999a), and the husk leaves protect the grain from variations of external temperature. Thus, a gradient of temperature may ex-ist between kernel and air surrounding the ear (Ledent, 1988).
Studies of the thermal behaviour of the ear have been done (e.g. Woodams and Nowrey, 1968; Polley et al., 1980; Singh, 1982; Ledent et al., 1993; Khabba et al., 1999a). There have been attempts to model
ternal temperature (Gaffney et al., 1980; Di Pentima and Güemes, 1987; Khabba et al., 1995) but these were not done under field conditions, with ears attached to the plants within the plant canopy.
We propose here a simple three-dimensional model, based on the energy balance of each elementary vol-ume on the ear surface and on heat conduction within the ear, to predict ear temperatures under field condi-tions from meteorological observacondi-tions.
2. Description of the model
Internal ear temperature may be important when variations of air temperatures are rapid. Extreme dam-aging temperatures may last only short periods. We thus estimated ear temperature using a small step time (1 min). The ear is simulated as an inclined cylinder, consisting of three concentric layers with a variable
Fig. 1. Schematic presentation of longitudinal section of maize ear and representation of the short wave radiation balance of the ear. Rb
and Rd are, respectively, direct and diffuse downward solar radiation measured on a horizontal plane; Rbd, Rddand Rsr are direct, diffuse
and scattered radiation reaching ear surface;φsandφe are solar azimuth and ear azimuth, respectively;βis solar elevation. A, B, and C
identify locations on the ear.
cross-section, attached to the peduncle (Fig. 1). The variation of the section radius along the ear (generat-ing AB or CB) was described by the parabolic equa-tion developed by Khabba et al. (1999a). Heat can be transferred from the stem to the ear by sap flow. A rough estimate indicates that this flux may represent only 2–4% of incident radiation. We then assumed that heat exchanges through the lower end of the ear AC was negligible.
external husk surface will be made with respect to the field conditions. The temperature of each elementary volume on the external husk surface (Fig. 1, generating ABC) was calculated using the energy conservation law (Saatdjian, 1993):
Energy input=Energy absorbed from solar reaction
+Energy exchange by long wave radiation
+Energy lost by convection
+Energy lost by evaporation
+Energy exchange by conduction into ear
The first four variables of the right-hand term are the external boundary conditions on the husk surface. The nature of each variable is discussed in turn below.
2.1. Short wave radiation,Rsw
Eq. (1) describes net short wave radiation Rsw, for
each element of the external surface area of the ear, in terms of its components (Fig. 1):
Rsw=(1−ae)(Rbd+Rdd+Rsr) (1)
where ae is the ear albedo, Rbd the downward
di-rect beam solar radiation (W m−2) normal to the ear surface, Rdd the downward diffuse solar radiation
Fig. 2. Schematic representation of the subdivision of the canopy: (1) horizontal layer; (2) vertical slice; (3) cell of vegetation. The canopy is divided into Nzvertical layers and Nx vertical slices (between two rows) parallel to the direction of the row. The canopy structure is assumed symmetric on both sides of row and on both sides of inter-rows line.
(W m−2) and Rsrthe scattering radiation (W m−2). The
components of this equation are not measured directly, but estimates are made using measured weather data.
2.1.1. Description of the canopy
In the case of the row crops, the canopy structure of the vegetation is divided into Nz horizontal layers
and Nx vertical slices parallel to the direction of the
row (Fig. 2). The intersection of the slice and the layer gives a cell, k, of vegetation. The thickness of the layers was chosen to be of the same order of magnitude as ear length. Each ear is situated in one cell and this cell is referred by, e. Each cell, k, may be characterised by its leaf area densities ak and its leaf inclination
distributiongk(α) (random leaf azimuth distribution
was assumed).
2.1.2. Radiation interception by vegetation
For a given directionΩ (i.e. heightβ and azimuth
φ,φ = 0◦ for the row direction), the probability of
non-interception in a cell k is given by the classical negative exponential law:
Pk =exp(−K[gk(α), Ω]akδz) (2)
K[gk(α), Ω] is the projection of an unit area of leaf,
with an inclination distribution gk(α), on the plane
normal toΩ.K[gk(α), Ω] has been described by
sev-eral authors (e.g. Ledent, 1977; De Castro and Fetcher, 1998). Nilson (1971) and Lemeur and Blad (1974) compared different functions for calculating the prob-ability of interception for distributions of foliage: ran-dom, regular or clumped. The negative exponential function used here assumes a random distribution of foliage within the cell.
