Many experts say "It doesn,t make sense to calculate the time of thermal equilibrium establishing since its time is infinity" (see exponential law of cooling or heating). On the other hand, the main postulate of thermodynamics says that the thermal equilibrium establishes within the finite period of time, and it is a basis for the next three laws of thermodynamics. How should be the time of thermal equilibrium existence in different systems easily and correctly calculated?
You mean classical or quantum? No one has anything but a formulaic approach for quantum. Should be a Millennium problem. There are countless "unsolved" problems (in that no one even knows how to start them) in this case. Classically the rate of equilibrium can be very fast! Villani has some recent work here.
Thermal equilibrium is normally assumed for process design, although the time to reach the thermal equilibrium may not be negligible.
In some cases a thermal imbalance is even beneficial and is used for particular purpose. See e.g. The Ranque-Hilsh vortex tube, a static device which separates a compressed gas stream into hot and cold streams.
http://en.wikipedia.org/wiki/Vortex_tube
Thanks, Mr. Clifford Chafin, for your comments. Both classical and quantum can be used to solve the problem. We used classical aproach, considering parabolic and hyperbolic heat conductivity equations and taken into account existence of temperature fluctuations.
Dear Giorgio,
Thanks for interesting comments. We used the Ranque-Hilsh vortex tube to design special fluidized bed where tools, made of high carbon steel, were hardened at the temperature deep below zero. Here we calculated cooling time at the core of tools and were waiting for thermal equilibrium establishing :).
Solve which problem? You can formally plug into density matrices but that doesn't give you temporal fluctuations. There a ad hod methods like fluctuation electrodynamics and the Caldeira-Leggett approach. If you start with a noneigenstate then you have temporal transitions but there is no equilibration. The explosion in SM results like C-G and Jarzinskii results are still looking for a proper quantum expression. If you have a new quantum approach I REALLY want to know it!!!
Kubo's approach is useful sometimes but no one has ever addressed the fundamental problems or determined the extent of its validity. If I just needed to get a number, I'd try this.
Dear Clifford,
I mean solving the contradiction between postulate of thermodynamics and infinity of heating (cooling). A standard for practical use should be developed showing what is thermal equilibrium with the accuracy, for example, + (-) 1%.
So do you mean the rate of convergence ala Cedric Villani? I think he has pretty hard results for gases.
Nikolai, the "infinity" stems from math whereas the "postulate of thermodynamics" or "postulate" in general is a truth which is based on observations and therefore has the frames of applicability. Clifford Chafin's question about "classical or quantum" was right about these frames. Thereby, the time to detect (establish) a thermal equilibrium is always limited by the accuracy of temperature difference measurements (DeltaT). Thus, the cooling time is limited by t ~ -Ln(DeltaT)/k.
Samvel, thank you very much for your comments ! Thanks also to All for interesting discussion. Hope for more discussion when a manuscript "Universal correlation for heating (cooling) time calculation suitable for steel parts of any configuration during their hardening" will be published.
Thermal equilibrium is established in finite period of time when exponential temeperature difference is comparable with the values of temperature fluctuations. In the practice, for process design it can be considered as dimensionless temperature decrease, for example, 1000 times.
Thanks ! More detailed discussion will be provided in the paper which we prepared for publication.
I checked your company site. Please post a link to your paper when ready.
I found a funny way to estimate the time required to change concentration from c1 to c2 in volume V. If you assume for simplicity that the volume is a cube, then you may calculate unit volume per molecule by dividing volume by the number of molecules. Then if you take a cube root, you will get an elementary cube related with each molecule. Each edge of the elementary cube has the same length a1, as in equilibrium we assume, that the molecules are equidistant as the volume is isotropic (recalls Born-von Karman boundary conditions used in solid state physics). After concentration change using the same argument we obtain a2. The difference delta_a between a1 and a2 is a distance each molecule has to pass in order to be located in a new equilibrium position. Of course it is kind of "far field" approximation, when we assume that we are far enough from the source of concentration change. Then if we have delta_a, we may use diffusion constant to estimate time as t=x^2/2D. I obtained 1E-8 ns for normal experimental conditions.
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