For something – anything – to acquire a temperature above absolute zero (0 K), it somehow needs to be able to warm. The only real requirement for something to be able to warm is for it to possess a ‘thermal mass’, or simply ‘mass’. A thermal mass provides the thing in question with what is (a bit awkwardly) called a ‘heat capacity’, meaning a capacity to absorb and store energy from some energy source (external or internal).
We already know, from basic thermodynamic principles, how energy can be transferred to (or from) an object. It can be transferred in the form of ‘heat’ [Q] or in the form of ‘work’ [W]. Whenever energy is transferred to an object, the ‘internal energy’ [U] of that object increases as a result, which simply means that the object in question has absorbed (energy isn’t ‘transferred’ to a system until it’s actually become ‘absorbed’ by it) the energy to store it inside its mass, as microscopic kinetic and potential energy of its atoms and molecules.
We already know, from the first post in this series, how system ‘internal energy’ [U] relates to system ‘temperature’ [T]. We know that a system with a high ‘heat capacity’ will warm more slowly than a system with a low ‘heat capacity’, both systems absorbing equal energy inputs, the high-heat-capacity system simply storing a larger portion of the absorbed energy as internal/molecular PE rather than as internal/molecular KE (determining the temperature). Both systems, however, will warm, only at different rates. U and T invariably move in the same direction. Unless there is an ongoing phase transition. Then U will increase and T will not. There is no process, though, where U increases and T decreases. The two correspond.
OK. We know that to make an object warm, we must make it accumulate ‘internal energy’. If it doesn’t, it cannot warm.
Before we reach the steady state (dynamic equilibrium), which is the point above which the temperature of a warming object will no longer rise, some of the incoming energy (the energy transferred to the object) will always accumulate within the mass of the object, raising its internal energy. This energy is then held inside the object, it stays there, in the end making up the object’s static, equilibrated fund of energy, its baseline U, determining its final, steady-state temperature.
The corollary of this storing up process of incoming energy for the object to warm, is that not as much energy will be able to leave the system as what comes in. Until such time as you actually do reach the steady state. The reason you will have no more warming upon reaching the steady state is, after all, that energy OUT finally equals energy IN. If there is no ‘work’ involved, then we have a situation where Qout finally equals Qin. Remember this. It is the heats that need to balance.
At this point, it is time to discuss Willis Eschenbach’s so-called ‘Steel Greenhouse’, as a suitable model to demonstrate and utilise the basic thermodynamic concepts we have learned.
Remember first that the ‘Steel Greenhouse’ model is a strictly radiative one. Except from the ultimate, constant energy source, the nuclear one inside the central sphere, heating it through conduction, there is no heat loss or heat gain by other means than by ‘thermal radiation’ between systems/surroundings in this setup.* All arrows represent radiative heat only. Meaning, no air (mass) and hence no real temperature surrounding or between our systems. This is very much a hypothetical situation simply designed to be instructive on certain basic concepts. Eschenbach and I very much disagree on exactly what these concepts are …
*However, there is also the point of conduction through the shell. Which we will ignore for the moment, but will get back to in the next post or the post after that. It is, after all, an essential point to how the real Earth system actually works …
What I very much want to be perfectly clear on, is that it’s not the described EFFECT that I’m objecting to. It is a very real, very natural effect and violates no thermodynamic laws whatsoever. No, it’s strictly Eschenbach’s cringeworthy attempt at an EXPLANATION of how the effect comes about, that I object to. And that quite frankly anyone in their right, critical mind should object to. At once, instinctively, loudly. I’m honestly amazed how people can seriously buy into his utterly inane fabulations, trying to pass them off as well-established Truths of ‘basic physics’.
Well, let’s get to it.
We start off with the basic setting – the central sphere without a surrounding shell:
We assume first that the sphere is a perfect blackbody and that its surroundings are a perfect vacuum at (an equivalent) 0 K.
