A guest post by Ken Coffman and Mikael Cronholm
In clicking around on the Internet, I found an outstanding paper called Thermodynamics of Furnace Tubes – Killing Popular Myths about Furnace Tube Temperature Measurement written by Mikael Cronholm. The paper was clever and wise…and made a lot of sense. Clearly Mikael knows a lot about infrared radiation and I’m a guy with questions. A match made in heaven?
We exchanged e-mails. I want to be clear about this…Mikael corrected some of my wrong ideas about IR. I’ll repeat that for the slow-witted. Some of my ideas about infrared radiation were wrong. I am considered a hard-headed, stubborn old guy and that’s completely true. However, I want to learn and I can be taught, but not by knuckleheads spewing nonsense and not by authoritarians who sit on thrones and toss out insults and edicts.
Ken Coffman (KLC) is the publisher of Stairway Press (www.stairwaypress.com) and the author of novels that include Hartz String Theory and Endangered Species.
Mikael Cronholm (MC) is an industry expert on infrared radiation, a licensed, level III Infrared Training Center Instructor and holds two Bachelor of Science degrees (Economics and Business Administration).
The following is a summary of our conversation.
KLC: Hello Mikael. I found your paper called Thermodynamics of Furnace Tubes and I found it very informative, practical and interesting. I hope you’ll bear with me while I ask a couple of dumb questions. I am an electrical engineer, so I have some knowledge about thermodynamics of conduction and convection, but not so much about IR radiation. In return for your time, I would be happy to make a donation to the charity of your choice.
If I take an inexpensive IR thermometer outside, point it at the sky and get a temperature reading of minus 25°C, what am I actually measuring? Is there anything valid about doing this?
MC: Just as a matter of curiosity, how did you find my paper? I checked your website and I guess this has to do with the Dragon, no? If you want to make a donation I would be happy to receive that book. If you can, my postal address is at the bottom. I don’t follow the debate more than casually, but I am a bit skeptical to all the research that is done on climate change…it seems that the models are continuously adjusted to fit the inputs, so that you get the wanted output…and they argue “so many scientists agree with this and that”…well, science is not a democracy…anyway…
About radiation, then. There is more to this than meets the eye. Literally!
Looking at the sky with an infrared radiometer you would read what is termed “apparent temperature” (if the instrument is set to emissivity 1 and the distance setting is zero, provided the instrument has any compensation). Your instrument is then receiving the same radiation as a blackbody would do if it had a temperature of -25°C, if that is what you measure. It is a quasi-temperature of sorts, because you don’t really measure on a particular object in any particular place, but a combination of radiation, where that from outer space is the lowest, close to absolute zero, and the immediate atmosphere closest to you is the warmest. (I have once measured -96°C on the sky at 0°C ground temperature.) What we have to realize though, is that temperature can never be directly measured. We measure the height of a liquid in a common thermometer, a voltage in a thermocouple, etc, and then it is calibrated using the zeroth law of thermodynamics and assuming equilibrium with the device and the reference.
KLC: Global warming (greenhouse gas) theory depends on atmospheric CO2 molecules absorbing IR radiation and “back radiating” this energy back toward the earth. If you look at the notorious Ternberth/Keihl energy balance schematic (as shown in Figure 1 of this paper: http://www.cgd.ucar.edu/cas/Trenberth/trenberth.papers/TFK_bams09.pdf ), you see the back radiation is determined to be very significant…more than 300W/m2. From your point of view as an IR expert, does this aspect of the global warming theory make any sense?
MC: The paper you sent me mentions Stefan-Boltzman’s law, but it does not talk about Planck’s law, which is necessary to understand what is happening spectrally. I suggest you read up on Planck and Stefan-Boltzman at Wikipedia or something. Wien’s law would be beneficial as well—they are all connected.
Planck’s law describes the distribution of radiated power from a blackbody over wavelength. You end up with a curve for each blackbody temperature. The sun is almost a blackbody, so it follows Planck quite well, and it has a peak at about 480nm, right in the middle of visual (Wien’s law determines that). The solar spectrum is slightly modified as it passes through the atmosphere, but still pretty close to Planckian. When the radiation hits the ground, the absorbed part heats it. The re-radiated power is going to have a different spectral distribution, with a peak around 10um (micrometer). Assuming blackbody radiation it would also follow Planck’s law.
