E1C05 09/14/2010
14:36:33 Page 205
Location:
1
2
3
4
d (in.):
4.494
4.499
4.511
4.522
s d :
0.006
0.009
0.010
0.003
5.36 The pressure in a large vessel is to be maintained at some set pressure for a series of tests. A
compressor is to supply air through a regulating valve that is set to open and close at the set pressure.
A dial gauge (resolution: 1 psi; uncertainty: 0.5 psi) is used to monitor pressure in the vessel. Thirty
trial replications of pressurizing the vessel to a set pressure of 50 psi are attempted to estimate
pressure controllability and a standard deviation in set pressure of 2 psi is found. Estimate the
uncertainty to be expected in the vessel set pressure. Compare to the uncertainty in estimating the
average set pressure.
5.37 A handbook value for the thermal expansion coefficient of a particular pure metal is a ¼ 16:52 Â
10
À6
C
À1 and states that the possible error in this value should not exceed 0:4 Â 10
À6
C
À1 .
Assuming that the population for a follows a rectangular distribution, estimate the standard
uncertainty to be assigned.
5.38 The cooling of a thermometer (e.g., Exs. 3.3 and 3.4) can be modeled as a first-order system with
G ¼ e
Àt=t . If G can be measured to within 2% and time within 1%, determine the uncertainty in t over
the range 0 G 1.
5.39 A J-type thermocouple monitors the temperature of air flowing through a duct. Its signal is
measured by a thermostat. The air temperature is maintained constant by an electric heater whose
power is controlled by the thermostat. To test the control system, 20 measurements of temperature
were taken over a reasonable time period during steady operation. This was repeated three times
with the results as:
Run
N
T
C
½
s T [
C]
1
20
181.0
3.01
2
20
183.1
2.84
3
20
182.1
3.08
The thermocouple itself has an instrument error with uncertainty Æ1
C (95%) with n ¼ 30. It has a
90% rise time of 20 ms. Thermocouple insertion errors are estimated to be Æ1.2
C (95%). What
information is found by performing replications? Identify the elemental errors that affect the
system’s control of the air temperature. What is the uncertainty in the set temperature?
5.40 Based on everyday experience, give an estimate of the systematic uncertainty you might expect in the
following measuring instruments: bathroom scale, metal tape ruler, micrometer, kitchen window
bulb thermometer, and automobile speedometer.
5.41 A tank is pressurized with air at 25 Æ 2
C (95%). Determine the relative uncertainty in the tank’s air
density if tank pressure is known to within Æ 1 % (95 %). Assume ideal gas behavior.
5.42 The density of air must be known to within Æ0.5%. If the air temperature at 25
C can be determined
to within Æ1
C (95%), what uncertainty can be tolerated in the pressure measurement if air behaves
as an ideal gas?
5.43 In pneumatic conveying, solid particles such as flour or coal are carried through a duct by a moving
air stream. Solids density at any duct location can be measured by passing a laser beam of known
Problems 205
14:36:33 Page 205
Location:
1
2
3
4
d (in.):
4.494
4.499
4.511
4.522
s d :
0.006
0.009
0.010
0.003
5.36 The pressure in a large vessel is to be maintained at some set pressure for a series of tests. A
compressor is to supply air through a regulating valve that is set to open and close at the set pressure.
A dial gauge (resolution: 1 psi; uncertainty: 0.5 psi) is used to monitor pressure in the vessel. Thirty
trial replications of pressurizing the vessel to a set pressure of 50 psi are attempted to estimate
pressure controllability and a standard deviation in set pressure of 2 psi is found. Estimate the
uncertainty to be expected in the vessel set pressure. Compare to the uncertainty in estimating the
average set pressure.
5.37 A handbook value for the thermal expansion coefficient of a particular pure metal is a ¼ 16:52 Â
10
À6
C
À1 and states that the possible error in this value should not exceed 0:4 Â 10
À6
C
À1 .
Assuming that the population for a follows a rectangular distribution, estimate the standard
uncertainty to be assigned.
5.38 The cooling of a thermometer (e.g., Exs. 3.3 and 3.4) can be modeled as a first-order system with
G ¼ e
Àt=t . If G can be measured to within 2% and time within 1%, determine the uncertainty in t over
the range 0 G 1.
5.39 A J-type thermocouple monitors the temperature of air flowing through a duct. Its signal is
measured by a thermostat. The air temperature is maintained constant by an electric heater whose
power is controlled by the thermostat. To test the control system, 20 measurements of temperature
were taken over a reasonable time period during steady operation. This was repeated three times
with the results as:
Run
N
T
C
½
s T [
C]
1
20
181.0
3.01
2
20
183.1
2.84
3
20
182.1
3.08
The thermocouple itself has an instrument error with uncertainty Æ1
C (95%) with n ¼ 30. It has a
90% rise time of 20 ms. Thermocouple insertion errors are estimated to be Æ1.2
C (95%). What
information is found by performing replications? Identify the elemental errors that affect the
system’s control of the air temperature. What is the uncertainty in the set temperature?
5.40 Based on everyday experience, give an estimate of the systematic uncertainty you might expect in the
following measuring instruments: bathroom scale, metal tape ruler, micrometer, kitchen window
bulb thermometer, and automobile speedometer.
5.41 A tank is pressurized with air at 25 Æ 2
C (95%). Determine the relative uncertainty in the tank’s air
density if tank pressure is known to within Æ 1 % (95 %). Assume ideal gas behavior.
5.42 The density of air must be known to within Æ0.5%. If the air temperature at 25
C can be determined
to within Æ1
C (95%), what uncertainty can be tolerated in the pressure measurement if air behaves
as an ideal gas?
5.43 In pneumatic conveying, solid particles such as flour or coal are carried through a duct by a moving
air stream. Solids density at any duct location can be measured by passing a laser beam of known
Problems 205