Heat Transfer and the Earth’s Surface


Previously I showed an equation that showed the ratio of convective and radiative heat transfer.  That equation is derived from the ratio of the Nusselt and the radiative heat transfer (RHT) rates.  From a practical standpoint the amount of energy transferred from the surface to the atmosphere by each method should be comparable for the situation of the Earth’s surface.

Warning:  this article contains significant engineering language and terminology.

Fans are effective at cooling because they induce what is called forced convection.  That increases the rate of heat transfer (speeds up cooling) from the warm object to the cool object.  Natural convection is what happens without a fan.  For instance a warm cookie placed on a counter top.  The cookie cools off by radiating heat away and by warming the air that it touches.  This warm air rises from the cookie to be replaced by cool air.  That is natural convection.  That is what happens to the surface of the Earth when it is warmed by the sun.  The warm air raises up because it is lighter and cool air replaces it.

Starting Point

There are many situations that the primary equations and dimensionless numbers that are used to determine the convective heat transfer from one object to another.  Since air is the medium of transfer from the surface to the atmosphere, that greatly simplifies the situation.  For air the Prandtl number is 0.713 (20 °C).  When dealing with air, the buoyancy forces will always dominate the viscosity forces (liquids will generally be opposite).  This results in a high Rayleigh number.  The transition point between laminar and turbulent natural convection will be used to lock the Rayleigh number down.  That value for air is 109.

Convective Transfer

That provides the starting point to derive the ratio between convective and RHT.  I will use the same methods as shown by Dr. Physics when he was demonstrating the ratio between the two transfers for the human body and the air.  The main difference is that he used the vertical cylinder to represent the human body.  The science is the same for transfer from a human as it is from the surface of the Earth.  Energy is leaving the warm body to the cooler air.

Nusselt number is the ratio of convective to conductive rates of heat transfer.

This is easily rearranged to show the energy transfer over the area as a function of the Nusselt number like this.

Energy as a function of Nusselt

Since the area will be the length and the size of interest is 1m and the thermal conductivity of air is 0.257 W/m/K the result is:

Energy per m2 by convective heat transfer

Since the Nusselt number is a function of the Rayleigh number it is easy to chart the amount of convective heat transfer over a range of air flows.

Radiative Heat Transfer

Since I want to compare the ratio of the RHT it will also need to be re-arranged from the normal Stefan-Boltzmann transfer I have shown before to the delta form.

Radiative heat transfer at surface

Ratio

Since the area is the same for both, it is only the ratio of the heat transfers that remain.

Nusselt Dependent Ratio of convective to radiative heat transfer at the Earth's surface.

This is in a different form than the one I used before because I want to show how important the turbulence is to the situation.  Air is always moving and in the atmosphere there is very little laminar flow.  So the lower region of the turbulent region should be the realm that determines the heat transfer ratio.  This would be the region where the Rayleigh Number is 109 < Ra < 1011.  The relationship for the Nusselt number and the results for this situation are as follows:

Relationship of the Nusselt number to the Rayleigh number for a horizontal surface convecting upwards

The Inconvenient Skeptic

Ratio of convective to radiative heat transfer for a warm surface to cool air above the surface for transition from laminar to turbulent.

This provides the general guideline for how much energy should be transferred by each method.  For the situation of transfer from the Earth’s surface to the atmosphere above, there is not a great temperature difference, as a result the amount of energy transferred by each method is comparable.  This is something that anyone with any practical experience in heat transfer should immediately recognize.  Radiative is not a dominant method unless there is vacuum or the temperature differences are large.  Neither of these situations is true for the surface of the Earth.

When the net LW (23 W/m2) and convective (17 W/m2) are compared in KT97, KT08 or any other energy balance of the Earth, the resultant ratio approximately 0.74.  That is exactly what is expected for natural convection at the laminar to turbulent transition.  That ratio corresponds to a Rayleigh number of 3.8E9 which is right in the range it should be.  Conversely if the convective were to be 4% like warmists that don’t understand heat transfer would have people believe, then the Rayleigh number would be 3.2E4.  That is an absurd proposition for the atmosphere.  The estimated value for the full atmosphere is 1017 < Ra < 1020.  For the sake of a limited scope of a few meters above the surface where the natural convection exists the estimated value of 3.8E9 is very reasonable.