We then assume that there is an energy source of some sort (most suitably a nuclear one, but it doesn’t really matter) at the core of the sphere supplying it with a constant input of energy equivalent to a 240 J/s/m2 flux. This input would be defined as the Qin of the central sphere. There is no point in assigning a particular temperature to the energy source. Only its energy output is of interest. (The same can be said about the Sun as the energy source of the Earth’s surface.) The energy thus transferred at a constant rate from the central source to the sphere around it is absorbed and slowly accumulates within the mass of the sphere. It spreads out through conduction, from molecule to molecule, until the energy accumulating is evenly distributed throughout the sphere, from its core (where the energy enters) to its outer surface (from where energy is finally able to escape the sphere). In this state, the sphere has become isothermal, and its surface temperature has reached a level where it is finally able to radiate away a flux equal to the incoming at its core: 240 W/m2. At this point, the sphere cannot warm any further. It has reached its steady-state temperature, a state of dynamic equilibrium – as much energy exits the system per unit of time as enters.
Since the sphere in this initial situation directly abuts a perfect vacuum at 0 K, it is to be considered a ‘pure emitter’, an object with its entire heat loss effectuated through the emission of EMR. Since it is also defined as a blackbody (and ‘ideal emitter’), an object with an emissivity of unity, its steady-state, omnidirectional radiative heat loss flux from its surface can be directly determined by the most simplified version of the Stefan-Boltzmann equation:
q = σT4
Where q is the radiative ‘heat flux’ (equal to Q̇ [Q dot]/A, radiative ‘heat transfer rate’ or ’emissive power’ [P] over area); J/s/m2 (W/m2). Whenever the pure emitter/blackbody conditions above are fulfilled, a q of 240 W/m2 demands a surface T of 255K or -18°C:
Figure 2. Graph originally from ‘The Engineering ToolBox’. The red circle, however, is my addition. It marks the spot where a surface temperature of -18°C would converge with the blue radiative heat emission flux curve, at an intensity of 240 W/m2.
The thing to bear in mind here is of course that some fraction of the energy constantly being transferred to the mass of the sphere from its core energy source will always stay stored within it, gradually raising its ‘internal energy’ [U] towards its final value in the process, rather than being transferred straight back out again by radiation from its outer surface, on the way to this final (steady-state) temperature. If this didn’t happen, it would simply never be able to reach it. It wouldn’t and couldn’t warm at all.
And only upon actually reaching this temperature will the heat loss flux from the surface of the sphere have grown all the way to become 240 W/m2, finally equalling the incoming energy input.
It is the heats that need to balance. The intensity of the incoming heat flux (qin) minus the intensity of the outgoing heat flux (qout) at any one particular point in time determines the incremental net heat (δQ) transferred to the system, in our case the sphere. This value, together with the total ‘heat capacity’ of the system (how much goes into internal potential energy vs. kinetic), in turn determines its instantaneous heating rate, the rate with which its temperature rises towards its final steady-state value.
Only in the most basic state, surrounded by nothing but a perfect vacuum at absolute zero (Figure 1), will the steady-state radiative heat flux radiated isotropically from the blackbody surface of our sphere into its perfect heat sink (cold reservoir) be precisely determined – through the Stefan-Boltzmann equation – by its own steady-state temperature alone.
As soon as its cold reservoir warms above absolute zero, however, this direct, simple relationship is broken. Now the radiative heat flux emitted by the sphere at dynamic equilibrium will be determined by the difference between the steady-state temperatures of two separate, opposing systems (sphere vs. cold reservoir surrounding it). Which means that even though the final radiative heat flux emitted from the sphere’s surface might very well end up the same, its steady-state surface temperature emitting this flux in this case would NOT. Because now the steady-state temperature of its facing heat sink would have to be taken into account as well.
Hold this thought. (It’s much less complicated than it sounds 😛 )
Moving on …
We now place the steel shell around the central sphere.
At the first instant (Figure 3), the shell is still hypothetically at 0 K (its radiative heat flux to space therefore 0 W/m2), but as soon as it starts absorbing the incoming radiative heat flux from the sphere now trapped within it, energy starts accumulating inside its mass, and its internal energy [U] and its temperature [T] starts rising. And with an increasing temperature comes an increasing radiative heat output (to the void outside of it, NOT in any way back towards the sphere, its energy (heat) source!).