The probability that a beam light reaches the cell k,
Pk′, is expressed as
Pk′=P1, P2, . . . , Pn (3)
where n is the number of cells in the path of the beam to reach the cell of interest, k.
2.1.3. Direct radiation,Rbd
Direct radiation interception is computed from the above considerations applied to the sun directionΩs.
Hence, direct radiation reaching the ear cell (e) my be expressed as
RbΩs =RbPe′(Ωs) (4)
where Rbis the direct solar radiation flux density
mea-sured on a horizontal surface, and P′
e(Ωs) the mean
probability of a beam of directionΩsreaching the ear
(e.g. Allen, 1974; Fukai and Loomis, 1976; Sinoquet, 1989). At any time, a local and instantaneous value of Rbd can be expressed as follows:
Rbd =
β andΩs can be easily calculated using classical
as-tronomical formulae from the latitude of the site and the day of the year (e.g. Garnier and Ohmura, 1968; De Castro and Fetcher, 1998).θ is the polar angle be-tween solar beam direction and the normal to the sur-face;1φ=φs−φeis the difference between sun and
ear azimuths, andαthe ear inclination (Fig. 1). Both
φs andφeare calculated relative to the row direction. 2.1.4. Incident diffuse radiation,Rdd
Diffuse radiation comes from all directions with variable intensity depending on elevation of the radi-ation and other factors. It was treated as coming from
a set of directional radiation sources, i.e. integrating contributions from the whole sky. Therefore, the sky was divided into solid angle sectors dΩ according to class of heights and azimuth angles. The amount of incident diffuse radiation Rb(Ω) coming from each
angle sector dΩ was derived from the Standard Over-Cast sky (SOC) distribution (Moon and Spencer, 1942) or Uniform OverCast sky (UOC) distribution (Walsh, 1961). The mean diffuse radiation reaching the ear may be written as
Summing over solid angle was performed with class intervals of 10◦forβ and 20◦forφ.
2.1.5. Radiation scattering,Rsr
Maize ear is generally situated in the canopy at about half height. It can receive scattered radiation from all surrounding cells or soil strips. A precise com-putation of the flux density of scattering radiation, Rsr,
requires the use of a method treating all radiation ex-changes within the canopy. The radiosity method has been frequently used for that purpose (Neveu, 1984). Its use is possible because the soil–vegetation–sky forms a closed system. The calculation of flux den-sity Rsr was split into two steps. First, the flux Re
intercepted by the cell of the ear was calculated using the method fully described by Sinoquet (1989) and
Sinoquet and Bonhomme (1992). Second, Rsr is
de-duced from Re by
Rsr =Re−(Rbe+Rde) (7)
2.2. Long wave radiation,Rlw
up of three different zones: the soil surface, the sur-rounding cells of vegetation and the atmosphere. The soil surface can reasonably be considered as having a uniform surface temperature (Tsol), but the flux density
of long wave radiation coming from the atmosphere is strongly dependent on the angle of view. However, an important fraction of this radiation comes from a lim-ited solid angle and the ear was almost vertical so the sides point towards the horizons. We estimated that most of atmosphere radiation absorbed by the ear, Ra,
came from the lower atmosphere, which allows us to writeRa ≈σ Ta4(σ is Stephan–Boltzmann constant,
equal to 5.67×10−8W m−2K−4). Since the temper-atures of maize leaves vary linearly from the bottom to the top of vegetation, between Tsoland air
temper-ature Ta (Khabba et al., 1999b), long wave radiation
balance of ear can be simplified as
Rlw =12σ (Tsol4 −T 4
s)+12σ (T 4
a −Ts4) (8)
where Tsis the external temperature of the husk. Using
this equation, the overestimation of energy by the first part of the right-hand term was almost balanced by the second part (Khabba et al., 1999b).
