Black body Thermal Radiation - Thermal Pack error?

[Ángel Martin]

I've been looking at the Black body thermal radiation program in the Thermal Pac, and wonder if there's some errors in the equations used - or in the program itself.

The first issue is a typo in the manual: in page 49 it describes the Stefan-Boltzmann constant (from the total emissive power) with the wrong units dimensions. Where it says "W/m^2.T" it should say "W/m^2.T^4"

The second one is with the Plank's Equation itself. The manual shows the numerator of the formula as: "2.pi.c1", with c1= 5.9544 E-17 W*m^2

However the referenced literature indicates a numerator of: "8.pi.h.c", with h being Plank's constant and c the speed of light in vacuum. If you do the math those two values aren't equal: (using SI units: h=6,6260689633 E-34 J.s ; and c=2,99792458 E8 m/s

2.pi.c1 = 3.7412598 E-16
8.pi.h.c = 4,99248207512 E-24

As far as I can tell the assumptions are the same (Energy by unit of volume per unit of wavelength), but these definitions are always tricky (medium's constants, sphere's volumes?). Does anybody see the obvious mistake I'm making, or is this a genuine issue in the Thermal Pac?

BTW, I find it strange they use Energy units instead of Power units (which is what it's being calculated), and also it's surprising they didn't employ the instruction E^X-1 in the "EbL" subroutine (for the denominator of the emissive power formula).

PS. Funny how the "referenced literature" varies: I found another site where the formula is different and this one matches the one used by HP in the Thermal Pac !

According to this reference the numerator is: "2.pi. h.c^2"
and sure enough h.c^2 = 5.9552147-17 [J.m^2/s]

which perfectly agrees with the c1 value in page 49.

Oh well, this manual is not up to HP's standards for sure!

The morale of the story: "Don't trust blindly what you find on the internet"


Cheers,
ÁM

[ijabbott]

I think the formulae are for different things. One is for spectral radiance and the other is for spectral energy density.

[SlideRule]

Blackbody Radiation: A History of Thermal Radiation Computational Aids and Numerical Methods [ISBN-10: 1482263122], CRC Press (414 pgs) … "understanding the behavior of a blackbody is of importance to many fields including thermal and infrared systems engineering, pyrometry, astronomy, meteorology, and illumination. This book gives an account of the development of Planck's equation together with many of the other functions closely related to it. Particular attention is paid to the computational aspects employed in the evaluation of these functions together with the various aids developed to facilitate such calculations" …

Chapter 8 covers the "electronic aids", ie computers & calculators.

BEST!
SlideRule

[Ángel Martin]

Thanks for the inputs guys, this subject appears to be richer than it first meets the eye.
For example, reading some more about the derivation of Wien's displacement constant I've come up with an interesting duality - depending on which equation is used as the starting point.

But the novelty is an expression of the maximum wavelength using the Lambert W function, I've "distilled" it from a direct comparison between the two referenced sources and it is as follows:

L.T = (h.c/k) . [ 1/ (5 + W(-5.exp(-5)) ]

L = wavelength in m
T = temperature in K
h = Planck's constant
c = speed of light
k = Stefan-Boltzmann constant

This is the first time I've "seen" the Wien's displacement constant in its "closed" form, and not as a numerical value . So interesting!

ÁM

[rprosperi]

(10-28-2019 01:56 PM)Ángel Martin Wrote:  Thanks for the inputs guys, this subject appears to be richer than it first meets the eye.
For example, reading some more about the derivation of Wien's displacement constant I've come up with an interesting duality - depending on which equation is used as the starting point.

But the novelty is an expression of the maximum wavelength using the Lambert W function, I've "distilled" it from a direct comparison between the two referenced sources and it is as follows:

L.T = (h.c/k) . [ 1/ (5 + W(-5.exp(-5)) ]

L = wavelength in m
T = temperature in K
h = Planck's constant
c = speed of light
k = Stefan-Boltzmann constant

This is the first time I've "seen" the Wien's displacement constant in its "closed" form, and not as a numerical value . So interesting!

ÁM

Is there a reason 'dot notation' is used for multiplication while discussing this topic? Of course in "the literature" normal math with implied multiplication is used everywhere to present equations, and this even creeps into the HP manual, but it is not typical so I'm curious why you chose to use that here (versus either the implied or fully unambiguous "*").

[Ángel Martin]

(10-28-2019 03:44 PM)rprosperi Wrote:  Is there a reason 'dot notation' is used for multiplication while discussing this topic? Of course in "the literature" normal math with implied multiplication is used everywhere to present equations, and this even creeps into the HP manual, but it is not typical so I'm curious why you chose to use that here (versus either the implied or fully unambiguous "*").

Not really, just my personal preference - the asterisk adds noise to formulas and omitting the dot is ambiguous if variables have multiple letters...