The physics of heat transfer

When I was going to college thermodynamics was dreaded by many engineering students. Apparently, it involved concepts difficult for many to grasp. For some reason it made perfect sense to me. Compared to most of my other engineering classes it was easy. I got a very high A in the class. That was decades ago, and I have forgotten a lot of it but I do have a good recollection of heat transfer.

That bright engineering students find it challenging means it should come as no surprise that people with little or no training in the subject would have misconception about how certain thermodynamic related physical phenomena occur. There have been many times on this blog I have made statements, or linked to articles, which described perfectly obvious observations. Some commenters declared them obliviously false. I didn’t want to take the time to explain why they were in error. It was just too much work for that particular situation.

It is now time to attempt explain certain things to people in terms and examples that will help them understand the physics of heat transfer. There are other sources on the web as well. But I will include example directly applicable to material on this blog.

The three* classic methods of heat transfer are:

  1. Thermal conduction (also called diffusion).
  2. Thermal convection.
  3. Thermal radiation.

The first two are relatively well understood at an intuitive level by nearly all functional people. It is the thermal radiation that I most want to address because of the clear lack of understanding I see in the comments here. I will explain the items 1 and 2 first to make the distinction from thermal radiation clearer. Please either stick with me or skip ahead if excessive boredom occurs.

Thermal conduction occurs when two objects of different temperature touch. Your finger touching an ice cube initiates the transfer of heat from your finger to the ice. The ice warms and when it reaches the melting point it changes phase from solid to liquid water.

If your finger is in a glass of liquid water and an ice cube thermal convection occurs**. Via conduction your finger warms the water touching your finger and because the warm water is slightly less dense than cold water*** the warm water rises. If the warm water is rising, then the cold water must be sinking. This creates a loop of water flow in the glass. It is slow enough that you cannot easily see it or feel it. If you were to put a drop of food coloring in the water, you probably could. The coldest water is next to the ice cube and the warmest is next to your finger. The water leaving the ice cube is replaced by water that recently left your warm finger. The warm water touching the ice cube conducts heat to the ice cube. This warms and melts the ice and cools the water causing it to sink. Heat is thus transferred from your finger to the ice.

Thermal convection occurs in gases as well as liquids. If you open a hot oven door with your face over the opening, you will feel an almost blast of hot air. No similar blast occurs at the crack at the bottom of the oven because cool air is rushing in. The hot air rises near a wood stove and cool air near the floor replaces it and forms a slow-moving loop of air. Soaring birds “ride the thermals” when different portions of the earth absorb more energy from the sun that others (for example dirt versus plants). This creates an updraft of air over the hotter earth which allows the birds to stay aloft with greatly reduced effort.

Thermal radiation occurs at all temperatures above absolute zero (-273.15 C or –459.67 F). But in our normal earthly circumstances most people are unaware of it because conduction and convection tend to dominate everyday life thermal transfers. Thermal radiation is electromagnetic radiation generated by the thermal motion of particles in matter.

  • Thermal radiation occurs across empty space as in from the sun to the earth.
  • Thermal radiation occurs through gases as in from the sun to the surface of the earth.
  • Thermal radiation occurs though solids as in the sun through glass.

The sun is not some magical generator. Microwave ovens emit a very specific frequency of identical electromagnetic waves which also transfer heat. The difference is only in the means by which the EM waves are creates, not in the nature of the waves.

The leaves of plants will sometimes get colder than the nearby air because they radiate their heat into outer space on a cold clear night when the air is still. Hence the air may be 35 F but the plant leaf can lose heat and drop to below 32F and get frost damage.

Orchard owner sometimes protect their crops by “heaters” which produce smoke to block the thermal radiation. The heaters do not produce enough heat to significantly heat the air. Owners also sometimes use large fans which move warmer air over the leaves. Via conduction the warmer air restores the heat lost by radiation.

The temperature of the sun is extremely high, and the thermal radiation occurs at high levels across a broad spectrum which includes the visible spectrum. The visible spectrum is what we call light. At the low frequency end of light, we call this thermal radiation red and even lower infrared. At the high frequency of light, we call this thermal radiation blue and even higher is ultraviolet.