2.3. Sensible heat exchange,Hs
Considering the ear as a cylinder, with a variable section, placed in air at temperature Tas, the convective
heat flux density, Hs, can be written as Hs=
thermal diffusion resistance of husk leaves (s m−1). The expression of convective heat-transfer coeffi-cient, h, depends on the average ear radius, re, and on
the dimensionless Nusselt number Nu (Monteith and Unsworth, 1990):
h= κNu
2re
(10)
whereκis the air thermal conductivity (0.0257 W m−1 k−1at 20◦C). Nu can be expressed as a function of ei-ther the Reynolds number(Re=ue2re/υ), in the case
of forced convection or the Grashof number (Gr =
8re3gβ(Ts −Tas)/υ) in the case of free convection.
To estimate the possible sizes of transfer coefficients for these two regimes, the ear was treated as a cylin-der of an average radius of 2.8 cm. Empirical relations derived from literature (e.g. Kreith, 1958; Leontiev, 1979; Monteith and Unsworth, 1990; Cellier et al., 1993) gave
and 30◦C, h (free) is expected to fall between 0.19 and 0.30 W m−2K−1. We will therefore assume that free convection was insignificant most of the time for maize ears under field conditions. This conclusion is consistent with those of Smart and Sinclair (1976) for spherical fruit and of Cellier et al. (1993) in the case of the apex of maize during early growth stages.
The wind speeds at the ear level, ue, was derived
from the wind speed, Vs, measured at height zs using
logarithmic wind profile above the canopy and an ex-ponential wind profile within the canopy (Khabba et al., 1999b):
ue=Vs
exp(aL(ze/ hc−1))
log((zs−d)/z0)
(13)
where L is the leaf area index and a an empirical co-efficient. d and z0are, respectively, the zero plane
dis-placement and roughness length. They were estimated from canopy height hc (Armbrust and Bilbro, 1997;
Kustas et al., 1989; Zhang and Gillespie, 1990; Bus-sière and Cellier, 1994; Sauer et al., 1996):
z0=0.13hc (14)
d=0.67hc (15)
2.4. Evaporation,λE
The latent heat flux density can be expressed as
λE=ρCp(es−ea) γ (rex+ri)
(16)
whereγ is the psychrometric constant (≈66 Pa K−1), and ea and es are, respectively, the vapour pressure
which constitute the husk. The external resistance to the vapour transfer, rex, is a combination of both
tur-bulent exchange resistance between canopy and bulk air (aerodynamic resistance) and resistance to air mass exchange within the canopy referred to as canopy re-sistance. For turbulent convection, rexis the same as rs (Jones et al., 1983; Toole and Real, 1986; Cellier
et al., 1993). riis the husk stomatal resistance and was
assumed to be only due to the resistance of the stom-ata of the external face of the husk.
The vapour pressures ea and es were written as
a functions of Td (dew point temperature) and Ts,
respectively: ea = P (Td) and es = P (Ts). Using
Taylor’s first-order series, the vapour pressure differ-ence in Eq. (16) can then be linearised into
es−ea=1(Ts−Td) (17)
where 1 is the slope of saturation vapour curve. It will be estimated at air temperature Ta which is
al-ways included between Ts and Td, and can be taken
as 12(Ts+Td). The estimation of stomatal resistance is
of major concern in all the models involving relations between plant and atmosphere, and it has presently no universal solutions (Tolk et al., 1995). To our knowl-edge, no investigations have been made on the stomatal resistance in the case of maize ear. We had therefore to derive ourselves a relation to calculate rifrom our
ex-perimental measurements in Belgium (explained be-low). The model fitting method (Powell, 1984; Khabba et al., 1999a) was used, i.e. riwas adjusted to obtain a
good fit of simulated temperatures to measured tem-peratures. The temperature at chosen points (within the ear) as explained below was measured as a func-tion of time. riwas estimated by minimising the sums
of squares of the differences between observed and simulated temperatures (least-squares method).
