x - X - (GX + ∆T ) ( x - X )
2
[
]
T2 = Tf + G(δ + 2 X ) + 2∆T
(C7)
δ
δ2
where
α 21(∆T + GX )X
∆T1
a1X =
g=
+ 1.
[
]
,
δ G(δ + 2 X ) + 2∆T
g
In general, the simplest temperature profiles that will satisfy the boundary conditions should be chosen.
The accuracy of the method increases as the order of a polynomial temperature choice increases; however,
the use of high-order polynomials (third and higher) is often not justified since a small increase in accuracy
requires significantly more computational effort. Equation C5 can be used to find a relation between X and
δ. In nondimensional form this is
2ρ β(g - 1)
β
[
]
- k21 σ(β + 2) + 2φ = 21
(C8)
.
ST
g
The heat balance integral forms for eq C1 and C2 are as follows.
X
T ( X, t)
T (0, t)
- U[T1( X, t) - T1(0, t)] -
∫
d
dx
T1( x, t)dx + Tf
- 1
α1 1
=0
(C9)
x
x
dt
dt
0
T ( X + δ, t)
T ( X, t)
[
]
- U T2 ( X + δ, t) - T2 ( X, t) -
- 2
α2 2
x
x
(C10)
d( X + δ)
X+δ
∫
d
dX
T2 ( x, t)dx + T2 ( X + δ)
- Tf
= 0.
dt
dt
dt
X
These two energy equations are summed, along with eq C3, to yield an integrated equation for the entire
region undergoing temperature changes. The result is
d
X+δ
X
T2 ( x, t)dx - ρ1lX + (ρ2c2 - ρ1c1 )Tf X - ρ2c2 ( X + δ)To + ( X + δ)
G
ρ1c1 ∫ T1( x, t)dx + ρ2c2
∫
dt
2
X
0
T1(0, t)
[
]
+ k2G - ρ1c1U∆T1 - ρ2c2U ∆T + G( X + δ) .
= -k1
(C11)
x
Equation C11, the energy integral equation, can now be written nondimensionally as
σ
∫ K1dσ
τ=
(C12)
0
1
σg′
2
C21 σ + φ σβ′
(
)
b1 + b2β -
1 -
- C21σ β + 1 -
6g
g
3
3
K1 =
(C13)
1 1
{
]}
[
- 2 + k21 - ψ 1 + 1 / ST + C21 φ + σ(β + 1)
σg
where
U∆T1
ψ=
Gα1
30
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