Glass is not a magical solid conductor of thermal radiation. When exposed to thermal radiation all substances will do the following three things in various degrees with the incoming thermal radiation

  1. Transmit it through to the other side.
  2. Absorb it.
  3. Reflect it.

The amount of transmission, absorption, and reflection depend on the substance and frequency of the thermal radiation. These differing amounts are each described by a number between 0.0 and 1.0 inclusively. These numbers are called coefficients. The sum of all these numbers will always be equal to 1.0 (conservation of energy). Hence clear glass, for visible light, may have a transmission coefficient of 0.90. That is, 90% of the thermal radiation in the visible spectrum passes through the glass. The reflection coefficient may be 0.08 and the absorption coefficient 0.02 for a total of 1.0. Colored glass absorbs and/or reflects some energy at certain frequencies and transmits most of the energy at other frequencies.

Clean water has a high transmission coefficient for visible light but is highly absorbing of a certain frequency in the microwave region of the spectrum. This is why microwave ovens can heat a cup of water. The water absorbs nearly all the microwave frequency thermal radiation which the water intercepts.

Brick, wood, and other common house construction materials transmit thermal radiation at frequencies we know as radio waves. You can easily listen to your radio and make cell phone calls inside your brick building. At visible light frequencies and normal wall thickness there is no human perceptible transmission.

Thermal radiation is also why a vacuum is not a perfect insulator. Even in the hard vacuum of deep space, far from stars or any other object a warm object will radiate its heat into the surrounding empty space as lower and lower frequencies of electromagnetic radiation until it approaches a temperature of absolute zero.

This is also why the earth cools at night. It radiates heat into outer space. If it didn’t get rid of heat at the same average rate at it absorbed, it from the sun it would either get warmer or colder until the thermal radiation at night increased or decreased to match the visible light (as well as thermal) radiation absorbed from the sun.

Absorption and retransmission is where things get most obscure in our ordinary life and is where the commenters have been going astray.

Taking the example of an ordinary brick in the sunlight. It transmits none of the light, reflects some of the light in the red region of the spectrum and absorbs the rest. The absorbed the light causes the brick to warm. Some of that thermal energy is transferred to it’s surroundings via conduction and convection. But some of it is emitted as thermal radiation. This thermal radiation will be at various frequencies depending upon the exact chemical composition of the brick, but most will be in the infrared region of the spectrum.

This change of frequency is how certain gases get classified as “greenhouse gases”. This is how paint can actually cool the substance it is painted on below the ambient air temperature.

I’ll explain the paint first since it is simpler and has less emotional content.

The back side of the paint receives thermal energy via conduction. Suppose this paint is on a building at 75 F. It emits thermal radiation out into its exterior environment with the clear empty (sun and moon transmit their own thermal radiation) sky being a very cold (many degrees below zero) heat sink. Normal paints absorb significant light energy as well as conduction gains from the air. But what if the paint had very low conduction ability on the outside but high conduction ability on the inside, and the paint also reflecting almost all light? The outgoing thermal radiation would dominate the incoming heat transfer from the air and sunlight. Hence, the paint would literally cool the building it was painted on without the use of any external power source.

Now let’s consider the case of water vapor in the atmosphere. This is transparent to visible light. Clouds are condensation and/or ice. This water vapor transmits visible light to the earth which absorbs it and retransmits infrared thermal radiation just like our brick. The water vapor in the atmosphere, just like our colored glass, blocks the thermal radiation via reflection and absorption. If the incoming high frequency energy zips through the water vapor in the atmosphere and the retransmitted low frequency outgoing energy is reflected back to earth and/or absorbed, then the earth will get warmer.

That is the extremely simple version of greenhouse gases. Things get really complicated when you throw in things like clouds which reflect significant portions of visible light as well as whether they are clouds of ice crystals or water droplets and their presence during the day versus night, the latitude, the type of surface (earth, water, forest, ice, etc.) they are shading, and probably many other things. Does water vapor and/or CO2 really cause “global warming”? I don’t know. I am skeptical of manmade changes of CO2 in earth’s atmosphere causing heating and I think water vapor is complex enough that modeling it accurately is probably currently impossible.

Venus, almost for certain, is far hotter because of its mix of atmospheric gases than it would be if the composition were something like 80% nitrogen (earth) instead of about 3.5% nitrogen.and 96.5% CO2. So, I believe greenhouse gases can be a real thing.

