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Correction to the uranium equation of state
Journal of Nuclear North-Holland
Materials
297
185 (1991) 297-298
Letter to the Editors
Correction
to the uranium equation of state
R.I. Sheldon and R.N.R. Mulford Los Alamos National Laboratory, Nuclear Materials Technology Diukion, Los Alamos, NM 87.545, USA Received
8 August
1991; accepted
30 August
L991
Mulford and Sheldon [l] reported the density and heat capacity of liquid uranium up to 5500 K. In that work, a wire specimen was rapidly heated with an electric current pulse and measurments performed on the free standing liquid column before it collapsed, Brightness temperatures were converted to thermodynamic temperatures by assuming that the normal emissivity of uranium at its melting point, 0.32 at 650 nm, reported by Stephens [2] was constant. The emissivity of electromagnetically levitated liquid uranium has been measured by Krishnan et al. [3] at 633 nm for the temperature range 2092-2713 K by rotating analyzer ellipsometry. The normal emissivity, e633fim= 0.218 + 0.013 + (27 & 6) x lo-‘T, was found to be lower than Stephens’s value when extrapolated to the melting point and also to have a modest temperature dependence. The purpose of this Ietter is to use the emissivity determined by Krishnan et al., extrapolated to high temperatures when necessary, to correct the brightness temperatures measured by Mulford and Sheldon. The results are given in table 1 which contains the enthaipy increment, brightness temperature, emissivity, corrected temperature, relative volume and resistivity, respectivefy. For a pyrometer of sufficiently narrow bandpass centered on the wavelength A, the temperature, T, brightness temperature, T,, and directional, spectral emissivity, e,(T), are related by
where C, is the second radiation constant. The brightness temperature is a function of the same variables as the directional, spectral emissivity. Elsevier
Science
Publishers
B.V.
The pyrometer used by Mulford and Sheldon employed an Melles-Griot (no. 03FIB014) 650 nm, 80 nm FWHM red filter and a United Detector Technologies PIN 8LC Schottky barrier, silicon photodiode. The filter transmittance, TV, and the detector spectral response function, rA, are shown in fig. 1. When the pyrometer bandpass is too wide for application of eq. (11, the temperature must be obtained by matching the brightness integraIs in eq. (2) ~e~V’%,,(Tfr,r~
dh = ~tD’n)~,r,
dh,
(2)
where LbA is the spectral radiance of a black~dy given by the Planck equation [4]. In table 1, brightness
Table 1 Thermophysical properties of liquid uranium at 1.10 MPa. H 2YUK and v2Y8K = 12.48 cms/mol refer to a-uranium H- ff,Y,, (kJ/mol) 100
110 120 130 140 150 160 170 180 190 200 210 220 230 240 2.50
TB
~633Im
v/
w 2071 2238 2403 2563 2721 2876 3028 3178 3325 3470 3612 3752 3889 4025 4159 4290
V,Y,K
P
&il 0.282 0.287 0.293 0.299 0.304 0.310 0.315 0.321 0.326 0.332 0.337 0.342 0.347 0.352 0.357 0.362
2357 2569 2780 2988 3196 3401 3605 3807 4006 4204 4399 4.592 4782 4971 5156 5340
1.21 1.23 1.26 1.29 1.32 1.34 1.37 1.40 1.43 1.47 1.50 1.53 1.56 1.60 1.63 1.67
74.6 77.6 80.7 84.0 87.3 90.7 94.3 98.0 101.7 105.6 109.6 113.8 118.0 122.4 127.0 131.6
cm)
R.I. Sheldon, R.N.R. M&ford / Correction to the uranium equation of state
298
dent of wavelength. Correction of the data using eq. (1) yields temperatures which are smaller than those from eq. (2), but the difference is never more than 10 K. The heat capacity, C,, and thermal expansion coefficient, l/V(dV/dT),, of liquid uranium for the temperature range 2357-5340 K are 50.2 k 0.3 J/K mol and (1.00 f 0.03) x 10d4 KP ‘, respectively.
References
p
0.265
0.00
0.55
0.60
0.65
0.70
WAVELENGTH Fig. 1. Pyrometer
0.75
0.80
[ll R.N.R.
0.85
(m~.crons)
filter transmittance and detector response function.
spectral
temperatures were corrected to true temperatures by numerically evaluating the integrals in eq. (2) assuming that the emissivity given by Krishnan et al. is indepen-
Mulford and RI. Sheldon, J. Nucl. Mater. 154 (1988) 268. Dl H.P. Stephens, High Temp. Sci. 6 (1974) 156. [31 S. Krishnan, J.K.R. Weber, P.C. Nordine and R.I. Sheldon, Spectral emissivity, optical and thermodynamic properties of levitated liquid uranium at high temperatures, J. Nucl. Mater., to be published. and R.D. Lee, Theory and Methods of (41 H.J. Kostkowski Optical Pyrometry, NBS Monograph 41 (March 1962).