The shell will also, like the sphere, necessarily have a ‘thermal mass’. It needs to for it to be able to warm. A hypothetical shell with zero ‘heat capacity’ is a meaningless entity. It is also completely useless in a model like Eschenbach’s ‘Steel Greenhouse’, because such an object could not warm, it could accordingly never acquire a temperature, and would thus be totally helpless at demonstrating the concept of insulation, which is what the steel shell around the central sphere ultimately, as we shall see, turns out to be – an insulating layer.
So, this is where the fun begins …
After some time, the shell’s temperature has risen to 50K. It is thus emitting a radiative heat flux to space (its Qout) worth of 0.35 W/m2. Not by any means a whole lot. But still larger than zero.
What, then, happens to the radiative heat flux going from the central sphere to the surrounding shell as the latter starts warming?
If the central sphere weren’t itself heated by the constant input from its nuclear energy source, the sphere/shell heat flux would grow smaller and smaller as the shell grew warmer and warmer. In the end, it would reduce to zero, at which point the temperatures of the sphere and the shell would be exactly equal. What the shell would do in this case, is extend the sphere’s total cooling time, that is, reduce its cooling rate from start to finish. Like proper insulation.
With the central sphere being constantly heated by the input from its internal energy source, the end result turns out quite differently. Now, none of the bodies will cool. Temperatures will rise at both ends.
Remember what we said about individual heat transfers: ‘They can only ever result in heating at one end, cooling at the other.’
This still holds. It is the first situation outlined above. Switch off the internal energy source of the central sphere and you’re down to one heat transfer only, and the sphere inevitably starts cooling.
But now we have two heat transfers working in tandem.
Still, don’t forget what we said about one system being involved in two separate heat transfers: ‘You need to keep the individual heat transfers apart in order to see which one is doing what.’
That system here is the central sphere:
Always keep in mind that there is no transfer of energy going on from the sphere to its energy source at its centre; likewise that there is no energy transfer going on from the shell to the sphere. In thermodynamics, energy transfers are always unidirectional. In the case of heats [Q], they always move spontaneously from warmer to cooler only. If you want to claim that there is something moving the other way also, then this something would by definition NOT be heat.
Also, according to basic thermodynamic principles like the 1st Law, how do you change the internal energy [U] – and thus normally the temperature [T] – of a system like the sphere? There are only two ways. You transfer energy to or from it by way of heat [Q] or work [W]. No other way. In our situation here, there is no work being done anywhere, so the ONLY way to change the U and thus the T of the sphere system is through the transfer of energy as heat.
So what do we have? Disregarding work, if you transfer energy to the sphere and this energy transfer to the sphere ends up increasing its internal energy and consequently its temperature … then you have transferred HEAT to the sphere. There is absolutely no way around it. Now, take a look at Figure 5 above. Wherefrom would such an energy transfer possibly arrive? From one place only. From the left. From the energy source inside of it. (In the case of the surface of the Earth, it would come from the Sun and from the Sun only. From a hotter place!) It simply could not come from the right, from the shell. Because the shell is cooler than the sphere and thus cannot transfer any energy as heat to the sphere. The sphere is rather the one transferring energy as heat to the shell, warming it in the process. The shell is NOT an energy source to the sphere. Not in any way. It could never be! It is its energy recipient. Its cold reservoir. Its heat sink. The sphere is the energy source of the shell. And this relationship does not change even when the shell starts warming. It remains cooler than the sphere.
So in the particular heat transfer between the sphere and shell (the right-hand one in Figure 5), the sphere is always losing internal energy (−U), thus cooling, the shell always gaining (+U), thus warming, by way of the unidirectional transfer of energy from the former to the latter as heat.
People, you absolutely NEED to take this to heart!
Still, when the shell is in place and starts warming, the sphere starts warming too. Why? How? What’s going on?
This is where people get confused. And this confusion is exactly what the ‘climate establishment’ capitalises on.