Summary: Thermal radiation is not as well known by the general public as thermal conduction and convection. But it is real and easily observed if it is pointed out to you. Thermal radiation becomes the dominate heat transfer mechanism when long distances are involved. Thermal radiation exists at different frequencies. Substances have different absorption, reflective, and transmission characteristics at the different frequencies. Because of these different characteristics at different frequencies, it is possible to create one-way “heat valves”. Cooling paint and “greenhouse gases” are possible and exist because of these thermal dynamic “valves” utilizing thermal radiation.

* I won’t directly cover transfer of energy by phase changes or transfer of mass of differing chemical species.

** I’m not going to address the case of a zero-gravity environment.

*** Yes, I know, at temperatures between 0 and 4 C this is not true. Let’s not complicate things. But it is interesting to note this anomaly is why ice generally forms on the top of a body of water rather than on the bottom then floating to the surface.


30 thoughts on “The physics of heat transfer

  1. “Microwave ovens emit thermal radiation in the form we know as microwave (duh!) electromagnetic waves.”

    Could you please provide the definition of “thermal radiation” you use?

    • That was an error. Thanks for pointing that out. I have elaborated on EM waves generated by thermal action.

  2. Joe:

    As a mechanical engineer with a specialty in heat transfer and thermogoddamics I have to applaud your effort to make heat transfer a little more intelligible to non-engineering types. A couple of things I’d like to point out…

    There can be a huge difference in emissivity and absorptivity for a material at a given wavelength of radiation. I used to design air conditioners and heat exchangers for cooling electronics enclosures, and a question I frequently got was, “What is the best color for an outdoor enclosure to keep it cool in the sunshine?” The answer is that it should be painted a nice, dull, flat white. I would often hear, “Wouldn’t a more reflective surface like polished aluminum or SST be better for reducing solar gain?”

    The answer is “no”, because of the difference in emissivities at different wavelengths. Flat white paint (as compared to gloss) is an excellent reflector of solar radiation (on the order of 90% when clean), but much more importantly, it’s an excellent EMITTER of radiation at the blackbody temperatures typical of enclosure surfaces, again, on the order of 90%. Polished aluminum or stainless are also excellent reflectors of solar radiation, but are terrible emitters (10% to 20% depending on finish) at the typical enclosure blackbody temperature wavelengths.

    I was fortunate enough to have taken my heat transfer course from, quite literally, the author of the best book on the topic, “Radiation Heat Transfer” (Ephraim Sparrow, ME-PhD, may he rest in peace), and the graphs show emissivity versus surface temperature for different materials and finishes is fascinating (at least to heat transfer specialists).

    One other thing to mention is that the radiation heat transfer between two surfaces is at the fourth power of the ABSOLUTE temperature difference (°K or °R) rather than the simple temperature difference. This can make calculating exact answers by equation somewhat difficult since when the unknown temperature (say, the surface temperature of the enclosure) is in all three modes (convection, conduction and radiation) you’ll have that unknown in both simple and terms to the 4th power. It is easily solved by iteration (make a guess, plug it into the equations, get a different temperature, plug that back in and re-solve), the answer should converge to a solution within about a half-dozen iterations. I wrote a program on my (admittedly antique) HP11C to run a solution for me for enclosure surfaces in sunshine; it would do enough iterations to converge within a 0.1°F within a couple of seconds.

    Again, thank you for bringing a topic near and dear to my heart (and which was the basis for a 40-year engineering career) to your blog.

  3. Thermal radiation is the method by which thermal imaging works. The imaging system collects light in the infrared band, which is the light emitted through thermal radiation. My physics education is very limited, but I’m familiar with that due to my time in the military and I’m sure most people have seen the the images generated by such systems in the news or on TV. One interesting thing about it is that glass is opaque to Infrared radiation. When you see a thermal image of a car (they’re typically rendered in grayscale) the glass is always black because it blocks IR light.

    I’m no scientist, but I’m pretty sure that’s why the interior of a car sitting in the sun heats up so much. The visible light from the sun makes it through the windows and heats up the interior of the car, which then emits IR light in the form of thermal radiation, but the glass blocks that, preventing its escape, causing the interior of the car to heat up rapidly.

