E1C05 09/14/2010
14:36:29 Page 191
In a similar manner, we assign the uncertainty in the data-acquisition source error in temperature.
The instrument errors are considered as systematic only and assigned a systematic uncertainty based
on the manufacturer’s statement as
b T
À Á
1
¼ b T
À Á
1
=2 ¼ 0:3
R
s T
À Á
1
¼ 0
The temporal variation causes a random uncertainty in establishing the mean value of temperature
and this is calculated as
s T
À Á
2
¼ s T ¼
s T
ffiffiffiffi
N
p ¼
3:0
R
ffiffiffiffiffi
10
p
¼ 0:9
R
v s T ¼ 9
b T
À Á
2
¼ 0
The uncertainties in pressure and temperature due to the data-acquisition source errors are combined
using the RSS method,
b p ¼ 11:28
ð
Þ
2 þ 0
ð Þ
2
h
i 1=2 ¼ 11:28 psfa
s p ¼ 0
ð Þ
2 þ 37:39
ð
Þ
2
h
i 1=2 ¼ 37:39 psfa
similarly,
b T ¼ 0:3
R
s T ¼ 0:9
R
with degrees of freedom in the random standard uncertainties to be
v
ð Þ s p ¼ N p À 1 ¼ 19
v
ð Þ s T ¼ 9
The degrees of freedom in the systematic standard uncertainties, v b T and v b p , are assumed large.
The systematic and random standard uncertainties propagate through to the result, calculated
about the operating point as established by the mean values for temperature and pressure. That is,
s r ¼
qr
qT
s T
2
þ
qr
qp
s p
2
"
# 1=2
¼ 1:3 Â 10
À4
 0:9
À
Á 2 þ 3:26 Â 10
À5
 37:39
À
Á 2
h
i 1=2
¼ 0:0012 lb m /ft
3
and
b r ¼
qr
qT
b T
2
þ
qr
qp
b p
2
"
# 1=2
¼ 1:3 Â 10
À4
 0:3
À
Á 2 þ 3:26 Â 10
À5
 11:28
À
Á 2
h
i 1=2
¼ 0:0004 lb m /ft
3
5.8 Multiple-Measurement Uncertainty Analysis 191