The thermal diffusivity can change as a function of depth and time depending on the soil
layer structure. To simplify calculations, the vertical node increments, ∆zi, are held fixed.
The time increment, ∆t, is allowed to vary according to
⎡ 0.45 ( ∆zi )2 ⎤
∆t
< 0.5 → ∆t = min ⎢
⎥.
kth,i
(4.22)
( ∆zi )
2
kth,i
⎢
⎥
⎣
⎦
Ensuring that the conditions set forth in Equation (4.21) are met, a lower value of 0.45 is
used to calculate the time step. Determination of the soil thermal diffusivity, kth, and other
physical properties needed to solve Equation (4.16) are discussed in Section 4.3. At each
calculation time step, ∆t is updated as necessary depending on the current soil physical
properties.
4.3 Material Properties
To determine the vertical temperature profile in the ground as governed by Equation
(4.16), several physical characteristics of the material are needed. They are the surface
diffusivity, k. The latter two parameters are, among other things, dependent on the
temperature and moisture content of the soil.
Following the recommendations of Farouki (1981), Johanson's method (1975) is used to
calculate the soil thermal conductivity as a function of soil type, porosity (n), dry density
volume (ww), and the Kersten number (Ke). The governing equation is
κ = (κsat κdry)Ke + κdry
(4.23)
where
⎧ provided by user
⎪ 135γ + 64.7
⎪
⎪ 2700 - 947γ 20%
d
κ dry = ⎨
(4.24)
d
⎪
T j,i > 273.15K peat
0.05
⎪
T j,i ≤ 273.15K peat
0.55
⎪
⎩
⎧ 0.57n κ s(1-n )
T j,i > 273.15K
⎪ n (1-n )
⎪2.2 κ s 0.269
T j,i ≤ 273.15K
ww
κ sat = ⎨
(4.25)
T j,i > 273.15K peat
0.55
⎪
⎪
T j,i ≤ 273.15K peat
1.80
⎩
⎧0.7 log Sr + 1.0 T j,i > 273.15K coarse
⎪
Ke = ⎨ log Sr + 1.0 T j,i > 273.15K fine
(4.26)
⎪
Sr T j,i ≤ 273.15K
⎩
35
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