E1C08 09/14/2010
14:54:2 Page 360
for conduction errors. Conduction errors should be minimized through appropriate design and
installation of temperature probes. Usually, the physical situation is sufficiently complex to preclude
accurate mathematical description of the measurement errors. Additional information on modeling
conduction errors may be found in Sparrow (14).
Radiation Errors
Consider a temperature probe used to measure a gas temperature. In the presence of significant
radiation heat transfer, the equilibrium temperature of a temperature probe may be different from the
fluid temperature being measured. Because radiation heat transfer is proportional to the fourth power
of temperature, the importance of radiation effects increases as the absolute temperature of the
measuring environment increases. The error due to radiation can be estimated by considering steadystate thermodynamic equilibrium conditions for a temperature sensor. Consider the case where energy
is transferred to/from a sensor by convection from the environment and from/to the sensor by radiation
to a body at a different temperature, such as a pipe or furnace wall. For this analysis conduction is
neglected. A first law analysis of a system containing the probe, at steady-state conditions, yields
q c þ q r ¼ 0
ð8:28Þ
where
q c ¼ convective heat transfer
q r ¼ radiative heat transfer
The heat-transfer components can then be expressed in terms of the appropriate fundamental
relations as
q c ¼ hA s T 1 À T
ð
Þ q r ¼ FA s es T
4
w À T
4
À
Á
ð8:29Þ
Assuming that the surroundings may be treated as a blackbody, the first law for a system consisting
of the temperature probe is
hA s T 1 À T p
À
Á ¼ FA s es T
4
p À T
4
w
ð8:30Þ
A temperature probe is generally small compared to its surroundings, which justifies the assumption
that the surroundings may be treated as black. The radiation error e r is estimated by
e r ¼ T p À T 1
À
Á ¼
Fes
h
T
4
w À T
4
p
ð8:31Þ
where
s ¼ the Stefan-Boltzmann constant (s ¼ 5.669 Â 10
À8 W/m
2 K
4 )
e ¼ emissivity of the sensor
F ¼ radiation view factor
T p ¼ probe temperature
T w ¼ temperature of the surrounding walls
T 1 ¼ fluid temperature being measured
Again, if the sensor is small compared to the scale of the surroundings, the view factor from the
sensor to the surroundings may be taken as 1. Normally, the uncertainty due to radiation error is set
as u ¼ e r . The uncertainty interval is not symmetrical and might be modeled as a uniform
(rectangular) distribution with bounds of 0 and e r .
360 Chapter 8 Temperature Measurements
14:54:2 Page 360
for conduction errors. Conduction errors should be minimized through appropriate design and
installation of temperature probes. Usually, the physical situation is sufficiently complex to preclude
accurate mathematical description of the measurement errors. Additional information on modeling
conduction errors may be found in Sparrow (14).
Radiation Errors
Consider a temperature probe used to measure a gas temperature. In the presence of significant
radiation heat transfer, the equilibrium temperature of a temperature probe may be different from the
fluid temperature being measured. Because radiation heat transfer is proportional to the fourth power
of temperature, the importance of radiation effects increases as the absolute temperature of the
measuring environment increases. The error due to radiation can be estimated by considering steadystate thermodynamic equilibrium conditions for a temperature sensor. Consider the case where energy
is transferred to/from a sensor by convection from the environment and from/to the sensor by radiation
to a body at a different temperature, such as a pipe or furnace wall. For this analysis conduction is
neglected. A first law analysis of a system containing the probe, at steady-state conditions, yields
q c þ q r ¼ 0
ð8:28Þ
where
q c ¼ convective heat transfer
q r ¼ radiative heat transfer
The heat-transfer components can then be expressed in terms of the appropriate fundamental
relations as
q c ¼ hA s T 1 À T
ð
Þ q r ¼ FA s es T
4
w À T
4
À
Á
ð8:29Þ
Assuming that the surroundings may be treated as a blackbody, the first law for a system consisting
of the temperature probe is
hA s T 1 À T p
À
Á ¼ FA s es T
4
p À T
4
w
ð8:30Þ
A temperature probe is generally small compared to its surroundings, which justifies the assumption
that the surroundings may be treated as black. The radiation error e r is estimated by
e r ¼ T p À T 1
À
Á ¼
Fes
h
T
4
w À T
4
p
ð8:31Þ
where
s ¼ the Stefan-Boltzmann constant (s ¼ 5.669 Â 10
À8 W/m
2 K
4 )
e ¼ emissivity of the sensor
F ¼ radiation view factor
T p ¼ probe temperature
T w ¼ temperature of the surrounding walls
T 1 ¼ fluid temperature being measured
Again, if the sensor is small compared to the scale of the surroundings, the view factor from the
sensor to the surroundings may be taken as 1. Normally, the uncertainty due to radiation error is set
as u ¼ e r . The uncertainty interval is not symmetrical and might be modeled as a uniform
(rectangular) distribution with bounds of 0 and e r .
360 Chapter 8 Temperature Measurements