...

Boguslaw.pdf

by user

on
Category: Documents
5

views

Report

Comments

Transcript

Boguslaw.pdf
THERMAL CONDUCTIVITY OF SOILS—COMPARISON
OF EXPERIMENTAL RESULTS AND ESTIMATION METHODS
1
1
Usowicz Boguslaw, 2Usowicz Lukasz
Institute of Agrophysics, Polish Academy of Sciences, ul, Doświadczalna 4, 20-290 Lublin 27
e-mail: [email protected]
2
ul. Królowej Jadwigi 6/56, 20-282 Lublin, e-mail: [email protected]
Introduction
To predict heat transfer in soil under the conditions of steady and non-steady heat flow
requires the knowledge of the basic thermal properties of soil (Noborio et al. 1996). While it
is possible to determine the heat capacity per unit volume of soil with fairly good accuracy,
numerous problems are encountered in the determination of thermal conductivity (Kersten
1949; de Vries, 1963, 1986, 1987; Kohnke and Nakshabandi, 1964; Kasubuchi, 1975;
Kossowski, 1977; de Vries and Philip, 1986; Hopmans and Dane, 1986; Abu-Hamdeh, 2001,
2003, Abu-Hamdeh et al., 2001; Bachmann 2001; Gori and Corasaniti, 2003; Nusier and AbuHamdeh, 2003; Tarnawski and Leong, 2000; Usowicz, 1991, 1992).
The propose of this study is to presents an evaluation of method for the calculation of the
thermal conductivity of soil on the basis of comparison of the thermal conductivity of sand,
clay, sandy loam, clay loam, silty loam, silty clay loam and peat, measured and calculated
from the statistical-physical model, de Vries model and numerical method. For loam, the
evaluation was performed on the basis of comparison of results obtained experimentally,
models, Laplace transform, and the method of null-alignment, with and without taking into
account the thermal conductivity resulting from water vapour movement due to temperature
gradient in the soil.
Thermal conductivity
One of the components of the heat balance of the active surface is heat flux in the soil G
(W m-2). This flux for homogeneous medium can be described from Fourier law:
∂T
(1)
G = −λ
∂z
where λ is thermal conductivity (W m-1 K-1), ∂T / ∂z is temperature gradient (K m-1).
In the non-steady state heat flux in the soil is described by equation:
∂T
∂ ⎛ ∂T ⎞
Cv
= ⎜λ
(2)
⎟
∂t ∂z ⎝ ∂z ⎠
where Cv (J m-3 K-1) is heat capacity of soil, t (s) is a time. For isotropic medium in one
dimension the conductive heat transfer is given by
∂T
∂ 2T
=α 2
(3)
∂t
∂z
where α is the ratio of thermal conductivity to the heat capacity, called soil thermal
diffusivity:
α=
λ
Cv
.
(4)
Method 1
Experimental thermal conductivity used in this paper was published by Kersten (1949), de
Vries (1963), Kimball et al. (1976a, 1976b), Asrar and Kanemasu (1983), Sikora and
Kossowski (1993) and Ochsner et al. (2001). Thermal properties of soil can be determined by
measuring the distribution of temperature in response to heat input to a line sources or were
measured using a thermo-time domain reflectometry (Thermo-TDR) probe (Ochsner et al.,
001). Thermal conductivity of various soils was measured at different bulk density, water
content and temperature of soil.
Method 2
De Vries model (de Vries, 1963) based on the application to a granular medium of
potential theory. This model considers the soil as a continuous of water for moist and air for
dry soil in which granules of soil particles, air or water are dispersed. The thermal
conductivity of soil is calculated as the weighted average of the conductivities of various soil
components:
n
λ=
∑k λ x
i =1
n
i
i
∑k x
i =1
i
i
(5)
i
where n – number of components, λi is the thermal conductivity of each components (Table
1), and xi is the volume friction of each components. Volumes of ki are calculated from:
−1
⎞ ⎤
1 n ⎡ ⎛λ
k i = ∑ ⎢1 + ⎜⎜ i − 1⎟⎟ g i ⎥
(6)
3 i =1 ⎣ ⎝ λ0
⎠ ⎦
where gi represents the shape factors for i-th components granules considered as ellipsoids.