    At any rate CO2 is without a doubt a greenhouse gas, that’s well established scientifically. The problem with the doomsayers is that in our atmosphere, it’s basically a trace gas and cannot possibly be the “control knob of the climate” as the doomsayers claim. Even if they were correct, human contributions to CO2 are a tiny fraction of the CO2 emitted through natural processes.

    Is the Climate changing? Of course it is, it always has been and always will be. Are humans having in impact on it? Possibly, probably not a significant one if so, but it’s possible that the minor increase in CO2 that human activity contributes is having a minor impact on the climate. Can humans be realistically expected to be able to reduce their CO2 emissions enough to have an appreciable impact on their contributions to it in the atmosphere? Don’t make me laugh. Even if we were able to do so, would it stop the climate from changing? The climate is always changing, it always has and it always will.

    The reason homo sapiens have been able to survive for 200,000 years or so is not because we can bend the climate to our will, but because we are unusually adept at adapting to it by using our big monkey brains, opposable thumbs and ability to reason to utilize the resources around us.

    BTW: One thing I’m confused about with the “green” arguments. Didn’t human beings evolve through the same natural processes as, say, polar bears, sea smelt and arugula? So, aren’t we just another part of nature? When beavers build a dam, block off a creek, form a pond (and possibly cause a flash flood), why is that more natural than humans building a dam? Just because it’s smaller scale, they use inferior building materials, poor engineering and can’t explain their reasoning in the “Beaver Dam Monthly” journal? Why is a Termite Mound, a bee hive, a prairie dog tunnel, etc a natural thing, but a human city isn’t? Because we can use tools and concoct better building materials? BS. We are natural creatures, therefore the changes we make to the planet are just a part of evolution. If, as a part of those changes, we lack foresight and destroy ourselves…well, I guess that’s just a part of evolution too isn’t it? Everything that has a beginning, also has an end.

  4. LOL. When I was an employee of Uncle Sam’s Canoe Club, we had a class called “heat transfer and fluid flow” that students renamed to “fleet transfer and student flow.” It was the cause of many sleepless nights learning about heat transfer between solids and liquids inside of a nuclear reactor.

  5. Divemedic, which Nuc Power school did you go to? NPS MINSY class 7605 here.

    I will say one thing for the Navy, the way we learned the energy balance equation in HTFF made a LOT more sense than the way it was taught in ThermoGodDamnics for my engineering degree. Work done is negative?? Why, oh why do you create an easy error?

    I used the techniques I learned in the Navy to tutor an ME student from a D up to a B for his ME course. The best part is HTFF is in the first half so not classified. I still have my notes in storage.

    • I went to NTC Orlando in 1986. I honestly don’t remember my class number.
      You just reminded me of something. We were often chastised for saying “23 over 72” when naming fractions. We were required to say “23 divided by 72.” The round sidewalk in the center of the school complex was called Rickover circle. One of the chief instructors once told us to muster on Rickover circle, and a student replied “Don’t you mean Rick divided by circle?” The instructor was not amused.

      • Not a surprise at all. Nukes are all smart-asses and Malicious Compliance is a thing….

  6. Nice.

    Something in your post that I had not considered before was the possibility of an object to radiate enough in some circumstances to drop below ambient. That seems a bit counterintuitive at first, even as someone who’s read college physics books for entertainment. But then they were general physics books and not specifically thermal dynamics books.

    • It’s the process used to make ice in the desert before modern refrigeration methods.

      Dig a pit, insulate it and cover it. On clear nights open the top to radiate the heat out of what’s in the pit …

      The Persians called it a “yakhchāls” (ice pit)

  7. Radiation cooling is how you can get black ice on the road when if the air temperature is a little over 32F and the night is clear.

    The glass window in an oven door is a great example of selective transmission for IR or microwaves vs. visible light. The screen in the microwave oven door window gives you a clue of the wavelength of the microwaves inside, no need for a screen in a regular gas or electric oven. Note that they work by convective heat transfer not radiative (unless you use the broiler).

    From my readings on climate models, water vapor is the principle driving force for climate variations but it is extremely difficult to model because of the amounts of energy involved in evaporation and condensation and the small scales involved. A microburst that drops an inch of rain in a few minutes is far too small to be modeled at global scale yet it will have a huge effect on the local conditions. The carbon cycle that moves CO2 in and out of the atmosphere is a long-term process and CO2 is only a gas in the atmosphere so it CAN be modeled easily.