Materials
297
185 (1991) 297-298
Letter to the Editors
Correction
to the uranium equation of state
R.I. Sheldon and R.N.R. Mulford Los Alamos National Laboratory, Nuclear Materials Technology Diukion, Los Alamos, NM 87.545, USA Received
8 August
1991; accepted
30 August
L991
Mulford and Sheldon [l] reported the density and heat capacity of liquid uranium up to 5500 K. In that work, a wire specimen was rapidly heated with an electric current pulse and measurments performed on the free standing liquid column before it collapsed, Brightness temperatures were converted to thermodynamic temperatures by assuming that the normal emissivity of uranium at its melting point, 0.32 at 650 nm, reported by Stephens [2] was constant. The emissivity of electromagnetically levitated liquid uranium has been measured by Krishnan et al. [3] at 633 nm for the temperature range 2092-2713 K by rotating analyzer ellipsometry. The normal emissivity, e633fim= 0.218 + 0.013 + (27 & 6) x lo-‘T, was found to be lower than Stephens’s value when extrapolated to the melting point and also to have a modest temperature dependence. The purpose of this Ietter is to use the emissivity determined by Krishnan et al., extrapolated to high temperatures when necessary, to correct the brightness temperatures measured by Mulford and Sheldon. The results are given in table 1 which contains the enthaipy increment, brightness temperature, emissivity, corrected temperature, relative volume and resistivity, respectivefy. For a pyrometer of sufficiently narrow bandpass centered on the wavelength A, the temperature, T, brightness temperature, T,, and directional, spectral emissivity, e,(T), are related by
where C, is the second radiation constant. The brightness temperature is a function of the same variables as the directional, spectral emissivity. Elsevier
Science
Publishers
B.V.
The pyrometer used by Mulford and Sheldon employed an Melles-Griot (no. 03FIB014) 650 nm, 80 nm FWHM red filter and a United Detector Technologies PIN 8LC Schottky barrier, silicon photodiode. The filter transmittance, TV, and the detector spectral response function, rA, are shown in fig. 1. When the pyrometer bandpass is too wide for application of eq. (11, the temperature must be obtained by matching the brightness integraIs in eq. (2) ~e~V’%,,(Tfr,r~
dh = ~tD’n)~,r,
dh,
(2)
where LbA is the spectral radiance of a black~dy given by the Planck equation [4]. In table 1, brightness
Table 1 Thermophysical properties of liquid uranium at 1.10 MPa. H 2YUK and v2Y8K = 12.48 cms/mol refer to a-uranium H- ff,Y,, (kJ/mol) 100
110 120 130 140 150 160 170 180 190 200 210 220 230 240 2.50
TB
~633Im
v/
w 2071 2238 2403 2563 2721 2876 3028 3178 3325 3470 3612 3752 3889 4025 4159 4290
V,Y,K
P
&il 0.282 0.287 0.293 0.299 0.304 0.310 0.315 0.321 0.326 0.332 0.337 0.342 0.347 0.352 0.357 0.362
2357 2569 2780 2988 3196 3401 3605 3807 4006 4204 4399 4.592 4782 4971 5156 5340
1.21 1.23 1.26 1.29 1.32 1.34 1.37 1.40 1.43 1.47 1.50 1.53 1.56 1.60 1.63 1.67
74.6 77.6 80.7 84.0 87.3 90.7 94.3 98.0 101.7 105.6 109.6 113.8 118.0 122.4 127.0 131.6
cm)
R.I. Sheldon, R.N.R. M&ford / Correction to the uranium equation of state
298
dent of wavelength. Correction of the data using eq. (1) yields temperatures which are smaller than those from eq. (2), but the difference is never more than 10 K. The heat capacity, C,, and thermal expansion coefficient, l/V(dV/dT),, of liquid uranium for the temperature range 2357-5340 K are 50.2 k 0.3 J/K mol and (1.00 f 0.03) x 10d4 KP ‘, respectively.
References
p
0.265
0.00
0.55
0.60
0.65
0.70
WAVELENGTH Fig. 1. Pyrometer
0.75
0.80
[ll R.N.R.
0.85
(m~.crons)
filter transmittance and detector response function.
spectral
temperatures were corrected to true temperatures by numerically evaluating the integrals in eq. (2) assuming that the emissivity given by Krishnan et al. is indepen-
Mulford and RI. Sheldon, J. Nucl. Mater. 154 (1988) 268. Dl H.P. Stephens, High Temp. Sci. 6 (1974) 156. [31 S. Krishnan, J.K.R. Weber, P.C. Nordine and R.I. Sheldon, Spectral emissivity, optical and thermodynamic properties of levitated liquid uranium at high temperatures, J. Nucl. Mater., to be published. and R.D. Lee, Theory and Methods of (41 H.J. Kostkowski Optical Pyrometry, NBS Monograph 41 (March 1962).