and minimum temperature is proportional to the depth z, of the order of (zP/2pD).
Equation (6.10) indicates that the soil maximum temperature corresponds to the
phase angle p/2 and the minimum to −p/2. These results are valid for the propagation of maximum and minimum temperature waves through a homogeneous
medium, only if the thermal diffusivity of the substrate layer remains constant
throughout the entire period and that the surface temperature variation is sinusoidal.
The thickness of the damping layer for an annual thermal wave is of the order of the
product of 365
1/2 with the corresponding D. For example, dry sandy soil with depth
D is about 0.082 m for daytime and 1.57 m for the annual wave.
The theoretical heat flux is given e.g. by Arya (1988)
H G ¼ Àk
@T
@z
z¼0
¼ 2p
qck
P
1=2
A s sin
2p
P
t À t m
ð
Þþ
p
4
!
ð6:12Þ
showing that the amplitude of the heat flow is proportional to the square root of the
product of the heat capacity and thermal conductivity and inversely proportional to
the square root of the period. From Eq. (6.12), it can be deduced that the time
instant corresponding to the maximum surface temperature, lagged the corresponding to maximum energy flow by P/8, or by about 3 h during the daytime, and
1.5 months for the year. The real conditions in soil layers differ from theoretical
ones in heat transfer equations above the soil. The daytime variation of soil surface
temperature may deviate from the theoretical wave profile due to factors such as soil
moisture, plant root systems, or factors relating to the water regime such as irrigation, precipitation, and evaporation.
6.1.3 Thermal Properties of Soils
The thermal properties of the soil relevant to heat transfer through a particular
medium, and their effect on the temperature distribution are the density, specific
heat, heat capacity and thermal conductivity. Most soils consist of particles of
varying dimensions and materials with a high degree of porosity. This can be filled
by air or water so that the thermal properties vary depending on these factors.
Table 6.2 shows some thermal properties of constituent materials of the soil. The
determination of soil heat flux according to the Fourier Law (Eq. 6.2) is not
practical due to the high atmospheric temperature vertical gradient in the top thin
layer of the soil. The ground heat flux can, however, be estimated at the surface
based on methodologies relying on measurements of soil heat flux with flux plates
(Foken 2017).
The apparent soil density q′ is given by
q
0 ¼ q s x s þ q l x g þ q g x g
ð6:13Þ
6.1 Conduction
165
Equation (6.10) indicates that the soil maximum temperature corresponds to the
phase angle p/2 and the minimum to −p/2. These results are valid for the propagation of maximum and minimum temperature waves through a homogeneous
medium, only if the thermal diffusivity of the substrate layer remains constant
throughout the entire period and that the surface temperature variation is sinusoidal.
The thickness of the damping layer for an annual thermal wave is of the order of the
product of 365
1/2 with the corresponding D. For example, dry sandy soil with depth
D is about 0.082 m for daytime and 1.57 m for the annual wave.
The theoretical heat flux is given e.g. by Arya (1988)
H G ¼ Àk
@T
@z
z¼0
¼ 2p
qck
P
1=2
A s sin
2p
P
t À t m
ð
Þþ
p
4
!
ð6:12Þ
showing that the amplitude of the heat flow is proportional to the square root of the
product of the heat capacity and thermal conductivity and inversely proportional to
the square root of the period. From Eq. (6.12), it can be deduced that the time
instant corresponding to the maximum surface temperature, lagged the corresponding to maximum energy flow by P/8, or by about 3 h during the daytime, and
1.5 months for the year. The real conditions in soil layers differ from theoretical
ones in heat transfer equations above the soil. The daytime variation of soil surface
temperature may deviate from the theoretical wave profile due to factors such as soil
moisture, plant root systems, or factors relating to the water regime such as irrigation, precipitation, and evaporation.
6.1.3 Thermal Properties of Soils
The thermal properties of the soil relevant to heat transfer through a particular
medium, and their effect on the temperature distribution are the density, specific
heat, heat capacity and thermal conductivity. Most soils consist of particles of
varying dimensions and materials with a high degree of porosity. This can be filled
by air or water so that the thermal properties vary depending on these factors.
Table 6.2 shows some thermal properties of constituent materials of the soil. The
determination of soil heat flux according to the Fourier Law (Eq. 6.2) is not
practical due to the high atmospheric temperature vertical gradient in the top thin
layer of the soil. The ground heat flux can, however, be estimated at the surface
based on methodologies relying on measurements of soil heat flux with flux plates
(Foken 2017).
The apparent soil density q′ is given by
q
0 ¼ q s x s þ q l x g þ q g x g
ð6:13Þ
6.1 Conduction
165