From an engineering point of view it is wholly reasonable and expected that the amount of energy transferred from the surface to the atmosphere by both convective and radiative methods be comparable in magnitude.  The purpose of most engineering is to make complex problems solvable.  The different dimensionless numbers used in this article provide the method that engineers use to make the estimates that allow the precise complexity to be accounted for in a reasonable manner.  Bulk temperature for gases and liquids are used instead of an exact temperature profile.  The modern world works because these tried and true methods work.  From an engineering perspective, radiative and convective heat transfer from the surface to the atmosphere are comparable in scope and magnitude.  That limits CO2 to about 3% of the total and also limits the effect of changing CO2 levels, but more on that later.

The Inconvenient Skeptic

Thermal map of convection. The air at the bottom surface warms up and changes density. This causes the air to rise and starts convection.

Posted in Energy Balance and Radiative Heat Transfer by inconvenientskeptic on December 13th, 2010 at 11:52 am.

16 comments

This post has 16 comments

  1. Richard Sharpe Dec 13th 2010

    “That limits CO2 to about 3% of the total and also limits the effect of changing CO2 levels, but more on that later.”

    Can you explain that a little more? That is, the 3% number.

    Also, what is to stop the other constituents of the atmosphere from transferring some of their (thermal) energy to CO2 via collisions? I guess that since CO2 is such a small proportion of the atmosphere (0.03% or something) this would not happen all that frequently.

  2. inconvenientskeptic Dec 13th 2010

    Richard,

    I showed here that when it comes to net energy transfers, CO2 is responsible for about 3% of the total energy that causes warming in the atmosphere.

    Most of the warming in the atmosphere is the result of collisions. I have not really discussed that, but when CO2 does absorb energy, it is shared with other molecules almost instantly through collisions.

    Every CO2 molecule is surrounded by tens of thousands of N2 and O2 molecules. They ensure that CO2 doesn’t get warmer by roughing it up every time it gets some energy. Those collisions happen billions of times each second. CO2 is a small fraction of the atmosphere, so the odds of it colliding with itself is small, but the actual collision rate with other gases is billions of times per second.

    This is why gas is generally at the same temperature.

  3. Richard111 Dec 14th 2010

    The Maxwell-Boltzmann energy distribution curves indicate that in the upper, cooler atmosphere, below 0C, CO2 molecules are extremely unlikely to reach emission energy levels. This indicates to me that most CO2 molecules will absorb their fingerprint bandwidth of radiation energy from any source and pass that energy to the surrounding air molecules almost instantly. The CO2 is acting as a filter for a small portion of the total LWIR from solid surfaces such as land or sea and ice and water droplets in clouds. It would seem most of this radiation, some 8% of the total, will only penetrate a hundred metres or so of CO2 “polluted” air. The heat energy thus “trapped” becomes part of the normal lapse rate tranfer up the air column. Increasing or decreasing the level of CO2 will not effect this process.

  4. inconvenientskeptic Dec 14th 2010

    Richard,

    Once energy is in the atmosphere it doesn’t matter how it transfers from one molecule to another. That CO2 can reach transmission state only says it didn’t equilibrate by collision.

    Gas transmission in the atmosphere happens, but it is really just a very, very small part of keeping the atmosphere at thermal equilibrium. Collisions dominate and CO2 concentrations don’t matter for collisions. Internal CO2 transmission is basically high energy collisions. They don’t change anything unless the energy escapes the atmosphere.

  5. Richard Sharpe Dec 14th 2010

    Richard111 says:

    “The Maxwell-Boltzmann energy distribution curves indicate that in the upper, cooler atmosphere, below 0C, CO2 molecules are extremely unlikely to reach emission energy levels. ”

    Surely this does not matter all that much since the main argument of the AGW CO2 proponents is that CO2 absorbs some fraction of the outgoing LWR (and because of collision broadening, that fraction is larger than you think) and thermalises it because in the lower atmosphere the mean path length is shorter than the mean time to emission.

  6. inconvenientskeptic Dec 14th 2010

    Their argument is not the direct. They propose that the energy absorbed is converted to thermal energy, but then re-emitted upwards where the cycle repeats itself.