Table 1. Values and expressions for parameters used in calculating the thermal conductivity
of soils (T in oC).
Reference a Parameters b
Expression/value b
9.103 - 0.028 T
λq, W m-1 K-1
-1 -1
2
2.93
λm, W m K
-1 -1
2
0.251
λo , W m K
-1 -1
0.552 + 2.34*10-3 T - 1.1*10-5 T2
1
λw, W m K
1
0.0237 + 0.000064 T
λa, W m-1 K-1
-1 -1
2
λapp , W m K
λa + hλv
2
h, dimensional
exp(ψMw/ρwR(T+273.15))
-1 -1
2
λv, W m K
LDaν (dρo/dT)
1
L, J kg-1
2490317 - 2259.4 T
Da, m2 s-1
0.0000229*((T+273.15)/273.15)1.75
1
Da, m2 s-1
21.7*10-6 (101.325/P)((T+273.15)/273.15)1.88
2
1
ν, dimensional
P/(P-(hρο R (T+273.15)/1000Mw))
-3
-3
10
exp(19.819 - 4975.9/(T+273.15))
1
ρo kg m
-3 -1
1
dρo/dT, kg m K
4975.9 ρo/(T+273.15)2
a
1. Kimball et al. (1976a); 2. De Vries (1963), b ψ − soil water pressure head, kPa; Mw –
molecular weight of water, 0.018 kg mol–1; ρw – density of liquid water, 1.0 Mg m–3; R –
universal gas constant, 8.3143 J mol–1; h – relative humidity; L – latent heat of vaporisation;
Da – diffusion coefficient for water vapour in air; ν – mass flow factor; ρo – saturated vapour
density; P – barometric pressure, kPa, thermal conductivity of: quartz, λq, other minerals, λm,
organic matter, λo, water or salt solution, λw, and air, λa.
In a soil not saturated with water, and with the presence of a temperature gradient, a part
of the energy flowing through the soil is transmitted by water vapour. Water vapour flow
through soil causes an increase in the thermal conductivity of the soil. The thermal
conductivity component related to water vapour and soil air is referred to by de Vries
[1963] by the general term compound thermal conductivity and expressed as a sum of the
thermal conductivity of air λa and the thermal conductivity of water vapour λv:
λapp = λa + λv
(7)
The thermal conductivity of water vapour λv can be calculated from the equation
formulated by Kirscher and Rohnalter (after de Vries, 1963), or from the equation modified by
Philip and de Vries (de Vries, 1987) (Table 1). Relative humidity h can be calculated from the
Kelvin equation (Table 1) or from an empirical relation (Kimball et al. 1976a), De Vries considers two ranges of water moisture in which relative humidity equals one, or varies
according to the Kelvin equation. Within the range between field water capacity and full
saturation with water, the relative humidity of the soil air equals one. Below the field water
capacity the relative humidity can be calculated from the Kelvin formula, or from an
empirical (Kimball et al. 1976a) propose another division of the moisture range into two
intervals. The first of these is from zero to a moisture value corresponding to the wilting
point, and the second - from the wilting point to soil saturation with water.
Method 3
The null-alignment method (Kimball and Jackson, 1975, Pikul and Allmaras, 1984) is
based on measurements of moisture and temperatures over a small increments of time and
depth and initial estimation of the thermal conductivity at a given reference level. The thermal
conductivity of soil at the references depth, i = r was calculated as:
Gi
λi =
.
(8)
⎛ ∂T ⎞
⎜
⎟
⎝ ∂z ⎠ i
Soil heat flux density, G, for all depths above references depth was determined from:
z
⎛ ∂T ⎞
G (z ) = G(z r ) − ∫ ⎜ Cv
⎟dz
∂t ⎠
zr ⎝
(9)
where Cv is heat capacity. In the null- alignment method, soil thermal conductivity was
calculated by means of the calorimetric method (9) and temperature gradient method, the
calculations being started from the point where zero gradients occurred.
Method 4
The Laplace transform method (Asrar and Kanemasu, 1983) based on the numerical
integration of Laplace transform of heat conduction equation in soil (3):
d 2 L[T ( z , t )] s
− L[T (z , t )] = 0
(10)
dt
α
where the Laplace time transform is given by:
∞
L[T ( z , t )] = ∫ T ( z , t ) exp(− st )dt ,
(11)
0
and where s ≥ 5.0 / t max is the Laplace transform parameter, tmax is the maximum duration of
the experiment.
