α 21(∆T + GX )X
∆T1
a1 X =
,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 highorder polynomials (third and higher) is often not justified since a small increase in accuracy
requires significantly more computational effort. Equation 4 can be used to find a relation between X and d.
In nondimensional form this is
2ρ21β( g  1)
β
[
]
 k21 σ(β + 2) + 2φ =
.
(8)
g
ST
Equation 5, the energy integral equation, can now be written nondimensionally, using eq 6 and 7 as
σ
τ = ∫ Kdσ
(9)
0
1
σg′
2
C21 σ + φ σβ′
(
)
b1 + b2β 
 C21σ β + 1 
1 
6g
g
3
3
K=
(10)
1 1
 2 + k21
σg
where
1 1
b1 =  +
+ C21φ
3 ST
1
b2 =  C21φ .
3
The derivatives of β and g can be found from the following equations
a +a a
dβ
= β′ = 5 1 4 .
(11)
dσ
a3 + a2a4
dg
= g′ = a1  a2β′ .
(12)
dσ
where
(σ + φ) β(β + 2)
α 21
a1 =
1 
m
m
α 21(σ + φ)
[2σ(β + 1) + 2φ]
a2 =
m2
1 2ρ21(g  1)
a3 =

 k21σ
g
ST
2ρ
1
a4 = 21 + 2 β
ST
g
9