    CO2 does absorb some fraction of the energy and it does broaden slightly, but the total energy that the atmosphere absorbs from the surface is not sensitive to the concentration of CO2. The spectrum that CO2 absorbs overlaps with H2O. It is only in the 13-14 micron band that broadening matters. CO2 levels have always been high enough to cause broadening and the total energy absorbed doesn’t change as much as the distance it takes changes with concentration.

  7. Richard Sharpe Dec 14th 2010

    “Their argument is not the direct. They propose that the energy absorbed is converted to thermal energy, but then re-emitted upwards where the cycle repeats itself.”

    That is not how I understand what is being said.

    My understanding is that they say that the energy is converted to thermal energy and that half is re-emitted upwards (where eventually it would be lost to space, I suppose) but that the other half is re-emitted downwards, where it somehow causes runaway temperature increases or something.

    However, it seems to me that this re-emitted LWR would mostly cause more evaporation from the surface of the oceans (which make up 70% of the surface of the earth)

  8. inconvenientskeptic Dec 14th 2010

    The problem with that idea is the net energy transfer. The atmosphere is colder than the surface (generally speaking). The atmosphere is warmed by the surface. Emitted LW from a cold object cannot cause a warmer object to get warmer.

    Re-transmission downwards or sideways doesn’t accomplish anything of significance because the net flow of energy is upwards.

    Energy flows from warm to cold. Re-transmission does not bypass that behavior.

  9. Richard Sharpe Dec 14th 2010

    I agree with all you say. I am simply trying to understand where they are going wrong …

  10. inconvenientskeptic Dec 14th 2010

    The main place they are wrong is treating LW transmission as a transfer of energy. They doggedly insist that Stefan-Boltzmann transmission is the same as net energy transfer.

    Read this one again. They use the transmission and consider it transfer. Two objects at the same temperature have no net transfer and no change in temperature as a result. They can have immense transmission while not transferring energy.

  11. Richard111 Dec 14th 2010

    John, can you expand on this point please: “”It is only in the 13-14 micron band that broadening matters.””

    In my limited reading on the web I have read of laboratory experiments with flasks of pure CO2 and light sources in excess of 1,000K and yes, CO2 on CO2 molecular collisions do show line broadening. Can you point to any experimental evidence that shows CO2 line broadening when it is only CO2 on N2 or O2 molecular collisions with kinetic energy speeds limited mostly to about 100 metres per second?

    You see, what bothers me as that though CO2 can easily absorb energy it is not instantly excellerated to a higher kinetic speed. The CO2 vibration mode changes and this can impart an increase in kinetic energy speed to another molecule during collision. Although it is possible another molecule can strike the energised CO2 molecule and cause the CO2 to emit, this is a rare event. I simply cannot see how line broadening can happen with a single CO2 molecule. Multiple CO2 on CO2 collisions in the atmosphere at 400ppmv are simply going to be extremely rare events.

  12. inconvenientskeptic Dec 14th 2010

    Richard,

    I was planning on putting together a kinetic gas article that deals with this. Lets discuss if further when I get that out in the next week or so.

  13. Glenn Tamblyn Dec 17th 2010

    Let me pick up on a few points here.

    “They propose that the energy absorbed is converted to thermal energy, but then re-emitted upwards where the cycle repeats itself.”

    This is incorrect John. It is well understood by climate scientists that the largest mode of transfer within the bulk of the Atmosphere is convection/collisions. I referred you to Hulpert’s paper from the 1920’s on this.

    Re-radiation is a part of the mix but not the dominant one. Where Absorption and Re-radiation matters is at the top and bottom of the atmosphere. At the bottom, the re-radiation is able to reach the surface. And at the top, the re-radiation is able to reach space. Everthing in between is mixing.

    Sort of like a cloud. At the bottom you can see the ground, at the top you can see the sky. In between it is all just cloud,.

    “I showed here that when it comes to net energy transfers, CO2 is responsible for about 3% of the total energy that causes warming in the atmosphere.”

    And I showed that your reasoning was totally faulty. How can a process (not just CO2) that absorbs 90% of outgoing LWR only contribute 3% to the GH Effect?