S-B’s law is in principle the integral of Planck from zero to infinity wavelength. Instruments do not have equal response from zero to infinity, but they are calibrated against blackbodies, and whatever signal they output is considered to mean the temperature of the blackbody. And so on for a number of blackbodies until you have a calibration curve that can be fitted for conversion in the instrument.
That means that the instrument can only measure correctly on targets that are either blackbodies, or greybodies with a spectral distribution looking like a Planck curve, but at a known offset. That offset is emissivity, the epsilon in your S-B equation in that paper. It is defined as the ratio of the radiation from the greybody to that of the blackbody, both at the same temperature (and wave length, and angle…). Some targets will not be Planckian, but have a spectral distribution that is different. If you want to measure temperature of those, you need to measure the emissivity with the same instrument and at a temperature reasonably close to the one you will measure on the target later.
So, of course, the whole principle behind the greenhouse effect is that shorter wavelengths from the sun penetrates the atmosphere easily, whereas the re-radiated power—being at a longer wavelength—is reflected back at a higher degree. I have no dispute about that fact. It is reasonable. So I think the Figure 1 you refer to is correct in principle. My immediate question is raised regarding the numbers in there though. The remaining 0.9 W/m2 seems awfully close to what I would assume to be the inaccuracies in the numbers input to calculate it. You are balancing on a very thin knifes edge with such big numbers as inputs for reaching such a small one. An error of +/- 0.5% on each measurement would potentially throw it off quite a bit, in the worst case. But I don’t know what they use to measure this, only that all the instruments I use have much less accuracy than that. But with long integration times…well, maybe…but there may be an issue there.
KLC: I am interested in some rather expensive thermopile-based radiation detectors called pyrgeometers (an example is the KippZonen CGR 3 instrument http://www.kippzonen.com/?product/16132/CGR+3.aspx).
If a piece of equipment like this is pointed into the nighttime sky and reads something like 300W/m2 of downwelling IR radiation, what is it actually measuring? If I built a test rig from IR-emitting lightbulbs calibrated to emit 300W/m2 and placed this over the pyrgeometers, would I get the same reading?
MC: “What is it actually measuring?” Well, probably a voltage from those thermopiles…and that signal has to be calibrated to a bunch of blackbody reference sources to covert it either to temperature or blackbody equivalent radiation.
Your experiment will fail, though! If you want to do something like that, you have to look at a target emitting a blackbody equivalent spectrum, which is what the instrument should be calibrated to. IR light bulbs emitting 300W/m2 is simply impossible, because 300W/m2 corresponds to a very low temperature! Use S-B’s law and try it yourself. Like this: room temp, 20°C = 293K. The radiated power from that is 293K raised to the power of 4. Then multiply with sigma, the constant in S-B’s law, which is 5.67*10-8, and you get 419 W/m2 or something like that, it varies with how many decimals you use for absolute zero when you convert to Kelvin. For 300 W/m2 radiation I get -23.4°C at 300 W/m2 when I calculate it (yes, minus!). Pretty cool light bulb.
I don’t know what your point is with that experiment, but if it is to check their calibration you need a lot more sophisticated blackbody reference sources if you want to do it at that temperature. But you could do a test at room temperature though. Just build a spherical object with the inside painted with flat black paint, make a small hole in it, just big enough for your sensor, and measure the temperature inside that sphere with a thermocouple, on the surface. Keep it in a stable room temperature at a steady state as well as you can and convert the temperature to radiation using S-B’s law. You should get the same as the instrument. Any difference will be attributable to inaccuracy in the thermocouple you use and/or the tested instrument. Remember that raising to the power of 4 exaggerates errors in the input a lot!
I hope I have been able to clarify things a little bit, or at least caused some creative confusion. When I teach thermography I find that the more you learn the more confused you get, but on a higher level. Every question answered raises a few more, which grows the confusion exponentially. It makes the subject interesting, though.
Let me know if you need any more help with your project!
KLC: I found your paper because one of the FLIR divisions is local and I was searching their site for reference information about IR radiation. I know what a 100W IR lamp feels like because I have one in my bathroom. If someone tells me there is 300W/m2 of IR power coming from space, and I hold out my hand…I expect to feel it. What am I missing?