The transform is used to determine the thermal diffusivity of soil, while the volumetric
heat capacity was measured or calculated (18). Thermal conductivity was determined from the
relation between thermal diffusivity and heat capacity
λ = α ⋅ Cv .
(12)
Method 5
The statistical-physical model (Usowicz 1991, 1992) is based on terms of heat resistance
(Ohm’s law and Fourier’s law), two laws of Kirchhoff and polynomial distribution (Eadie et
al., 1989). The volumetric unit of soil in the model (fig. 1a) consists of solid particles, water
and air, is treated as a system made up of the elementary geometric figures, in this case
spheres, that form overlapping layers (fig. 1b).
c)
b)
a)
Fig. 1. Schematic diagram of the statistical model construction: a) unit volume of soil, b)
the system of spheres that form overlapping layers, c) parallel connection of resistors in the
layers and series between layers.
It is assumed that connections between layers of the spheres and the layer between
neighbouring spheres will be represented by the serial and parallel connections of thermal
resistors, respectively (fig. 1c). Comparison of resultant resistance of the system, with
consideration of all possible configurations of particle connections together with a mean
thermal resistance of given unit soil volume, allows estimating thermal conductivity of soil
λ (W m–1 K–1) according to the equation (Usowicz, 1991, 1992):
4π
(13)
λ= L
u∑
j =1
P ( x1 j ,..., x kj )
x1 j λ 1 (T )r1 + ... + x kj λ k (T )rk
where u is the number parallel connections of soil particles treated as thermal resistors, L is the
number of all possible combinations of particle configuration, x1, x2 ,..., xk – a number of particles
of individual
particles of a soil with thermal conductivity λ1, λ2 ,..., λk and particle radii r1, r2 ,..., rk,
k
where ∑ x ij = u , j=1,2,...,L, P(xij) – probability of occurrence of a given soil particle
i =1
configuration
calculated from the polynomial distribution:
u!
x
x
P ( x1 j ,..., x kj ) = !
f 1 1 j ... f k kj .
(14)
!
x1 j ...x kj
The condition:
∑ P(X = x )= 1 must also be fulfilled. The probability of selecting a given
L
j =1
j
soil constituent (particle) fi, i = s, c, g, in a single trial was determined based on fundamental
physical soil properties. In this case f s , f c , and f g are the content of individual minerals and
organic matter – f s = 1 − φ , liquid – f c = θ v and air – f g = φ − θ v in a unit of volume, φ –
soil porosity.
Parameters of the model were defined earlier based on
empirical data (Usowicz, 1992; 1995). Degrees of freedom
u characterizing the number of parallel combinations of
thermal resistors as a function of saturation of soil with
water (θv/φ ) are presented on Figure 2, where θv(m3 m–3) is
the soil water content, φ(m3 m–3) is the porosity.
Equivalent radius of the spheres rk (m) for all soil
components were defined according to the equation (16):
14
12
10
u
8
6
4
2
0
0
0.2
0.4
0.6
Water saturation (θv /φ)
0.8
rk = 0.036 f o + 0.044
1
(15)
Fig. 2. Number of parallel
connections of resistors as a
function of soil water
saturation.
where: fo(m3 m–3) is the volumetric organic matter content.
The radius increased with increasing soil organic matter
content, with maximum up to 0.08.
Sharp transition of the value u with water saturation of
soil (θv/φ) resulting in sharp increase in calculated values of soil thermal conductivity and thus
in greater evaluation error of the thermal conductivity. To avoid such transition and to reduce
the error a procedure was suggested that allow determining the thermal conductivity in the
range of any saturation using a linear interpolation of the conductivity in the range (Usowicz
and Usowicz, 2004).
In this procedure, the thermal conductivity of the medium was determined from general
equation of the thermal conductivity (1) for two successive values of u and u+1 (Fig. 2) and
corresponding water contents of the medium, θv(u), θv(u+1) and then thermal conductivity for
the searched water content of the medium, θv from the following equation:
λ =λ (u ) +
θ v − θ v (u )
(λ (u + 1) − λ (u )) .
