Estimation of Thermodynamic and Transport Properties …
239
θ f
˜
t
=
˜
t
C R + 1
+
˜
R c + 1/3
(C R + 1)
2
− 2
∞
m=1
e
−β
2
m
˜
t
˜
N m β 2
m
˜
t > 0
(3b)
where the dimensionless norm ˜
N m is
˜
N m =
C R ˜
R c
2 β
4
m +
C R + C R ˜
R c − 2 ˜
R c
C R β
2
m + C R + 1,
(3c)
and the β m eigenvalues satisfy the following eigen condition
C R β m cos(β m ) −
C R ˜
R c β
2
m − 1
sin(β m ) = 0
( 3 d )
Then, the temperature solution to the X62B50T00 problem is obtained by means
of the superposition principle for linear cases as shown below.
θ =
θ
˜
x, ˜
t
0 ≤ ˜
t ≤ ˜
t h
θ
˜
x, ˜
t
− θ
˜
x, ˜
t − ˜
t h
˜
t > ˜
t h
(4a)
θ f =
θ f
˜
t
0 ≤ ˜
t ≤ ˜
t h
θ f
˜
t
− θ f
˜
t − ˜
t h
˜
t > ˜
t h
(4b)
Also, when ˜
R c → 0 (perfect contact), the X62B10T00 problem tends to the
X42B10T00 case. Equations (3a)–(3d) reduce to [10]
θ
˜
x, ˜
t
=
˜
t
˜
N 0
+
˜
x
2
2 ˜
N 0
−
˜
x
˜
N 0
+
1
3 ˜
N
2
0
−
∞
m=1
cos(β m ˜
x) − C R β m sin(β m ˜
x)
˜
N m β 2
m
e
−β
2
m
˜
t
(5a)
θ f
˜
t
= θ
0, ˜
t
˜
t > 0
(5b)
where the non-dimensional norm ˜
N m and the eigenvalues β m are, respectively,
˜
N m =
C R + 1
m = 0
(CR β m )
2 +C R +1
2
m = 1, 2, . . .
(5c)
β m cot(β m ) = −C
−1
R
m = 0, 1, 2, . . .
(5d)
Note that the X42B50T00 solution can be obtained by using Eqs. (4a), (4b).