The “steel greenhouse” concept for demonstrating the radiative greenhouse effect has been debunked many times on this blog (the least reason of which its advocates attempt to conserve temperature instead of energy!), but the solution for it sitting in an ambient-temperature environment has never been demonstrated. We will do that now here and will present the maths for both of the scenarios where for a) the ambient-temperature is zero Kelvin, and b) the ambient-temperature is greater than zero.
Advocates of the steel greenhouse think that temperature is a conserved quantity, i.e., that temperatures add together. Their diagram above shows a “core” producing 235 W/m2 out of its surface area, and they think that this is conserved by the shell emitting at the same temperature at 235 W/m2 over a larger surface area. The shell emits more energy than the “nuclear core” produces given the shell’s larger surface area of emission at the same temperature and flux…and so steel greenhouse advocates literally believe that 5 = 1 conserves energy! The problem with these steel greenhouse advocates is that they have absolutely no clue whatsoever at all what they’re doing in any way, shape, form, or hope; they write things down without having the slightest idea of what the words they’re writing down mean and what their end goal even is, to the point that they believe that 5 > 1 is the proof for 5 = 1. That’s literally what they say openly say. They say that 5 > 1 is the proof that 5 = 1. Let that sink in. It’s worse than monkeys on typewriters because at least that would just produce unreadable gibberish. With these guys, they produce readable gibberish, but then they demand that you accept the gibberish for the simple fact that you can read it! They demand that you accept that 5 > 1 is the proof for 5 = 1 because you were able to read that. Energy is not conserved when their shell emits more total energy to the outside than is produced internally to the shell from the core sphere. QED.
The Correct Treatment
We begin with a sphere of radius Rsp internally producing its own power (i.e. energy) of Psp. If the sphere emits to a 0K environment then its internal power generation is given by its surface temperature Tsp.
1a) Psp = 4πRsp2σTsp4
If the sphere exists in a universal ambient-temperature environment at temperature T0, then the sphere’s minimum temperature would be T0 if it did not produce any internal power. If the sphere does produce any power then its temperature will rise above T0, and its energy production would then be given by
1b) Psp = 4πRsp2σ(Tsp4 – T04)
since 4πRsp2σ(T04) is the total energy already-present at the sphere’s surface due to the environment. Thus, for a given sphere temperature Tsp, less power is required to be produced by the sphere if the sphere already starts off in a universal ambient-temperature environment. Or identically, if the sphere produces the same amount of power, then in the ambient-temperature environment it will come to a higher temperature.
The ambient-temperature environment is an independent source of thermal energy pre-existent (or at least simultaneously existent) to the presence of the sphere; it is not dependent upon any power production from the sphere. If the unpowered sphere entered the environment at zero Kelvin, the sphere would then come to the ambient temperature of the environment. The ambient-temperature environment exists with a temperature greater than zero Kelvin because it either independently produced or received an independent deposit of energy to produce it. The ambient-temperature environment thus serves as an independent and additional power source for the sphere. At a minimum the sphere is sustained with the energy required for it to exist at the temperature of the environment even if the sphere started off at zero Kelvin, and if the sphere is taken away from the environment the environment does not have its temperature affected. The ambient-temperature environment is an infinite “heat sink” that can sustain any introduced object’s temperature at T0, and whose temperature is not affected by any introduced object.