θ v (u + 1) − θ v (u )
(16)
In data pertaining to a specific soil, five main components were distinguished, of the
following values of thermal conductivity: quartz, λq, other minerals, λm, organic matter, λo,
water or salt solution, λw, and air, λa (Usowicz and Usowicz, 2004). The values of coefficients
of thermal conductivity and their relation to temperature, pressure, and soil water potential are
presented in Table 1. For a soil non-saturated with water, with high salt concentration
((Kimball et al. 1976a; Noborio and McInnes, 1993) and with high soil temperature gradient
in the soil the coefficient of air thermal conductivity is replaced by apparent thermal
conductivity of soil air, λapp, composed of the air thermal conductivity, λa, and the thermal
conductivity of water vapour, λv, (de Vries, 1963; Usowicz, 1993, 1995).
Method 6
Numerical method (Sikora and Kossowski, 1993) based on solution of the one
dimensional of heat conduction equation with it finite-difference form:
Ti , j − Ti , j −1
Ti −1, j −1 − 2Ti , j −1 + Ti +1, j −1
=α
(17)
∆t
(∆z )2
where T is soil temperature, i and j indices of depth z (cm) and time t(s). Numerical method
was used to calculated thermal diffusivity of soil. In this case the thermal diffusivity was
2
assumed to be constant through out the soil and stability criterion α ⋅ (∆z ) ≤ 0.5 is ensured
(Sikora, Kossowski 1993). Volumetric heat capacity of soil was calculated from (19), thermal
conductivity from (12).
Heat capacity
Volumetric heat capacity Cv (MJ m–3 K–1) was calculated using empirical formulae
proposed by Vries de (1963):
C v = (2.0 x s + 2.51xo + 4.19 x w ) ⋅ 10 6
(18)
or by Sikora and Kossowski (1993):
C v = (c s + 4.19 ⋅ 10 3 θ w )ρ
(19)
3
–3
where: xs , xo , xw (m m ) are volumetric contributions of mineral and organic components
and water, respectively, cs – specific heat of soil solid (713 J kg–1 K–1), θw – soil water content
(kg kg–1), ρ (Mg m–3) – soil bulk density.
Validation of methods
The agreement between predicted and measured results was determined with a mean
square error (σb) and relative maximum error (ηb):
n
σb =
∑( f
i =1
mi
− f ci )
2
,
(20)
k
where: fmi is the measured value, fci is the calculated value, k = n – 1 if n < 30 and k = n, n >
30, n – number of data.
The relative maximum error was calculated using the following equation:
⎧ f − f ci
⎫
⋅ 100%⎬ .
η b = max ⎨ mi
(21)
i =1, 2 ,L, n
f
mi
⎩
⎭
Also linear regression equations of the thermal conductivity and determination coefficient
R2 were developed.
Results and discussion
The comparisons of the thermal conductivity values of various soils (Fairbanks sand,
Healy clay, Felin silty loam, Fairbanks peat and Avondale loam), measured and calculated
from the models, are presented in Table 2 and Figure 3. Also comparison results obtained
from de Vries model and from the null-alignment method, statistical-physical model and
numerical method are presented in Figure 4. The data in Figure 3a indicate that the results
obtained from the de Vries model and the statistical-physical models are on a similar level of
agreement with the measured values. Regression coefficients were close to unity, however
permanent factors in the equation were close to zero. Determination coefficients of the linear
regression were high 0.987 and 0.985 respectively. Mean square errors σ (W m–1 K–1) and
relative maximum errors η (%) ranged from 0.057 to 0.123 (W m–1 K–1) and from 12 to
38.3% for Stat model and ranged from 0.017 to 0.127 (W m–1 K–1) and from 22 to 70.7% for
de Vries model. In the case of data in Figure 3b the calculated and measured thermal
conductivity generally agreed wall, R2 = 0.872 for Stat model and 0.938 for de Vries model.
Calculated values from de Vries model are notably higher then measured values at low
thermal conductivity. In the case of Stat model calculated values are notably lower then
measured at low thermal conductivity. Mean square errors and relative maximum errors were
0.238 (W m–1 K–1) and 58.3% for Stat model and 0.149 (W m–1 K–1) and 62.5% for de Vries
model.
