Slaying Your Own Argument

In a post entitled "CO2 as a radiation valve contravenes the laws of thermodynamics" our old friend and "Dragon Slayer" Alan Siddons analogises the atmosphere as a "radiation" valve. He imagines such a valve letting in solar radiation, but preventing any escaping to space. Of course, in such a system, the Earth's surface would continue to heat up because there's no exit for the heat generated in the surface. He claims that this disproves the "greenhouse effect" because "back-radiation" from the "valve" (the atmosphere) is somehow magnified and therefore causes a "runaway" heating cascade. But with no path for the heat to escape, such an effect is inevitable anyway, "greenhouse effect" or not. It's nosensical, of course, and proves nothing.

The second part of the post describes a "valve" which actually lets some radiation to escape, so is worth examining. However, in summary this is another example of a poorly labelled diagram, which in this case seems to have confused the author himself;

I urge you to notice that the valve's efficiency doesn't actually matter, either, because physical laws are violated even in a modest case. In some sense, in fact, the crimes get worse. For instance, let's install a 20% valve, so that 80% of the infrared escapes and 20% back-radiates. 

In this case, 0.8 exits while 0.2 is "retained" by the surface. But 0.2 also radiates back to the surface, so it gains 0.4 in total (again, as a minimum: further back-radiation effects must arbitrarily be halted). In other words, even when the oft-mentioned "net flow" favors the outward movement of thermal energy (a modeling effort to satisfy the Second Law), the alleged heating effect still contradicts the First Law because you're getting more energy than you put in. Any furnace manufacturer would eagerly exploit such a loophole in the law if it existed. 

"In this case, 0.8 exits while 0.2 is 'retained' by the surface" - yet the upward red arrow at the top reads 0.8, meaning that the red arrow below it must represent 1 unit in order for the "valve" to have absorbed 0.2 units. Not a good start, since if the surface "retains" 0.2 it can only radiate 0.8 with a corresponding 0.64 exiting the "valve". The 0.8 shown escaping plus the 0.2 "retained " by the surface plus the 0.2 absorbed by the valve add up to 1.2 and not 1. Earlier in his post he said "You can't obtain more energy than you put in", yet he's just done it! Apart from this fundamental error, if the surface retains a net anything the system can never reach equilibrium - it has built-in positive feedback from the word go. He says earlier "for every unit of sunlight going in", so if for every unit going in 0.2 is "retained" the surface will continue to heat up, and thus far in the process the the "valve" has played no part in any "feedback". So the diagram doesn't actually represent what the description says it does. In reality the surface never does actually radiate 1 unit, as implied in the diagram, unless the"valve" is initially set to allow all radiation through, and then activated. All that's needed (once again, as in "A Simpe Solution") is to consider the energy actually lost from the system to get a properly thought-out analogy or "model".

At equilibrium, 1 unit escapes through the valve, which means that the surface is radiating 1/0.8 = 1.25, and the "valve" is absorbing and re-radiating 0.25 downwards. But where does that "extra" energy come from? It comes of course from energy absorbed during the equilibrium process. initially the surface acquires 1 unit, absorbs that energy and heats up until it radiates 1 unit. The "valve" absorbs 0.2 of that, and it heats up until it radiates the same down. The process continues with diminishing increments - the 0.2 is absorbed by the surface, increasing its heat content until it radiates 1.2 units, with 0.96 escaping and a total of 0.24 absorbed by the "valve" and so on. Of course, the process doesn't actually have discrete steps - it's smooth and continuous. The heat content of surface and valve increase until radiation in and out of the system balances and equilibrium is obtained. No physical laws are violated, and "furnace maufacturers" are well aware that it takes time for their products to heat up to operating temperature when turned on.

There's the nub of the matter - the "Slayer's" misunderstanding of the difference between heat flow amd heat content. The surface of the Earth and the atmosphere don't radiate because they are absorbing radiation, they radiate because they contain heat. Radiation from the sun merely tops up the heat content. At night there is no radiation from the sun, yet of course the Earth/atmosphere system cotinues to radiate heat to space, slowly cooling. It's surely a simple concept to grasp, yet it appears to elude some.

A Simpe Solution

In Zen and the Second Law of Thermodynamics I discussed claims of the non-existence of the so-called "greenhouse effect". Some of those claims use analogies or examples to "prove" the point. Most of these are misconceived at best and erroneous at worst. Most scientists and informed non-scientists accept the basic physics. I won't say "99%" or similar because I'm aware that 87.37% of statistics are made up on the spot, and in any case I don't know what the figure might be. This is not an "appeal to consensus" - the basic physics involved is well understood and founded on fundamental physical laws.

Here's such an analogy designed to "prove" that the infrared radiation ("back radiation") from the atmosphere to the Earth's surface either doesn't exist or cannot have the claimed effect. Not surprisingly, it's from one of the many authors of "Slaying the Sky Dragon" who thinks he's illustrated "perpetual feedback":

Say you have a blackbody plate (think of an electric heater) radiating 1000 W/m² toward another plate which, because of distance, absorbs half of that intensity, i.e., 500 W/m². At equilibrium, the receiving plate thus radiates 250 W/m² toward the 1000 W/m² plate. Question: Does the 1000 W/m² plate thereby rise to 1250 W/m²? If so, then, by raising the radiator's temperature without adding more energy, you've disproved the first law of thermodynamics. Effectively, you've made the radiator heat itself. Moreover, now at 1250 W/m², the radiator will heat the other plate still more, absorb another dose of back-radiated energy, and will reach 1562 W/m². And so on, ad infinitum.

