-
0
h4 (Tσ , α1) = σ - (η / α o ) fo ∫ [1/ K1(s)]ds < h3 (α1)] .
^
(82)
Tσ
Therefore, there exists a unique T1 such that Ts < T1 < Tσ and eq 58 holds true. u
Differentiating eq 76 with respect to α1, we obtain
T1
0
= -αoδo (Ko η)-1 - ∫ [1/ K1(s)]ds .
(L / k1)P′(δ-)
(83)
α`
T1
From eq 83 we find on l(α o )
^
T1
T1
as α1 → α1s .
< 0 , and
→-∞
(84)
α1
α1
As α1 increases from α1e to α1s on l(α o ) , the flux fo increases from zero to fs, while T1
^
decreases from Tσ to Ts. In Proposition 3 α o is an arbitrary positive number. Hence, we may
^
conclude that an ice layer grows in the region Si and on Cs. Below we will study the region
Sp defined as
{
}
Sp = (α1 , α o ): k1[ko + η-1LK2 {Ts (α o )}]-1α1 > α o > 0 .
(85)
FROST PENETRATION
We will seek solutions with a positive Vo in Sp. If such solutions exist, by eq P7 P′ (δ )
vanishes and eq P2 is reduced to
f (δ-) = -K2 (T1)T ′(δ-).
(86)
Using eq P1, 43 and 86, and neglecting sensible heat terms, we obtain
eoρ10 (1 - ν1wo1 )Vo = Y - fo
-
(87)
where
-1
-1
eo = s2 (1 - η-1s3 y),
s3 = ko L / (d1d2 - 1)
(88)
ν1 = ν(T1),
Y = y α o / η,
y = K2 (T1).
(89)
We will write eq P4 as
ρ10 (1 - ν1wo1 )Vo = Q - fo
-
(90)
where
Q = (k1α1 - koα o ) / L.
(91)
It should be noted that eo and Y are functions of T1 because y is a function of T1.
Using eq 87 and 90, we will express Vo and fo in terms of Y and Q as
e1ρ10 (1 - ν1wo1)Vo = Q - Y
-
(92)
e1 fo = Y - eoQ
(93)
where e1 is a positive function of T1 defined as
13
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