2.5. Energy exchange by conduction into the ear,Hc
For each elementary volume at husk surface, the heat flux exchanged by conduction into the ear was described by
Hc= −κsgrad(T ) (18)
whereκsis the thermal conductivity of husk layer and T the temperature. The thermal exchange within each
one of the three layers of the ear; cob, grain or husk, was described by Fourier’s law
∂T
conductivity, density and heat capacity of each layer of the ear. Eq. (19) take into account heat transfer in the radial (r), polar (θ) and longitudinal (z) direction. At the interface between joining layers, husk–grain, husk–cob or grain–cob, heat conduction was assumed purely conductive:
TX−ε/2=TX+ε/2 (20)
κegrad(T )|X−ε/2=κegrad(T )|X+ε/2 (21)
where X is the interface level andεa very small real number compared with step interval1r and1z(ε=
0.1 mm).
Eqs. (19)–(21) (with appropriate external boundary conditions Eqs. (1), (8), (9) and (16)) were solved us-ing a finite difference scheme and alternatus-ing direc-tion method (Samorsky and Gulin, 1973). A mesh of
(r, θ, z)= (27,36,41)was found to be sufficient to model the problem accurately.
2.6. Procedure of model testing
Eq. (13) was adjusted by Khabba et al. (1999b) using the wind speed measurements as those made in Belgium (described below). The value obtained for the empirical coefficient a was 0.51(r2 =0.96). In the absence of an estimate of stomatal resistance, the test of the model was made as follows:
• Before estimating ear stomatal resistance with the model, the equations used to calculate ear radiation balance were tested. This was done by comparing measured and calculated downward and upward diation received at the ear position. The flux of ra-diation was calculated in the model by
RbΩs+Rdd+Rsr+
• The stomatal resistance ri was estimated using
the model (method described above). The values obtained were used to determine a relationship between ri and micrometeorological factors. The
• Finally the model was tested using experimental observations made in Morocco (June 1997) and in Belgium (September 1998).
3. Experiment
Data to test the model were collected from two field experiments: the first was in Marrakech, Morocco (31◦37′N, 7◦52′W, altitude = 643 m) and
mea-surements were taken between 11 and 15 June 1997, 75 days after planting. The second field was in Louvain-la-Neuve, Belgium (50◦40′N, 4◦40′E, altitude=130 m) and observations were made from 12 to 23 September 1998, 138 days after planting. Ear development was assessed by grain moisture content M (wet basis). In Belgium, it varied from 52.8±1.5% (at the beginning of observations) to 49.7±2.4% (at the end). In Morocco, the variation in moisture content was negligible, M =67.6±1.8%. The up-permost 11 (from 16) leaves in Morocco, and 9 (from 15) leaves in Belgium were still green at the time of measurements. Canopy height, hc, and leaf area
index were, respectively, 1.5 m and 3.6 in Morocco and 2.2 m and 5.2 in Belgium.
In Morocco, the field dimensions were 70 m×40 m and the plant density 70 000 plants ha−1. The rows
were oriented north–south. Micrometeorological ob-servations of total and diffuse downward solar ra-diation flux density on a horizontal surface as well as of wind speed at 6 m were made near the maize field (≈300 m from maize field). The following mea-surements were made within the field: air tempera-ture at ear level and at 2 m above the crop using two thermocouples TTC10; air relative humidity at 2 m above the field using a capacitive hygrometer (Vaisala, Helsinki, Finland); soil surface temperature, average of two chromel alumel thermocouples (1.5 mm diam-eter); ear temperatures were measured at mid-length: three thermocouples were inserted in the middle of grains at three polar positions (north-east, south and north-west). These measures were made on three ears for which the heights above the ground, ze, were
mea-sured. The thermocouples wires were connected to a multichannel digital electronic thermometer with an accuracy of 0.1◦C. All these data were recorded ev-ery 30 min from 6 to 19 h (UT) (27 observations per day).