MC: Yeah, you put your hand in front of a 100W bulb, but how big is your hand…not a square meter, I’m sure. It is per area unit, that is one thing you are missing. The 100W of the bulb is the electrical power consumption, not the emitted power of the visual light from it. That’s why florescent energy-saving lamps as opposed to incandescent bulbs give much more visual light per electrical Watt, because they limit the radiation to the visual part of the spectrum and lose less in the IR, which we cannot see anyway. The body absorbs both IR and visual, but a little less visual.
And, here is the other clue. Your light bulb radiation in your bathroom is added to that of the room itself, which is 419 W/m2, if the room is 20°C. Your 300 W/m2 from space is only that. You will feel those 300 W/m2, sure. It will feel like -25°C radiating towards your hand. But you don’t feel that cold because your hand is in warmer air, receiving heat (or losing less) from there too.
Actually, we cannot really feel temperature—that is a misconception. Our bodies feel heat flow rate and adjust the temperature accordingly. It is only the hypothalamus inside the brain that really has constant temperature. If you are standing nude in your bathroom, your body will radiate approximately 648 W/m2 and the room 419 W/m2, so you lose 229 W/m2. That is what you feel as being cooled by the room, from radiation only. Conduction and convection should be added of course. The earth works the same way—lose some, gain some. It is that balance that is being argued in the whole global warming debate.
KLC: I still feel like I’m missing something. IR heat lamps are pretty efficient, maybe 90%? Let’s pick a distance of 1 meter and I want to create a one-square meter flooded with an additional 300W/m2. It must be additional irradiation, doesn’t it? That’s going to take a good bunch of lamps and I would feel this heat. However, I go outside and hold out my hand. It’s cold. There’s no equivalent of 300W/m2 heater in addition to whatever has heated the ambient air.
Perhaps I’m puzzled by something that is more like a flux…something that just is as a side-effect of a temperature difference and not really something that is capable of doing any work or as a vehicle for transporting heat energy.
It’s a canard of climate science that increasing atmospheric CO2 from 390PPM to 780PPM will raise the earth’s surface temperature by about 1°C (expanded to 3°C by positive feedbacks). From my way of thinking, the only thing CO2 can do is increase coupling to space…it certainly can’t store or trap energy or increase the earth’s peak or 24-hour average temperature.
Any comments are welcome.
MC: Efficiency of a lamp depends on what you want, if heat is what want then they are 100% efficient, because all electrical energy will be converted to heat, the visible light as well, when it is absorbed by the surrounding room. If visible light is required, a light bulb loses a lot of heat compared to an energy saving lamp. Energy cannot be created or destroyed—first law of thermodynamics.
When you say W/m2 you ARE in fact talking about a flux (heat flow is what will be in W). If you have two objects radiating towards each other, the heat flow direction will be from the hotter one, radiating (emitting) more and absorbing less, to the cooler one, which radiates less and absorbs more (second law of thermodynamics). The amount of radiation emitted from each of them depends on two things ONLY, the temperature of the object and its emissivity. So radiation is not a side effect to temperature, it is THE EFFECT. Anything with a temperature will radiate according to it, and emissivity. (If something is hotter than 500°C we get incandescence, emission of visible light.) Assuming an emissivity of unity, which is what everyone seems to do in this debate, the radiation (flux. integrated from zero to infinity) will be equal to what can be calculated by Stefan-Boltzmann’s law, which is temperature in Kelvin, raised to the fourth power, multiplied by that constant sigma. It’s that simple!
With regard to your thought experiment, it is always easier to calculate what an object emits than what it absorbs, because emission will be spreading diffusely from an object, so exactly where it ends up is difficult to predict. I am not sure where you are aiming with that idea, but it does not seem to be an easy experiment to do in real life, at least not with limited resources.
CO2 is a pretty powerful absorber of radiated energy, that fact is well known. Water vapor is an even stronger absorber. In the climate debate it is also considered a reflector, which probably also true, because that is universal. Everything absorbs and reflects to a degree. So I guess that the feedback you mention has to do with the fact that increasing temperature increases the amount of water vapor, which increases absorption, and so on. But my knowledge is pretty much limited to what happens down here on earth, because that is what matters when we measure temperature using infrared radiation. However, it is important to remember, again, that we talk about different spectral bands, the influx is concentrated around a peak in the visual band and the outgoing flux is around 10 micrometer in the infrared band, and the absorption may not be the same.
