Abstract
In this article, calculations of the spectral and total cavity emissivity of a high-temperature fixed-point radiator by means of the Monte–Carlo technique in conjunction with calculations of the temperature drop across its back wall using the finite-element approach are presented. The temperature drop across the back wall of a fixed-point cavity radiator is influenced by the heat exchange within the cavity and between the cavity and the front end of the associated furnace. The special effects of these influences were virtually neglected in earlier estimates of the temperature drop for high-temperature fixed-point radiators, resulting in an overestimation of the value of this parameter. These same effects have a non-negligible influence on the cavity emissivity. Even though the heat exchange between the furnace and cavity enhances the temperature uniformity within the cavity, it appears that the cavity cannot be assumed to be isothermal for the case considered, as is usually taken for granted when dealing with fixed-point cavity radiators. Since the temperature drop and total emissivity are affected by the same thermophysical processes, there exists a correlation between these parameters, which might find practical application. To provide experimental evidence to the findings inferred from the calculations, results of measurements of the cavity radiance-temperature of two high-temperature fixed-point cells are presented, enclosing an ingot of eutectic Re-C, for which the cavities are provided with different apertures. For λ = 650 nm, the measured differences in cavity radiance-temperature are shown to be compatible with the differences in radiance temperature calculated for these cavities.
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Abbreviations
- Symbol:
-
Description
- T :
-
Temperature (K)
- T FP :
-
Fixed-point temperature
- T cav :
-
Areal average of the cavity- bottom temperature over the field of view of the thermometer
- ΔT :
-
T FP−T cav
- T E :
-
Eutectic temperature of Re-C
- λ:
-
Wavelength of observation
- T λ :
-
Radiance temperature at λ
- T λ (3):
-
Radiance temperature, measured for a cavity aperture of 3 mm
- T λ (8):
-
Radiance temperature, measured for a cavity aperture of 8 mm
- x f :
-
Distance from furnace center in the direction of the thermometer
- T f(x f):
-
Furnace-temperature profile
- T f,low(x f):
-
Lower bound to T f(x f) from the point of view of the cavity
- T f,high(x f):
-
Upper bound to T f(x f) from the point of view of the cavity
- T cav(high):
-
T cav corresponding to T f,high(x f)
- T cav(low):
-
T cav corresponding to T f,low(x f)
- ΔT(high):
-
ΔT corresponding to T f,high(x f)
- ΔT(low):
-
ΔT corresponding to T f,low(x f)
- x cyl :
-
Distance with respect to the rim of the conical bottom of the cavity along the cavity cylinder
- T cyl (x cyl):
-
Cavity-temperature profile
- T cyl,B :
-
T cyl close to the cavity bottom: T cyl,B > T cav due to the heat exchange between cavity and furnace
- T isoc (λ):
-
Apparent cavity temperature assuming the cavity temperature is uniform, derived from the associated emissivity \(\varepsilon_{\rm isoc} (\lambda)\) , defined below.
- ΔT isoc(λ):
-
T isoc (λ)−T cav
- ΔT isoc(λ;high):
-
ΔT isoc (λ) associated with T f,high(x f)
- ΔT isoc (λ;low):
-
ΔT isoc (λ) associated with T f,low(x f)
- Δ{ΔT isoc (λ)}:
-
ΔT isoc(λ;high) −ΔT isoc(λ;low)
- T isoc :
-
Apparent cavity temperature assuming the cavity temperature is uniform, derived from the associated total emissivity \(\varepsilon_{\rm isoc}\) , defined below.
- ΔT isoc :
-
T isoc−T cav
- ΔT isoc(high):
-
ΔT isoc associated with T f,high(x f)
- ΔT isoc(low):
-
ΔT isoc associated with T f,low(x f)
- Δ{ΔT isoc}:
-
ΔT isoc(high)−ΔT isoc(low)
- \(\varepsilon (\lambda)\) :
-
Spectral emissivity cav+furn: areal average of the local spectral emittance averaged over the FOV of the thermometer, with T cav as the reference temperature.
- \(\rho (\lambda)\) :
-
Spectral reflectivity: \(\rho (\lambda)= 1- \varepsilon (\lambda)\)
- \(\varepsilon\) :
-
Total emissivity cav+furn: areal average of the local total emittance averaged over the FOV of the thermometer, with T cav as the reference temperature.