Is the shell now somehow the thing that’s doing the extra warming? The sphere, after all, was already equilibrated with its energy source at its core, at 255K, before the shell was brought in, its total steady-state content of internal energy all originating directly from this source. So one would think that this is the maximum temperature (and hence, the maximum content of internal energy) that could ever be achieved by the sphere with only the 240 W/m2 input from its inner energy source.
And one would seemingly be right.
However, does this in any way mean that, when the sphere evidently starts warming beyond this maximum temperature with the shell up around it, it is somehow energy from the shell adding on to that equilibrated internal energy content of the sphere, further raising its temperature?
NO!!!! Most certainly NOT!
The ‘extra’ energy now accumulating inside the sphere, raising its temperature in the process, is of course still directly drawn from the core energy source, and only therefrom. It could only ever be. Nothing returns to accumulate a second time!
We’ll get back to this in the next post …
The interesting circumstance here is that, when the sphere has achieved heat balance with its energy source (its Qout equals its Qin), it wants to stay there. At all costs. Any process disrupting this balance will have to be dealt with immediately. And it is, by physical necessity.
Since, as we pointed out earlier, the conceptual model of the ‘Steel Greenhouse’ is basically a purely radiative one, the Stefan-Boltzmann equation for radiative heat transfer applies. We already found the equilibrated radiative heat loss for the blackbody sphere alone in space. Now that the blackbody steel shell is emplaced around it, we need to expand the formula a bit:
q = σ(Th4 – Tc4)
Since q (the sphere’s Qout) needs to remain at 240 W/m2 (to balance the sphere’s Qin), we immediately realise that, in order to pull this off, the temperature of the sphere (Th) will have to rise as the temperature of the shell surrounding it (Tc) rises. This is what we discussed a bit earlier.
But by how much? We rearrange to solve for Th:
Th = 4√((q/σ) + Tc4)
Th = 4√((240/σ) + 504) = 255.1K
A rise in sphere temperature of a mere 0.1K (!) will suffice to retain balance with a shell temperature of 50K.
So why and how does the sphere’s temperature increase? It all happens automatically and instantly as the opposing shell temperature rises.
We know from before that as long as a system’s Qin is larger than its Qout, then some of the energy supplied by the energy source (providing the Qin) will always naturally accumulate inside the system, increasing its internal energy [U] and, accordingly, its temperature [T]. This will in turn progressively increase the system’s output (Qout), up to the point where the outgoing energy finally balances the incoming and the accumulation stops.
So what happens if you then, after initial equilibration, manage to reduce the outgoing heat flux back again from its attained steady, maximum level, somehow retard its energy loss per unit of time, while keeping the incoming constant?
Well, the same thing would happen as during the original warming process. Same general mechanism. Only now with the final, equilibrated U and T turned from end points to starting points. In other words, there would be further warming.
This is the basics of how insulation works.
The process is technically a stepwise one (reduced Qout → accumulation of internal energy from the constant (and now once again larger) Qin → raised T → Qout increases, back to balance), but outwardly it appears as a continuous, smooth rise without discernible lag. As long as the surrounding (insulating) layer keeps warming, this process will go on.
When the shell temperature (Tc) rises, its ‘radiative heat potential’ (σTc4) increases, and so the difference in heat potentials between the two opposing surfaces – the sphere surface and the shell surface – is reduced [refer to the radiative heat transfer equation just above]. This in turn leads to a smaller actual transfer of energy as radiative heat per unit time per area [q] from the warmer sphere to the (still) cooler shell than before (when the shell was even cooler).
Notice how the shell’s σTc4 is a mere radiative heat potential when facing the warmer surface of the sphere inside (heats do not oppose each other!), but an actual radiative heat flux (a real transfer (output) of energy) from the outside of the shell, facing its cold reservoir, the 0 K vacuum of space.
As the shell continues to warm from (most of) the energy continuously transferred to it as heat from the warmer sphere piling up inside its mass, the sphere is also forced to warm to maitain its heat balance with its own energy source:
With a shell temperature of 100K, how much warmer than its initial equilibrated temperature is the sphere forced to become?