    And you continue to use vague language about ‘warming’ – KT is a snapshot of the energy flows. It is not about warming – CHANGE of temperature – but what the temperature is as a RESULT of the flows.

    “The main place they are wrong is treating LW transmission as a transfer of energy. They doggedly insist that Stefan-Boltzmann transmission is the same as NET energy transfer.” (my emphasis)

    You are putting words in peoples mouths here John. Can you give us one single reference/citation/quote for ANY CLIMATE SCIENTIST saying this. NO Physicist, Engineer or Climatologist/Meteorologist would say something so absurd. I have NEVER seen anyone say such a thing. So, Sources Please!

    “CO2 on CO2 molecular collisions do show line broadening”

    Richard. My understanding of Line Broadening is that it is based on two effects. And these aren’t dependent on what the molecules around the CO2 (or others) are.

    Density effect. If a molecule that might radiate a photon is surrounded by more other molecules of ANY variety then collisions and even near passes at the time it is radiating will alter the exact quantum mechanics of what frequency it radiates at.

    Doppler effect. In a higher temperature gas, some of the molecules are moving faster. So the Doppler effect will impact on the frequency at which they radiate.

    So line broadening isn’t specific to CO2. It is a behaviour of any molecule that is part of a gas.

    Both these effects impact not just on ABSORPTION frequency but also just as much on RADIATION frequency.

    “The atmosphere is warmed by the surface. Emitted LW from a cold object cannot cause a warmer object to get warmer.”

    This is very confused John. You started out looking at K&T which is a snapshot of energy flows,. A static picture. Then, as I have commented on before but you have never responded to, you talk about WARMING – Change of Temperature.

    So, “Emitted LW from a cold object cannot cause a warmer object to get warmer”. NO, BUT IT CAN DETERMINE WHAT TEMPERATURE THAT OBJECT IS AT.

    Your argument is totally confusing WARMTH with WARMING. Absolute value vs Change of that value.

    If you want to discuss the GH Effect, AGW etc, you first need to discuss what sets the steady state values. Only then discuss what might cause changes in that steady state value.

  14. inconvenientskeptic Dec 19th 2010

    Glenn,

    There is a pattern to your postings. You start off saying how wrong I am about something by using a strawman argument.

    In this case your strawman is:

    “This is incorrect John. It is well understood by climate scientists that the largest mode of transfer within the bulk of the Atmosphere is convection/collisions. I referred you to Hulpert’s paper from the 1920’s on this.”

    You end by saying that you need to determine steady state and determine the impact of a change. You ignore that basis of the article which is how convection and LW absorption are comparable.

    Textbook case of ignoring the point while making it sound like the person you disagree with is wrong.

    The point of this article is that the amount of energy transferred by convection and LW absorption from the surface to the atmosphere is comparable. Since you failed to discuss that… you should be moderated off topic. But I prefer not to do that.

  15. Glenn Tamblyn Dec 21st 2010

    Actually John, I wasn’t responding to your post. I was responding to the follow on comments you made. And my comments were relevent to those comments and therefore on-post.

    As for my first point being a strawman – no it wasn’t John. I was actually pointing out that the comment you had made was a strawman; misrepresent an argument in order to try and knock it down. Similarly with the comment about SB and net transfer.

    Both those statements are simply not true.

    One thing that really gets my goat when responding to many sceptic arguments is how often misrepresentations of the science are put forward as a basis for a supposed rebuttal of the science. By all means, disagree with the science but at least accurately describe the thing you are disagreeing with.

    My last paragraph was a summary position based on the fact that this discussion is stretching over many, many separate posts and the occassional summary is needed. I understand the need to break this up into smaller posts but it makes it more difficult to keep track of the big picture of the arguments.

    Now to the body of your post. More in the form of open questions:

    Your eqn for Convective transfer is using a Delta T between the Earths surface and the ground level atmosphere since this is the only basis for analysing convection. Similarly you have reduced the SB eqn to a Delta T form. But are these the same Delta T? Convection is based on the VERY near surface temperature whereas SB is based on the temperature of the larger air mass in radiative exchange with the surface. What height are you assuming this to be and thus what delta T are you assuming? If the same temp, why are you assuming that radiative exchange only occurs with the VERY near surface air? Since the Lapse Rate for the atmosphere is around 6.5 DegC/Km, air higher in the atmosphere in radiative exchange with the ground will be at varying temperatures. When does the simplification applied to get your delta form of the SB eqn break down?

