Sakuma–Hattori equation
In physics, the Sakuma–Hattori equation is a mathematical model for predicting the amount of thermal radiation, radiometric flux or radiometric power emitted from a perfect blackbody or received by a thermal radiation detector.
History[edit]
The Sakuma–Hattori equation was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1982.[1] In 1996, a study investigated the usefulness of various forms of the Sakuma–Hattori equation. This study showed the Planckian form to provide the best fit for most applications.[2] This study was done for 10 different forms of the Sakuma–Hattori equation containing not more than three fitting variables. In 2008, BIPM CCT-WG5 recommended its use for radiation thermometry measurement uncertainty budgets below 960 °C.[3]
General form[edit]
The Sakuma–Hattori equation gives the electromagnetic signal from thermal radiation based on an object's temperature. The signal can be electromagnetic flux or signal produced by a detector measuring this radiation. It has been suggested that below the silver point,[a] a method using the Sakuma–Hattori equation be used.[1] In its general form it looks like[3]
- is the scalar coefficient
- is the second radiation constant (0.014387752 m⋅K[6])
- is the temperature-dependent effective wavelength (in meters)
- is the absolute temperature (in K)
Planckian form[edit]
Derivation[edit]
The Planckian form is realized by the following substitution:
Making this substitution renders the following the Sakuma–Hattori equation in the Planckian form.
Discussion[edit]
The Planckian form is recommended for use in calculating uncertainty budgets for radiation thermometry[3] and infrared thermometry.[7] It is also recommended for use in calibration of radiation thermometers below the silver point.[3]
The Planckian form resembles Planck's law.
However the Sakuma–Hattori equation becomes very useful when considering low-temperature, wide-band radiation thermometry. To use Planck's law over a wide spectral band, an integral like the following would have to be considered:
This integral yields an incomplete polylogarithm function, which can make its use very cumbersome. The standard numerical treatment expands the incomplete integral in a geometric series of the exponential
The Sakuma–Hattori equation shown above was found to provide the best curve-fit for interpolation of scales for radiation thermometers among a number of alternatives investigated.[2]
The inverse Sakuma–Hattori function can be used without iterative calculation. This is an additional advantage over integration of Planck's law.
Other forms[edit]
The 1996 paper investigated 10 different forms. They are listed in the chart below in order of quality of curve-fit to actual radiometric data.[2]
Name | Equation | Bandwidth | Planckian |
---|---|---|---|
Sakuma–Hattori Planck III | narrow | yes | |
Sakuma–Hattori Planck IV | narrow | yes | |
Sakuma–Hattori – Wien's II | narrow | no | |
Sakuma–Hattori Planck II | broad and narrow | yes | |
Sakuma–Hattori – Wien's I | broad and narrow | no | |
Sakuma–Hattori Planck I | monochromatic | yes | |
New | narrow | no | |
Wien's | monochromatic | no | |
Effective Wavelength – Wien's | narrow | no | |
Exponent | broad | no |
See also[edit]
Notes[edit]
References[edit]
- ^ a b Sakuma, F.; Hattori, S. (1982). "Establishing a practical temperature standard by using a narrow-band radiation thermometer with a silicon detector". In Schooley, J. F. (ed.). Temperature: Its Measurement and Control in Science and Industry. Vol. 5. New York: AIP. pp. 421–427. ISBN 0-88318-403-6.
- ^ a b c Sakuma F, Kobayashi M., "Interpolation equations of scales of radiation thermometers", Proceedings of TEMPMEKO 1996, pp. 305–310 (1996).
- ^ a b c d Fischer, J.; et al. (2008). "Uncertainty budgets for calibration of radiation thermometers below the silver point" (PDF). CCT-WG5 on Radiation Thermometry, BIPM, Sèvres, France. 29 (3): 1066. Bibcode:2008IJT....29.1066S. doi:10.1007/s10765-008-0385-1. S2CID 122082731.
- ^ J Tapping and V N Ojha (1989). "Measurement of the Silver Point with a Simple, High-Precision Pyrometer". Metrologia. 26 (2): 133–139. Bibcode:1989Metro..26..133T. doi:10.1088/0026-1394/26/2/008. S2CID 250764204.
- ^ "Definition of Silver Point - 962°C, the melting point of silver". Retrieved 2010-07-26.
- ^ "2006 CODATA recommended values". National Institute of Standards and Technology (NIST). Dec 2003. Retrieved Apr 27, 2010.
- ^ a b MSL Technical Guide 22 – Calibration of Low Temperature Infrared Thermometers (pdf), Measurement Standards Laboratory of New Zealand (2008). Updated: Version 3. July 2019, [1]
- ^ ASTM Standard E2758-10 – Standard Guide for Selection and Use of Wideband, Low Temperature Infrared Thermometers, ASTM International, West Conshohocken, PA, (2010). Updated: ASTM E2758-15a(2021), https://www.astm.org/e2758-15ar21.html