This seems to prove a feedback causing a runaway heating in the system. However, the author (Alan Siddons) is confusing an infinite geometric series (with a finite sum) to a never-ending sequence of feedback. Let's look closer at the diagram. Something is missing (par for the course in these analogies). In this case it's the radiation lost from the system, and examining that will directly give us the equilibrium conditions.

At equilibrium, the two-plate system is receiving the equivalent of 1000 W/m² via the heated plate, so the system must lose that energy to the surroundings. If the heated plate is now radiating x W/m², then x/2 W/m² is being lost to the surroundings. The receiver is absorbing x/2 W/m², so must be radiating the same amount, with half of that radiation, x/4 W/m² lost to the surroundings. We have a simple expression to solve:

x/2 + x/4 = 1000   so 3x/4=1000  and x = 1,333.3

The heater radiates 1,333.3/2 = 666.7 (actualy 666.66 recurring) to the receiver, which radiates half, or 333.3 back:

Balance is restored and no runaway heating. But where does the "extra" radiation come from, which as claimed above "violates the First Law"? It comes from the heat energy stored in the system during the equilibrium process, and the amount depends on the specific heat of heater and receiver. There is no violation of the First Law, and no violation of the Second Law either. Net flow between the two plates is from hotter to cooler, with both plates heating up until overall equilibrium is restored. Al the heat energy in the system came from the heated plate, which has some of that returned to it from the receiver. The returned heat reduces the net heat lost from the heater, so because the input remains the same, it heats up until the heat lost from the system equals the heat input.

Does the cooler receiver "heat" the hotter heated plate? No, it simply replaces some of the energy radiated (and therefore lost) from the heater, so it's the continual 1000 W/m² energy input which does the heating. Mr. Siddons has himself said "a cooler body cannot heat a hotter body, it just slows the rate of cooing". This is a perfect example - initially the heater radiates 500 W/m² to the receiver, reducing to a net 666.7 - 333.3 or 333.3 W/m² at equilibrium, and the result is a hotter radiator, radiating a total of 1333.3 W/m².

The misconception arises from only considering the instantaneous input to the system and ignoring the heat stored during the equilibrium process, a common thread in such discussions. If you intend waving a big stick, make sure you've got hold of the right end of it first.

A Question of Scale (3) - Latent Heat

Latent (hidden) heat plays an important role in climate and weather, but its importance is often overlooked or misunderstood. Heat is simply the energy of vibration of atoms or molecules in solids, liquids and gases. Heat is motion; motion is heat. More motion equals more heat - hotter. Less motion equals less heat - colder.

In a solid such as ice, water molecules are fixed relative to one another. They can move a small amount relative to their neighbours, but can't move freely through the solid; it's as if they were inter-connected by springs or stiff elastic. These notional "springs" are the strong attractive force which operates at the relatively short distance between the molecules.

In a liquid, molecules vibrate much more, and are free to move about, like table-tennis balls in a bucket. Shake the bucket, and the balls move about. Shake it more; balls from the bottom can move to the top, and vice-versa. They collide with one another, thus transferring energy of motion (kinetic energy). More vigorous shaking may result in a ball leaping above the others - it may even leap out of the bucket. This is a simple illustration of evaporation - balls moving through the air have effectively become a gas.

In a gas, the molecules have much greater energy of motion, and are free to move in all directions like the balls that escaped from the bucket. They collide one with another, transferring and sharing their kinetic energy. As a consequence they are much farther apart than molecules in a liquid or solid.

To transform a solid into a liquid, enough heat has to be added for the kinetic energy of the atoms or molecules to overcome the strong attraction between them. The amount of heat per unit of mass of the solid is termed the latent heat of fusion for that solid. It is many times greater than the specific heat, which is the amount of heat per unit of mass to raise its temperature by one degree Celsius (or Kelvin). Why is this important? It takes far more heat to melt ice than it does to warm ice to its melting point.

The specific heat of ice is 2.108 kJ/kgK (thousand joules per kilogram per degree Kelvin); the latent heat of fusion of ice is 334 kJ/kg. It takes 158 times as much heat to melt a given mass of ice as to raise its temperature by one degree. It appears to me that it's a common misconception that once ice is raised to its melting point it will melt rapidly. It would take 158 times longer to melt ice than to raise its temperature the last degree to the melting point, given the same rate of heat input.

We use this property of ice almost daily. A couple of ice cubes in a glass of juice or spirit lowers the temperature rapidly, but that cooling reduces the heat available for the ice to melt, which must then come from the surroundings - the sun, the air or a warm hand. A large ice-floe or iceberg can drift for months before disappearing completely. Some of the much larger ones can endure for years. The ice that does melt removes a lot of heat from the surrounding air and water, reducing the temperature difference and slowing the transfer of heat and therefore the melting rate.

Enough about ice - I can see the bottom of my glass. Conversion of a liquid into a gas, as in water to water vapour (or steam) requires an even greater input of heat energy as does melting. The figure of 334 kJ/kg for ice to water compares with 2,270 kJ/kg for water to vapour - nearly 7 times as much. This is important because it's the mechanism by which vast quantities of heat energy are transported aloft from the surface of the Earth into the atmosphere. It's a major factor in climate - some would argue (including myself) that it's the major factor.

Evaporation of water cools land and more importantly ocean surfaces, the resulting upward convection drives surface winds, and the water vapour condenses to forms clouds which block sunlight from reaching the surface. The condensation into clouds releases the latent heat into the upper atmosphere, where it's better placed to finally radiate into space and balance the incoming dynamo of the climate; energy from the sun. Water and its latent heat therefore has a major impact in cooling the Earth's surface, a direct result of its unusual properties compared with other common substances.

See this page for the properties of dihydrogen monoxide (or hydroxyl acid - it's nasty stuff, thousands of people die because of it, or a lack of it, or in it, every year).