- ρ :
-
Total reflectivity: \(\rho= 1-\varepsilon\)
- \(\varepsilon_{\rm isoc} (\lambda)\) :
-
\(\varepsilon (\lambda)\) calculated for cav+furn, assuming the cavity temperature to be uniform at the temperature T cav,defined above
- \(\rho_{\rm isoc} (\lambda)\) :
-
\(1-\varepsilon_{\rm isoc}(\lambda)\)
- \(\varepsilon_{\rm isoc}\) :
-
\(\varepsilon\) calculated for cav+furn, assuming the cavity temperature to be uniform at the temperature T cav, defined above
- \(\rho_{\rm isoc}\) :
-
\(1-\varepsilon_{\rm isoc}\)
- Δρ(λ):
-
\(\rho (\lambda) - \rho_{\rm isoc}(\lambda)\)
- \(\Delta\rho\) :
-
\(\rho - \rho_{\rm isoc}\)
- ρ{λ;T f(x f)}:
-
ρ(λ) corresponding to T f(x f), defined above
- ρ isoc{λ; T f(x f)}:
-
ρ isoc(λ) corresponding to T f(x f)
- ρ{λ;T f,high(x f)}:
-
ρ(λ) corresponding to T f,high(x f), defined above
- ρ{λ;T f,low(x f)}:
-
ρ(λ) corresponding to T f,low(x f), defined above
- ρ av (λ):
-
[ρ{λ;T f,high(x f)} + ρ{λ;T f,low(x f)}]/2
- T cav{ρ av (λ)}:
-
T cav, calculated for ρ av (λ)
- ρ isoc (λ;cell):
-
ρ isoc (λ), calculated for the cell only, assuming the cavity temperature is uniform
- T isoc (λ;cell):
-
Apparent cavity temperature, cell only, derived for the associated reflectivity ρ isoc(λ;cell)
- ΔT isoc (λ;cell):
-
T cav{ρ av (λ)}− T isoc(λ;cell)
- T f,ref(x f):
-
{T f,high(x f) + T f,low(x f)}/2
- ρ{λ;T f,ref(x f)}:
-
ρ(λ)corresponding to T f,ref(x f)
- ρ isoc{ λ; T f,ref(x f)}:
-
ρ isoc (λ) corresponding to T f,ref(x f)
- Δρ{λ; T f,ref(x f)}:
-
ρ{λ; T f,ref(x f)} − ρ isoc{λ; T f,ref(x f)}
- M :
-
Exitance
- σ:
-
Stefan-Boltzmann constant
- c2 :
-
Second constant of Planck
- A :
-
Defined via ΔT = A · ρ
- A isoc :
-
Defined via ΔT = A isoc · ρ isoc
- A ref :
-
10−9 T E/4 (mK)
- ΔT corr :
-
[A isoc−A ref] ρ isoc
References
Woolliams E.R., Machin G., Lowe D., Winkler R. (2006). Metrologia 43, R11
P. Jimeno-Largo, Y. Yamada, P. Bloembergen, M.A. Villamanan, G. Machin, in Proceedings of TEMPMEKO 2004, 9th International Symposium on Temperature and Thermal Measurements in Industry and Science, ed. by D. Zvizdić, L. G. Bermanec, T. Veliki, T. Stašić (FSB/LPM, Zagreb, Croatia, 2004), pp. 335–340
P. Bloembergen, Y. Yamada, B.B. Khlevnoy, P. Jimeno-Largo, in Proceedings of the 9th International Conference on New Developments and Applications in Optical Radiometry, Davos, Switzerland, ed. by J. Groebner (Weltstrahlungszentrum, Davos, Switzerkland, 2005), pp. 283–284
P. Bloembergen, Y. Yamada, P. Jimeno-Largo, B.B. Khlevnoy, ibid., pp. 287–288.
Y. Yamada, N. Sasajima, H. Gomi, T. Sugai, in Temperature: Its Measurement and Control in Science and Industry, Vol. 7, ed. by D. C. Ripple (AIP, New York, 2003), pp. 985–990
Prokhorov A.V. (1998). Metrologia 35, 465
Fischer J., Jung H.J. (1989). Metrologia 26, 245
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Bloembergen, P., Khlevnoy, B.B., Jimeno Largo, P. et al. Spectral and Total Effective Emissivity of a High-Temperature Fixed-Point Radiator Considered in Relation to the Temperature Drop Across its Back Wall. Int J Thermophys 29, 370–385 (2008). https://doi.org/10.1007/s10765-007-0323-7
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DOI: https://doi.org/10.1007/s10765-007-0323-7