E1C05 09/14/2010
14:36:27 Page 180
Example 5.5
A stopwatch is to be used to estimate the time between the start and end of an event. Event duration
might range from several seconds to 10 minutes. Estimate the probable uncertainty in a time
estimate using a hand-operated stopwatch that claims an accuracy of 1 min/month (95%) and a
resolution of 0.01 s.
KNOWN u 0 ¼ 0:005 s ð95%Þ
u c ¼ 60 s=month ð95% assumedÞ
FIND u d , u N
SOLUTION The design-stage uncertainty gives an estimate of the suitability of an instrument
for a measurement. At 60 s/month, the instrument accuracy works out to about 0.01 s/10 min of
operation. This gives a design-stage uncertainty of
u d ¼ u
2
o þ u
2
c
À
Á 1=2 ¼ Æ0:01 s 95%
ð
Þ
for an event lasting 10 minutes versus Æ0.005 s (95%) for an event lasting 10 s. Note that instrument
calibration error controls the longer duration measurement, whereas instrument resolution controls
the short duration measurement.
But do instrument resolution and calibration error actually control the uncertainty in this measurement? The design-stage analysis does not include the data-acquisition error involved in the act of physically
turning the watch on and off. But a first-order analysis might be run to estimate the uncertainty that enters
through the procedure of using the watch. Suppose a typical trial run of 20 tries of simply turning a watch on
and off suggests that the uncertainty in determining the duration of an occurrence is
u 1 ¼ t n;p s x ¼ 0:05 s
The uncertainty in measuring the duration of an event would then be better estimated by the Nthorder uncertainty of Equation 5.23,
u N ¼ u
2
1 þ u
2
c
À
Á 1=2 ¼ Æ0:05 s 95%
ð
Þ
This estimate holds for periods of up to about two hours. Clearly, procedure controls the uncertainty,
not the watch. This uncertainty estimate could be further improved by considering how well the
operator can synchronize the watch action with the start and finish-line action.
Example 5.6
A flow meter can be calibrated by providing a known flow rate through the meter and measuring the
meter output. One method of calibration with liquid systems is the use of a catch and time technique
whereby a volume of liquid, after passing through the meter, is diverted to a tank for a measured
period of time from which the flow rate volume/time, is computed. There are two procedures that
can be used to determine the known flow rate Q in, say, ft
3 /min:
1. The volume of liquid, 8, caught in known time t can be measured.
Suppose we arbitrarily set t ¼ 6 s and assume that our available facilities can determine
volume (at Nth order) to 0.001 ft
3 . Note: The chosen time value depends on how much
liquid we can accommodate in the tank.
180 Chapter 5 Uncertainty Analysis
14:36:27 Page 180
Example 5.5
A stopwatch is to be used to estimate the time between the start and end of an event. Event duration
might range from several seconds to 10 minutes. Estimate the probable uncertainty in a time
estimate using a hand-operated stopwatch that claims an accuracy of 1 min/month (95%) and a
resolution of 0.01 s.
KNOWN u 0 ¼ 0:005 s ð95%Þ
u c ¼ 60 s=month ð95% assumedÞ
FIND u d , u N
SOLUTION The design-stage uncertainty gives an estimate of the suitability of an instrument
for a measurement. At 60 s/month, the instrument accuracy works out to about 0.01 s/10 min of
operation. This gives a design-stage uncertainty of
u d ¼ u
2
o þ u
2
c
À
Á 1=2 ¼ Æ0:01 s 95%
ð
Þ
for an event lasting 10 minutes versus Æ0.005 s (95%) for an event lasting 10 s. Note that instrument
calibration error controls the longer duration measurement, whereas instrument resolution controls
the short duration measurement.
But do instrument resolution and calibration error actually control the uncertainty in this measurement? The design-stage analysis does not include the data-acquisition error involved in the act of physically
turning the watch on and off. But a first-order analysis might be run to estimate the uncertainty that enters
through the procedure of using the watch. Suppose a typical trial run of 20 tries of simply turning a watch on
and off suggests that the uncertainty in determining the duration of an occurrence is
u 1 ¼ t n;p s x ¼ 0:05 s
The uncertainty in measuring the duration of an event would then be better estimated by the Nthorder uncertainty of Equation 5.23,
u N ¼ u
2
1 þ u
2
c
À
Á 1=2 ¼ Æ0:05 s 95%
ð
Þ
This estimate holds for periods of up to about two hours. Clearly, procedure controls the uncertainty,
not the watch. This uncertainty estimate could be further improved by considering how well the
operator can synchronize the watch action with the start and finish-line action.
Example 5.6
A flow meter can be calibrated by providing a known flow rate through the meter and measuring the
meter output. One method of calibration with liquid systems is the use of a catch and time technique
whereby a volume of liquid, after passing through the meter, is diverted to a tank for a measured
period of time from which the flow rate volume/time, is computed. There are two procedures that
can be used to determine the known flow rate Q in, say, ft
3 /min:
1. The volume of liquid, 8, caught in known time t can be measured.
Suppose we arbitrarily set t ¼ 6 s and assume that our available facilities can determine
volume (at Nth order) to 0.001 ft
3 . Note: The chosen time value depends on how much
liquid we can accommodate in the tank.
180 Chapter 5 Uncertainty Analysis