With so many scientists arguing about the effects of CO2 I am not the one to think I have the answers. I really don’t know what the truth is. And the problem that all these scientists have is that they will never be able to test if their theories are correct, because the time spans are too long. For a theory to be scientifically proven, it has to be stipulated and tested, and the test must be repeatable and give the same results in successive tests for the theory to be proven.
If not, it is not science, it is guessing.
More like a horoscope…
Jim D says:
February 14, 2011 at 4:59 pm
Bill Illis, nitrogen and oxygen heat and cool by conduction and convection only, not by radiation.
—————
Accordingly to your theory, these are meaningless concepts. They play no part in the greenhouse effect. Why do you bring them up now?
@ur momisugly cba. Yes, I know it is a restricted definition, which I kind of pointed out, implicitly. I just though a simplified answer would be good enough, since the main point I was making was that radiation in itself is normally not defined as heat. But there are two definitions of heat, differing only in the semantics.
mikael,
heat is energy in transit. loosely, radiation readily fits the definition. One too easily tends to forget that there is more to energy states than kinetic energy of the motion of individual molecules.
cba says:
“Bill Illis
N2 and O2 are at the same temperature in a small parcel of air, just like the co2. The co2 absorbs some energy and the average time it takes to radiate it away is more than enough time for it to be transferred away by collisions, most likely n2 followed by o2. just because a co2 molecule excited by a photon is likely to have its energy reduced by a collision, so too is the likelihood that a co2 molecule will be excited by a collision and capable of emitting a photon or capable of being ‘defused’ by yet another collision. net result is that a certain fraction of these molecules will emit a photon of a particular energy (wavelength) based upon the temperature and upon the proclivity of the co2 molecule to absorb or emit that energy. absorption doesn’t really depend much on temperature but the emission is highly dependent. The blackbody curve for a give temperature is actually a portrayal of the fraction of molecules at particular energies that are capable of emitting photons. The bb curve must be a solid or liquid – or an optically thick enough gas to be thick at all wavelengths. For the sun to essentially emit a 5800k BB curve from the photosphere, one has many heavier elements present that are ionized and so are capable of emitting a continuum rather than merely a spectrum.”
I think this is all correct.
But, I would like to posit another “world” for the warmistas. Suppose the atmosphere of Earth consisted of ONLY N2 and O2. Now, these gases could not cool by IR emissions, but they would warm by conduction from the surface. And they could not radiate to space. So would they continually warm for millions of years?? Would they melt the planet? WTF, folks?
Bill Illis, you are saying conduction and convection are meaningless when they explain the temperature profile of the atmosphere and a large part of the heat flux from the ground. They are vital to any complete theory. Some people have trouble with the concepts of conduction, convection and radiation all being important at the same time, I guess.
@ur momisugly cba. Yes, that is the other definition. Both are equally valid, but when I teach heat transfer and IR temperature measurement to engineers, I find the definition that implies that heat is contained in objects to be easier to work with. I accept both definitions and the systems of nomenclature that they create, I just prefer one because of simplicity. And following this discussion it seems to me that the definition I prefer for practical reasons also seems to be the one that people use here.
If I define heat as being the thermal energy flow I run into problems explaining heat capacity for example. The term “heat transfer” becomes a bit funny if “heat” in itself is a transfer, a bit like saying”free gift”.
http://en.wikipedia.org/wiki/Heat#Semantic_misconceptions
Great experts for this place. An electrical engineer and a business/economics major. And these musings disprove global warming and the properties of CO2? Lol. Check any temperature charts lately? Any Mars missions planned from your garage? How about no Watts in the bulb?
marky48,
And what is your expertise?
No one here is trying to disprove global warming. It is the continuation of the natural warming cycle from the LIA. And no one is arguing the properties of CO2. It is a trace gas that delays the emission of radiation to space.
The question is whether a minor trace gas controls the climate. If you believe you have solid evidence that it does, provide it.
@ur momisugly marky48. Great contribution! Very informative! Naah, not really…
When you find something interesting to post that questions or discusses the facts presented instead of arguing ad hominem and attempting to ridicule me, you can try again.
