E1C08 09/14/2010
14:53:57 Page 328
The measured values of b in Example 8.5 are different at each value of temperature. If b were
truly a temperature-independent constant, and these measurements had negligible uncertainty, all
three measurements would yield the same value of b. The variation in b may be due to a physical
effect of temperature, or may be attributable to the uncertainty in the measured values.
Are the measured differences significant, and if so, what value of b best represents the behavior
of the thermistor over this temperature range? To perform the necessary uncertainty analysis,
additional information must be provided concerning the instruments and procedures used in the
measurement.
Example 8.6
Perform an uncertainty analysis to determine the uncertainty in each measured value of b in Example 8.5,
and evaluate a single best estimate of b for this temperature range. The measurement of b involves the
measurement of voltages, temperatures, and resistances. For temperature there is a random error
associated with spatial and temporal variations in the oven temperature with a random standard uncertainty
of s T ¼ 0:19
C for 20 measurements. In addition, based on a manufacturer’s specification, there is a
known measurement systematic uncertainty for temperature of 0.36
C (95%) in the thermocouple.
The systematic errors in measuring resistance and voltage are negligible, and estimates of the
instrument repeatability, which are based on the manufacturer’s specifications in the measured
values and assumed to be at a 95% confidence level, are assigned systematic uncertainties of 1.5%
for resistance and 0.002 V for the voltage.
KNOWN Standard deviation of the means for oven temperature, s T ¼ 0:19
C; N ¼ 20. The
remaining errors are assigned systematic uncertainties at 95% confidence assuming large degrees of
freedom, such that B x ¼ 2b x :
B T ¼ 2b T ¼ 0:36
C
B R =R ¼ 2b R
ð Þ=R ¼ 1:5%
B E ¼ 2b E ¼ 0:002 V
FIND The uncertainty in b at each measured temperature, and a best estimate for b over the
measured temperature range.
SOLUTION Consider the problem of providing a single best estimate of b. One method of
estimation might be to average the three measured values. This results in a value of 3609 K.
However, since the relationship between R T =R 0
ð
Þand 1=T À 1=T 0
ð
Þis expected to be linear, a leastsquares fit can be performed on the three data points, and include the point (0, 0). The resulting value
of b is 3638 K. Is this difference significant, and which value best represents the behavior of the
thermistor? To answer these questions, an uncertainty analysis is performed for b.
For each measured value,
b ¼
ln R T =R 0
ð
Þ
1=T À 1=T 0
Uncertainties in voltage, temperature, and resistance are propagated into the resulting value of b for
each measurement.
328 Chapter 8 Temperature Measurements
14:53:57 Page 328
The measured values of b in Example 8.5 are different at each value of temperature. If b were
truly a temperature-independent constant, and these measurements had negligible uncertainty, all
three measurements would yield the same value of b. The variation in b may be due to a physical
effect of temperature, or may be attributable to the uncertainty in the measured values.
Are the measured differences significant, and if so, what value of b best represents the behavior
of the thermistor over this temperature range? To perform the necessary uncertainty analysis,
additional information must be provided concerning the instruments and procedures used in the
measurement.
Example 8.6
Perform an uncertainty analysis to determine the uncertainty in each measured value of b in Example 8.5,
and evaluate a single best estimate of b for this temperature range. The measurement of b involves the
measurement of voltages, temperatures, and resistances. For temperature there is a random error
associated with spatial and temporal variations in the oven temperature with a random standard uncertainty
of s T ¼ 0:19
C for 20 measurements. In addition, based on a manufacturer’s specification, there is a
known measurement systematic uncertainty for temperature of 0.36
C (95%) in the thermocouple.
The systematic errors in measuring resistance and voltage are negligible, and estimates of the
instrument repeatability, which are based on the manufacturer’s specifications in the measured
values and assumed to be at a 95% confidence level, are assigned systematic uncertainties of 1.5%
for resistance and 0.002 V for the voltage.
KNOWN Standard deviation of the means for oven temperature, s T ¼ 0:19
C; N ¼ 20. The
remaining errors are assigned systematic uncertainties at 95% confidence assuming large degrees of
freedom, such that B x ¼ 2b x :
B T ¼ 2b T ¼ 0:36
C
B R =R ¼ 2b R
ð Þ=R ¼ 1:5%
B E ¼ 2b E ¼ 0:002 V
FIND The uncertainty in b at each measured temperature, and a best estimate for b over the
measured temperature range.
SOLUTION Consider the problem of providing a single best estimate of b. One method of
estimation might be to average the three measured values. This results in a value of 3609 K.
However, since the relationship between R T =R 0
ð
Þand 1=T À 1=T 0
ð
Þis expected to be linear, a leastsquares fit can be performed on the three data points, and include the point (0, 0). The resulting value
of b is 3638 K. Is this difference significant, and which value best represents the behavior of the
thermistor? To answer these questions, an uncertainty analysis is performed for b.
For each measured value,
b ¼
ln R T =R 0
ð
Þ
1=T À 1=T 0
Uncertainties in voltage, temperature, and resistance are propagated into the resulting value of b for
each measurement.
328 Chapter 8 Temperature Measurements