Not very much:
Th = 4√((q/σ) + Tc4)
Th = 4√((240/σ) + 1004) = 256.5K
1.5 degrees warmer.
At this point, the shell emits an isotropic radiative heat flux to the 0 K vacuum surrounding it worth of 5.67 W/m2. Still not hugely impressive; a flux 16.2 times as intense as the one at 50K, but still 42.3 times less intense than the incoming heat flux from the sphere on the inside.
But, the shell grows ever warmer:
It is now at 200K. Its Qout has grown to be 90.73 W/m2. We’re getting there. The slope of its output is ever steepening. And as the slope of its output steepens, its warming rate is progressively declining. Because a smaller and smaller portion of the energy absorbed (from the 240 W/m2 of Qin) manages to remain inside the mass of the shell from one second to the next. Because more and more is shed (Qout) at the same time. The amount staying after each ’round’ moves towards zero. The steady state of dynamic equilibrium is drawing near.
So how warm is the sphere at this point? The difference in opposing radiative heat potentials needs to stay the same. So when the shell temp goes up, the sphere temp naturally must go up, from the resulting internal piling up of energy supplied by its central source (its Qin). But the temperature rise is by no means equal in absolute terms:
Th = 4√((q/σ) + Tc4)
Th = 4√((240/σ) + 2004) = 276.3K
The sphere is now 21.3 degrees warmer than originally. The shell, though, is 200 (!) degrees warmer.
Finally we reach the steady state, when neither the shell nor the sphere can warm any longer:
At this final stage, the shell has reached the initial temperature of the sphere and thus emits a similar flux to space (240 W/m2), the same size as the incoming flux equivalent from the ultimate energy source at the core of the central sphere. We have established radiative heat balance.
So, when the shell is at 255K in the steady state, how warm must the sphere be?
Th = 4√((q/σ) + Tc4)
Th = 4√((240/σ) + 2554) = 303.3K
The sphere at radiative equilibrium has warmed a grand total of 48.3 degrees, from 255 to 303.3K. This is its new steady-state temperature.
The final temperature of the sphere is close to 1.19 times its initial temperature. It might sound like a random observation, but in fact, this would always be the case with a blackbody arrangement such as this. It doesn’t matter how hot or cold it was to begin with. It doesn’t matter how large or small the energy input. The central, heated object, insulated by a surrounding one, will always reach its steady-state at a temperature 1.19 times higher than its original one. Meaning, it will also always end up being 1.19 times warmer (in absolute temperature) than its surrounding shell.
1.19 (or 1.189207115…..) is simply equal to 4√2.
At radiative equilibrium in this particular situation (disregarding differences in radii*), the ‘radiative heat potential’ of the sphere (σTh4) is, per the Stefan-Boltzmann equation, exactly twice the size of the ‘radiative heat potential’ of the shell (σTc4), meaning, the actual radiative heat flux moving from the sphere to the shell is at this point of exactly the same intensity as the actual radiative heat flux moving from the shell to space (as can be seen from Figure 8 above).
*The inverse square law says that the intensity of a radiative flux diminishes the further away from its source it gets. Also, different radii means different overall areas from which the energy is emitted, so the fluxes would actually not be the same, only the total amount of energy. We have ignored any such mathematical complications to our conceptual model, simply because they would only distract from the concepts we want to explore.
This relationship leads us to the conclusion that the final (radiative equilibrium) temperature of the sphere raised to the fourth (Tf4), is exactly twice the initial temperature of the sphere raised to the fourth (Ti4):
Tf4 = 2 * Ti4
Which in the end takes us here:
Tf = 4√2 * Ti
In the upcoming post, we will explain exactly why and how Willis Eschenbach’s “back radiation” explanation of the extra warming of the sphere must be wrong. How to properly resolve the workings of ‘radiative heat transfer’?
And we will start exploring the real mechanisms responsible for heating a planetary surface under a massive atmosphere. Recall that point about conduction through the shell …