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 was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1987.[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 pointTemplate:Cref2, a method using the Sakuma–Hattori equation be used.[1] In its general form it looks like:[3]

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

where:

 ${\displaystyle C}$ Scalar coefficient ${\displaystyle c_{2}}$ Second Radiation Constant (0.014387752 m⋅K[4]) ${\displaystyle \lambda _{x}}$ Temperature dependent effective wavelength in meters ${\displaystyle T}$ Temperature in Kelvin

Planckian form

Derivation

The Planckian form is realized by the following substitution:

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

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

Discussion

The Planckian form is recommended for use in calculating uncertainty budgets for radiation thermometry[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.

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

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:

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

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

${\displaystyle \int _{0}^{\lambda _{2}}{\frac {c_{1}}{\lambda ^{5}[\exp({\frac {c_{2}}{\lambda T}})-1]}}d\lambda =c_{1}({\frac {T}{c_{2}}})^{4}\int _{c_{2}/(\lambda _{2}T)}^{\infty }{\frac {x^{3}}{\exp(x)-1}}dx}$
${\displaystyle J(c)\equiv \int _{c}^{\infty }{\frac {x^{3}}{\exp x-1}}dx=\int _{c}^{\infty }{\frac {x^{3}\exp(-x)}{1-\exp(-x)}}dx=\int _{c}^{\infty }\sum _{n\geq 1}x^{3}\exp(-nx)dx}$
${\displaystyle =\sum _{n\geq 1}\exp(-nc){\frac {(nc)^{3}+3(nc)^{2}+6nc+6}{n^{4}}}}$

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