In the last post we examined the equations conserving energy, defining heat flow, and thermodynamic equilibrium for a power-generating sphere enclosed by a shell. We examined the equations for when the system either existed in an ambient-temperature environment above 0K, or not. The objection by the climate alarmists is to say that the shell serves the system in the exact same way that a universal ambient-temperature environment does, and therefore if the existence of such an environment means that the sphere will attain a higher temperature for a given fixed power production by the sphere, then the shell will make the sphere get to a higher temperature too.
The equation for a given fixed power PspO emitted by the sphere in an ambient temperature environment at TA was
1) PspO = 4πRsp2σ(Tsp4 – TA4)
Equation 1 tells us that if the temperature of the sphere is equal to the temperature of the universal ambient-temperature environment, then the sphere is not producing any power of its own. If it’s not obvious, the equation is derived by taking the power which would be emitted by the sphere without reference to background, 4πRsp2σTsp4, but then subtracting the amount of power that is already present due to the universal ambient-temperature background at the radius of the sphere which is 4πRsp2σTA4.
The climate alarmists believe that the shell is the same thing as a universal ambient-temperature environment, so let us proceed in that manner. The shell’s interior energy flux is σTsh4 if the universal ambient-temperature is 0K (which we will assume for sake of simplicity in the equations), and if the shell is supposed to now provide an ambient-temperature environment for the sphere in a manner as a universal independent ambient temperature, then since the shell provides an energy at the surface of the sphere equal to 4πRsp2σTsh4 the new equation for the fixed power emitted by the sphere is simply
2) PspO = 4πRsp2σ(Tsp4 – Tsh4)
To do anything more with the equation requires understanding what the temperature of the shell is. What we will not do however is commit a gross violation of the law of conservation of energy by demanding that the larger shell emits at the same temperature and surface flux as the smaller sphere would by itself, as Willis Eschenbach does; rather, the shell must emit outwards the original raw power produced by the sphere PspO, and so
3) Psh = PspO = 4πRsh2σ Tsh4
and the temperature of the shell is
4) Tsh4 = PspO/4πRsh2σ
We can now take equation 4 and put it back into equation 2 in order to simplify terms, which gives
5) Psp0 = 4πσTsp4 * Rsp2Rsh2/(Rsp2 + Rsh2)
for the power emitted by the sphere, and for the temperature of the sphere
6) Tsp4 = (Psp0/4πσ)((Rsp2 + Rsh2)/Rsp2Rsh2)
If we consider the two limits for the shell radius, where Rsh → ∞ and Rsh → Rsp, then the temperature of the sphere goes to that given simply by its own power output over its surface area since the temperature and local flux of the shell goes to zero as from equation 4 when the shell radius is large, whereas the internal power generation of the sphere is half of its total output and its temperature increases by the fourth-root of 2 when the shell radius is near the sphere radius. The latter result is the one the climate alarmists wish to focus on.
We readily grant that the solution to the equations show what they show, when solved in this manner. One must also readily grant that the solution to the equations showed what they showed in the previous post where the sphere didn’t need to increase in temperature.
Two sets of equations claiming to describe the exact same problem, but ending with different results. The one commonality between the solutions is that they are both conserving energy externally: that is, the raw unique power produced by the sphere is conserved by the power emitted outward by the shell.
The specific difference between the two solutions which leads to the different results is that one solution also utilizes the equations for heat flow and the concept of thermodynamic equilibrium and the formal statement of the First Law of Thermodynamics, while the other solution signally does not use any equations for heat flow or thermodynamic equilibrium or the formal statement of the First Law; the climate alarmist solution is the latter, and is referred to generally as the radiative greenhouse effect.
The climate alarmist replacement for a consideration of heat flow and thermodynamic equilibrium is to require that any secondary emission, no matter its source or the nature or characteristics of its source nor the nature of the emission, will add with the energy of the original power source and thus cause the original power source to rise in temperature. This would indeed conserve energy if the energy from the secondary emissions behaved and were conserved this way, and this is what leads to the climate alarmist RGHE solution.
However, if one incorporates into the model the equations for heat flow and the concept of thermodynamic equilibrium and the formal statement of the First Law of Thermodynamics (conservation of energy), then secondary emission from cooler objects which gained their thermal energy from the original power source does not add back with the power source to increase its temperature as this would violate the definition and directionality of heat flow, and it would also be inconsistent with the formal formulation of the 1st Law of Thermodynamics which states that an object can only increase in temperature if it receives heat. The climate alarmist solution treats all emission to function as heat, whereas the definition of heat flow and the concept of thermodynamic equilibrium states that emission can only act as heat if it is flowing from a more intense source to a less intense source, i.e. from warmer to cooler.
The climate alarmist RGHE solution for the sphere-shell problem is thermodynamically incomplete. It doesn’t utilize all of the physics that it should. It conserves energy externally, but in a way that is internally actually inconsistent and contradictory to the 1st Law of Thermodynamics which is the law that is about conservation of energy. That is, the Laws of Thermodynamics tell us to conserve energy, but to conserve it in a specific dynamic way where heat only flows from hot to cold and where thermodynamic equilibrium is the preferred end-state for any system (powered or not) and where thermodynamic equilibrium has the definition of heat flow having been reduced to zero. The climate alarmist RGHE solution disallows thermal equilibrium to ever exist even conceptually because the inner sphere always emits more total power from its surface (greater than its actually-internally-generated power) than the surface of the shell emits; this conserves the internally generated power from the sphere on the surface of the shell, but it disallows that thermal equilibrium should ever exist between the sphere and the shell. The climate alarmist RGHE solution does not incorporate any statements let alone equations about heat flow anywhere at all, and this is quite unlike any textbook on radiant heat transfer that exists.
The Slayer solution does use the definition of heat flow, does use the concept of thermodynamic equilibrium, and does use the 1st Law of Thermodynamics as formally stated. The climate alarmist RGHE solution signally does not, and hence is ontologically incomplete, and hence does not connect to reality. The climate alarmist solution is wrong, whether the shell radius is extremely large or extremely close to the sphere’s radius…it is wrong, always. The correct solution starting off from the exact same physical scenario but also incorporating the definition of heat flow and the formal statement of the Law of Conservation of Energy is the Slayer solution in the previous post.
Make your choice: the full set of laws and equations of thermodynamics, or just one of the laws incorrectly utilized as it is.