In Belgium, the experiment was performed over a large maize field (1.5 ha) located near the Catholic University of Louvain in Louvain-la-Neuve. The plant density was about 90 000 plants ha−1. The rows 75 cm apart were oriented east–west. Micrometeoro-logical measurements of total and diffuse downward solar radiation flux density, wind speed at 3.5 m, wet and dry bulb temperatures were made at a nearby meteorological station (≈500 m from the field). In the field, the following observations were made: (1) air temperature at ear level using copper–constantan thermocouple (AWG 24) placed in a double shielded aspirated screen; (2) soil temperature with infrared thermometer (model Everest 4003) from 14 to 23 September 1998; (3) net radiation, at ear level, with one net radiometer (TRL, Delta-T Devices, England), and upward radiation with one radiometer (TSL, Delta-T Devices, England); the sum of these two mea-surements gives the downward and upward all-wave radiation at ear level: (4) ear temperatures were also measured at mid-length of the ear: four thermocouples (AWG 30) were inserted in the middle of grains at polar positions corresponding to north, east, south and west. These measurements were made on four ears for which the heights about the ground, ze, were
mea-sured. All these data were recorded on a data-logger (Campbell Scientific, Shepshed, UK) every minute (1440 observations per day). The values of Rs were
used to calculate the day length (when Rsis positive)
and length of night (Rs is zero). The soil was sandy
in Marrakech and a clay loam in Louvain-la-Neuve. Soil reflectances were, respectively, 0.16 and 0.14 (0.13 on rainy days). Following Davies and Buttimor (1969), ear albedo was chosen asae=0.29.
The geometrical structure of the canopy was mea-sured on two (10 and 14 June 1997) and three (14, 18 and 21 September 1998) occasions during the experimental period in Morocco and in Belgium, respectively. In total, 12 and 22 plants, randomly chosen, were used in Morocco and Belgium, respec-tively. Structure parameters (leaf area density and leaf inclination distributions) of each cell were esti-mated by the plant silhouette method modified for a two-dimensional description (Sinoquet and Bon-homme, 1989). Vertical and cross-row distributions of these two parameters are shown in Figs. 3 and 4, respectively. The number of horizontal layers Nz
Fig. 3. Vertical and a cross-row distributions of leaf area density. The number Nzof layers was 6 in Morocco and 10 in Belgium, the number Nx of slices was 5. The geometrical structure of the canopy is assumed symmetrical on both sides of inter-rows line. The layer numbers refer to heights from the bottom to the top. The slice numbers refer to a cross-row, from the row to the inter-row (Fig. 2).
slices were taken between two rows for the two ex-periments. The geometrical structure of the canopy was assumed symmetric on both sides of the row.
4. Results and discussion
4.1. Comparisons between radiation measured and simulated on horizontal plane at ear position
Before using the proposed model to estimate the ear stomatal resistance, the method used to calculate ear radiation balance was tested. Measured and simulated downward and upward radiation, at ear position, are presented in Fig. 5. The agreement between calculated and observed values was tested with linear regression. For daytime measurements, slope of the regression lines was 1.06, the intercept 36.7 andr2=0.87, and
Fig. 4. Vertical and cross-row distributions of mean leaf angle. The number Nz of layers was 6 in Morocco and 10 in Belgium, the number Nx of slices was 5. The geometrical structure of the canopy is assumed symmetrical on both sides of inter-rows line. The layer numbers refer to heights from the bottom to the top. The slice numbers refer to a cross-row, from the row to the inter-row (Fig. 2).
for night-time (Rs = 0), the values were 1.03, 18.2
and 0.90, respectively.
4.2. Stomatal resistance
In the absence of reference values for the stomatal resistance, our estimation was made by inverting the model using measurements of ear temperatures in day-time conditions. We used ear temperatures data from alternate days, i.e. 13, 15, 17, 19, 21 and 23 Septem-ber 1998 to determine mean hourly values of stom-atal resistance of the husk leaves (method described above). The data on ear temperatures from the other days were used to test the model.
Stomatal resistance is affected by environmental factors among which solar radiation, Rs, and water
vapour deficit are the most important. Water vapour deficit was represented by the difference between dew point and air temperatures. The difference Ta −Td
is partly related to Rs. Since, stomatal resistance is
inversely proportional to the solar radiation (Norman, 1979), ri was plotted against (Ta−Td)R−s1 (Fig. 6).