    How do you take account of the fact that around the world, temperatures are different at the surface and in the atmosphere. Since the SB eqn in particular is based on Temp’s to the 4th power, local variations can be significant.

    How have you evaluated the impact of negative convection – warm air masses that heat the underlying ground. Since the figure for thermals is a NET figure, some of this will be convection in reverse.

    Your calculations based on Rayleigh number are assuming that you are dealing with Free Convection. In reality weather systems involve winds over substantial parts of the world. So you need to look at Forced or Mixed convection to do a more realistic analysis. You then need to factor in coriolis force in how it alters flow patterns.

    You state “Bulk temperature for gases and liquids are used instead of an exact temperature profile. The modern world works because these tried and true methods work” How do pressure changes with altitude affect this analysis? How do you allow for temperature variation with altitude. Does the Adiabatic Lapse Rate change your conclusions. Dry or Saturated version? Maybe the “tried and true methods” used by engineers in more microscopic contexts are simply inadequate for the real Real world of the macroscopic scale.

    And the big elephant in the room as far as your analysis is concerned is EvapoTranspiration. You are trying to compare Radiation vs Convection as if this is a 2 process Heat Transfer problem when in fact it involves 3 processes. Is the Thermals aspect of KT totally independent of ET or does it include convective transfer as a consequence of ET? I Dunno, I haven’t read the source papers KT cite. Have you? How do your ‘expected’ ratios adjust when dealing with the ratios of 3 processes, particularly when two of them, Thermals & ET, are closely linked?

    John. I am not trying to be a smart-arse. I haven’t the faintest notion what the answers to these questions (and dozens of others could be added) actually are. And I doubt you do either.

    My point is that any attempt to estimate/calculate what the ‘expected’ ratio of radiative to convective heat transfer ‘should’ be is meaningless. You are trying to apply Engineers rule-of-thumb insights from their normal working life, small scale, small world experience, to subjects that have a totally different context and scale. Engineers deal with pumps, power plants, pipelines. Not planets. Their day-to-day working experience isn’t automatically transferable.

    If you want to understand how these things work you need to go back to the basics. Remember, Engineering is Applied Science. Scientists work it out, Engineers apply it. Ask the Scientists.

  16. inconvenientskeptic Dec 24th 2010

    Glenn,

    The reason that the temperature drops with altitude is that most energy is transferred near the surface. That lower boundary layer of the atmosphere is the surface and since the atmosphere and the surface are in near steady state equilibrium is the best indication that much of the heat transfer takes place at that boundary.

    You are correct that there are times where the air is warmer than the surface. That happens on a regular basis. That is easy to see during the winter when snow is melting when the temp is 3-4 C (37-40F). This does happen on a regular basis and is the one of the driving factors of weather, but in the end it doesn’t matter on average because the atmosphere absorbs far more energy overall. Walking on asphalt during the summer is a good example of the ground warming the air up.

    By definition energy transfer from the surface to the atmosphere is very near the surface. That is why I used such a low Rayleigh number. The transition from laminar to turbulent convection is a good starting point for the limited natural convection at the surface. Certainly higher in the atmosphere there is greater turbulence, but that doesn’t affect the surface convection.

    You should notice that the higher the Rayleigh number the more convection transfer there would be. That the KT balance would generate a Rayleigh number around 4e9 is a good indication that the assumption of the Rayleigh number is accurate. When the results of the theory and the observations fit it is usually a good sign.

    The purpose of this article was to show that it is easy (relatively) to show that convection and radiative heat transfers should be comparable in scope for the Earth’s surface and the atmosphere. Both of these are dwarfed by the latent heat transfer of water vapor to the atmosphere.

    My point is that a small change in a small piece of the puzzle isn’t going to cause a big change in the Earth’s climate. CO2 is responsible for about 15% of the greenhouse gas absorption in the atmosphere. That makes it about 3% of the total energy transfer from the surface to the atmosphere. If it goes from 3% to 3.3% that could possibly result in a temperature change of 0.1 C.

    CO2 doesn’t affect enough energy for a change to result in a significant climate shift.

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