From eq 41 we have
[
]
- ρ ν{T(ξ)} V
f (ξ) = f + s ρ
, 0 < ξ < δ.
(P1)
o
2
10
30
o
From eq 36 we have
f (ξ) = -K1{T(ξ)}P′(ξ) - K2 {T(ξ)}T ′(ξ) ,
0 < ξ < δ.
(P2)
From eq 32 and 49 we have
kT ′(ξ) = -koα o - d1 1d2 [ρ10 - ρ30 ν{T(ξ)}]LVo ,
-
0 < ξ < δ.
(P3)
From eq 33 and 42 we have
k1α1 - koα o = Lfo + [ρ10 - ρ30 ν{T(δ)}]LVo .
(P4)
Boundary conditions are given as
T (0) = 0
(P5)
P (δ ) = Pa
(P6)
P′(δ-) ≥ 0,
Vo ≥ 0
Vo P′(δ-) = 0.
(P7)
and
We will rewrite eq P3 as
ηT ′ = -π1 + πo ν
(43)
where πo and π1 are given as
-
-
πo = d1 1d2 ko 1 ρ30LVo , π1 = α o + πo wo ,
= ρ10 / ρ30 .
(44)
wo
Since ν (T) < wo for T < 0C and αo ≥ 0, T′(ξ) is strictly negative. Hence the function T (ξ) is
invertible for δ ≥ ξ > 0. Integrating eq 43 by using eq P5, we obtain
]-1dT.
T
T(ξ) = -(π1 / η)ξ - (πo/π1)∫ ν[1 - (πo/π1)ν
(45)
0
Integrating eq P2, we obtain
P[ξ(T )] - Pn + (δo / Ko ) fo = - ∫ (K2 / K1)dT -∫
f [ξ(T )](K1T ′)-1 dT.
T
T
(46)
0
0
Setting T = T1 in (46) and using eq P6, we obtain
σ + (δo / Ko ) fo = -∫ (K2 / K1)dT + ∫ f [ξ(T )](K1T ′)-1 dT.
0
0
(47)
T1
T1
Equation 47 provides the functional dependence of T1 on αo and α1 that specifies a given
thermal condition as well as on δo and σ that specifies a given hydraulic condition in terms
of functions and parameters, such as K1, K2, and ν, etc., describing the properties of a given
soil.
8
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