E1C05 09/14/2010
14:36:28 Page 190
Example 5.13
The density of a gas, r, which follows the ideal gas equation of state, r ¼ p=RT, is estimated through
separate measurements of pressure p and temperature T. The gas is housed within a rigid impermeable
vessel. The literature accompanying the pressure measurement system states an instrument uncertainty to within Æ1% of the reading (95%), and that accompanying the temperature measuring system
indicates Æ0.6
R (95%) uncertainty. Twenty measurements of pressure, N p ¼ 20, and 10 measurements of temperature, N T ¼ 10, are made with the following statistical outcome:
p ¼ 2253:91 psfa s p ¼ 167:21 psfa
T ¼ 560:4
R
s T ¼ 3:0
R
where psfa refers to lb/ft
2 absolute. Determine a best estimate of the density. The gas constant is
R ¼ 54.7 ft-lb/lb m -
R.
KNOWN p; s p ; T; s T
r ¼ p/RT; R ¼ 54:7 ft-lb/lb m -
R
ASSUMPTIONS Gas behaves as an ideal gas.
FIND r
0
¼ r Æ u r 95%
ð
Þ
SOLUTION The measurement objective is to determine the density of an ideal gas through
temperature and pressure measurements. The independent and dependent variables are related
through the ideal gas law, r ¼ p=RT. The mean value of density is
r ¼
p
RT
¼ 0:0735 lb m =ft
3
The next step must be to identify and estimate the errors and determine how they contribute to the
uncertainty in the mean value of density. Since no calibrations are performed and the gas is
considered to behave as an ideal gas in an exact manner, the measured values of pressure and
temperature are subject only to elemental errors within the data-acquisition error source group
(Table 5.2): instrument errors and temporal variation errors.
The tabulated value of the gas constant is not without error. However, estimating the possible
error in a tabulated value is sometimes difficult. According to Kestin (8), the systematic uncertainty
in the gas constant is on the order of Æ(0.33 J/kg-K)/(gas molecular weight) or Æ0.06 (ft-lb/lb m -
R)/
(gas molecular weight). Since this yields a small value for a reasonable gas molecular weight, here
we assume a zero (negligible) systematic error in the gas constant.
Consider the pressure measurement. The uncertainty assigned to the temporal variation (data
scatter) error is based on the variation in the measured data obtained during presumably fixed operating
conditions. The instrument error is assigned a systematic uncertainty based on the manufacturer’s
statement, which is assumed to be stated at 95% confidence:
b p
À Á
1
¼ B p =2
À
Á
1
¼ 0:01 Â 2253:51=2Þ ¼ 11:28 psfa
s p
À Á
1
¼ 0
À
where the subscript keeps track of the error identity. The temporal variation causes a random
uncertainty in establishing the mean value of pressure and is calculated as
s p
À Á
2
¼ s p ¼
s p
ffiffiffiffi
N
p ¼
167:21 psfa
ffiffiffiffiffi
20
p
¼ 37:39 psfa
v s p ¼ 19
and assigning no systematic uncertainty to this error gives
b p
À Á
2
¼ 0
190 Chapter 5 Uncertainty Analysis
14:36:28 Page 190
Example 5.13
The density of a gas, r, which follows the ideal gas equation of state, r ¼ p=RT, is estimated through
separate measurements of pressure p and temperature T. The gas is housed within a rigid impermeable
vessel. The literature accompanying the pressure measurement system states an instrument uncertainty to within Æ1% of the reading (95%), and that accompanying the temperature measuring system
indicates Æ0.6
R (95%) uncertainty. Twenty measurements of pressure, N p ¼ 20, and 10 measurements of temperature, N T ¼ 10, are made with the following statistical outcome:
p ¼ 2253:91 psfa s p ¼ 167:21 psfa
T ¼ 560:4
R
s T ¼ 3:0
R
where psfa refers to lb/ft
2 absolute. Determine a best estimate of the density. The gas constant is
R ¼ 54.7 ft-lb/lb m -
R.
KNOWN p; s p ; T; s T
r ¼ p/RT; R ¼ 54:7 ft-lb/lb m -
R
ASSUMPTIONS Gas behaves as an ideal gas.
FIND r
0
¼ r Æ u r 95%
ð
Þ
SOLUTION The measurement objective is to determine the density of an ideal gas through
temperature and pressure measurements. The independent and dependent variables are related
through the ideal gas law, r ¼ p=RT. The mean value of density is
r ¼
p
RT
¼ 0:0735 lb m =ft
3
The next step must be to identify and estimate the errors and determine how they contribute to the
uncertainty in the mean value of density. Since no calibrations are performed and the gas is
considered to behave as an ideal gas in an exact manner, the measured values of pressure and
temperature are subject only to elemental errors within the data-acquisition error source group
(Table 5.2): instrument errors and temporal variation errors.
The tabulated value of the gas constant is not without error. However, estimating the possible
error in a tabulated value is sometimes difficult. According to Kestin (8), the systematic uncertainty
in the gas constant is on the order of Æ(0.33 J/kg-K)/(gas molecular weight) or Æ0.06 (ft-lb/lb m -
R)/
(gas molecular weight). Since this yields a small value for a reasonable gas molecular weight, here
we assume a zero (negligible) systematic error in the gas constant.
Consider the pressure measurement. The uncertainty assigned to the temporal variation (data
scatter) error is based on the variation in the measured data obtained during presumably fixed operating
conditions. The instrument error is assigned a systematic uncertainty based on the manufacturer’s
statement, which is assumed to be stated at 95% confidence:
b p
À Á
1
¼ B p =2
À
Á
1
¼ 0:01 Â 2253:51=2Þ ¼ 11:28 psfa
s p
À Á
1
¼ 0
À
where the subscript keeps track of the error identity. The temporal variation causes a random
uncertainty in establishing the mean value of pressure and is calculated as
s p
À Á
2
¼ s p ¼
s p
ffiffiffiffi
N
p ¼
167:21 psfa
ffiffiffiffiffi
20
p
¼ 37:39 psfa
v s p ¼ 19
and assigning no systematic uncertainty to this error gives
b p
À Á
2
¼ 0
190 Chapter 5 Uncertainty Analysis