Unqualified Offerings

By Thoreau

1)  Long bull sessions with a chemistry professor over greasy Chinese food in a near-empty campus food court.
2)  Coming up with my own derivation of the Carnot efficiency.  As some of you may recall, in June I put up a post praising Carnot’s work on the maximum efficiency of an engine, one of the Big Fundamental Results of theoretical physics.  Well, recently I decided to see if I could derive the Carnot efficiency without any mention of gases.  I hear that Caratheodory did something along those lines, but I want to do this myself.  So far, I get the following result (the proof of which cannot fit in the margins of this post):

Max Efficiency = 1 – (Tc/Th)^alpha

where Tc = the temperature of the environment that you’re dumping waste heat into, Th = the temperature of your heat source, and alpha is some unspecified number.  I got this result by assuming that heat only flows from high temperature to low temperature (unless, of course, you do some work to push it in the other direction) and then using Carnot’s line of reasoning to prove that the most efficient engine is a reversible engine.  Then I assumed that you have a bunch of intermediate reservoirs, so that you take heat from something at Th, dump the waste into something at a temperature T1 (where Th>T1>Tc), then take the heat from the T1 reservoir, do work with it, dump the waste into something at a tempertature T2 (where T1>T2>Tc) and so forth.  I also assumed that the maximum efficiency only depends on Tc/Th.

By analyzing the efficiency of a cascade of reversible engines working between those reservoirs, it’s easy to prove that the maximum efficiency is 1 if Tc=0 and the efficiency is a smooth function of Tc/Th.  You can always get some work from the waste heat and then dump the waste from that process into something at some arbitrarily low temperature, and keep going to smaller and smaller temperatures until you’ve consumed all of the waste heat.

Once you have that, you can do some math to prove that 1-Max Efficiency = (Tc/Th)^alpha.  Basically, the power law comes about because you can assume that Th/T1 = T1/T2 = T2/T3 ….. = Tn/Tc, i.e. this is self-similar, so if you do the math it has to follow a power law.

Anyway, I do all that, and without any notion of temperature being related to energy I get the result above.  Which is pretty neat.  I can’t think of a rigorous way to prove that alpha = 1 without talking about isothermal gas expansions and doing some dimensional analysis.  Maybe that’s the way to go.  But for now, I’m trying to stay as “pure” as possible in this.  I suspect that somebody has already done this, but I’m following Feynman’s advice and NOT reading the literature (well, except that excerpt from Feynman saying not to read the literature).

Why am I doing all this?  Because I’ve been working on the fundamental limit to a new imaging technique.  In doing so, I got some nice results, some of which have been accepted at a journal (old version here) and some of which will be written up soon.  (I want my student to do a simulation to include in the paper.)  And while working on these results, I felt myself using the same mental muscles that I use when I do thermodynamics.  I even came up with a result that involves random processes and algorithms (not quite information theory, but very close).  So I decided to re-read Carnot’s work, to see how another person went about working on fundamental limits.  I really wish he were alive today, because I like the way he thinks and I suspect it would be fun to discuss physics with him over foie gras.  (He is French, after all.)  After Carnot, I’m going to re-read some Feynman.  I know, the results he discussed there had been worked out by others, but it’s a nice survey.