The values less than 30 s m−1 correspond to rainy
days (13 and 15 September 1998). These low values of riare linked to the presence of liquid water on the
ear surface. Such increasing trends have often been observed (Carlson, 1991; Turner, 1991; Collatz et al., 1991; Cellier et al., 1993) but the most surprising features of this figure are the low values taken by ri
Fig. 6. Relation between ear stomatal resistance ri and the
rela-tionship(Ta−Td)R−s1. Each point is an hourly average. The
equa-tion of the line is used to estimate ri in the model. Ta, Td and Rs
are air temperature, dew point temperature and solar irradiance, respectively.
and the presence of an upper threshold. Some expla-nations can be put forward. (1) First, water vapour could come from the inner of the ear and even from the gaps between the husk leaves; as a consequence, the source surface for water vapour would be much larger than the simple external surface of the ear. (2) Second, the presence of dew accumulated inside the ear whose evaporation consumes a noticeable part of incident radiation. (3) Third, particular physiological characteristics of the husk leaves may interfere; unfor-tunately, we have no precise information on this point; (4) Fourth, neglecting the energy fluxes by stem flow in and from the ear overestimates the available energy. Stomatal resistance was assumed to follow a linear trend up to(Ta−Td)R−s1 = 2.4 and having a
con-stant value above that limit. The regression line drawn through the experimental points of Fig. 6 was
ri =49.2(±1.1)(Ta−Td)R−s1−22.4(±2.7) (r2=0.96,d.d.l.=57) (23) This relation was used in the model to calculate the diurnal ear temperatures. During the night, a value of
ri = 3000 s m−1 was chosen in order to account for the stomatal closure (Norman, 1979).
4.3. Comparison between predicted ear temperature and measured ear and air temperatures
Fig. 7 shows this comparison for the days 14, 16, 18, 20 and 22 September 1998. The calculated accu-racy of the model estimations are practically identi-cal for different ear orientations (east, south, west and north). Agreement between computed and measured ear temperature is good: for daytime(Rs 6= 0) and night-time (Rs = 0), mean residuals (difference, d, between simulated and measured values) were 0.5 and 0.3◦C, respectively, standard deviations of d were 0.7 and 0.5◦C, respectively, and r2values were 0.94 ((n=
13 020;5 days×4 polar positions≈651 min per day)
Fig. 7. Comparison between air temperature measured at a weather station (dashed line) and ear temperature (mid-length of ear and at centre of grain) measured (solid line), and calculated (dotted line) from our model of ear temperature for the data collected in Belgium on 14, 16, 18, 20 and 22 September 1998.
Model predictions were also compared with values observed in Morocco. We used Eq. (23) to estimate stomatal resistance. The results, plotted in Fig. 8, are similar to those obtained in Belgium: mean residual, d, standard deviations and r2values were 0.6, 0.9 and
0.87◦C (n = 405), respectively. This indicated that Eq. (23) was valid for this other data set. The data were used to estimate the relationship between riand (Ta−Td)Rs−1, and was found to be almost identical
Fig. 8. Comparison between air temperature measured at a weather station (dashed line) and ear temperature (mid-length of ear and at centre of grain) measured (solid line) and calculated (dotted line) from our model of ear temperature for the data collected in Morocco on 11, 12, 13, 14 and 15 June 1997.
ri=48.3(±2.1)(Ta−Td)Rs−1−21.0(±1.8) (r2=0.94,d.d.l.=61) (24) Statistical analysis (Coursol, 1983) showed that the differences between the coefficients of the two corre-lations were not significant(P ≤0.05).
Table 1
Values of the parameters used for the sensitivity testa
Parameter Constant values Variable values
Leaf area density, ak β=60◦, 1φ=180◦ a
kB×(0.5,0.75,1,1.25,1.5)
Sun elevation,β akB,1φ=180◦ 40◦,50◦,60◦,70◦,80◦
1φ=φs−φe akB,β=60◦ 0◦,45◦,90◦,135◦,180◦
aak
B: distribution of the leaf area density shown in Fig. 3 (in Belgium),β: sun elevation, and1φ: difference between sun and ear
azimuths.
difference, standard deviations and r2, respectively. Values of mean difference between measured ear and air temperatures are generally higher between solar noon and sunset (mean of d was 1.8◦C in Belgium
and 2.4◦C in Morocco). After reaching a maximum, the temperature of the grains decreased at a slower rate than air temperature. During the two periods of measurements, maximum difference between ear and air temperatures were found between 13 and 16 h (UT); the largest differences were 2.1 and 3.6◦C in Belgium and Morocco, respectively. This can be ex-plained by the considerable thermal inertia of maize ear (Ledent, 1988; Khabba et al., 1999a). These re-sults show clearly that our three-dimensional model gives a better estimate of grain temperature.