The results presented in Table 2, without taking into account the effect of water vapour,
indicate that the statistical-physical model gives the best approximation of the calculated
values to the measured ones (Usowicz, 1995). The other methods also provide fairly good
approximations. However, their mean square error is nearly twice as high, or even higher than
in the case of the statistical-physical model. When the effect of water vapour is taken into
account, the de Vries model significantly increases thermal conductivity while the statisticalphysical model only slightly, though the thermal conductivity due to water vapour was calculated
identically for both the models (Kimball et al. 1976a). The high contribution of thermal conductivity
due to water vapour in the overall thermal conductivity of soil in the de Vries model was caused,
most probably, by poor selection of the shape coefficient in the function of moisture. And the
question of how significantly the shape coefficient affects the calculated values of thermal
conductivity of soil is discussed in the papers by Kimball et al. (1976a,b).
Table 2. Average soil temperature, moisture content, heat capacity, and thermal conductivity
for 0 to 10 cm of Avondale loam at an average bulk density of 1.4 Mg m–3, quartz fraction of
0.373 g g–1 and particle density of 2.71 Mg m–3 (Usowicz, 1995)
Date
Thermal conductivity (W m–1 K–1)
Average Moisture Heat
temp.*
content* capacity*.
Methods
°C
17 May 1973
11 July 1973
20 Sept 1973
21 Sept. 1973
2 Oct. 1973
7 Dec. 1973
25.2
32.3
26.9
26.9
27.2
11.6
3
–3
m m
0.259
0.282
0.276
0.266
0.225
0.261
–3
–1
MJ m K
1
2
2#
3
4
5
5#
2.34
2.43
2.40
2.36
2.20
2.34
1.13
1.21
1.55
1.13
1.04
1.15
1.17
1.20
1.34
1.63
0.96
1.09
1.17
1.19
1.18
1.34
1.59
1.13
1.14
1.17
1.19
1.15
1.26
1.55
1.05
1.01
1.16
1.17
1.00
1.00
1.51
0.88
1.01
1.12
1.14
1.14
1.17
1.46
1.17
1.09
1.16
1.17
0.456
0.131
0.094
0.057
0.067
σb (W m–1 K–1) 0.114
ηb (%)
13.6
51
20
12.2
12
14
Data: 2,2#, 3 from Kimball et al. (1976b) and *, 1,4 from Asrar and Kanemasu (1983). Method: 1 – Measured in the laboratory
(Asrar and Kanemasu, 1983); 2 – de Vries model without vapour and 2# – de Vries model with vapour (Kimball et a.l. (1976b);
3 – Null-alignment (Kimball et al. (1976b)); 4 – Laplace transform (Asrar and Kanemasu, (1983)) 5 – statistical-physical
model without vapour, 5# – statistical-physical model with vapour.
As follows from the comparisons and analyses, the values of thermal conductivity of soil
obtained from the models (Tables 2 and Fig. 3a), without taking into account the effect of water
vapour, from the method with Laplace transform, and from the null-alignment method, are in fairly
good agreement over a wide range of moisture and soil density values. Greater differences were
observed at very low moisture values. If no high temperature gradients occur in the soil and it can
be assumed that the effect of water vapour on the overall effect of conductivity is negligible, the
methods presented can be applied alternatively for the determination of thermal conductivity.
As it was to be expected, the values of thermal conductivity of soil obtained from the
null-alignment method and from the method with Laplace transform are somewhat underestimated with relation to values calculated from the statistical-physical model and considerably
underestimated with relation to values obtained from the de Vries model with the conductivity
due to the effect of water vapour taken into account (Table 2). This situation is caused by the
fact that a part of the energy is transmitted through the latent heat which is hidden in water
vapour. The null-alignment method and the method with Laplace transform do not fully
provide for that fact in their calculations. It is known, however, that when water vapour flows
through the soil some of the latent energy is released during vapour condensation, which is
manifested in an increase in the temperature of the soil layer in which the vapour condensed.
That part of the latent heat which passes through the object freely has a decreasing effect on the
total thermal conductivity of soil. And thus, with high temperature gradients and high water
vapour flow rates it is recommended to use the statistical-physical model and the de Vries
2.5
K-1)
2
1.5
1
de Vries - y = 0.963x + 0.061
2
R = 0.987
0.5
Stat - y = 0.979x + 0.028
R2 = 0.985
0
0
deVries
-1
de Vries
Stat
Linear (de Vries )
Linear (Stat)
Modeled thermal conductivity (W m
-1
-1
Calculated conductivity (W m K )
model, with special emphasis on correct weight determination. The application of the nullalignment method and the method with Laplace transform is not recommended in this case.
