Why use Stefan's Law to measure temperature?

[sophiecentaur]

TL;DR

Why is radiant energy flux used to estimate temperatures in thermographic cameras

Measuring the temperatures of bright (visible spectrum) cosmic objects would use spectral analysis. But temperatures of IR (warm / hot) radiators is done using Stefan's Law with radiometric cameras. There seems no reason why suitable filters couldn't be used to find the black body temperature using the ratio of two measured intensities after the fashion of colour TV cameras. Why?
Could it be to do with the actual size of IR filters to mount on a two channel image sensor array? I'm sure there is a good reason for this but I can't think of one.

 

[Charles Link]

I don't know that I can answer what you are asking, but for a laboratory type blackbody at approximately 1000 degrees Centigrade, one of the standard methods to measure its temperature about 30-40 years ago was to use an optical pyrometer where the color/brightness was matched between an internal heated filament and the source being measured.

The commercially available calibration blackbody sources also often had internal thermocouples to monitor their temperature around this same time.

Back about 40 years ago, I was successful at determining the temperature with an alternative measurement where a pyroelectric detector with a very broad and uniform spectral response measured the total energy collected, using Stefan's law. The pyroelectric detector was an instrument that was sold commercially, (my company had purchased the one that I used for my experiment), and was calibrated electrically with a resistor in contact with the pyroelectric detector. It was known as the ECPR=electrically calibrated pyroelectric radiometer. There was minimal atmospheric absorption over the one foot path from source to detector. The measurement had an accuracy comparable to that of the optical pyrometer, plus or minus 3 degrees or thereabouts.

Since that time, I think such a measurement might have become fairly routine, where cameras with pyroelectric sensors can use Stefan's law to measure the temperature.

 

[sophiecentaur]

I don't know that I can answer what you are asking,

That was a pretty good effort! Thanks. That's the power of PF; we bump into islands of expertise all over the place in these forums.
You are confirming that spectral analysis isn't used these days so why change?. I guess that the terrestrial situation is different from astronomical measurements. I guess it's probably to do with signal to noise ratios and actual received power levels from distant stars. Moreover, astronomers are very interested in spectra anyway so they use the high quality spectral information they have already.

 

[Charles Link]

Perhaps we should include a little more detail.

Stefan's law is $M=\sigma T^4$ where $M$ is $\pi$ times the integral of the Planck function over the entire spectrum: $M= \pi \int\limits_0^{\infty} L(\lambda,T) \, d \lambda$.

I think generally they don't use Stefan's law like I did with a very broadband essentially flat spectral response over all wavelengths.

Instead they use something such as HgCdTe photodiodes with a single wide bandpass infrared filter using the Planck function integrated over a limited range, e.g. with limits on the previous integral, matching the bandpass filter, that may be from 8 microns to 12 microns or thereabouts. The HgCdTe doesn't respond beyond 12 microns. In some cases, they may even have two or more infrared bandpass filters for an essentially two color measurement, but the extra spectral detail isn't necessary for many applications.

 

[sophiecentaur]

a little more detail.

So it's still a single colour analysis except for some special applications? I guess that way is chosen because the 'numbers' allow / demand it. It still surprises me because more than one filter would allow straightforward ratios of filter outputs - just like the colour telly.

Another schoolday today!
Cheers

 

[Charles Link]

I think it is probably impractical and perhaps almost impossible to use Stefan's law, (which is the power radiated over the complete spectrum), except over very short distances because of atmospheric absorption bands, but using the Planck blackbody function integrated over a wide bandpass that is relatively free of atmospheric absorption is normally how the temperature assessment would be done. It could be necessary to also assign an estimated surface emissivity to the temperature measurement.

 

[sophiecentaur]

perhaps almost impossible to use Stefan's law,

Which is why I wonder why more detailed spectral analysis is not common. Probably because near enough has to be good enough.

 

[Andy Resnick]

So it's still a single colour analysis except for some special applications? I guess that way is chosen because the 'numbers' allow / demand it. It still surprises me because more than one filter would allow straightforward ratios of filter outputs - just like the colour telly.

Another schoolday today!
Cheers

I think the problem is more difficult than that- the emissivity can vary wildly over the relevant waveband (especially either MWIR or LWIR bands) and so, regardless of the spectral sampling you want to do, you end up having to assume the emissivity is constant, which defeats the purpose of complexifying the measurement.

My $0.03 (b/c inflation).

 

[sophiecentaur]

the emissivity can vary wildly over the relevant waveband

I imagine the surface electron activity in 'cool condensed matter' will be subject to that much more than surfaces of stars at thousands of degrees so you can expect black body type radiation from basically hot hydrogen.

This all suggests that radiometric measurement may be not very accurate.
[Edit: I Quick Look at Google suggests that +- 5 degrees or +-55% is what you can expect. So what you can gather from a Thermographic camera is problem just to tell you 'warmer or cooler'. It claims nothing more.]

 

[Gleb1964]

I think it is probably impractical and perhaps almost impossible to use Stefan's law..

It is impossible to use Stefan's law for unresolved source.

 

[Charles Link]

It is impossible to use Stefan's law for unresolved source.

I was able to use it to determine the temperature of a calibration blackbody source with known aperture size at close range, ( distance of one foot). In general, bandpass measurements with the Planck blackbody function are necessary.