The comparison of the measured grain temperatures against the temperature of the air surrounding the ear shows that the model estimates of grain temperature are a greater improvement than using air temperature; the average difference obtained were 1.3◦C in Bel-gium and 2.1◦C in Morocco. Standard deviations of the difference were 1.7 and 2.5◦C and r2values were 0.86 and 0.87, respectively.
4.4. Sensitivity analysis of ear temperature
The parameters included in the sensitivity were: the values of the distribution of leaf area density ak,
the sun elevation β and the difference between sun and ear azimuths 1φ. The two first variables were chosen because they have a significant influence on the probability of radiation interception (Sinoquet and Bonhomme, 1992; De Castro and Fetcher, 1998). The third parameter,1φ, was chosen because it is an important factor in the absorption of solar radiation (Neveu, 1984). The values assigned to each parameter are given in Table 1. The values of each parameter were varied, one at a time, while the others were
maintained constant, and the model was run for each combination of values. The model calculates the vari-ation in grain temperature for 4 h, at mid-length of ear. The ear was assumed initially to be at a uniform temperature of 15◦C. The other parameter values were: ue = 0.5 m s−1, Rs = 600 W m−2, Rd/Rs =
0.3, Ta = 18◦C,Tas = 20◦C,Td = 12◦C,Tsol =
10◦C,α = 30◦, φs = 0◦. These values represent the
optimum conditions, observed in Belgium, for ear heating. The canopy characteristics considered were those for the experiment performed in Belgium.
4.4.1. Effect of the leaf area density
Fig. 9 shows that ear temperature is highly influ-enced by ak. The relationship is hyperbolic for all
values used for the leaf area density. Using the val-ues of akB(distribution measured in Belgium, Fig. 3),
Fig. 9. Relationship between simulated ear temperature, at mid-length of ear in the middle of grain, and the values of the dis-tribution of the leaf area density ak (indicated in the box). akBis
the distribution of leaf area density measured in Belgium (Fig. 3). Values of solar elevation,β, and difference between sun and ear azimuths,1φ, were fixed at 60◦ and 180◦, respectively. The ear
was a uniform temperature (15◦C) at time zero, and it was
Fig. 10. Relationship between simulated ear temperature, at mid-length of ear in the middle of grain, and sun elevation β. Values of leaf area density, ak, are those shown in Fig. 3 and dif-ference between sun and ear azimuths,1φ, is fixed at 180◦. The
ear was a uniform temperature (15◦C) at time zero, and it was
submitted to temperature and radiation conditions given in Section 4.4.
ear temperature initially at 15◦C reached 21.5◦C after 4 h under the climatic conditions quoted below. When ak was increased or decreased by half (i.e. 1.5akBor
0.5akB) the temperature estimated after 4 h was 19.2
and 22.7◦C, respectively. The effect of a
k values on
model results requires the use of accurate estimations of the real values. However, measurements or estima-tions ak are time consuming (Myneni, 1991).
4.4.2. Effect of the sun elevation
Calculated ear temperature increase with time was greater for higher sun elevations: β = 70 and 80◦ (Fig. 10). After 4 h, calculated ear temperature was 20.2, 21.5 and 22.2◦C forβ=40,60 and 80◦, respec-tively. Higher sun elevation promotes good solar ra-diation penetration within maize stands (Sinoquet and Bonhomme, 1992).
