PSI recently had a paper submitted where the author claimed that there are solutions to thermodynamics with heat flowing from cold to hot, as long as heat was flowing from hot to cold via a different mechanism at the same time. Paper is here; author’s name has been removed.
Here we will present the review, just because it’s great reading for math and physics Übermenschen.
For specific criticism and some reasons why your paper was not accepted.
PL: “The First Law of Thermodynamics, conservation of energy for the radiator says: Rate of energy out/in by conduction at T to/from surroundings at Ts = rate of energy in/out by radiation from/to surroundings minus/plus the rate of energy accumulation/depletion within the radiating body.”
Firstly, that is one of the most convoluted, unreadable statements I’ve ever come across. I’d ask for a re-write edit based on this sentence alone. Thankfully you then wrote it as an equation:
“PL: Rate Out = Rate In – Rate of Accumulation
Qo(t) = Qi(t) – m Cp dT(t)/dt”
Since you’re using Q then these terms should mean heat, although you don’t call them heat in your text, and so again it needs editing and a re-submission based on that alone. However, then something gets much worse. The right-most term is indeed called heat and one will typically find it as dQ/dt = m Cp dT/dt or more simply dQ = m Cp dT. But…this *is* the rate of heat energy in for an object, and it *is* the rate of heat energy out for an object, depending on the sign of dT, and, this *is* the rate of accumulation for an object. Your equation is literally stating:
Rate of Accumulation = Rate of Accumulation – Rate of Accumulation
m Cp dT(t)/dt = m Cp dT(t)/dt – m Cp dT(t)/dt
which is senseless.
However, given that convoluted preliminary sentence, it seems that in this equation you’re actually trying to separate conduction from radiation, but that isn’t clear at all from the terms in the equation written since they simply denote heat out and in. Your Qo(t) is actually the heat from conduction, and Qi(t) is the heat from radiation, given your convoluted intro sentence. But given the way that you have inverted in writing in that convoluted sentence almost every single term in its relationship to what should be positive and negative, because you go from in/out to out/in several times, it becomes exceedingly difficult to track what you’re actually doing especially when you factor in that the individual terms drop any notational reference to the conduction and radiation that they’re supposed to correspond to. It’s a hair away from being totally unreadable…strictly it is actually unreadable, and totally inconsistent. And I’ll also note that that first equation which you start from has no reference stated for it from source material.
At this point, without going any further, your paper is being sent back for major revisions asking you to use proper English sentence structure, and proper & consistent mathematical and physics notation. If the author didn’t know what that meant, and the paper then came back with the same English and mathematical grammatical syntax convolution, then the paper would immediately be recommended for outright rejection. The author would need to go find somewhere else to publish.
Going further, for the sake here, not that I would have if this was a paper I was reviewing (it would have been sent back by now), you state:
“PL: Qo(t) = Qi(t) – m Cp dT(t)/dt
At steady state, T is constant, dT/dt = 0, out/in = in/out and Qo = Qi. Let Qc be rate by conduction and Qr rate by radiation. Qo is Qc if Qc > 0. Qi is Qr if Qr > 0. Qc = Qr”
If dT/dt = 0, this means that there is no heat flow, and so all Q’s should be equal to zero. dT/dt = 0 defines all Q’s equal to zero. Instead you are saying that the conductive heat input must be equal to the radiative heat loss, or in other words, that heat is entering and leaving at the same time. There is some sort of underlying ambiguity which has been set up here but for now we will go with it given that heat is said to be leaving and entering at the same time via different mechanisms but at equal rates.
However, since you’ve stated that dT(t)/dt = 0, then since the object is in thermal equilibrium, Kirchhoff’s Law will be in effect; all of your subsequent discussion and where you arrive at heat flowing from cold to hot depends upon Kirchhoff’s Law not being in effect, but you started the analysis with the very conditions under which Kirchhoff’s Law is defined to be in effect…i.e. when dT(t) = 0, i.e. in thermal equilibrium. So there’s a logical error here. You’ve set up the conditions under which Kirchhoff’s Law is *defined* to be in effect, i.e. dT(t)/dt = 0, thermal equilibrium, but then you go on to dispense with it.
Your equation and text itself is inconsistent because your preliminary convoluted sentence is referring to the Q’s as those from convection and radiation, but then you write the equation as Q’s in and out. So now I must correct this and sort it out:
m*Cp*dT refers to an object, and so any Q’s relating to this must also refer to same object, so that if dT is positive, then dQ is positive, i.e. if the object has risen in temperature then it has taken in heat – positive temperature change = positive heat input. So getting rid of the heat in and out notation which makes no sense, and using heat from conduction and heat from radiation notation, then
dQc + dQr = m Cp dT
So now if dT = 0, then dQc = – dQr. The heat input from conduction equals the heat output from radiation, or vice-versa. And this must be a general result applicable to all situations. If we very carefully read your text, it eventually becomes somewhat clear that you state the same thing.