  8. The real job was always trying to explain it in simple enough terms for my lizard brain to grasp. Thank you!
    And I will admit wave dynamics are very tough for me.
    I wouldn’t even consider arguing with a thing you’ve said, Joe. It’s all been proven or you wouldn’t have said it.
    But how much of an effect are “greenhouse gasses” having on this planet?
    That’s the political football. And the power grab for a bunch truly ignorant people.
    All that aside.
    The real amounts of heat being transferred compared to the amount of radiation both transferred and absorbed. Is the question.
    Does CO2 hold or radiate heat more than say N2? Might very well. But not enough to make a real difference? Especially in the mixture of gasses that are our atmosphere?
    As for water, Heat and pressure were used to drive it 35 miles into the upper atmosphere. But what keeps it there?
    The human body that is 60% water drops at over 1100 feet per second back toward earth from that elevation. The atmosphere at that elevation is not dense enough to support it.
    Why would a condensed water droplet be any different?
    In the lower atmosphere. Pressure levels are enough that liquid water can float around. Mix in and absorb/release gasses according to all the laws of physics. Thermodynamics, radiation, and a lot of other things the understanding of which has doomed me to be a laborer all my life.
    Are there “greenhouse gasses”? yes. But as far as Bill Gates being correct as to they’re effect. No, there ain’t no such thing as a “greenhouse gas”. The way their being used is pure hyperbole.
    And it leaves out masses of other processes that keep those things in check.
    In order for a wooly mammoth to be frozen so fast that the food in it’s stomach was un-rotted. The Birds-eye frozen foods engineer said it would have to have been subjected to near -300 degreesF. IIRC. As cold traps heat. Deeper cold is required to penetrate that deep into a given object, in the amount of time allotted.
    The real question is how did we get back to the temp where we are today?
    The answer is there are a lot of things happening at once.
    And I don’t feel alone in thinking we might never grasp it all mentally. Tech is great. I truly won’t want to live without it. But no one can deny it’s use as a political tool also.
    I feel that has perhaps been the real misunderstanding in the comments?
    Not the real science, But the emotion the science is being used to generate.
    And thanks again for all you do Joe.

    • MTHead, gaseous water in the air (humidity) isn’t the same as clouds or fog (microdroplets of condensed water) so it isn’t subject to the same density limits that control maximum cloud height. Simple diffusion will drive H2O molecules to all the way into space.

      Further confusing things, terminal velocity for a microdroplet is FAR, FAR lower than that for a 200lb human body, the speed you quoted above isn’t applicable.

      Consider that rain drops are falling at their terminal velocity and they come down a lot slower than hailstones from the same clouds.

      • I’m good with that. But are you forgetting the pressure those gas/liquid/solids are being subjected to?
        The only reason water is able to float down here is because of the density of atmosphere. And the temperature its being subjected to.
        But gravity is.
        Terminal velocities are going to be determined by the external pressure around it. The density of what it’s going through. Which are not present at 35 miles up.
        And why would it take so much heat and pressure to push it up there in the first place? If water would just float up and get sucked into space?
        Something tells me we would be very dry down here if that could happen.
        As soon as water cools to a liquid state. Gravity is going to have its way until external heat and pressure change that.
        I could be all wet on this one. But something tells me no, neighbor.

        • Water IS a gas below the boiling point and can be a gas below the freezing point to. You can find tables on the vapor pressure of water in various conditions of pressure and temperature that will give you numerical values, a good part of HVAC for housing and people is dealing with the water vapor in the air from breathing and people sweating so Mechanical Engineers have to care about the different. Heck, the Triple Point is where solid, liquid, and gaseous water all coexist in equilibrium.

          Relative humidity is NOT water droplets in the air, it is water in gas form displacing the other gases in the atmosphere in the mix. Water droplets in clouds, mare’s tails, and fog are a different animal entirely. Diffusion is a relatively slow process and diffusion against gravity is even slower but it does occur.

          This is why we call it ThermoGoddamnics.. Lots of non-intuitive processes taking place.