In absence of degrees on the subject, I have worked with and studied IR thermography, as it is applied in industry and research, for over 20 years, and I have been teaching it for 15 years or so, and written books about it. This discussion is on the fringes of my expertise, and I don’t claim to be a climate scientist or expert at all, but I am quite enjoying learning about it here. And if you look a little carefully you will find that I am on neither side in the debate. You have apparently taken sides though, based on what, I don’t know. If you would attempt to enlighten me you will have to show a little more intelligence than you just did.
jae says:
February 14, 2011 at 7:59 pm ………………..
Suppose the atmosphere of Earth consisted of ONLY N2 and O2. Now, these gases could not cool by IR emissions, but they would warm by conduction from the surface. And they could not radiate to space. So would they continually warm for millions of years?? Would they melt the planet? WTF, folks?
———————-
jae, there was a good thread on this several months ago. Although I’d like to track it down for you, I think somebody will recall it better than I do and save me the trouble.
Obviously, the surface could still radiate and accept heat from the atmosphere through collisions. Convection would arise.
“For a theory to be scientifically proven, it has to be stipulated and tested, and the test must be repeatable and give the same results in successive tests for the theory to be proven.”
Try http://en.wikipedia.org/wiki/John_Tyndall. And http://en.wikipedia.org/wiki/Svante_Arrhenius. Late to the party as usual, sport.
Then try these nuggets on for size. Only fools and not even oil companies dispute man’s contribution to the greenhouse effect and the observed effects of global warming. If you are smarter than NASA you haven’t shown it here. Get out or lose your reputation. If you value the one you claim to have, that is.
“The question is whether a minor trace gas controls the climate. If you believe you have solid evidence that it does, provide it.”
See the above smokestack. Ever heard of exponents? It sounds like you’ve had too much of the emissions already though like the rest of these toasties.
@ur momisugly cba
That CO2 imparts warmth to N2 and O2 seems straight-forward, but it also seems at odds with the notion that it can heat the terrestrial surface at the same time by radiation.
If CO2 happily re-radiates (I know some don’t like that expression) the energy it absorbs, why is the surface not simply tossing it right back out when it receives it?
Sure, the minerals of the surface will briefly become hotter but then they will radiate more vigourously and a percentage of that radiation will be at frequencies incompatible with CO2 absorption, so it will be gone, gone.
What does “back-radiation” have to do with anything? Isn’t it like radiation within an iron bar that I heat with an oxy-acetylene flame? Overwhelmed?
@marky48
Including your IQ as part of your screen-name puts you at a bit of a disadvantage.
Very few will take your incredibly astute comments seriously and the planet will slip a little further into catastrophe.
Yes, it’s somewhat your fault but don’t beat yourself up too much!
marky48, right on que with insults and wiki references no less, ……….. here we go again.
marky48.
Arhenius having the wrong bandwidths I guess doesn’t really matter. Believers KNOW they are right so it is OK to adjust things to fit their incorrect models based on 100 year old incorrect science.
HAHAHAHAHAHAHAHAHAHA
@All…
Imagine you have 450 boxes on ground and each box can contain only one cat. Imagine you have only one box buoying in the air and it only can contain one cat. Now, imagine the Sun is sending 900 cats towards the ground, i.e. towards the place where the 450 boxes are placed. Remember, only one cat per box. From the 900 cats, only 450 cats hit on the ground, where the boxes are placed. 45 cats are “bounced” towards the outer space and 405 cats go into their respective boxes. There will be 45 empty boxes. But… ~365 cats get scaping from their correspondent boxes and they want to go back to the source of cats. They find that one box is obstructing their passage to the glory and one cat is “trapped”, momentaneously, into the box floating in the air. Question:
How many cats can be “absorbed” by the empty box that is floating in the air?
Let’s continue:
The cat inside the floating box tries to scape to the glory and jumps out from the box. Question:
Would the cat go back to the ground looking for another empty box? For answering this question, remember that there are not empty boxes on the ground because the Sun is sending cats continuously, therefore, the empty boxes on ground are occupied immediately by another cat incoming from the Sun.
Let’s continue: The cat finds another empty box floating few meters far away from the first box from which it had scaped and try to jump toward it. It fails and feels a strong cats current from the ground pushes it towards the outer space; additionaly, it finds that there is a very strong pressure exerted from the surface against which it cannot struggle. Questions:
Will the cat be re-absorbed by any of the occupied boxes on the ground?
Could it be that other kinds of containers do exist in the air, at shorter distances than the prior box where it was “trapped”?
