Estimation of Thermodynamic and Transport Properties …
241
˜
C
+
kk =
k
2 C kk /N
2(T max − T in )
2
≈
1
2θ 2
max
˜
t N
2
s=1
˜
t N
0
˜
X
+
s,k
2
d˜ t
(8b)
˜
C
+
CC =
C
2 C CC /N
2(T max − T in )
2
≈
1
2θ 2
max
˜
t N
2
s=1
˜
t N
0
˜
X
+
s,C
2
d˜ t
(8c)
˜
C
+
kC =
kCC kC /N
2(T max − T in )
2
≈
1
2θ 2
max
˜
t N
2
s=1
˜
t N
0
˜
X
+
s,k
˜
X
+
s,C d˜ t
(8d)
The factor of 2 appearing on the denominator of the equations listed before
accounts for the number of sensors utilized when performing the experiment, that
is another practical experimental condition. Also, ˜
X
+
s,k = X
+
s,k /(q
f,0 L/k) and
˜
X s,C = X
+
s,C /(q
f,0 L/k) denote the non-dimensional scaled sensitivity coefficients
of temperature with respect to k and C, respectively, at the s-th sensor position (s =
1 or 2), with X
+
s,k = k X s,k and X
+
s,C = C X s,C , and θ max is computed as follows
θ max =
T max − T in
q
f,0 L/k
=
˜
T f
˜
t N
for ˜
t N < ˜
t h
˜
T f
˜
t h
for ˜
t N ≥ ˜
t h
(9)
The optimal experiment aimed at estimating k and C under X62B50T00 (imperfect contact) and X42B50T00 (perfect contact) models is defined by Figs. 3 and 4,
respectively, when C R = 0.05.
Fig. 3 + criterion for simultaneous estimation of k and C under X62B50T00 model ( ˜
R c = 0.5)