 

[Gleb1964]

It's quite simple: if the source's angular size is below the resolution limit, it becomes diluted into the point spread function. If the object's angular size is unknown, its temperature cannot be determined.

 

[Charles Link]

It's quite simple: if the source's angular size is below the resolution limit, it becomes diluted into the point spread function. If the object's angular size is unknown, its temperature cannot be determined.

In that case, one would need additional spectral detail, such as measuring with two or more bands.

 

[Andy Resnick]

I imagine the surface electron activity in 'cool condensed matter' will be subject to that much more than surfaces of stars at thousands of degrees so you can expect black body type radiation from basically hot hydrogen.

I'm not entirely sure what you mean- for example, MWIR (3-5 micron band) is well suited for imaging 'hot' internal combustion engines while LWIR (8-12 micron band) is good for living objects and room temperatures. In both of those wavebands, an object's emissivity is most definitely not constant. I'm less familiar with SWIR (0.9- 1.7 micron waveband) imaging systems, tho.

Edit- maybe a better answer is to procure a sample of the material you want to image and characterize the spectral emissivity?

 

[sophiecentaur]

Edit- maybe a better answer is to procure a sample of the material you want to image and characterize the spectral emissivity?

That would make sense but it's a few steps higher than just pointing a camera at an unknown material and estimating its temperature with no special filtering. But the 'accepted' accuracy is apparently only +-5% so maybe it doesn't matter.

Also, the actual surface temperature is not particularly relevant in cases where it's the rate of heat loss from an insulated building. For that measurement, there's no 'fourth root' problem conversion from flux to temperature. Star magnitudes used to be assessed by comparison of 'brightness'. Star temperature is measured from spectral profile.

 

[Andy Resnick]

That would make sense but it's a few steps higher than just pointing a camera at an unknown material and estimating its temperature with no special filtering. But the 'accepted' accuracy is apparently only +-5% so maybe it doesn't matter.

But that's sort of my point- without any 'ground truth' data, it's not possible to accurately determine the temperature of an unknown material just by remotely detecting the IR radiation emitted from it. And I would go further and say +-5% is wildly optimistic.

Also, the actual surface temperature is not particularly relevant in cases where it's the rate of heat loss from an insulated building. For that measurement, there's no 'fourth root' problem conversion from flux to temperature. Star magnitudes used to be assessed by comparison of 'brightness'. Star temperature is measured from spectral profile.

Well.... that depends entirely on your application. Most of the IR imaging applications I was involved with had to do with defeating camoflauge.

 

[sophiecentaur]

+-5% is wildly optimistic.

Those seem to be the numbers in the catalogues of everyday instruments but adverts say all sorts of nonsense.
Camoflauge!! yes I remember the stories of spotting VC on the river banks 'Nam . There was a method of using radar and looking for harmonics from metal structures but none from organic material.

Application is everything in such a fuzzy subject.

 

[verb]

The Wikipedia article on Planck's law, https://en.wikipedia.org/wiki/Planck's_law, has a section called Percentiles that I wrote from scratch in November 2009. The table of percentiles has three boldface entries at 25.0%, 41.8%, and 64.6%. These are the respective peaks when the radiation is plotted as a function of wavelength, wavelength-frequency neutral, and frequency. Wien's Law is determined using the first of these, and its peak λ yields color temperature T = 2898/λ. For the Sun, the black body peak is at λ = 0.502 μm, hence the color temperature is T = 2898/.502 = 5773K. But if you use Christian Gueymard's model, its 25% peak is at λ = 0.520 μm giving T = 2898/520 = 5573K.

 

[Andy Resnick]

Those seem to be the numbers in the catalogues of everyday instruments but adverts say all sorts of nonsense.

Well.... if you know the emissivity, then I would agree that a commercial IR detector/camera can hit that 5% accuracy. The problem is if you don't know the emissivity.....

 

[sophiecentaur]

Well.... if you know the emissivity, then I would agree that a commercial IR detector/camera can hit that 5% accuracy. The problem is if you don't know the emissivity.....

Looking at a given object or similar objects, it would be easier. Hot and cold spots on a human body or on an engine would be suitable subjects of thermography. A mixture of polar bears and humans (taken together ) would be a bigger problem.

 

[Gleb1964]

For example, FLIR: Infrared Camera Accuracy and Uncertainty in Plain Language

Infrared cameras are typically calibrated using multiple blackbody sources at different temperatures. The camera is mounted on a rotating platform and sequentially pointed at these reference sources.

Camera Accuracy Specs and the Uncertainty Equation​

You’ll notice that most IR camera data sheets show an accuracy specification such as ±2ºC or 2% of the reading. This specification is the result of a widely used uncertainty analysis technique called “Root‐Sum‐of‐Squares”, or RSS. The idea is to calculate the partial errors for each variable of the temperature measurement equation, square each error term, add them all together, and take the square root. While this equation sounds complex, it’s fairly straightforward. Determining the partial errors, on the other hand, can be tricky.

“Partial errors” can result from one of several variables in the typical IR camera temperature measurement equation, including:

• Emissivity
• Reflected ambient temperature
• Transmittance
• Atmosphere temperature
• Camera response
• Calibrator (blackbody) temperature accuracy

Once reasonable values are determined for the “partial errors” for each of the above terms, the overall error equation will look like this:

Where the ΔT1, ΔT2, ΔT3, etc are the partial errors of the variables in the measurement equation.