2.5
b)
a)
3
y = 0.798x + 0.232
1
2
-1 -1
Measured conductivity (W m K )
Stat
2.0
R2 = 0.938
y = 1.150x - 0.186
R2 = 0.872
1.5
1.0
deVries
Stat
1:1
Linear (deVries)
Linear (Stat)
0.5
0.0
3
0.0
0.5
1.0
1.5
2.0
2.5
Measured thermal conductivity (W m -1 K-1)
Fig. 3. Comparisons of the thermal conductivity values of various soils measured and
calculated from the de Vries and statistical-physical (Stat) models. Measured and calculated
from the de Vries model data presented in figure (a) and (b) were published by de Vries
(1963) and Ochsner et al. (2001).
Thermal conductivity, others models
3
Stat - y = 1.011x - 0.030 Null - y = 0.814x + 0.095
2
2
R = 0.989
R = 0.792
2.5
Numerical - y = 0.938x - 0.063
2
R = 0.272
2
1.5
1
Null-alignment
Numerical method
Model Stat
Model Stat
Null-alignment
Numerical method
0.5
0
0
0.5
1
1.5
2
2.5
3
Thermal conductivity, de Vries model
Fig 4. Comparisons of thermal conductivity obtained from de Vries model and from the nullalignment method, numerical method and statistical-physical model (Stat). Predicted data
from de Vries model were published in papers: de Vries (1963), Sikora and Kossowski
(1993); Null-alignment data, Asrar and Kanemasu (1983) and Sikora and Kossowski (1993);
Numerical data, Sikora and Kossowski (1993).
Generally, similar values of thermal conductivity were obtained from de Vries model and
from Stat model (Fig. 4). Calculated thermal conductivity from the de Vries model and the
statistical-physical models were higher then those determined by numerical method and nullalignment method.
Conclusion
The methods presented can be used alternatively for the determination of thermal
conductivity if no high temperature gradients occur in the soil and if it can be assumed that
the effect of water vapour on the overall effect of conductivity is slight. With high
temperature gradients and high rates of water vapour flow in the soil, the statistical-physical
and the de Vries models should be used, the latter requiring considerable care, especially in
the determination of weight values.
References
1. Abu-Hamdeh N.H.: SW—Soil and water: measurement of the thermal conductivity of
sandy loam and clay loam soils using single and dual probes. Journal of Agricultural
Engineering Research. 80(2), 209-216, 2001.
2. Abu-Hamdeh N.H., Khdair A.I., Reeder R.C.: A comparison of two methods used to
evaluate thermal conductivity for some soils. International Journal of Heat and Mass
Transfer. 44, 5, 1073-1078, 2001.
3. Abu-Hamdeh N.H.: Thermal properties of soils as affected by density and water content.
Biosystems Engineering. 86(1), 97-102, 2003.
4. Asrar G., Kanemasu E.T.: Estimating thermal diffusivity near the soil surface using
Laplace transform: Uniform initial conditions. Soil Sci. Soc. Am. J., 47, 397-401, 1983.
5. Bachmann J., Horton R., Ren T. and van der Ploeg R. R.: Comparison of the thermal
properties of four wettable and four water-repellent soils. Soil Sci. Soc. Am. J. 65, 16751679, 2001.
6. de Vries D.A.: Thermal properties of soil. In: Physics of Plant Environment (ed. W.R. van
Wijk) North-Holland, Amsterdam, 210-235, 1963.
7. de Vries D.A.: A critical analysis of the calorimetric method for determining the heat flux
in soils. Proc. 8–th Int. Heat Transfer Conf. Hemiphen Publ. Corp. Washington, vol.2,
473–476. 1986.
8. de Vries D.A., Philip J.R.: Soil heat flux, thermal conductivity and the null-alignment
method. Soil Sci. Soc. Am. J., 50,12-18,1986.
9. de Vries D.A.: The theory of heat and moisture transfer in porous media revisited. Int. J.
Heat Mass Transfer. Vol.30, No.7, pp. 1343–1350,.1987.