We now introduce a shell of radius Rsh concentric with the sphere, with Rsh > Rsp. The shell produces no power of its own and will only be heated by the environment and/or the sphere inside it. With no internal power production from the sphere the shell would likewise be sustained at the temperature of the environment, either zero Kelvin, or T0. Whatever temperature Tsh the shell is, it emits power Psh to a 0K environment of
2a) Psh = 4πRsh2σTsh4
or if the environment is greater than 0K then to it goes
2b) Psh = 4πRsh2σ(Tsh4 – T04)
What we want to determine for conservation of energy is whether the power emitted outward at the surface of the shell is equal to the power generated by the sphere inside the shell. That is, equations 2 should be equal to equations 1 for conservation of energy: Psp = Psh. The shell’s surface would emit on its interior as well, however, internal emission by the shell will always meet another interior side of the shell (or the sphere), and hence will not leave the shell. Internal emission by the shell’s surface hence does not lead to a loss of energy for the shell, and hence the energy produced by the sphere will be conserved with the outward emission of the shell to the environment. Thus for conservation of energy
3a) 4πRsp2σ Tsp4 = 4πRsh2σTsh4
3b) 4πRsp2σ(Tsp4 – T04) = 4πRsh2σ(Tsh4 – T04)
Equations 3 cannot be proven without knowing the temperature of the shell. While the shell is warming it is storing energy as internal thermal energy which original came from the sphere, and so during this warming period energy would not be conserved as emission outward from the shell since some energy is being “lost” internally to the shell (energy is conserved if one considers the outward emission of the shell, plus its internal thermal energy). That is, Tsh is changing while the shell is warming. Therefore what we want to know is the temperature of the shell after it has finished warming, which state is called thermal equilibrium. Thermal equilibrium is defined or established when the heat flow between two objects reduces to zero, and the heat flow between two objects is defined as the net difference of the power emitted by either object. It is important to note here that heat is defined only as the net of the difference between the power emissions, and that therefore either power emission by itself is generally not heat. This goes to the quote from Schroeder in “Thermal Physics” (Addison Wesley Longman, 2000) that: “Much of thermodynamics deals with three closely related concepts: temperature, energy, and heat. Much of students’ difficulty with thermodynamics comes from confusing these three concepts with each other.” And so note once again that energy is generally not heat.
Heat flow, which is typically denoted as “Q”, is produced by the difference in energy emission between two objects. If we wish that positive Qsp-sh means heat flow from the sphere to the shell, then the heat flow equations are
4a) Qsp-sh = 4πRsp2σTsp4 – 4πRsh2σTsh4
4b) Qsp-sh = 4πRsp2σ(Tsp4 – T04) – 4πRsh2σ(Tsh4 – T04)
The first term on the right-hand-side in equations 4 is the power produced and emitted outward by the sphere, and the second term in equations 4 is the power emitted internally by the shell. The heat equations go to zero at thermal equilibrium and so solving for the temperature of the shell where Qsp-sh = 0
5a) Tsh4 = Rsp2/Rsh2Tsp4
5b) Tsh4 = Rsp2/Rsh2(Tsp4 – T04) + T04
Given that equations 4 originate from equations 1 and 2, then equations 3 can do nothing but reduce to an identity when incorporating equations 5. That is, if we substitute equation 5 for Tsh into equation 3 for conservation of energy, we have
3a’) 4πRsp2σ Tsp4 = 4πRsh2σRsp2/Rsh2Tsp4 = 4πσRsp2Tsp4
3b’) 4πRsp2σ(Tsp4 – T04) = 4πRsh2σ(Rsp2/Rsh2(Tsp4 – T04) + T04 – T04) = 4πRsp2σ(Tsp4 – T04)
The identity between the sphere and the shell’s externally-emitted power means that conservation of energy has occurred.
Note that in equations 5, the shell temperature is always less than the sphere temperature. It would be impossible under conservation of energy for the shell to emit at the same surface temperature and surface flux of the sphere but over a larger surface area. The reduction in shell temperature to that of the sphere goes as Rsp1/2/Rsh1/2, which given the fourth-power dependency of surface flux on temperature is indicative of the inverse square law of radiant intensity with distance. The surface of the shell cannot emit the same energy density as the surface of the sphere because the shell never experiences that same energy density in the first place, due its internal surfaces’ distance from the sphere; it also cannot because that would violate the First Law of Thermodynamics (conservation of energy).
Why is there a difference in how the sphere responds when the sphere’s ambient-temperature environment is provided by an independent environment rather than when provided by the shell? The difference is that the shell’s temperature above T0 is sustained entirely by the sphere’s power. If the sphere’s energy production is halted, the shell will immediately begin to cool. If the sphere’s energy production is then restarted at the same rate as previous, the shell will then be induced to warm back up to the temperature it was previously at, and then will likewise be maintained there given that the shell is losing energy on its exterior.