So if we now look at your later equation (4) or just insert terms to dQc = -dQr…
dQc = k(Ts – T) –> that makes sense because the object is the reference; if there is positive heat +dQc into the object from conduction then we expect Ts to be greater than T, and k is always positive.
dQr = sigma * (α εs Ts4 – αs ε T4) –> that makes sense (in this writing) because if absorptivities and emissivities are unity, then positive heat dQr into the object from radiation is because Ts is greater than T.
So putting it together, and this is the same equation (4) as found in your text:
k(Ts – T) = -sigma * (α εs Ts4 – αs ε T4)
and given that your dQc = – dQr is a general result for the thermal equilibrium you’ve defined it thus applies to all situations of such, and so we can then look at an ideal case where absorptivities and emissivities are all unity:
k(Ts – T) = -sigma*(Ts4 – T4)
Thus, if Ts was less than T, then the left hand side would be a negative number, -x say.
-x = -sigma*(Ts4 – T4)
+x = +sigma*(Ts4 – T4)
The only way that the right hand side can be a positive number is if Ts is greater than T…however, this is in contradiction to the defined requirement that Ts was less than T.
Thus, there is a fundamental, general error, lurking somewhere in your creation here, and so it is no wonder that you could find heat flowing from cold to hot…once the initiating error is set up, subsequent errors can only follow. And we have now proven that there is a general error embedded somewhere in your creation. In other words it is mathematically impossible for dQc = – dQr. This isn’t a possible expression in physics or the mathematics. Certainly it can lead to heat flowing from cold to hot…because that is impossible…an impossibility can come from an impossible expression.
The impossibility must arise when we say that
dQc + dQr = m Cp dT
can have a condition where dT(t) = 0 without both dQc and dQr being zero. In other words, the only condition where dT(t) = 0 is if both dQc & dQr equal zero. If dQc and dQr were not zero, then it is not possible for dT(t) = 0. dQc = – dQr only when they both equal zero. Why would that be?
Is there a non (Ts – T) = 0 solution for
k(Ts – T) = -sigma*(Ts4 – T4)?
(Ts – T) = -(sigma/k) * (Ts4 – T4)
(sigma/k) is always positive and is just a scaling factor so let’s just remove it so that we have focus on the variable terms:
(Ts – T) = – (Ts4 – T4)
Well, again, this is actually just the same thing as above: if (Ts – T) > 0, and since all T > 0, then (Ts4 – T4) > 0, and so the negative sign is impossible. The error is indeed in saying that
dQc + dQr = m Cp dT
can have a condition where dT(t) = 0 without both dQc and dQr being zero. It’s mathematically impossible. Probably related to the 2nd Law of Thermodynamics.
As I said at the beginning, there was something strange about saying that the heat loss from conduction must equal the heat input from radiation under thermal equilibrium as a general condition. It’s much more sensible to say that, at thermal equilibrium, both of the heat exchanges from conduction and radiation must each be zero. Notwithstanding that thermal equilibrium was defined in the setup, but then the defined condition of thermal equilibrium, i.e. Kirchhoff’s Law, was dispensed with.
If we look at the full general equation:
k(Ts – T) + sigma(α εs Ts4 – αs ε T4) = m Cp dT
and given that we have proven that there are no mathematical solutions for dT = 0 without both terms on the left hand side also being zero, then that condition can only arise when both terms of the left hand side always have the same sign. That is, either temperature is increasing and both terms contribute positively, or temperature is decreasing and both terms are negative. They both approach zero from the same side of positive or negative as dT approaches zero. Why?
If k(Ts – T) was positive, then that means that heat was flowing from the warmer surrounding Ts to the cooler object T. However, if the other term was negative due to the effects of absorptivity and emissivity, then that would mean that, for that radiative mechanism, heat was flowing from cold to hot. That is a violation of the 2nd Law of Thermodynamics because the mathematical law of entropy increase does not allow two objects to spontaneously diverge in temperature by any mechanism, since that would mean that there existed a mechanism to spontaneously and passively decrease entropy. The objects diverge in temperature and entropy decreases if heat flows from cold to hot because that would mean that the cold object loses thermal energy thus decreases in temperature, while the hot object gains that said energy and thus rises in temperature; this is a decrease in entropy.
I’ve said it elsewhere, but to repeat: the Laws of Thermodynamics are actually laws of mathematics, i.e. ontological mathematics or the mathematics of existence. Such mathematics is a self-consistent and complete system of logic. If you create an error of mathematics, then you will find the same error popping up other places. That’s what happened here. The logic of the physics was violated by setting up thermal equilibrium and then dispensing with the conditions of thermal equilibrium, i.e. Kirchhoff’s Law. It was then found that the equations under those conditions lead to self-contradiction. It was then found that such a contradiction arises due to a violation of the 2nd Law of Thermodynamics. It’s all in the math, and in this case the math of the 2nd Law.
The reason why papers are simply sent back without further perusal when there are early indications that there are problems with the paper, is because it takes so much time to figure out where people eventually really go wrong. And it is never of any use asking the author to agree with you.