where x 1 ¼ e n ; x 2 ¼ J n ; x 3 ¼ P n ðp s Þ; x 4 ¼ ~
T n :
Then using the function T S (x 1 ,. . .,x 4 ) to the Tailor expansion and ignoring items
containing dx 1 is larger in power then unit the expression is derived for estimating
the absolute error of the temperature T S :
DT s
j
j ¼
X 4
i¼1
@T s
@x i
x i ¼ x i ;
i¼1;:::;4
Á Dx i
j j ;
(8.10)
where Dx i is the uncertainty of the parameter x i measurements;
x i is the exact value
of the i-th parameter.
Calculation of corresponded derivatives leads to the relation:
DT s
j
j ¼ L n P n p s
ð ÞT s De n
j jþ
L n
K n
e n P n p s
ð ÞT s DJ n
j j
þ L n e n 1 À P n p s
ð Þ
B n; ~
T n
Â
Ã
K n
"
#
T DP n p s
ð Þ
j
j
þ e n P n p s
ð Þ 1 À P n p s
ð Þ
½
Šexp
bn
~
T n
B n; ~
T n
Â
à 2 T
2
s D ~
T n
D n K 2
n
~
T 2
n
(8.11)
Following notations (symbols) are introduced in the Eq. 8.11:
D n ¼ 1 þ e n P n p s
ð Þ
an
3
K n
;
(8.12)
L n ¼
an
3
K n D n ln D n
ð Þ
:
(8.13)
Different approaches might be used for estimating the value DT s
j
j.
The first approach. It is necessary to specify for estimating the influence of a
certain parameter a on the value DT s
j
j the exact value of the parameter a ¼ a 0 and
the measured value containing an uncertainty (error)
a 1 ¼ a 0 À Da 0
(8.14)
After calculation T s (a 0 ) and T s (a 1 ) the desired uncertainty DT s
j
j might be found
from the relation:
DT
j j a 0 ;Da
¼ T s a 1
ð Þ À T s a 0
ð Þ
j
j :
(8.15)
8.2 Analytical Approaches to the Estimation of Uncertainty of the Surface Temperature
75