10. Eadie W.T., Drijard D., James F.E., Roos M., Sadoulet B.: Metody statystyczne w fizyce
doświadczalnej. PWN. W–wa, 63–64, 1989.
11. Gori F., Corasaniti S.: Experimental Measurements and Theoretical Prediction of the
Thermal Conductivity of Two- and Three-Phase Water/Olivine Systems. International
Journal of Thermophysics. 24(5), 1339-1353, 2003.
12. Hopmans J.W., Dane J.H.: Thermal conductivity of two porous media as a function of
water content, temperature and density. Soil Sci., 142 (4), 187-195, 1986.
13. Kasubuchi T.: The effect of soil moisture on thermal properties in same typical Japanese
upland soils. Soil Sci. Plant Nurt. 21, 107–112, 1975.
14. Kersten M.S.: Thermal properties of soils. Bull. 28. Uniwersity of Minnesota. Inst.
Technology, Enginering Experiment Station, 52, 21, 1949.
15. Kimball B.A., Jackson R.D.: Soil heat flux determination: A null-alignment method.
Agric. Meteorol., 15, 1-9, 1975.
16. Kimball B.A., Jackson R.D., Reginato R.J., Nakayama F.S., Idso S.B.: Comparison of
field–measured and calculated soil-heat fluxes. Soil Sci. Soc. Am. J., 40, 18–25, 1976a.
17. Kimball B.A., Jackson R.D., Nakayama F.S., Idso S.B., Reginato R.J.: Soil-heat flux
determination: Temperature gradient method with computed thermal conductivities. Soil
Sci. Soc. Am. J., 40, 25–28, 1976b.
18. Kohnke H., Nakshabandi A.G.: Heat transfer in soils. 8-th Intern. Congress of Soil
Science, Bucharest, 183–193, 1964.
19. Kossowski J.: Charakterystyka cieplnych właściwości warstwy ornej gleby pola
doświadczalnego Felin. Raport z tematu MR II. 08.02.8. Badanie stosunków cieplnych
środowiska glebowego. ( Maszynopis w IA PAN), 1977.
20. Noborio K., McInnes K.J.: Thermal conductivity of salt-affectd soils. Soil Sci.Soc.Am.J.,
57, 329–334. 1993.
21. Noborio K., McInnes K.J., Heilman J.L.: Two-dimensional model for water, heat, and
solute transport in furrow-irrigated soil: I. Theory, II Field evaluation. Soil Sci. Soc. Am.
J., 60, 1001-1021, 1996.
22. Nusier O. and Abu-Hamdeh N. Laboratory techniques to evaluate thermal conductivity for
some soils. Heat and Mass Transfer. 39, 2, 2003, 119 - 123.
23. Ochsner T.E., Horton R. and Ren T.: A new perspective on soil thermal properties. Soil
Sci. Soc. Am. J. 65, 1641-1647, 2001.
24. Pikul J.L. Jr., Allmaras R.R.: A field comparison of null-alignment and mechanistic soil
heat flux. Soil Sci. Soc. Am. J., 48,1207-1214,1984.
25. Sikora E., Kossowski J.: Thermal conductivity and diffusivity estimations of uncompacted
and compacted soils using computing methods. Polish J. Soil Sci., 26(1), 19-26, 1993.
26. Tarnawski V.R., Leong W.H.: Thermal conductivity of soils at very low moisture content
and moderate temperatures. Transport in Porous Media. 41(2):137-147, 2000.
27. Usowicz B.: Studies on the dependence of soil temperature on its moisture in field. Ph.D.
Thesis (in Polish).Academy of Agriculture, Lublin, 1991.
28. Usowicz B.: Statistical - physical model of thermal conductivity in soil. Pol. J. Soil Sci.,
XXV/1, 25–34, 1992.
29. Usowicz B.: A method for the estimation of thermal properties of soil. Int. Agrophysics,
7(1), 27-34, 1993.
30. Usowicz B.: Evaluation of methods for soil thermal conductivity calculations. Int.
Agrophysics, 9(2), 109–113, 1995.
31. Usowicz B., Usowicz L.: ThermalWin beta (Soil thermal properties software package).
Copyright 2004, Institute of Agrophysics, Polish Academy of Sciences, Lublin.
Fly UP