E1C08 09/14/2010
14:54:2 Page 361
Example 8.13
A typical situation where radiation would be important occurs in measuring the temperature of a
furnace. Figure 8.36 shows a small temperature probe for which conduction errors are negligible,
which is placed in a high-temperature enclosure, where the fluid temperature is T 1 and the walls of
the enclosure are at T w .
Convection and radiation are assumed to be the only contributing heat transfer modes at steady
state. Develop an expression for the equilibrium temperature of the probe. Also determine the
equilibrium temperature of the probe and the radiation error in the case where T 1 ¼ 800
C,
T w ¼ 500
C, and the emissivity of the probe is 0.8. The convective heat transfer coefficient is 100
W/m
2 C.
KNOWN Temperatures T 1 ¼ 800
C and T w ¼ 500
C, with h ¼ 100 W/m
2 C. The emissivity
of the probe is 0.8.
FIND An expression for the equilibrium temperature of the probe, and the resulting probe
temperature for the stated conditions.
ASSUMPTIONS The surroundings may be treated as black, and conduction heat transfer is
neglected.
SOLUTION The probe is modeled as a small, spherical body within the enclosed furnace; the
view factor from the probe to the furnace is 1. At steady state, an equilibrium temperature may be
found from an energy balance. The first law for a system consisting of the temperature probe from
Equation 8.30 is
h T 1 À T p
À
Á þ se T
4
w À T
4
p
¼ 0
ð8:32Þ
The probe is attempting to measure T 1 . Equation 8.32 yields an equilibrium temperature for
the probe of T p ¼ 642.5
C. The predicted radiation error in this case can be as large as (from
Equation 8.31) À157.5
C. This result indicates that radiative heat transfer can create significant
measurement errors at elevated temperatures.
COMMENT Because the radiation error in this case can only lower the indicated
temperature, it is an example of a nonsymmetrical error (see Section 5.10). We can use
Temperature
probe
Radiation to
enclosure walls
Convection
from fluid
Surrounding
fluid
T ∞ = 800 C
Enclosure
walls
T w = 500 C
Figure 8.36 Analysis of a temperature
probe in a radiative and convective
environment.
8.7 Physical Errors in Temperature Measurement 361
14:54:2 Page 361
Example 8.13
A typical situation where radiation would be important occurs in measuring the temperature of a
furnace. Figure 8.36 shows a small temperature probe for which conduction errors are negligible,
which is placed in a high-temperature enclosure, where the fluid temperature is T 1 and the walls of
the enclosure are at T w .
Convection and radiation are assumed to be the only contributing heat transfer modes at steady
state. Develop an expression for the equilibrium temperature of the probe. Also determine the
equilibrium temperature of the probe and the radiation error in the case where T 1 ¼ 800
C,
T w ¼ 500
C, and the emissivity of the probe is 0.8. The convective heat transfer coefficient is 100
W/m
2 C.
KNOWN Temperatures T 1 ¼ 800
C and T w ¼ 500
C, with h ¼ 100 W/m
2 C. The emissivity
of the probe is 0.8.
FIND An expression for the equilibrium temperature of the probe, and the resulting probe
temperature for the stated conditions.
ASSUMPTIONS The surroundings may be treated as black, and conduction heat transfer is
neglected.
SOLUTION The probe is modeled as a small, spherical body within the enclosed furnace; the
view factor from the probe to the furnace is 1. At steady state, an equilibrium temperature may be
found from an energy balance. The first law for a system consisting of the temperature probe from
Equation 8.30 is
h T 1 À T p
À
Á þ se T
4
w À T
4
p
¼ 0
ð8:32Þ
The probe is attempting to measure T 1 . Equation 8.32 yields an equilibrium temperature for
the probe of T p ¼ 642.5
C. The predicted radiation error in this case can be as large as (from
Equation 8.31) À157.5
C. This result indicates that radiative heat transfer can create significant
measurement errors at elevated temperatures.
COMMENT Because the radiation error in this case can only lower the indicated
temperature, it is an example of a nonsymmetrical error (see Section 5.10). We can use
Temperature
probe
Radiation to
enclosure walls
Convection
from fluid
Surrounding
fluid
T ∞ = 800 C
Enclosure
walls
T w = 500 C
Figure 8.36 Analysis of a temperature
probe in a radiative and convective
environment.
8.7 Physical Errors in Temperature Measurement 361