        • Maxwell’s equations for Electromagnetics are ugly but solvable. The Navier-Stokes equations the govern fluid flow are massively ugly and only allow complete solutions for unrealistically simple cases. My last roommate got his PhD working on solving them for flow in a pipe… Terminal velocity is a function of pressure but also the mass, cross sectional area, and coefficient of drag for the medium. A water droplet has milligrams of mass and a thousandth of a square millimeter in cross section so it’s terminal velocity will be very low.

          • Whatever state it’s in. At 180,000 feet it’s pretty much frozen solid. At least -20C. possibly a lot colder.
            What will that actually do to the climate way down here? Like the Zen master said, We’ll see.

    • CO2 is a greenhouse gas, and it traps heat.
      Part of the problem, though, is that it is both a small fraction of the atmosphere composition, and the heat-trapping function is logarithmic.
      Most panic-attack GlowBull warmers assume it’s linear: double the CO2, double the warming effect.
      Most scientific glow-bull warmers think it’s down low to the left in the steep part at current concentrations, or maybe a little higher up near the 45-degree slope part where it passes through x-1.
      But a while back Lord Moncton showed it was much further to the right, so doubling the CO2 concentration only increases warming a little bit, and the more we add the less that addition does. We are into the “it makes almost no measurable difference” part of the graph.
      So yes, it absorbs and re-radiates… but not enough to really matter.

      • Exactly, CO2 is less than 300 PPM. Of that humans are only responsible for less than 20 PPM.
        And as Joe is pointing out. We need to make sure nature gets the credit for what’s going on with the climate. Not humans.

        • CO2 is now around 400 PPM.
          Below 200 and there start to be serious problems with plants growing, and grasses can seriously out-compete many of our food-crop plants.
          Many crops would do much better with it up around 1000 PPM.
          One of the great things to see is the graphs with error bars on natural CO2 emissions: they are about the same size as the entirety of all human emissions; that’s why most warmists don’t include them.

          • Even if the error bars and human emissions are similar in magnitude the CO2 in our atmosphere is increasing. There are multiple questions to be asked giving that fact:

            1) Is the increase due to human activity?
            2) What is the impact on humanity and the planet if we don’t stop the CO2 increase?
            3) What is the impact on humanity if we were decrease our contribution to the point the atmosphere CO2 stops increasing or decreases?

            I don’t believe there are answers to the first two question which can be defended with any certainty.

            The answer to the third question is fairly certain and is clearly extremely negative. Hence, looking at a risk matrix with the various actions and certainty of consequences it appears to me that with the available information the safest path is that we don’t cut our own throats trying to reduce CO2. That path is the way to certain doom, while the other paths are unknown and is of, probably, manageable impact. And if we want to error on the side of safety, global warming is far better than global cooling. Sea level rises of a few feet over decades is easier to handle that a mile of ice over everything north of Portland Oregon 100 years from now.

  9. I spent several years of my life managing a project to model chemical contaminates in near surface ground water at the 25 square mile Rocky Mountain Arsenal. The results had a such a large variance that the managers chose another contractor to repeat the analysis using different model that also resulted in just as disappointing results.

    No matter how good the model and our understanding of the physics, models of an in-situ processes are difficult to fit to sampled data. They do help understand the processes involved, but, like most models, are poor prediction tools. Adding to the complexity are the interactions and random inputs.

    A prime example is weather forecasting. We are now able to forecast the weather out to about two weeks using multiple supercomputers with different models. But because of the butterfly effect – a metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier – we will likely not see much more improvement in weather forecasts. While climate is a more stable process it is not immune to these limits. Likewise for living beings and human creations.

    Heisenberg’s uncertainty principle applies much more widely than most proponents of science think and that even includes many scientists.

  10. The killer class for me was Chemical Thermodynamics. Energy, Entropy, Gibbs Free Energy – c’mon man. It was partial derivative hell.

  11. People have trouble with the basics of this? I don’t recall exactly when in elementary school we covered convection, conduction, and radiation, but I know it was before HS physics class. I guess I don’t really want to read the comments on the Tonga volcano water vapor.

    Now I’m wondering what percentage of Congressmen have no grasp at all of how this stuff works. Anyone? Bueller?

      • The real deal was the admiral’s response. Never cracked a smile. “We don’t anticipate that.”
        To funny!
        The only thing that would have made it better was to ask the congressman. Could you repeat that?
        That video must be part of the Pentagon’s officer training. This is what you’re up against men! Know your enemy!

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