Sakuma–Hattori equation

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[]

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 uncertainty budgets below 960 °C.^[3]

General form[]

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]

S(T)={\frac {C}{\exp \left({\frac {c_{2}}{\lambda _{x}T}}\right)-1}},

where:

$C$ is the scalar coefficient
$c_{2}$ is the second radiation constant (0.014387752 m⋅K^[4])
$\lambda _{x}$ is the temperature-dependent effective wavelength in meters
$T$ is the absolute temperature in kelvins

Planckian form[]

Derivation[]

The Planckian form is realized by the following substitution:

\lambda _{x}=A+{\frac {B}{T}}

Making this substitution renders the following the Sakuma–Hattori equation in the Planckian form.

Sakuma–Hattori equation (Planckian form): $S(T)={\frac {C}{\exp \left({\frac {c_{2}}{AT+B}}\right)-1}}$
Inverse equation ^[5]: $T={\frac {c_{2}}{A\ln \left({\frac {C}{S}}+1\right)}}-{\frac {B}{A}}$
First derivative ^[6]: ${\frac {dS}{dT}}=\left[S(T)\right]^{2}{\frac {Ac_{2}}{C\left(AT+B\right)^{2}}}\exp \left({\frac {c_{2}}{AT+B}}\right)$

Discussion[]

The Planckian form is recommended for use in calculating uncertainty budgets for ^[3] and infrared thermometry.^[5] It is also recommended for use in calibration of radiation thermometers below the silver point.^[3]

The Planckian form resembles Planck's law.

S(T)={\frac {c_{1}}{\lambda ^{5}\left[\exp \left({\frac {c_{2}}{\lambda T}}\right)-1\right]}}

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:

S(T)=\int _{\lambda _{1}}^{\lambda _{2}}{\frac {c_{1}}{\lambda ^{5}\left[\exp \left({\frac {c_{2}}{\lambda T}}\right)-1\right]}}d\lambda

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

\int _{0}^{\lambda _{2}}{\frac {c_{1}}{\lambda ^{5}\left[\exp \left({\frac {c_{2}}{\lambda T}}\right)-1\right]}}d\lambda =c_{1}\left({\frac {T}{c_{2}}}\right)^{4}\int _{c_{2}/(\lambda _{2}T)}^{\infty }{\frac {x^{3}}{\exp(x)-1}}dx

after substituting

\lambda =c_{2}/(xT)

,

d\lambda =-c_{2}/\left(x^{2}T\right)dx

. Then

{\begin{aligned}J(c)&\equiv \int _{c}^{\infty }{\frac {x^{3}}{\exp x-1}}dx=\int _{c}^{\infty }{\frac {x^{3}\exp(-x)}{1-\exp(-x)}}dx\\&=\int _{c}^{\infty }\sum _{n\geq 1}x^{3}\exp(-nx)dx\\&=\sum _{n\geq 1}\exp(-nc){\frac {(nc)^{3}+3(nc)^{2}+6nc+6}{n^{4}}}\end{aligned}}

provides an approximation if the sum is truncated at some order.

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[]

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	$S(T)={\frac {C}{\exp \left({\frac {c_{2}}{AT+B}}\right)-1}}$	narrow	yes
Sakuma–Hattori Planck IV	$S(T)={\frac {C}{\exp \left({\frac {A}{T^{2}}}+{\frac {B}{2T}}\right)-1}}$	narrow	yes
Sakuma–Hattori – Wien's II	$S(T)=C\exp \left({\frac {-c_{2}}{AT+B}}\right)$	narrow	no
Sakuma–Hattori Planck II	$S(T)={\frac {CT^{A}}{\exp \left({\frac {B}{T}}\right)-1}}$	broad and narrow	yes
Sakuma–Hattori – Wien's I	$S(T)=CT^{A}{\exp \left({\frac {-B}{T}}\right)}$	broad and narrow	no
Sakuma–Hattori Planck I	$S(T)={\frac {C}{\exp \left({\frac {c_{2}}{AT}}\right)-1}}$	monochromatic	yes
New	$S(T)=C\left(1+{\frac {A}{T}}\right)-B$	narrow	no
Wien's	$S(T)=C\exp \left({\frac {-c_{2}}{AT}}\right)$	monochromatic	no
Effective Wavelength – Wien's	$S(T)=C\exp \left({\frac {-A}{T}}+{\frac {B}{T^{2}}}\right)$	narrow	no
Exponent	$S(T)=CT^{A}$	broad	no

Notes[]

^

Silver point, the melting point of silver 962°C [(961.961 ± 0.017)°C^[7]] used as a calibration point in some temperature scales.^[8]

It is used to calibrate IR thermometers because it is stable and easy to reproduce.

References[]

^ ^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. vol. 5. New York: AIP. pp. 421–427. ISBN 0-88318-403-6. {{cite book}}: |volume= has extra text (help)
^ ^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.
^ "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).
^ ASTM Standard E2758-10 – Standard Guide for Selection and Use of Wideband, Low Temperature Infrared Thermometers, ASTM International, West Conshohocken, PA, (2010).
^ 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.
^ "Definition of Silver Point - 962°C, the melting point of silver". Retrieved 2010-07-26.

[Sakuma1-1] 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. vol. 5. New York: AIP. pp. 421–427. ISBN 0-88318-403-6. {{cite book}}: |volume= has extra text (help)

[Sakuma2-2] Sakuma F, Kobayashi M., "Interpolation equations of scales of radiation thermometers", Proceedings of TEMPMEKO 1996, pp. 305–310 (1996).

[Fischer-3] 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.

[4] "2006 CODATA recommended values". National Institute of Standards and Technology (NIST). Dec 2003. Retrieved Apr 27, 2010.

[MSLNZ-5] MSL Technical Guide 22 – Calibration of Low Temperature Infrared Thermometers (pdf), Measurement Standards Laboratory of New Zealand (2008).

[6] ASTM Standard E2758-10 – Standard Guide for Selection and Use of Wideband, Low Temperature Infrared Thermometers, ASTM International, West Conshohocken, PA, (2010).

[7] 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.

[8] "Definition of Silver Point - 962°C, the melting point of silver". Retrieved 2010-07-26.

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