4.4.3. Effect of the difference between sun and ear azimuths
Fig. 11 shows that this relationship was also hyper-bolic, but the effect of1φon ear temperature was less important in comparison with ak andβ. For 1φ =
90,135 and 180◦, the difference between ear tempera-tures was not significant(P ≤0.05). Values for these angles were higher than those for 0 and 45◦. For1φ
between 90 and 180◦, direct radiation is almost
per-Fig. 11. Relationship between simulated ear temperature, at mid-length of ear in the middle of grain, and1φ. Values of leaf area density, ak, are those shown in Fig. 3 and sun elevation,β, is fixed at 60◦. The ear was a uniform temperature (15◦C) at time
zero, and it was submitted to temperature and radiation conditions given in Section 4.4.
pendicular to the ear surface, giving higher radiation and a consequent increase in ear temperature. After 4 h, calculated ear temperatures were 20.6, 21.2 and 21.5◦C for1φ=0,90 and 180◦, respectively.
5. Conclusion
The model presented above was used to calculate the temperature within maize ear under field condi-tions. It is based on energy conservation equations applied to the ear surface. The model accounts for radiation scattering from all surrounding vegetation or soil. The simplest estimate of grain temperatures can be obtained by assuming that they were equal to air temperature measured at a weather station, this give an average difference between the estimated and measured grain temperature of 1.1◦C in Belgium and 1.8◦C in Morocco. Whereas using our current model
gave smaller average differences of 0.5◦C in Belgium
and 0.6◦C in Morocco — a significant improvement.
We derived a relationship between ri and (Ta − Td)Rs−1 using the model. This relationship needs to
be tested under different environmental conditions. The distribution of leaf area density ak had a
a large extent. The sun elevation is also an important parameter of the model, but it can be calculated or measured accurately, and thus it is not subject to es-timation errors. The effect of the difference between sun and ear azimuths is less important compared to ak andβ effects.
The main potential use of the model is as a tool to understand, explain and predict the effect of different characteristics of the canopy or climatic conditions on ear temperature. Different situations can be simulated and the model can be applied to establish the relation-ship of with ear temperature.
Nomenclature
ae ear albedo
ak leaf area density of cell k (m2m−3)
Cp air specific heat (J kg−1K−1)
Cpe ear specific heat (J kg−1K−1) d zero plane displacement (m)
E water vapour flux density (kg m−2s−1) ea partial pressure of water vapour in air (Pa) es vapour pressure inside substomatal
chambers of the husk (Pa)
gk leaf inclination distribution in cell k
h convective heat transfer coefficient (W m−2K−1)
hc canopy height (m)
Hc flux exchanged by conduction (W m−2) Hs sensible heat (W m−2)
L leaf area index (m2m−2)
P′
e mean probability that a light
beam reaches the ear
Pk probability of non-interception in cell k
Pk′ probability that a light beam reaches cell k Rb incident direct solar radiation (W m−2) Rbd direct solar radiation at ear surface
(W m−2)
Rbe direct short wave radiation intercepted
by cell of ear (W m−2)
RbΩs vertical direct solar radiation at ear level (W m−2)
Rd incident short wave diffuse
radiation (W m−2)
Rdd diffuse solar radiation at ear
surface (W m−2)
Rde diffuse radiation intercepted by cell of
vegetation containing an ear (W m−2)
Rd(Ω) diffuse short wave radiation coming from directionΩ at ear level (W m−2)
Re radiation intercepted by cell of vegetation
containing an ear (W m−2) re ear radius (m)
rex external resistance to the vapour transfer
(s m−1)
ri internal resistance, husk stomatal resistance
(s m−1)
Rlw long wave radiation at ear surface (W m−2) rs thermal diffusion resistance of husk
leaves (s m−1)
Rs total solar radiationRb+Rd(W m−2) Rsr scattered short wave radiation at ear
surface (W m−2)
Rsw short wave radiation at ear surface (W m−2)
t time (s)
Ta reference temperature of the air, above the
field (K)
Tas temperature of air surrounding the ear (K) Td dew point temperature (K)
Ts temperature of the husk surface (K) Tsol temperature of soil surface (K) ue wind speed at ear level (m s−1) Vs wind speed at height zs(m s−1) z0 roughness length (m)
ze ear level (m)
zs weather station level (m) Greek letters
α ear inclination (◦)
β elevation angle (◦)
γ psychrometric constant (66 Pa K−1)
δ partial derivative
φe ear azimuth angle (◦) φs sun azimuth angle (◦) Ω radiation direction
Ωs sun direction
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