As for the shell’s interior emission, it cannot act as heat and thus warm the sphere since the sphere is always producing more power or at most (at equilibrium) an equal amount of power that the shell emits. To raise the sphere’s temperature from emission from the shell would require positive heat flow from the shell to the sphere, but this is never possible because at most the shell emits the same power as the sphere, and never more than the sphere. To raise an object’s temperature requires either work performed on it or heat transferred into it, and the shell doesn’t perform work on the sphere and it can never send a net difference positive balance of power as heat to the sphere.
The steel greenhouse is one of the best models for demonstrating the impossibility and hence non-existence of the radiative greenhouse effect of climate alarmism. The steel greenhouse has every foundation necessary for it to demonstrate a radiative greenhouse effect, yet, when the model is solved utilizing the law of conservation of energy and the definition of heat and thermal equilibrium, no alarming greenhouse effect manifests. Resting upon this basis, climate politics and a large part of climate science itself is rendered defunct.
The standard derivation of the radiative greenhouse effect, by the climate alarmists, arrives at a different solution because it doesn’t utilize the definition and equation for heat flow and thermal equilibrium; it does utilize conservation of energy, and by this it seeks to claim a sufficient foundation for legitimacy, but without utilizing the definition of heat flow and thermal equilibrium the laws of thermodynamics are incomplete and thus a solution ignoring them is likewise incorrect. Their error seems to rest solely upon the problem as stated in the thermodynamics text by Schroeder, of confusing energy with heat.
The consensus derivation of the radiative greenhouse effect has been demonstrated as debunked, but an alternative description which is often utilized at this point to attempt to resurrect the same effect is called “slowed-cooling”. This alternative offer is paradoxical in that slowed-cooling still wouldn’t explain how a temperature higher than the input forcing is achieved in the first place, as this is the foundational element of the standard consensus derivation of the radiative greenhouse effect. That is, if a body cools down from a -180C forcing input in a slower manner than otherwise, this can’t explain why a temperature higher than -180C would ever be found. What climate alarmism and its standard radiative greenhouse effect requires is that an input forcing power equivalent to -180C of radiant flux is amplified to a higher absolute temperature by the presence of a shell (i.e. the atmosphere in the case of the Earth). It doesn’t matter that slowed cooling from an input of -180C results in a higher average closer to (but still below) -180C of a rotating system, what matters is that the -180C forcing input is amplified to a much higher absolute temperature. And so the “slowed-cooling” offer to resurrect the radiative greenhouse effect is insufficient for its purported obligation, and its development in this post-hoc manner is scientifically disingenuous in any case.
That being said, the shell can indeed lead to the sphere cooling at a slower rate. If the shell had a high thermal energy capacity then when the power from the sphere is halted the shell would cool but would only lose a small part of its total internal thermal energy over time, thus maintaining a higher (but dropping) temperature over time as compared to if it had a smaller thermal energy capacity and smaller total internal thermal energy. In this case the minimum temperature of the sphere would be set by the current temperature of the shell given that the shell provides an ambient (but lower-temperature and cooling) environment for the sphere; eventually the entire system still ends at the environment temperature.
The fact is that none of the alarming claims from alarmist climate science rest upon the “slowed-cooling” idea but the basis of the science is on the amplification model, and further, the model of sphere and shell is purely radiative whereas the Earth’s “shell” atmosphere is in contact with the Earth’s surface and the continuous convection of the generally-cooler atmosphere over the general-warmer surface allows the atmosphere to continuously cool that surface.
The full nature of the relations between Earth’s atmosphere and surface is of course quite complex, but one thing is undoubtedly established: the atmosphere cannot and hence does not raise the surface temperature of the Earth above that of the solar input power forcing by radiative means. There is no radiative greenhouse effect, and therefore the swath of climate alarmism and its politics is rendered invalid.