Could the cat prevail over that strong current of cats and the powerful pressure, which push it towards the outer space, if it doesn’t bump into another empty container?
Keys:
Cats = IR quantum/waves.
Cats from the Sun = short wavelength IR quantum/waves.
Cats from the ground = long wavelength IR quantum/waves.
Cats jumping from the boxes floating in the air = very low energy density long wavelength IR quantum/waves.
Boxes on ground = limited and limiting configurations the energy absorbed by the ground can adopt.
Boxes floating in the air = limited and limiting configurations the energy absorbed by CO2 molecules can adopt.
Other empty containers floating in the air = limited and limiting configurations that the energy can adopt without minimizing the free energy.
Cats’ current = short and long wavelength IR quantum/waves outgoing from the emitters.
Pressure = radiation pressure.
Conclusion: Through spontaneous processes of heat transfer, it is impossible for a colder system to do work on another warmer system. In other words, it is impossible for a colder system to transfer energy towards a warmer system.
😐
” Observed effects of global warming”. You mean like all the hurricanes ? Maybe you mean the cold and snow. Oh, and flooding, cause thats never happened before. It’s all covered by the theory as well. Global warming, it’s whats for dinner, ……
I can’t resist. Those were some mighty big nuggets you had there marky48.
[Smoking Frog]The Inclusion-Exclusion Principle requires that we know the overlap(s) to begin with, so it can’t give us the answer.
[Jim Masterson] But it does give us an answer that is almost the same:
.49 + .06 + .20 – (.49)*(.06) – (.06)*(.20) – (.49)*(.20) + (.49)*(.06)*(.20) = 0.61648.
It can’t do it by itself, and you’re not using it by itself. You’re assuming random overlap without knowing it. (It didn’t occur to me that a random-overlap calculation could be that simple, so I thank you for the insight.)
Let the cloud layers be called A, B, C, corresponding to the 49%, 6%, 20%. Assume that the patches that make up each layer are infinitesimally small and randomly distributed in the spherical shell in which the layer resides. Then the most likely (A,B) overlap is (0.49)(0.06), the most likely (A,C) overlap is (0.49)(0.20), and so forth for (B,C) and (A,B,C).
Naturally, the patches are not infinitesimally small, but I doubt that this changes the result by much. Non-random distribution could change it by much, but it would take a meteorologist to deal with that problem.
Anyway, your answer is better than mine. I can see that, generally speaking, the two methods will give different answers, but I don’t see how to explain that the random-overlap answer is not the average of the two extremes.
I have read through this thread and have not yet seen a reasonable answer to the question raised in the comment –
richard verney says:
February 13, 2011 at 4:11 pm
What’s up with all the boxes of cats?
The simplest way I know to understand the greenhouse effect is to realize that when you’re standing on the ground in daytime the Sun is shining on you in visible light, and the atmosphere is shining on you in infrared. If you change the atmosphere so it shines more (e.g. adding more CO2 and other greenhouse gases) you will be warmer on the surface because there is more electromagnetic radiation coming down on you than otherwise. That’s it. More CO2 = more radiation on you.
All the complex logic might be important if you want to understand the details of radiative transfer (e.g. if you were grad student in physical oceanography or atmospheric science), but the previous description is good enough.
“I just say there are no scientific proof that increased CO2 emission causes climate change, or that it does not. And I am not on any side in the debate, for that very reason.”
Think for a bit. Is there scientific *proof* that Miami’s climate next winter is going to be warmer than Minneapolis’s?
Well, sure, because you can rely on basic physics. Because of the inclination of Earth’s axis, in the winter there is more electromagnetic radiation hitting Miami than Minneapolis, and this will be the case next winter too. You can handwave about advection and chaos and weather and unpredictability and yadda blah blah blah, but the overall number one physics controlling the temperature is the amount of EM radiation. After all, what else is there to make you above absolute zero? (only a tiny amount from radioactive decay of Uranium in the Earth’s center, and this can be measured and it is miniscule)
More E&M radiation = warmer climate.
And likewise, more CO2 in the atmosphere = more E&M radiation. This is not a theoretical prediction, this is a measured and observed *fact*.
With more CO2 in the atmosphere it is physically impossible for the climate *not* to change.
Next step is quantifying the amount, and then it gets complicated, but the people who have been doing this for 50 years are reasonably good at it, and you should believe them.
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